Calculus

🧭 Overview

🧠 One-sentence thesis

Calculus is fundamentally about pairs of functions—an original function (height or distance) and its growth rate (slope or speed)—and the first step is understanding how to compute that growth rate.

📌 Key points (3–5)

• Calculus studies pairs of functions: Function (1) is the original (e.g., height, distance), and Function (2) is its growth rate (e.g., slope, speed).
• Three fundamental examples: linear (y = 2x), quadratic (y = x²), and exponential (y = 2ˣ) functions grow at different rates—exponential eventually wins.
• Two methods to find growth rates: Method 1 uses limits (change in y / change in x as Δx → 0); Method 2 uses rules to build new rates from known ones.
• Common confusion: average slope vs. instantaneous slope—average slope is Δy/Δx over an interval; calculus aims to find the slope at a single point by taking the limit.
• Why it matters: the growth rate (dy/dx) tells you how fast a function is changing, which applies to real-world problems like speed (rate of distance change).

📊 Three core functions and their growth rates

📈 Linear function: y = 2x

• Formula: y(x) = 2x
• Growth rate: dy/dx = 2 (constant)
• The graph is a straight line with constant slope.
• Ratio of "up" to "across" is always 2, no matter which two points you pick.
• Example: between x₁ = 1 and x₂ = 2, Δy = 4 − 2 = 2 and Δx = 1, so Δy/Δx = 2.

📈 Squaring function: y = x²

• Formula: y(x) = x²
• Growth rate: dy/dx = 2x (linear, not constant)
• The graph is a parabola with increasing slope.
• Example: between x₁ = 1 and x₂ = 2, Δy = 4 − 1 = 3 and Δx = 1, so average slope = 3; between x₁ = 0 and x₂ = 2, average slope is different.
• Don't confuse: the average slope over an interval is not the same as the instantaneous slope at a point.

📈 Exponential function: y = 2ˣ

• Formula: y(x) = 2ˣ
• Growth rate: dy/dx = 2ˣ · (ln 2)
• The graph is an exponential curve with exponentially increasing growth rate.
• At first (near x = 0), the linear function grows fastest, but the exponential eventually overtakes both others.
• Example: at x = 10, y = 2¹⁰ = 1024, far exceeding y = 10² = 100 for the quadratic.
• The exponential "wins" in the long run.

🔍 What is a function?

🔍 Definition and notation

A function has inputs x and outputs y(x). To each x it assigns one y.

• Domain: the set of allowed inputs x.
• Range: the set of resulting outputs y.
• Example: for y = 2x with x ≥ 0, the range is y ≥ 0; for y = 2ˣ with x ≥ 0, the range is y ≥ 1.

🔍 Three ways to describe a function

1. Formula: y(x) = 2x
2. Graph: shows x (horizontal) and y (vertical)
3. Input-output pairs: the set of all (x, y) pairs

• The high-level definition: a function is the set of all input-output pairs, or the rule that assigns an output to every input.
• Practically, we learn by examples first, then refine the definition.

🧮 Two methods to compute growth rates

🧮 Method 1: Limits

• Write the ratio: (change in y) / (change in x) = Δy / Δx.
• Take the limit as Δx → 0.
• This gives the instantaneous rate of change (the slope at a single point).
• Example: for y = x², the average slope between x₁ and x₂ is Δy/Δx; the limit as Δx → 0 gives dy/dx = 2x.

🧮 Method 2: Rules

• Use known growth rates and combine them with rules.
• Constant factor rule: if y = C · f(x), then dy/dx = C · (df/dx).
  • Example: y = 5x² has growth rate 5 · (2x) = 10x.
• Sum rule: if y = y₁ + y₂, then dy/dx = dy₁/dx + dy₂/dx.
  • Example: y = 5x² + 2x has growth rate 10x + 2.
• Linear combination: the growth rate of C₁y₁ + C₂y₂ is C₁(dy₁/dx) + C₂(dy₂/dx).
• Don't confuse: these rules only work when you already know the growth rates of the building-block functions.

📐 Slope of a graph

📐 Average slope

Average slope = (change in y) / (change in x) = (y₂ − y₁) / (x₂ − x₁) = Δy / Δx

• Δ (delta) is the symbol for "change."
• The average slope is the ratio of "distance up" to "distance across" between two points.
• Example: for y = x² between x₁ = 1 and x₂ = 2, average slope = (4 − 1) / (2 − 1) = 3.

📐 Constant vs. changing slope

Function	Slope behavior	Example
y = 2x	Constant slope = 2	Δy/Δx = 2 for any interval
y = x²	Changing slope	Average slope = 3 between x = 1 and x = 2; different elsewhere

• For y = 2x, the ratio Δy/Δx = (2x₂ − 2x₁) / (x₂ − x₁) = 2 always.
• For y = x², the slope increases as x increases (the graph gets steeper).

📐 Function (1) and Function (2)

• Function (1): height of the graph (or distance traveled).
• Function (2): slope of the graph (or speed of the car).
• Example: if Function (1) is distance traveled y = C·t, then Function (2) is speed dy/dt = C (constant).
• The core task of differential calculus: given Function (1), find Function (2).

🚗 Distance and speed analogy

🚗 Real-world interpretation

• Function (1): distance traveled = C·t (linear function of time).
• Function (2): speed of the car = C (constant).
• The growth rate of distance with respect to time is speed.
• Example: if you drive at constant speed C, your distance increases linearly, and the slope of the distance-time graph is C.

🚗 Why calculus matters

• When speed is not constant (e.g., accelerating car), the distance function is not linear (e.g., y = t²).
• Calculus finds the instantaneous speed (slope) at any moment by taking the limit Δy/Δx as Δx → 0.
• This generalizes to any rate of change: growth rate, slope, speed, etc.

🧭 Overview

🧠 One-sentence thesis

The slope at a single point is defined as the limit of the average slope ratio Δy/Δx as the step Δx becomes infinitesimally small, which for y = x² yields the instantaneous slope 2x.

📌 Key points (3–5)

• Average slope vs instantaneous slope: average slope is Δy/Δx over an interval; instantaneous slope at a point is the limit of this ratio as Δx approaches zero.
• The central problem of differential calculus: defining "rate of change" at a single moment when nothing actually changes in that moment.
• Algebraic technique: expand y(x + Δx) − y(x), cancel leading terms, divide by Δx, then let Δx shrink to find the limit.
• Common confusion: do not try to define dy, dx, or 0/0 separately; instead, recognize that the ratio Δy/Δx has a well-defined limit.
• First-order vs second-order terms: after cancellation, the first-order term (proportional to Δx) dominates; second-order terms (proportional to (Δx)²) vanish when divided by Δx and Δx shrinks.

📐 Understanding slope: from average to instantaneous

📏 Average slope between two points

Average slope = (change in y) / (change in x) = (y₂ − y₁) / (x₂ − x₁) = Δy / Δx

• The Greek letter Δ (delta) means "change."
• For y = x², the average slope depends on which two points you choose.
• Example: Between x₁ = 1 and x₂ = 2, y₁ = 1 and y₂ = 4, so average slope = (4 − 1)/(2 − 1) = 3.
• Between x₁ = 0 and x₂ = 2, the average slope is different (not 3), showing that the slope is changing along the curve.

🎯 The problem: slope at a single point

• At one point, the distance across is Δx = 0 and the distance up is Δy = 0.
• Formally, the ratio looks like 0/0, which has no meaning.
• The inspiration of calculus: give this ratio a useful meaning by taking a limit.
• Don't confuse: we do not define dy and dx separately; we define the limit of the ratio Δy/Δx.

🔬 The algebraic method for y = x²

🧮 Step-by-step calculation

The excerpt shows how to compute Δy/Δx for y = x²:

1. Set up the ratio: Δy/Δx = [y(x + Δx) − y(x)] / Δx
2. Expand: (x + Δx)² − x² = x² + 2x·Δx + (Δx)² − x²
3. Cancel leading terms: x² and −x² cancel
4. Divide by Δx: [2x·Δx + (Δx)²] / Δx = 2x + Δx

🔍 First-order vs second-order terms

Term	Form	Behavior after dividing by Δx	Importance
First-order	2x·Δx	Becomes 2x	Dominates; responsible for most of Δy
Second-order	(Δx)²	Becomes Δx (still small)	Vanishes as Δx shrinks

• The first-order term 2x·Δx is proportional to Δx; after division, it leaves 2x.
• The second-order term (Δx)² is proportional to (Δx)²; after division, it becomes Δx, which disappears as Δx approaches zero.
• Example: if Δx = 0.01, then (Δx)²/Δx = 0.01, which is negligible compared to 2x.

🎓 The limit definition of slope

📉 Slope at x as a limit

The slope at x is the limit of Δy/Δx = [y(x + Δx) − y(x)] / Δx as Δx becomes very small.

• The distance across (from x to x + Δx) is Δx.
• The distance up (from y(x) to y(x + Δx)) is Δy.
• As Δx shrinks, the ratio Δy/Δx approaches a definite number: the instantaneous slope.
• For y = x², the ratio 2x + Δx approaches 2x as Δx approaches zero.
• Therefore, the slope at any point x is 2x.

⚠️ What not to do

• Do not try to define dy and dx separately as "infinitesimally small numbers."
• Do not try to give meaning to 0/0 directly.
• The successful plan: recognize that the ratio Δy/Δx is clearly defined for any nonzero Δx, and this ratio can approach a limit.

🌟 Generalizing to y = xⁿ

🔢 The pattern

• The excerpt mentions "y = xⁿ" in the title, indicating that the same method applies to higher powers.
• The algebraic expansion will again produce leading terms that cancel, a first-order term that dominates, and higher-order terms that vanish.
• The slope (derivative) will be a function of x, computed by the same limit process.

🧭 Overview

🧠 One-sentence thesis

The slope at a point is defined as the limit of the ratio Δy/Δx as Δx becomes very small, and for y = x², this limit equals 2x.

📌 Key points (3–5)

• What slope at a point means: the limit of the average slope Δy/Δx as the horizontal step Δx shrinks toward zero.
• How algebra leads to calculus: compute the ratio (y(x + Δx) − y(x))/Δx algebraically, then examine what happens as Δx becomes very small.
• Key algebraic pattern for y = x²: the leading x² terms cancel, leaving 2x·Δx as the "first-order term" and (Δx)² as the "second-order term."
• Common confusion: we are not dividing 0 by 0; instead, the ratio Δy/Δx is well-defined for small Δx, and we look at its limit.
• Why this matters: this two-step process (algebra then limit) extends to all power functions y = xⁿ and forms the foundation of differential calculus.

🔍 The central question of differential calculus

🔍 What "slope at a point" means

• At a single moment or single point, nothing actually changes—so what does "rate of change" mean?
• The excerpt poses this as the crucial question: how to define and compute slope when we are at one point on a curve like y = x².

🎯 The two-step answer

1. Algebra step: compute the average slope Δy/Δx over a small interval from x to x + Δx.
2. Calculus step: let Δx become very small and find the limit of that ratio.

Slope at x: the limit of (Δy/Δx) = (y(x + Δx) − y(x))/Δx as Δx shrinks.

• The excerpt emphasizes that we do not try to define dy, dx, or 0/0 individually.
• Instead, the ratio Δy/Δx is clearly defined for any nonzero Δx, and we examine what value it approaches.

🧮 Algebraic calculation for y = x²

🧮 Setting up the ratio

• Horizontal distance: from x to x + Δx, so the step is Δx.
• Vertical distance: from y(x) to y(x + Δx), so Δy = (x + Δx)² − x².

The ratio becomes:

• Δy/Δx = ((x + Δx)² − x²)/Δx

🔢 Expanding and simplifying

Expand (x + Δx)²:

• (x + Δx)² = x² + 2x·Δx + (Δx)²

Substitute into the ratio:

• Δy/Δx = (x² + 2x·Δx + (Δx)² − x²)/Δx

Cancel the leading x² terms:

• Δy/Δx = (2x·Δx + (Δx)²)/Δx = 2x + Δx

🎯 First-order vs second-order terms

Term	Name	Role
2x·Δx	First-order term	Responsible for most of Δy; survives division by Δx as 2x
(Δx)²	Second-order term	After dividing by Δx, becomes Δx, which disappears as Δx → 0

• The excerpt highlights that the "leading terms" x² and −x² cancel.
• The important term is 2x·Δx, which after division gives 2x.
• The second-order term (Δx)²/Δx = Δx is "still small" and will vanish in the limit.

🚀 Taking the limit

🚀 What happens as Δx becomes very small

• After simplification, Δy/Δx = 2x + Δx.
• As Δx shrinks toward zero, the ratio approaches 2x.
• Therefore, the slope at x is 2x.

Example: At x = 3, the slope is 2·3 = 6; at x = 5, the slope is 2·5 = 10.

⚠️ Don't confuse: ratio vs individual terms

• We are not defining dy or dx separately.
• We are not computing 0/0.
• We are computing a well-defined ratio for small but nonzero Δx, then examining its limit.
• The excerpt warns: "Trying to define dy and dx and 0/0 is not wise, and I won't do it."

🔗 Connection to general power functions

🔗 Extension to y = xⁿ

• The excerpt's title mentions "y = x² and y = xⁿ," indicating this algebraic method generalizes.
• The same two-step process (expand, cancel leading terms, divide, take limit) applies to any power function.

🔗 Broader context from practice questions

The excerpt includes practice questions that reinforce:

• Drawing slope graphs from function graphs (when f is increasing, slope s is positive; when f is at max/min, slope is zero).
• Recovering the function f from its slope df/dt requires knowing a starting height f(0).
• The area under the slope graph relates back to the change in the function.

These questions show that the slope concept applies to any function f(t), not just polynomials.

🧭 Overview

🧠 One-sentence thesis

This section provides summaries and practice problems for video lectures on differential and integral calculus, covering derivatives, integrals, optimization, and key functions like exponentials and trigonometric functions.

📌 Key points (3–5)

• Purpose: Companion material for "Highlights of Calculus" video lectures available on MIT OpenCourseWare.
• Structure: Contains summaries and practice questions for 15 video lectures covering differential and integral calculus topics.
• Key topics: Maximum/minimum problems using derivatives, the big picture of integrals, derivatives of sine/cosine, product/quotient/chain rules, and growth rates.
• Common confusion: Understanding when to use first derivative (slope = 0 for max/min) versus second derivative (bending direction to confirm max vs min).
• Format: Each lecture summary includes key concepts, worked examples, and practice questions for self-study.

📐 Maximum and Minimum Problems

📐 Finding extrema using derivatives

To find maximum and minimum values of a function y(x): solve dy/dx = 0 to find points x* where slope = zero, then test each x* for a possible minimum or maximum.

• The derivative equals zero at peaks and valleys of a curve.
• Not every zero-slope point is a max or min; further testing is needed.
• Example: For y(x) = x³ - 12x, solving 3x² = 12 gives x* = 2 and x* = -2.

🔍 Second derivative test

The second derivative d²y/dx² tells us about bending:

Second derivative	Meaning	Type of point
d²y/dx² > 0	Curve bends upward	Minimum
d²y/dx² < 0	Curve bends downward	Maximum
d²y/dx² = 0	Bending changes direction	Point of inflection

• At x* = 2: second derivative = 6(2) = 12 > 0 → minimum (bending up).
• At x* = -2: second derivative = 6(-2) = -12 < 0 → maximum (bending down).
• Example: y = sin x + cos x has maximum at x = π/4 radians (45 degrees) where y = √2.

🎯 Word problem strategy

• Choose a suitable variable x to represent the quantity you can control.
• Express the quantity to optimize as a function of x.
• Example: Minimum of y = (x-1)² + (x-2)² + (x-6)² occurs at x = 3, the average of 1, 2, and 6.

🔗 The Big Picture of Integrals

🔗 Integration as reverse differentiation

Key problem: Recover the integral y(x) from its derivative dy/dx.

• Integral calculus reverses differential calculus: if you know the slope history, you can find the original function.
• Notation: y(x) = ∫(dy/dx)dx adds up the whole history of slopes.
• Example: If dy/dx = x³, then y(x) = (1/4)x⁴ + C (any constant C works).

📊 Three interpretations of integrals

Method 1 - Recognition: Recognize dy/dx as the derivative of a known function.

• If dy/dx = e^(2x), then y = (1/2)e^(2x) + C.

Method 2 - Area under curve: The integral equals the area under the graph of the derivative.

• For s(t) = 6t (speed), the distance y(t) = 3t² equals the triangular area under the speed graph.

Method 3 - Sum of small steps: Add up rectangles of height s(t*) and width Δt.

• As Δt → 0, the sum of rectangles → exact integral.
• Each rectangle: s(t*)Δt = height × base = area.

🎓 Fundamental Theorem of Calculus

Two parts connect derivatives and integrals:

1. Derivative of integral: If f(x) = ∫ₐˣ s(t)dt, then df/dx = s(x).
2. Integral of derivative: If df/dx = s(x), then ∫ₐᵇ s(x)dx = f(b) - f(a).

• The integral from start to end equals the total change.
• Example: Area under s(t) = eᵗ from 0 to t is y(t) = eᵗ - 1 (not just eᵗ, because y(0) must equal 0).

🌊 Derivatives of Sine and Cosine

🌊 Key derivative formulas

d/dx(sin x) = cos x and d/dx(cos x) = -sin x (when x is measured in radians).

• Radians are essential: 2π radians = 360 degrees = full circle.
• On a unit circle (radius 1), x radians corresponds to arc length x.
• At x = 0: sin x has slope 1 = cos 0; cos x has slope 0 = -sin 0.

📏 Proving the sine derivative

The key limit: (sin x)/x → 1 as x → 0.

Geometric argument:

• Draw a right triangle with angle x on a unit circle.
• Straight piece (sin x) < curved arc (x) < longer straight piece (tan x).
• This gives cos x < (sin x)/x < 1, which "squeezes" the limit to 1.

Using trigonometry:

• sin(x + Δx) = sin x cos Δx + cos x sin Δx.
• Δy = sin(x + Δx) - sin x = (sin x)(cos Δx - 1) + (cos x)(sin Δx).
• Divide by Δx and let Δx → 0: dy/dx = (sin x)(0) + (cos x)(1) = cos x.

🔄 Important properties

• The derivative of (sin x)² + (cos x)² equals zero (confirming this sum always equals 1).
• Second derivative of sin x is -sin x; second derivative of cos x is -cos x.
• Maximum of y = sin x + cos x occurs where cos x = sin x, at x = π/4.

⚙️ Product, Quotient, and Chain Rules

⚙️ Product Rule

The derivative of f(x)g(x) is f(x)(dg/dx) + g(x)(df/dx).

• Geometric picture: Change in area = top strip + side strip.
• Δy = f(x + Δx)Δg + g(x)Δf (approximately).
• Example: For y = x² sin x, dy/dx = x²(cos x) + (sin x)(2x).

➗ Quotient Rule

If y = f(x)/g(x), then dy/dx = [g(df/dx) - f(dg/dx)]/g².

• Numerator: "bottom times derivative of top minus top times derivative of bottom."
• Denominator: bottom squared.
• Example: d/dx(sin x / cos x) = d/dx(tan x) = 1/cos²x = sec²x.
• Example: d/dx(1/x⁴) = -4/x⁵, matching the power rule for negative exponents.

🔗 Chain Rule

For z = f(g(x)), the derivative is dz/dx = (dz/dy)(dy/dx) where y = g(x).

• Think of it as "canceling dy" in the notation.
• Must substitute back: Replace y with g(x) in the final answer.
• Example: z = (x⁵)⁴ = x²⁰ has dz/dx = 4y³ · 5x⁴ = 20(x⁵)³x⁴ = 20x¹⁹.
• Example: z = cos(4x) has dz/dx = -sin(4x) · 4 = -4sin(4x).

Don't confuse: The chain rule applies when one function is "inside" another, not when they're multiplied (that's the product rule).

📈 Growth Rates and Special Functions

📈 Comparing growth rates

Functions ordered from slow to fast growth as x gets large:

Type	Examples	Growth speed
Logarithmic	log x, √x	Very slow
Polynomial	x, x², x³	Moderate
Exponential	2ˣ, eˣ, 10ˣ	Fast
Factorial	x!, xˣ	Very fast

• At x = 1000: log 1000 = 3, but 1000¹⁰⁰⁰ = 10³⁰⁰⁰.
• Logarithms are exponents: log(10⁹) = 9.
• For negative powers: 1/x² decays much slower than e⁻ˣ.

📊 Logarithmic scales

Log scale: Plot x = 1, 10, 100, 1000 equally spaced (no zero point).

• Equal spacing represents equal ratios, not equal differences.
• Example: x = 1, 2, 4, 8 are equally spaced on a log scale.

Log-log graph: Both axes use log scale.

• If y = Axⁿ, then log y = log A + n log x (a straight line).
• The slope of the line equals the exponent n.

Semilog graph: Only y-axis uses log scale.

• If y = Abˣ, then log y = log A + x log b (a straight line).
• Useful for exponential growth or decay.

🔄 Inverse functions and logarithms

If y = eˣ, then the inverse function is x = ln y (natural logarithm).

• ln y is the exponent in y = eˣ.
• Key property: ln(yY) = ln y + ln Y (add logarithms because you add exponents).
• ln(yⁿ) = n ln y (multiply exponent).
• Change of base: ln y = (ln 10)(log y), where log means base-10 logarithm.

🎯 Practical Applications

🎯 Linear approximation

Near x = a, the function f(x) ≈ f(a) + (x - a)f'(a) follows its tangent line.

• The tangent line has known height f(a) and slope f'(a).
• Example: eˣ ≈ 1 + x near x = 0 (the linear part of the series).
• Example: x¹⁰ ≈ 1 + 10(x - 1) near x = 1, so (1.1)¹⁰ ≈ 2.

🔍 Newton's Method

Iterative method to solve f(x) = 0:

• Start with guess a where f(a) and f'(a) are known.
• Follow the tangent line to where it crosses zero: x ≈ a - f(a)/f'(a).
• Use this new x as the next starting point and repeat.

Example: Solve x² = 1.2.

• Start at a = 1: f(1) = -0.2, f'(1) = 2, so x ≈ 1 - (-0.2)/2 = 1.1.
• Start at a = 1.1: f(1.1) = 0.01, f'(1.1) = 2.2, giving even better x.

🌱 Differential equations of growth

Exponential growth: dy/dt = cy has solution y(t) = y(0)eᶜᵗ.

With constant source: dy/dt = cy + s has solution y(t) = -s/c + Aeᶜᵗ.

• Constant solution y = -s/c when spending s balances income cy.
• Choose A so y(0) matches the starting value.

Logistic equation: dP/dt = cP - sP² models population with competition.

• Transform to linear equation using y = 1/P.
• Population approaches limit c/s as t → ∞ (S-curve growth).

This section provides foundational material for self-study alongside video lectures, with emphasis on understanding concepts through examples and practice.

🧭 Overview

🧠 One-sentence thesis

Graphs reveal the essential behavior of functions—growth, bending, maxima, minima—through visual inspection of signs (positive/negative/zero) in the function itself, its slope, and its second derivative, whether or not a formula is available.

📌 Key points (3–5)

• What graphs reveal without formulas: from a rough graph of y(x), you can extract information about slope (dy/dx), bending (d²y/dx²), and area, especially by identifying where these are positive, negative, or zero.
• Special situations at zero: when dy/dx = 0 (local max/min), when d²y/dx² = 0 (inflection point), or when y(x) = 0 (crossing the axis), the function exhibits critical behavior.
• Common confusion—which graph are you looking at?: if the given graph is y(x), dy/dx, or d²y/dx², the interpretation of maxima, minima, and inflection points changes completely.
• Chain rule discovery via calculator: graphing sin(kx) and its numerical derivative reveals that the derivative is proportional to k·cos(kx), leading to the general chain rule pattern.
• Why computing matters: computer-based graphics let you see functions, adjust viewing windows, zoom in/out, and discover mathematical ideas experimentally rather than passively.

📊 Reading graphs without formulas

📊 What a graph of y(x) tells you directly

• The excerpt emphasizes extracting "basic information about the growth rate (the slope) and the minimum = maximum and the bending (and area too)."
• Growth rate (slope): where y is increasing, dy/dx > 0; where y is decreasing, dy/dx < 0.
• Maxima and minima: occur where dy/dx = 0 (the slope changes sign).
• Bending (concavity): controlled by d²y/dx² (the slope of the slope); where d²y/dx² changes sign, there is an inflection point.
• Example: a smooth hill on the graph of y(x) has a local maximum at the peak (dy/dx = 0) and an inflection point where the curve changes from bending upward to bending downward.

🔄 When the graph is dy/dx instead of y(x)

"Suppose this is the graph of dy/dx, the derivative of y(x). Answer the following questions about y(x), the original function."

• Local minimum of y(x): occurs where dy/dx crosses from negative to positive (dy/dx = 0 at that point).
• Local maximum of y(x): occurs where dy/dx crosses from positive to negative.
• Inflection point of y(x): occurs where dy/dx has a local max or min (i.e., where d²y/dx² = 0).
• Don't confuse: if you see a peak in the graph and it's labeled dy/dx, that peak is not a maximum of y; it's where the slope of y is largest (an inflection point of y).

🔄 When the graph is d²y/dx² (second derivative)

"Suppose this is the graph of the second derivative d²y/dx² (slope of the slope)."

• Inflection point of y(x): occurs where d²y/dx² = 0 (the graph crosses the x-axis).
• Local min/max of y(x): cannot be determined from d²y/dx² alone; you need information about dy/dx.
• The excerpt notes: "If any of these questions can't be answered, explain why."
• Example: if d²y/dx² is always positive, y is concave up everywhere, but you cannot tell where y has a minimum without knowing where dy/dx = 0.

🔗 Discovering the chain rule with a calculator

🔗 Experimenting with sin(kx)

The excerpt walks through graphing Y₁ = sin(X) and its numerical derivative Y₂ = nDeriv(Y₁, X, X):

1. Y₁ = sin(X): Y₂ appears to be cos(X).
2. Y₁ = sin(2X): Y₂ appears to be 2·cos(2X).
3. Y₁ = sin(3X): Y₂ appears to be 3·cos(3X).

• Conjecture: "If k is some constant, then the derivative of sin(kx) is [k·cos(kx)]."

🧩 Chains (compositions) and the inner/outer pattern

"Those functions are chains (also called compositions). They can be written in the form Y = f(g(x))."

• For sin(kx), the outer function is f(x) = sin(x) and the inner function is g(x) = kx.
• The excerpt then generalizes: "If y = sin(g(x)), then dy/dx = [cos(g(x))·g'(x)]."
• This is the chain rule: differentiate the outer function, leave the inner function alone, then multiply by the derivative of the inner function.

🧪 Testing non-linear inner functions

• The excerpt suggests Y = sin(√x), so g(x) = √x.
• Conjecture: dy/dx = cos(√x)·(derivative of √x).
• "Check your conjecture by graphing Y and comparing to the graph of the numerical derivative."

📝 General chain rule statement

"Whenever we have a composition of an outer and an inner function, the chain rule applies."

Examples to predict and check:

1. y = (2x + 4)³ → dy/dx = 3(2x + 4)²·2
2. y = cos²(x) = (cos x)² → dy/dx = 2·cos(x)·(−sin x)
3. y = cos(x²) → dy/dx = −sin(x²)·2x
4. y = [sin(x² + 1)]³ → dy/dx = 3[sin(x² + 1)]²·cos(x² + 1)·2x

Don't confuse: cos²(x) means (cos x)² (outer function is squaring), while cos(x²) means cos applied to x² (outer function is cosine).

💻 Computer-based graphics and learning

💻 Why graphics matter

"For calculus, the greatest advantage of the computer is to offer graphics. You see the function, not just the formula."

• The excerpt emphasizes: "The power to see this subject is enormous, because it is adjustable."
• You can change the viewing window, zoom in/out, and experiment with the domain and range.
• "The computer offers the experience of actually working with a function."

🔍 Example 1: Do x³ and 3ˣ meet again?

• At x = 3, x³ = 3ˣ = 27.
• At x = 2, 2³ = 8 < 3² = 9; at x = 4, 4³ = 64 < 3⁴ = 81.
• "If x³ is always less than 3ˣ we ought to know—these are among the basic functions of mathematics."
• The computer can solve x³ = 3ˣ numerically or plot both functions.
• Key insight: "If the graphs cross once, they must cross again—because 3ˣ is higher at 2 and 4." A crossing point near 2.5 is found by zooming in.
• The excerpt notes: "I am less interested in the exact number than its position—it comes before x = 3 rather than after."

🧠 Learning by doing

The excerpt draws three conclusions:

1. "A supercomputer is not necessary."
2. "High-level programming is not necessary."
3. "We can do mathematics without completely understanding it."

Rewritten: "We can learn mathematics while doing it."

• "The hardest part of teaching calculus is to turn it from a spectator sport into a workout. The computer makes that possible."

🔢 Example 3: Finding the special b

"Find the number b for which xᵇ = bˣ has only one solution (at x = b)."

• When b = 2, the second solution is 4 (above 2); when b = 3, the second solution is below 3.
• "If we move b from 2 to 3, there must be a special 'double point'—where the graphs barely touch but don't cross."
• At this special b, the curves are tangent (same slope).
• "This special point b can be found with computer-based graphics. In many ways it is the 'center point of calculus.'"
• The excerpt notes: "This number can be discovered first by experiment."

📐 Example 4: Graphing e^x − x^e

"Graph y(x) = eˣ − xᵉ. Locate its minimum. Zoom in near x = e."

• From the derivatives of eˣ and xᵉ, show that dy/dx = 0 at x = e.
• "Can you see why d²y/dx² > 0 at x = e?"
• This confirms a minimum (concave up at the critical point).

🔎 Zooming and viewing windows

🔎 The power of zoom

"The use of the zoom is the best part of graphing. Not only do we choose the domain and range, we change them."

• The viewing window is controlled by four numbers: limits A ≤ x ≤ B and C ≤ y ≤ D, or opposite corners (A, C) and (B, D), or center (a, b) and scale factors c and d.
• "Clicking on opposite corners of the zoom box is the fastest way."

📉 Example 5: Solving x⁴ − 11x³ + 5x − 2 = 0

• "The first tool is algebra—try to factor the polynomial. That succeeds for quadratics, and then gets extremely hard."
• Two good choices:
  1. (Mathematics) Use the derivative. Solve by Newton's method.
  2. (Graphics) Plot the function and zoom in.
• "This particular function is zero only once, in the standard window from −10 to 10."
• "The graph seems to be leaving zero, but mathematics again predicts a second crossing point. So we zoom out before we zoom in."

🌀 Example 6: Oscillating functions

"Zoom out and in on the graphs of y = cos(40x) and y = x·sin(1/x). Describe what you see."

• These functions oscillate rapidly or have unusual behavior near certain points.
• Zooming reveals structure that is invisible at default scales.

⚠️ Example 7: Limits and machine precision

"What does y = (tan x − sin x)/x³ approach at x = 0?"

• "For small x the machine eventually can't separate tan x from sin x. It may give y = 0."
• "Can you get close enough to see the limit of y as x → 0?"
• Don't confuse numerical error with mathematical truth: the computer has finite precision.

🧮 Symbolic computation and ideas

🧮 What symbolic computation does

"In symbolic computation, answers can be formulas as well as numbers and graphs."

• The derivative of y = x² is seen as "2x"; the derivative of sin(t) is "cos(t)."
• "The computer does more than substitute numbers into formulas—it operates directly on the formulas."

💡 Mathematics is ideas, not just formulas

"Mathematics is not formulas or computations or even proofs, but ideas."

• "The symbols and pictures are the language."
• "The book and the professor and the computer can join in teaching it."
• "Your part is to learn by doing."

🧩 Example 8: Factorials and counting

A computer algebra system quickly finds 100! (158 digits, last 24 are zeros).

• Question 1: How many digits (approximately) are in N!?
  • The excerpt notes: "It will never show more than N² digits, because none of the N terms can have more than N digits."
  • "A much tighter bound would be 2N, but is it true? Does N! always have fewer than 2N digits?"
• Question 2: How many zeros (exactly) are at the end of N!?
  • For 10! = 3,628,800, there are two zeros: one from 10, the other from 5×2.
  • "Can you explain the 24 zeros in 100!? An idea from the card game blackjack applies here too: Count the fives."
  • Hard question: How many zeros at the end of 200!?

✍️ Writing in calculus

"May I emphasize the importance of writing? We totally miss it, when the answer is just a number."

• "A one-page report is harder on instructors as well as students—but much more valuable."
• "You can't write sentences without being forced to organize ideas—and part of yourself goes into it."
• Proposed exercise: "Follow through on Examples 1–4 above and report. Without a computer, pick a paragraph from this book that should be clearer and make it clearer."
• "Ideas are like surfaces—they can be seen many ways. Mathematics can be learned by talking."

🧭 Overview

🧠 One-sentence thesis

Calculus connects velocity and distance through two fundamental operations—differentiation (finding velocity from distance by computing slope) and integration (finding distance from velocity by computing area under the graph)—which are inverse processes that reveal how these two quantities depend on each other.

📌 Key points (3–5)

• The central relationship: velocity v and distance f are paired functions where each can be recovered from the other through calculus operations.
• Differentiation (slope): the velocity at any time is the slope of the distance graph; this process goes from f to v.
• Integration (area): the distance traveled is the area under the velocity graph; this process goes from v to f.
• Common confusion: negative velocity means backward motion (negative slope on the f-graph), and area below the axis in the v-graph counts as negative distance.
• Functions as the language: v(t) and f(t) are functions that map each input time t to output values (velocity or distance), with domain (all valid times) and range (all possible outputs).

🚗 The fundamental velocity-distance relationship

🚗 The speedometer and odometer analogy

• A car's dashboard shows two instruments measuring related but different quantities:
  • Speedometer: measures velocity v (miles per hour or km/hr)—a rate involving time.
  • Odometer: measures total distance f (miles or km)—no time unit.
• The core question: if you have a complete record of one, can you recover the other?
• Example: if the speedometer record is complete but the odometer is broken, the distance information is still recoverable through calculus (without re-driving the car).

🔄 Two directions of calculus

The entire subject of calculus is built on converting between v and f:

Direction	Name	What it does	Symbol
f → v	Differentiation	Finds velocity from distance record	Differential calculus
v → f	Integration	Finds distance from velocity history	Integral calculus

📏 Constant velocity: the simplest case

📏 Linear distance from constant speed

When velocity is fixed at v = 60 (mph), distance increases at a constant rate:

• After 2 hours: f = 120 miles
• After 4 hours: f = 240 miles
• General formula: f = vt (distance equals velocity times time)
• The graph of f is a straight line—this is called "linear" growth.

Don't confuse: this example assumes the car starts at full velocity instantly (a "step function") and begins from f(0) = 0.

📐 Slope equals velocity

For constant velocity, the relationship is purely algebraic (no calculus needed yet):

Slope = (change in distance) / (change in time) = vt / t = v

• Geometrically: the velocity is the slope of the distance graph.
• Example: from f₁ = 120 at t₁ = 2 to f₂ = 240 at t₂ = 4, the ratio is 120/2 = 60 mph at both points.

📐 Adjusting the starting point

• If distance starts at 20 instead of 0: formula becomes f = 20 + 60t.
• The constant 20 cancels when computing change in distance, so slope is still 60.
• Negative velocity (v = -30) produces a downward-sloping graph (f = -30t), representing backward motion.

🔲 Area under the velocity graph

🔲 Integration as area computation

The opposite of slope is area—this is the key geometric insight:

The distance f is the area under the v-graph.

• When v is constant, the region under the graph is a rectangle.
• Rectangle area = height (v) × width (t) = vt = distance traveled.
• This is integration: going from v to f by computing area.

🔲 The two central facts

1A: The slope of the f-graph gives the velocity v. The area under the v-graph gives the distance f.

These are inverse operations—differentiation and integration undo each other.

🔄 Forward and backward motion

🔄 Handling negative velocity

Example: a car goes forward at speed V, then backward at speed -V:

• Forward part (0 ≤ t ≤ 3): velocity = +V, distance climbs to f(3) = 3V.
• Backward part (3 < t ≤ 6): velocity = -V, distance decreases back to f(6) = 0.
• The car returns to its starting point.

🔄 Negative area interpretation

• Negative velocity makes the distance graph go downward (negative slope).
• Area below the axis in the v-graph is counted as negative.
• Total area "under" the v-graph = 0 when forward and backward motions cancel.

Don't confuse: the velocity function is discontinuous at t = 3 (the needle jumps), and v(3) is undefined. But the distance function f is continuous—no jump in distance, just a corner where the slope changes.

📊 Functions: inputs, outputs, and notation

📊 What a function is

A function assigns one output to each input; v(t) is the value of function v at time t.

• Input: the time t (read "v of t" for v(t)).
• Output: the velocity or distance at that time.
• The function is like a memory bank with a record at each t.

Example formula for forward-back motion:

• v(t) = +V if 0 ≤ t < 3
• v(t) = -V if 3 < t ≤ 6
• v(3) is undefined (discontinuous jump).

📊 Domain and range

Domain: the set of all inputs (valid times).
Range: the set of all outputs (possible distances or velocities).

• Forward-back example: domain of f is 0 ≤ t ≤ 6; range is 0 ≤ f(t) ≤ 3V.
• Range of v contains only two values: +V and -V.

A function is only allowed one value at each input time t.

📊 Linear functions

Every linear function has the form f(t) = vt + C:

• Graph is a straight line.
• v is the slope.
• C is the constant that adjusts the starting point (vertical shift).

🔧 Transforming functions

🔧 Adding/subtracting constants

• f(t) - 2: subtract 2 from all distances → moves graph down (affects range).
• f(t - 2): subtract 2 from time → moves graph right (affects domain).

Example: starting from f(t) = 2t + 1:

• f(t) - 2 = 2t - 1 (shifted down)
• f(t - 2) = 2(t - 2) + 1 = 2t - 3 (shifted right)

🔧 Multiplying (scaling and zooming)

• 2f(t): doubles all distances → stretches graph vertically (doubles the slope).
• f(2t): speeds up time by factor of 2 → everything happens twice as fast, graph compressed horizontally (also doubles the slope).

Don't confuse: these four transformations—f(t) ± C, f(t ± C), kf(t), and f(kt)—each affect the graph differently. The first two are shifts; the last two are zooms/stretches.

🧭 Overview

🧠 One-sentence thesis

The sum of differences between consecutive numbers always equals the last number minus the first, and this algebraic fact—when applied to slopes and areas—reveals the Fundamental Theorem of Calculus without requiring limits or curved graphs.

📌 Key points (3–5)

• Core algebraic insight: For any sequence of numbers f, the differences v add up to f_last − f_first, because middle terms cancel out.
• Connection to slopes: The differences v represent the slopes of a piecewise linear graph of f.
• Connection to areas: The sum of differences equals the total area under the piecewise constant v-graph.
• Common confusion: Don't confuse the slope (velocity v, tax rate) with the accumulated quantity (distance f, total tax)—slopes tell you the rate of change at each point, while f accumulates those changes.
• Why it matters: This algebraic version of the Fundamental Theorem works for step-by-step data before calculus extends it to smooth curves.

🔢 The fundamental difference pattern

🔢 How differences add up

Start with any list of numbers f = 0, 2, 6, 7, 4, 9. Their differences are v = 2, 4, 1, −3, 5.

When you add the differences:

• 2 + 4 + 1 − 3 + 5 = 9
• This equals the last f (which is 9) minus the first f (which is 0)

🧮 Why middle terms cancel

Write out the sum in full:

• (5 − 1) + (12 − 5) + (7 − 12) + (10 − 7)
• The 5's cancel, the 12's cancel, the 7's cancel
• Only 10 − 1 remains

Key principle 1B: The differences of the f's add up to (f_last − f_first).

Example: For f = 1, 5, 12, 7, 10, the differences are 4, 7, −5, 3, which sum to 9 = 10 − 1.

📊 Connecting to slopes and graphs

📈 Piecewise linear f-graph

When you plot the f numbers and connect them with straight lines, you get a piecewise linear graph.

• The slope of each piece equals the corresponding difference v
• Slope = (change in f) / (change in t) = v / 1 = v
• At breakpoints the slope changes suddenly

📉 Piecewise constant v-graph

The v numbers are plotted as horizontal segments (constant over each interval).

• This resembles velocity graphs where speed is constant between time steps
• The v-graph looks like a staircase

Don't confuse: The f-graph connects points with slanted lines; the v-graph uses horizontal steps.

🎯 The area-slope connection

🟦 Area under the v-graph

The total area under the piecewise constant v-graph equals f_last − f_first.

• Each rectangle has base 1 and height v
• Total area = sum of all v's
• This works even if you stop partway through (e.g., at t = 3.5, you count half the last rectangle)

🎓 Fundamental Theorem preview

Key principle 1C: The v's are slopes of f(t). The area under the v-graph from t_start to t_end equals f(t_end) − f(t_start).

This is the Fundamental Theorem of Calculus, proven here using only algebra for piecewise linear functions. Chapter 5 will extend it to smooth curves using limits.

🔬 Pattern examples

🔢 Linear growth (constant velocity)

f = 2, 3, 4, 5, 6, 7
v = 1, 1, 1, 1, 1

• Constant differences → straight line
• Sum of v's = 5, which equals 7 − 2
• Formula: f = t + 2

🟪 Quadratic growth (odd number differences)

f = 0, 1, 4, 9, 16 (perfect squares)
v = 1, 3, 5, 7 (odd numbers)

• The j-th square is f_j = j²
• The j-th difference is v_j = 2j − 1
• Beautiful fact: first j odd numbers sum to j²
• Example: 1 + 3 + 5 + 7 = 16 = 4²

📈 Exponential growth (powers of 2)

f = 1, 2, 4, 8, 16
v = 1, 2, 4, 8

• Both sequences are powers of 2
• f_j = 2^j and v_j = 2^(j−1)
• The smooth version f(t) = 2^t has slope proportional to itself: v(t) = c·2^t

Don't confuse: 2j (linear), j² (quadratic), and 2^j (exponential) grow at very different rates.

🌊 Oscillating motion (discrete sine/cosine)

f = 0, 1, 1, 0, −1, −1, 0
v = 1, 0, −1, −1, 0, 1

• Both sequences repeat with period 6
• The f-graph resembles a sine wave (piecewise linear)
• The v-graph resembles a cosine wave (piecewise constant)
• Sum of v's = 0, confirming f_last − f_first = 0 − 0

💰 Real-world application: income tax

💵 Tax brackets and rates

1991 single filer tax structure:

• Bracket 1 (0 to $20,350): 15% rate
• Bracket 2 ($20,350 to $49,300): 28% rate
• Bracket 3 (above $49,300): 31% rate

The tax function f(x) is piecewise linear with slopes equal to the tax rates.

📐 Point-slope equation

For income x in the top bracket:

• f(x) = $11,158.50 + 0.31(x − $49,300)
• Starting tax (at $49,300) + rate × extra income
• The number $11,158.50 is the tax at the end of the middle bracket

⚠️ Marginal vs average rates

Marginal rate (v): the slope at point x; tax on each additional dollar
Average rate: total tax f(x) divided by total income x

Don't confuse: A 31% marginal rate does not mean 31% of all income is taxed—only the income in that bracket is taxed at that rate. The average rate is always lower than the top marginal rate.

Example: Tax is like accumulated distance (f), tax rate is like velocity (v) at each point.

🔧 Notation and subscripts

🔤 Subscript system

• Numbers in sequence: f₀, f₁, f₂, ..., f_j
• First difference: v₁ = f₁ − f₀
• General difference: v_j = f_j − f_(j−1)
• Differences start at j = 1 (there is no v₀)

📝 Function notation

Instead of discrete numbers, use f(t) for any time t:

• Domain: all possible input values (times)
• Range: all possible output values (distances)
• Example: v(t) = V for t ≤ T, then v(t) = 0 for t > T

⚡ Special cases

🚀 Burst of speed

A car travels at speed V until distance 1, then stops:

• v(t) = V up to t = T, then v(t) = 0
• f(t) = Vt up to t = T, then f(t) = 1
• Area under v-graph = V × T = 1, so T = 1/V

📶 Step function (extreme case)

When V → infinity, the car jumps instantly to distance 1:

• The f-graph becomes a vertical step (unit step function U(t))
• The slope is infinite at the jump point
• The v-graph becomes a "delta function" spike with area 1

Note: This is beyond ordinary calculus—it involves impulses rather than continuous velocities.

🧭 Overview

🧠 One-sentence thesis

Calculus solves the problem of finding instantaneous velocity from a distance curve by taking the limit of average velocities over shrinking time intervals, and conversely finds distance from velocity by computing area under the velocity curve.

📌 Key points (3–5)

• The two central problems: (1) finding distance when velocity changes, and (2) finding velocity (slope) when the distance graph is not a straight line.
• Average vs instantaneous velocity: average velocity is the slope of a straight line between two points on the curve; instantaneous velocity is the limit as those points come together.
• The limiting process: as the time interval h approaches zero, the average velocity (2t + h) approaches the instantaneous velocity (2t).
• Common confusion: instantaneous velocity requires only local information near a point, but finding distance requires the entire velocity history up to that time.
• The Fundamental Theorem connection: if the slope of f(t) produces v(t), then the area under v(t) produces f(t) back again.

🚗 The core problem setup

🚗 Two opposite questions

Calculus addresses two inverse problems:

• Question 1: If velocity v(t) is changing, how do you compute the distance traveled?
• Question 2: If the distance graph f(t) is not a straight line, what is its slope (velocity)?

The excerpt emphasizes these go in opposite directions and require different approaches.

📈 The example pair

The section uses a specific pair throughout:

• Distance: f(t) = t²
• Velocity: v(t) = 2t

When velocity is v(t) = 2t, a physicist says acceleration is constant (equals 2). The speedometer reading goes steadily up.

Example: After 10 seconds the speed is 20 feet per second; after 44 seconds it's 88 feet/second (which equals 60 miles/hour).

📊 Average velocity: the stepping stone

📊 What average velocity means geometrically

Average velocity = (change in f) / (change in t)

• It is the slope of a straight line connecting two points on the distance curve.
• When computing an average, we pretend velocity is constant—returning to the easiest case.
• Example: From t = 10 to t = 11, distance goes from 100 to 121, so average velocity = (121 - 100) / 1 = 21 feet/second.

⚖️ The "Mean Value Theorem" intuition

The excerpt gives a legal analogy:

• If you enter a highway at 1:00 and exit 150 miles away at 3:00, your average speed is 75 mph.
• Even if police never clocked you at exactly 75, "they would have a definite feeling that you must have been doing 75 sometime."

Don't confuse: This is about average over an interval, not the reading at any single instant.

🔬 Finding instantaneous velocity

🔬 The shrinking interval method

To find velocity at exactly t = 10:

1. Compute average over 1 second: 21 ft/s
2. Compute average over 0.5 seconds: 20.5 ft/s
3. Keep reducing the time interval h

The general formula for average between t = 10 and t = 10 + h:

• v_average = [(10 + h)² - 10²] / h = (100 + 20h + h² - 100) / h = 20 + h

As h shrinks toward zero, the average approaches 20.

Conclusion: The velocity at t = 10 is v = 20.

🧮 The general algebraic computation

For any time t (not just t = 10):

• v_ave = [f(t + h) - f(t)] / h
• For f(t) = t²: v_ave = [(t + h)² - t²] / h = (t² + 2th + h² - t²) / h = 2t + h

Key result: As h approaches zero, the average velocity 2t + h approaches v(t) = 2t.

The excerpt notes: "This is the key computation of calculus." It requires algebra (to handle the variable t) and then a limit (as h → 0).

🎯 Why the limit is easy here

The excerpt reassures: "The general theory of limits is not particularly simple... but here we don't need it."

• In this example, the limiting value is easy to identify.
• The average 2t + h clearly approaches 2t as h → 0.

🔄 The opposite direction: velocity to distance

🔄 Integration as the inverse problem

If v(t) = 2t increases linearly with time, what is the distance?

• This goes in the opposite direction (it is integration).
• The Fundamental Theorem of Calculus says: if the slope of f(t) leads to v(t), then the area under the v-graph leads back to f(t).

📐 Computing area under the velocity curve

For v(t) = 2t, the graph forms a triangle:

• Base = t
• Height = 2t
• Area = ½ × base × height = ½ × t × 2t = t²

This matches f(t) = t², confirming the relationship.

🧩 Extended examples and variations

🧩 Time delays (Example 1)

If the car doesn't start until t = 1:

• v = 0 and f = 0 up to that time
• After starting: v = 2(t - 1) and f = (t - 1)²
• The time delay of 1 enters the formulas and shifts the graphs.

🧩 Changing acceleration (Example 2)

If acceleration changes from 2 to another constant a:

• Velocity changes from v = 2t to v = at
• Distance becomes f = ½at²

Quantity	Formula	Note
Acceleration	a	Constant
Velocity	v = at	Slope of velocity curve
Distance	f = ½at²	Notice the factor ½

Special case: If a equals gravitational constant g, then v = gt is the velocity of a falling body (tested by Galileo at the Leaning Tower of Pisa).

🧩 Adding initial velocity (Example 3)

For f(t) = 3t + t²:

• The average velocity calculation includes an extra 3h in the numerator
• This adds 3 to the velocity: v = 3 + 2t
• If Galileo had thrown a weight instead of dropping it, the starting velocity v₀ would add v₀t to the distance.

🤔 Local vs global: a crucial distinction

🤔 Finding slope (local problem)

To find the slope of the f-graph at a particular time t, you don't have to know the whole history.

• Point B moves toward point A on the curve.
• The problem is local—speed is completely decided by f(t) near point A.

🤔 Finding area (global problem)

To find the area under the v-graph up to time t, you do have to know the whole history.

• A short record of speed is not enough to recover total distance.
• We can only know the increase in mileage without earlier information, not the total.

Don't confuse: These are fundamentally different operations—differentiation is local, integration is global.

🤔 Conceptual questions from the excerpt

The excerpt includes diagnostic questions (from Steve Monk, University of Washington):

Secant line question: As point B approaches A on a curve, what happens to the slope of the line connecting them?

• The excerpt notes 57% of students answered correctly.
• This tests understanding of how average slopes approach instantaneous slope.

Two-car comparison: Given velocity graphs for cars C and D:

• Which is going faster at a specific time? (Look at height of velocity graph)
• Which has greater acceleration? (Look at slope of velocity graph)
• If they start together, which is ahead at the end? (Requires imagining the area/distance)

The excerpt warns: "More than half the class got it wrong" on the distance question—you must look at the speed graph and imagine the distance graph.

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🧭 Overview

🧠 One-sentence thesis

Circular motion provides a geometric way to understand sine and cosine functions, revealing that the slope of the sine curve is the cosine curve and the slope of the cosine curve is the negative sine curve.

📌 Key points (3–5)

• Circular motion setup: A ball travels counterclockwise on a unit circle with constant speed 1, where angle equals time t (measured in radians).
• Position and velocity relationship: The ball's coordinates are x = cos t and y = sin t; its velocity is tangent to the circle with upward component cos t and horizontal component −sin t.
• Harmonic oscillation: The "shadow" of the ball moving up and down creates simple harmonic motion with distance f = sin t and velocity v = cos t.
• Key derivative pairs: The slope of sin t is cos t; the slope of cos t is −sin t.
• Common confusion: Speed vs. velocity—the ball has constant speed 1, but the shadow (oscillating mass) has changing velocity that slows to zero at the top and bottom.

🔄 Setting up circular motion

🎯 The basic setup

A ball travels on a circle of radius 1, centered at the origin, where the angle equals the time t.

• The ball moves counterclockwise starting from the point (1, 0) on the x-axis.
• At any time t, the ball is at position x = cos t, y = sin t.
• The angle is measured in radians, not degrees.
• A full circle is completed at t = 2π (not t = 360).

📏 Radians vs. degrees

Measure	Full circle	Conversion
Radians	2π	1 radian ≈ 57.3 degrees
Degrees	360°	1 degree ≈ 0.01745 radians

• Radians are preferred because they make the speed exactly 1.
• The circumference of a unit circle is 2πr = 2π(1) = 2π.
• Traveling distance 2π in time 2π gives speed = 1.

⚡ Speed is constant

• The ball travels at constant speed 1 around the circle.
• Distance = 2π, time = 2π, so speed = 1.
• Don't confuse: speed is constant, but velocity (which includes direction) is always changing.

🎯 Finding velocity from geometry

🔺 The velocity triangle

• When you "let go" of the ball (like releasing a ball on a string), it flies off tangent to the circle.
• The velocity direction is perpendicular to the radius from the center.
• The velocity triangle has the same shape as the position triangle, but rotated 90°.

⬆️ Upward velocity component

The upward component of velocity is cos t, when the upward component of position is sin t.

• This comes from the geometry of the velocity triangle.
• The angle t appears in the velocity triangle as the angle with the vertical.
• Example: At t = 0, the ball is at (1, 0) moving straight up with velocity cos 0 = 1.
• Example: At t = π/2, the ball is at (0, 1) at the top with upward velocity cos(π/2) = 0 (momentarily stopped vertically).

↔️ Horizontal velocity component

• The horizontal velocity is −sin t (negative because initially moving left).
• The velocity satisfies (cos t)² + (−sin t)² = 1, matching the speed of 1.

🔃 Simple harmonic motion

🎭 The shadow concept

• Instead of watching the ball go around, watch its shadow moving up and down on the y-axis.
• The shadow's height matches the ball's height: f(t) = sin t.
• This creates simple harmonic motion—smooth oscillation between +1 and −1.

🌊 Oscillation characteristics

• The mass oscillates between y = 1 and y = −1.
• Unlike a "bang-bang" bouncing motion with constant velocity ±1, this motion slows down to zero at the extremes.
• The mass speeds up in the middle and slows at the top and bottom (like a spring).
• Time for one complete cycle: 2π.

📊 Distance and velocity graphs

• Distance: f(t) = sin t (the height of the shadow).
• Velocity: v(t) = cos t (the upward velocity of the ball, which equals the shadow's velocity).
• The velocity and distance curves are "out of phase"—shifted by π/2.

📐 Slopes of sine and cosine

📈 Slope of the sine curve

When the distance is f(t) = sin t, the velocity is v(t) = cos t.

• The slope of the sine curve at any time t equals cos t.
• At t = 0: sin 0 = 0 and slope = cos 0 = 1 (steepest upward).
• At t = π/2: sin(π/2) = 1 (maximum) and slope = cos(π/2) = 0 (flat).
• At t = π: sin π = 0 and slope = cos π = −1 (steepest downward).
• Key insight: At maximum or minimum, the slope is zero (curve levels off).

📉 Slope of the cosine curve

When the distance is f(t) = cos t, the velocity is v(t) = −sin t.

• This is the "twin pair" of the first relationship.
• Think of starting the time clock at the top of the circle (t = 0 at angle π/2 in the original setup).
• Or watch the ball's horizontal motion: distance across is cos t, velocity across is −sin t.

⚡ Speeded-up motion

• If the ball travels twice as fast, reaching angle 2t at time t:
  • Position: x = cos 2t, y = sin 2t.
  • Speed is now 2 (completes circle in time π).
  • Upward velocity: 2 cos 2t (not just cos 2t).
  • The velocity triangle is twice as big.
• General pattern: if f = sin(kt), then v = k cos(kt).

🔗 Connecting derivatives and integrals

🔄 The fundamental relationship

• Differential calculus: Compute v from f (find slopes).
• Integral calculus: Compute f from v (find areas).
• These are inverse operations.

📊 Area under the cosine curve

• Since the slope of sin t is cos t, the area under cos t equals the change in sin t.
• Example: Area from t = 0 to t = π/2 equals sin(π/2) − sin 0 = 1 − 0 = 1.
• This demonstrates the power of the Fundamental Theorem of Calculus.

📝 Summary of pairs

Distance f	Velocity v	Notes
vt	v	Constant velocity
½at²	at	Constant acceleration
sin t	cos t	Harmonic motion
cos t	−sin t	Harmonic motion (shifted)

🎓 Key formulas and relationships

🔢 The Pythagorean identity

• (cos t)² + (sin t)² = 1
• This comes from x² + y² = 1 for points on the unit circle.
• Applies to both position and velocity triangles.

🎯 Special values to remember

• At t = 0: cos 0 = 1, sin 0 = 0
• At t = π/2: cos(π/2) = 0, sin(π/2) = 1
• At t = π: cos π = −1, sin π = 0
• At t = 2π: cos 2π = 1, sin 2π = 0 (back to start)

🧭 Overview

🧠 One-sentence thesis

Trigonometry extends from right-triangle ratios to circular motion by measuring angles in radians and using fundamental identities derived from the Pythagorean theorem, providing essential formulas needed throughout calculus.

📌 Key points (3–5)

• Six basic ratios: cosine, sine, tangent, secant, cosecant, and cotangent come from the three sides of a right triangle (x, y, r) and remain constant when the triangle is scaled.
• From triangles to circles: circles allow angles beyond 90°, with radians replacing degrees as the natural unit (2π radians = 360°, distance around = rθ).
• Pythagorean identities: cos²θ + sin²θ = 1 and related identities (1 + tan²θ = sec²θ, cot²θ + 1 = csc²θ) flow from x² + y² = r².
• Addition formulas: cos(s - t) = cos s cos t + sin s sin t, derived from equal distances in rotated circles, leads to double-angle and other formulas.
• Common confusion: distinguish cos 2t (cosine of double angle) from cos²t (square of cosine); also sin(-θ) = -sin θ (odd) but cos(-θ) = cos θ (even).

📐 The six trigonometric functions

📐 Basic ratios from a right triangle

The six functions are ratios of the three sides x, y, r of a right triangle:

• cos θ = x/r (near side / hypotenuse), sec θ = r/x = 1/cos θ
• sin θ = y/r (opposite side / hypotenuse), csc θ = r/y = 1/sin θ
• tan θ = y/x (opposite side / near side), cot θ = x/y = 1/tan θ

• The ratios depend only on the angle θ, not the triangle's size—scaling doesn't change them.
• Three functions on the right are reciprocals of the three on the left.
• Tangent equals sine divided by cosine: tan θ = (y/r)/(x/r) = y/x.

📏 Size constraints

Function	Range
|cos θ|, |sin θ|	≤ 1
|sec θ|, |csc θ|	≥ 1
tan θ, cot θ	Any value

• When cos θ → 0, tan θ → infinity (the triangle becomes infinitely steep at 90°).
• Example: at θ = 2π/3, cos θ = -1/2, sin θ = √3/2, tan θ = -√3, sec θ = -2, csc θ = 2/√3, cot θ = -1/√3.

🔄 From triangles to circles

🔄 Why circles extend trigonometry

• Triangles work well up to 90° and are acceptable up to 180°, but fail beyond that (cannot put a 240° angle into a triangle).
• On a circle, angles are measured counterclockwise from the positive x-axis: 90° is straight up, 180° is left, 360° = 0°.
• Each angle yields a point on the circle; coordinates x and y can be negative (but never r).

🌀 Radians replace degrees

The angle θ is measured by the arc length rθ around the circle; when r = 1, the distance is simply θ.

• A full circle has circumference 2πr, so the angle is 2π radians = 360°.
• Key conversions: π radians = 180°, π/2 radians = 90°, 1° = 2π/360 radians, 1 radian ≈ 360°/2π.
• Example: a 45° angle is 1/8 of a circle = 2π/8 radians, and the arc length is 2π/8.

⚖️ Even and odd functions

• Negative angles go clockwise: -θ reverses y but not x or r.
• cos(-θ) = cos θ (even function, no sign change).
• sin(-θ) = -sin θ, tan(-θ) = -tan θ (odd functions, sign changes).
• Periodicity: adding 2π (one full rotation) doesn't change x, y, r, so all six functions have period 2π.

🔢 Fundamental identities

🔢 The Pythagorean identities

From x² + y² = r², dividing by r², x², or y² yields three key identities:

• cos²θ + sin²θ = 1
• 1 + tan²θ = sec²θ
• cot²θ + 1 = csc²θ

• The first identity (cos²θ + sin²θ = 1) is the most important and must be unforgettable.
• Dividing x² + y² = r² by r² gives (x/r)² + (y/r)² = 1, which is cos²θ + sin²θ = 1.
• Dividing by x² gives 1 + (y/x)² = (r/x)², which is 1 + tan²θ = sec²θ.
• These identities allow constant switching between sines and cosines throughout calculus.

🔗 Complementary angle connection

sin θ = cos(π/2 - θ) and cos θ = sin(π/2 - θ)

• The complementary angle is π/2 - θ because the two angles add to π/2 (a right angle).
• This connection allows formulas for cosine to be converted into formulas for sine.

➕ Addition and double-angle formulas

➕ Deriving cos(s - t) from equal distances

• By comparing distances in two identical circles (one rotated), the distance squared from (cos s, sin s) to (cos t, sin t) equals the distance squared from (cos(s-t), sin(s-t)) to (1, 0).
• Expanding both expressions and using cos²θ + sin²θ = 1 yields:

cos(s - t) = cos s cos t + sin s sin t

• Replacing t with -t (and using cos(-t) = cos t, sin(-t) = -sin t) gives:

cos(s + t) = cos s cos t - sin s sin t

🔁 Double-angle formulas

Setting s = t in the addition formula:

cos 2t = cos²t - sin²t = 2cos²t - 1 = 1 - 2sin²t

• The three forms come from substituting cos²t = 1 - sin²t or sin²t = 1 - cos²t.
• Rearranging: cos²t = (1 + cos 2t)/2 and sin²t = (1 - cos 2t)/2 (needed in calculus).

➕ Sine addition formulas

Using the complementary angle connection from cosine formulas:

sin(s - t) = sin s cos t - cos s sin t sin(s + t) = sin s cos t + cos s sin t sin 2t = 2 sin t cos t

• Don't confuse: sin 2x (sine of double angle) vs. 2 sin x (twice the sine).

📏 Distance formula application

📏 Distance between two points

Distance d = √[(x₂ - x₁)² + (y₂ - y₁)²]

• This Pythagorean formula measures the straight-line distance between (x₁, y₁) and (x₂, y₂).
• The x-distance is |x₂ - x₁|, the y-distance is |y₂ - y₁|, and d is the hypotenuse.
• Applying this formula to points on two identical circles (one rotated) leads directly to the addition formula for cos(s - t).

🧭 Overview

🧠 One-sentence thesis

When you plot discrete points sin n (for integer n) instead of the continuous curve sin x, the visual appearance changes dramatically with scale, revealing unexpected patterns—including what looks like 44 separate sine curves—because certain integers n bring sin n close to zero when n is near multiples of π.

📌 Key points (3–5)

• Discrete vs continuous: sin n (points at integers) looks completely different from sin x (continuous curve); the same 1000 points can appear as sine curves or hexagons depending on scale.
• Why patterns emerge: points cluster into apparent sine curves because certain integers n make sin n ≈ 0 (when n is close to multiples of π, like 22 ≈ 7π).
• Counting the curves: there are 44 apparent sine curves starting near heights sin 0, sin 1, …, sin 43; 22 start upward and 22 downward.
• Common confusion: scale matters—tilting or compressing the x-axis changes what you see; the "sine curves" are an illusion created by which points happen to be "close" in value.
• The middle curve: the curve starting at (0,0) returns to zero around n = 7810 because 44N ≈ 14N + π when N ≈ 177.5, giving a period of about 15,620.

📊 What changes with scale

📊 Continuous vs discrete graphs

• sin x (continuous): by x = 10,000 the curve oscillates 10,000/(2π) ≈ 1591 times; too crowded to see individual cycles.
• sin n (discrete points): picks 10,000 individual points from the curve; these points appear to lie on more than 40 separate sine curves.

🔍 The same points look different

The excerpt shows two graphs of the first 1000 points:

• First graph: points seem to lie on sine curves.
• Second graph: most people see hexagons.
• Key insight: "Tilt the second graph and look from the side at a narrow angle"—the narrow angle compresses the x-axis back to the first scale, and the first graph reappears.

"The effect of scale is something we don't think of."

• What is "close" depends on the scale you choose.
• Example: zooming in or out on a computer changes which features you notice.

Don't confuse: the graphs are identical data; only the viewing scale differs.

🔢 Which points are near (0,0)?

🔢 When is sin n close to zero?

A point near (0,0) means sin n ≈ 0.

• sin 1 = 0.84 (not close to zero; starts the seventh sine curve).
• sin 2 = 0.91, sin 3 = 0.14 (because 3 and π ≈ 3.14 are close, so sin 3 ≈ sin(π − 3) ≈ sin 0.14 ≈ 0.14).
• sin 4, sin 5, …, sin 21 are not especially close to zero.

🎯 The first close point: sin 22

• Why: 22/7 ≈ π, so 22 ≈ 7π.
• Then sin 22 = sin(7π − 22) ≈ sin(−0.01) ≈ −0.01.
• This is the first point to the right of (0,0) and slightly below; it begins a curve downward.

🎯 The next close points

• sin 44: 44 ≈ 14π + 0.02, so sin 44 ≈ sin 0.02 ≈ 0.02.
  • This point (44, sin 44) starts the middle sine curve.
• sin 88: similarly close to zero.

Pattern: multiples of 44 stay near zero because 44 ≈ 14π + 0.0177.

🌀 How many sine curves are there?

🌀 Counting the curves

There are 44 curves.

• They begin near the heights sin 0, sin 1, sin 2, …, sin 43.
• Of these 44 curves, 22 start upward and 22 start downward.

🕵️ Why only 42 are visible

The excerpt notes:

• sin 11 ≈ 0.99999 (very close to 1).
• sin 33 ≈ −0.9999 (very close to −1).
• These are so close to the top and bottom that their curves are invisible.

Why sin 11 ≈ 1: because sin 22 ≈ 0, so 11 ≈ (22)/2 ≈ 7π/2, which is near an odd multiple of π/2 where sine peaks.

🔄 Following a single curve

• "It is almost impossible to follow a single curve past the top—coming back down it is not the curve you think it is."
• The discrete points jump between different apparent curves.

🔁 The middle curve and its period

🔁 Points on the middle curve

The middle curve passes through points at n = 0, 44, 88, and every multiple 44N.

🔁 When does it return to zero?

Question: when does 44N come very close to a multiple of π?

• We know 44 ≈ 14π + 0.0177.
• So 44N = (14π + 0.0177)N = 14πN + 0.0177N.
• For 44N ≈ (some integer)π + π, we need 0.0177N ≈ π.
• Solving: N ≈ π / 0.0177 ≈ 177.5.

📐 The period calculation

• At N = 177.5, we have 44N = 7810.
• This is half the period of the sine curve.
• The sine of 7810 is very near zero.
• The actual points on the middle curve near this zero crossing are n = 44 × 177 and n = 44 × 178, with sines just above and below zero.
• Halfway between is n = 7810.

The equation for the middle sine curve is y = sin(x / 7810).

• Its full period is 15,620—beyond the 10,000-point graph.

Example: the fourth point on the middle curve is at n = 4 × 44 = 176, so the point is (176, sin 176).

🧩 Summary table

Concept	Value / Description
Total apparent curves	44
Curves starting upward	22
Curves starting downward	22
First point near (0,0)	sin 22 ≈ −0.01
Middle curve zero crossings	n = 44N, returns to zero around n = 7810
Middle curve period	≈ 15,620
Why patterns emerge	22 ≈ 7π and 44 ≈ 14π + 0.0177

🧭 Overview

🧠 One-sentence thesis

The derivative measures the instantaneous rate of change of a function by taking the limit of average change ratios as the time or space step shrinks to zero.

📌 Key points (3–5)

• What the derivative measures: the instantaneous rate of change (velocity, slope, marginal demand) at a single point, found by taking limits of average rates.
• The limit process is essential: you must first compute the difference f(t + Δt) − f(t), divide by Δt, and only then let Δt approach zero—setting Δt = 0 too early gives the meaningless 0/0.
• Notation flexibility: the derivative can be written as f′(t), df/dt, or v(t); the independent variable can be t (time) or x (position); df/dt is a single symbol, not a fraction you can cancel.
• Common confusion—when the derivative does not exist: at corners or jumps in the graph, the forward and backward average slopes differ, so there is no single limiting slope.
• Sign of the derivative: increasing functions have positive slope; decreasing functions have negative slope (e.g., 1/t has derivative −1/t²).

📐 The formal definition

📐 The derivative as a limit

Derivative: f′(t) = lim(Δt → 0) [f(t + Δt) − f(t)] / Δt

• The numerator f(t + Δt) − f(t) is the change in distance (or height), often written Δf.
• The denominator Δt is the change in time (or position).
• The ratio Δf / Δt is the average velocity (or average slope) over the interval.
• The limit of this ratio as Δt → 0 is the instantaneous velocity (or instantaneous slope).

🔤 Notation and terminology

Symbol	Read aloud	Meaning
f(t)	"f of t"	Value of function f at time t
Δt	"delta t"	A nonzero (usually short) time step
Δf	"delta f"	The change f(t + Δt) − f(t)
Δf / Δt	"delta f over delta t"	Average velocity
f′(t)	"f prime of t"	The derivative at time t
df/dt	"d f d t"	Same as f′(t); Leibniz notation
lim(Δt → 0)	"limit as delta t goes to zero"	The limiting process

• Important: Δf is not "delta times f"; it is the change in f. Similarly Δt is not "delta times t."
• Important: df/dt is a single notation; do not cancel the d's or treat it as a fraction during the definition.
• The step Δt (or h) can be positive or negative; the limit must be the same from both directions.

🔄 Alternative letters

• Independent variable: t (time) or x (position).
• Dependent variable: f, y, u, etc.
• Derivative: df/dt, dy/dx, f′(x), y′(x), etc.
• Example: y = f(x) has slope dy/dx = y′(x).

🧮 Computing derivatives: the three-step method

🧮 The standard procedure

1. Write out the difference: f(t + Δt) − f(t).
2. Divide by Δt: form the ratio [f(t + Δt) − f(t)] / Δt and simplify algebraically.
3. Take the limit: let Δt → 0 to find df/dt.

Critical rule: Do not set Δt = 0 in step 1 or you will get 0/0, which is meaningless. Algebra must happen first.

📊 Example: f(t) = t²

• Step 1: f(t + Δt) − f(t) = (t + Δt)² − t² = t² + 2t·Δt + (Δt)² − t² = 2t·Δt + (Δt)².
• Step 2: Δf / Δt = [2t·Δt + (Δt)²] / Δt = 2t + Δt.
• Step 3: As Δt → 0, the average 2t + Δt approaches the derivative 2t.

📊 Example: f(t) = 1/t

• Step 1: f(t + Δt) − f(t) = 1/(t + Δt) − 1/t.
• Use common denominator: = [t − (t + Δt)] / [t(t + Δt)] = −Δt / [t(t + Δt)].
• Step 2: Δf / Δt = −1 / [t(t + Δt)].
• Step 3: As Δt → 0, this approaches −1/t².

Interpretation: The function 1/t is decreasing, so its derivative is negative.

⚠️ When the derivative does not exist

⚠️ Corners and jumps

• Example: f(t) = 2t for t < 3, then f(t) = 6 (constant) for t ≥ 3.
• For t < 3: df/dt = 2.
• For t > 3: df/dt = 0 (derivative of a constant is zero).
• At t = 3: the graph has a corner; the forward average (0) and backward average (2) differ, so f′(3) does not exist.

Don't confuse: A function can be defined everywhere but fail to have a derivative at isolated points (corners, cusps, jumps).

⚠️ The derivative of a constant function is zero

• If f(t) = c (constant), then f(t + Δt) = c, so Δf = 0.
• Δf / Δt = 0 / Δt = 0 for any Δt ≠ 0.
• Limit: df/dt = 0.

🔢 Basic derivative facts

🔢 Constant velocity

• If f(t) = Vt (constant velocity V), then Δf = V·Δt.
• Δf / Δt = V always.
• Derivative: df/dt = V.
• Example: the derivative of 2t is 2.

🔢 Reciprocal function

• The derivative of 1/t is −1/t².
• The derivative of 1/x is −1/x².
• More generally: the derivative of 4/x is −4/x² (by the same method).

🔢 Increasing vs decreasing

• Increasing function (going upward): positive slope, df/dt > 0.
• Decreasing function (going downward): negative slope, df/dt < 0.
• Example: 1/t decreases as t increases, so its derivative −1/t² is negative.

🧩 The square rule (advanced)

🧩 Derivative of u(x)²

• If u(x) is a function with derivative du/dx, what is the derivative of f(x) = [u(x)]²?
• Not (du/dx)²—you do not square the derivative.
• Compute: Δf = [u(x + Δx)]² − [u(x)]² = [u(x + Δx) + u(x)] · [u(x + Δx) − u(x)].
• Divide by Δx: Δf / Δx = [u(x + Δx) + u(x)] · [Δu / Δx].
• As Δx → 0: u(x + Δx) → u(x) and Δu/Δx → du/dx.
• Square rule: d(u²)/dx = 2u · (du/dx).

🧩 Applications of the square rule

• u = x²: derivative of x⁴ is 2·x²·(2x) = 4x³.
• u = 1/x: derivative of 1/x² is 2·(1/x)·(−1/x²) = −2/x³.
• u = sin x (given du/dx = cos x): derivative of sin²x is 2·sin x·cos x.

Don't confuse: The derivative of u² is not the square of the derivative; it is twice u times the derivative of u.

🧭 Overview

🧠 One-sentence thesis

The derivative of x raised to any power n follows the pattern that the derivative of xⁿ is n times xⁿ⁻¹, and this rule extends to all real exponents and combinations of powers (polynomials), enabling us to solve differential equations and apply calculus to economics and biology.

📌 Key points (3–5)

• The power rule: The derivative of xⁿ is n·xⁿ⁻¹ for any real number n (positive, negative, or fractional).
• How the binomial formula reveals the pattern: Expanding (x + h)ⁿ produces xⁿ + n·xⁿ⁻¹·h + (terms with h², h³, ...), and after subtracting xⁿ and dividing by h, only n·xⁿ⁻¹ survives as h approaches zero.
• Linearity rules for combinations: The derivative of c times f(x) is c times the derivative of f(x); the derivative of f(x) + g(x) is the sum of their derivatives.
• Reversing the process (differential equations): Given dy/dx, we can find y(x), but only up to an arbitrary constant C (the starting value).
• Common confusion—going backward is not unique: If dy/dx = 3x², then y = x³ + C for any constant C, not just y = x³; constants disappear when differentiating but reappear when integrating.

🔢 The power rule and its derivation

🔢 The pattern for integer powers

The excerpt demonstrates the derivative of x³ step by step:

• Start with f(x) = x³, so Δf = (x + h)³ − x³.
• Expand: (x + h)³ = x³ + 3x²h + 3xh² + h³.
• Subtract x³: Δf = 3x²h + 3xh² + h³.
• Divide by h: Δf/h = 3x² + 3xh + h².
• As h → 0, the limit is 3x².

Why this works: The key term is the one with exactly one factor of h, which is n·xⁿ⁻¹·h; all other terms have h², h³, etc., and vanish after dividing by h and taking the limit.

🔢 The binomial formula for (x + h)ⁿ

The binomial formula: (x + h)ⁿ = xⁿ + n·xⁿ⁻¹·h + (other terms with h², h³, ...) + hⁿ.

• There are n parentheses in (x + h)(x + h)...(x + h).
• To get a term with exactly one h, choose h from one parenthesis and x from the rest: n ways to do this.
• Result: the coefficient of xⁿ⁻¹·h is n.
• After subtracting xⁿ, dividing by h, and letting h → 0, the derivative is n·xⁿ⁻¹.

Example: For n = 4, (x + h)⁴ = x⁴ + 4x³h + 6x²h² + 4xh³ + h⁴. The key term is 4x³h, so the derivative of x⁴ is 4x³.

🔢 Extension to all real exponents

The rule holds for:

• Negative powers: The derivative of x⁻¹ = 1/x is −1·x⁻² = −1/x² (from Section 2.1).
• Fractional powers: The derivative of x^(1/2) = √x is (1/2)·x^(−1/2) = 1/(2√x).
• Any real n: The derivative of x^(2.2) is 2.2·x^(1.2).

Don't confuse: x⁻² means 1/x², and x⁻¹/² means 1/√x; negative exponents produce decreasing functions with negative slopes.

🧮 Derivatives of polynomials

🧮 Linearity: multiplying by constants

Rule 2C: The derivative of c times f(x) is c times f'(x).

• If f(x) is multiplied by a constant c, so is f(x + h), so is Δf, and so is Δf/h.
• The limit (the derivative) is also multiplied by c.
• Example: The derivative of 6x³ is 6 times 3x², which is 18x².

🧮 Linearity: adding functions

Rule 2D: The derivative of f(x) + g(x) is f'(x) + g'(x).

• Adding f + g means adding Δf + Δg.
• Divide by h: (Δf + Δg)/h = Δf/h + Δg/h.
• The limit of a sum is the sum of limits: f' + g'.
• Example: The derivative of 6x³ + (1/2)x² is 18x² + x.

🧮 Constants vanish

• The derivative of any constant is zero (a constant just shifts the graph up or down without changing slope).
• Example: The derivative of 9 + 2x − x⁵ is 0 + 2 − 5x⁴ = 2 − 5x⁴.

Why this matters: In differential calculus, constants disappear; in integral calculus (going backward), constants reappear and must be determined from initial conditions.

🔄 Reversing the process: differential equations

🔄 Finding y from dy/dx

• Forward: Given y = x³, we find dy/dx = 3x².
• Backward: Given dy/dx = 3x², we deduce y = x³ + C for any constant C.
• The constant C is the starting value of y when x = 0.

Example: If dy/dx = 2x + 1, then y = x² + x + C (any function of the form x² + x shifted vertically).

🔄 Two challenges in differential equations

1. No systematic method for all slopes: Given dy/dx = (sin x)/x, we have no formula for y(x) yet.
2. Equations mixing y and dy/dx: Many real-world equations involve both the function and its derivative:
   • Economics: dy/dx = 3y/x.
   • Geometry: dy/dx = 3y^(2/3).
   • These require new techniques (Chapter 6).

Roadmap: Chapters 2–4 compute derivatives; Chapter 5 (integrals) finds y from dy/dx; Chapter 6 solves differential equations where y and dy/dx are mixed.

🔄 Application: modeling an epidemic

The excerpt mentions AIDS data through 1988 fitting a cubic polynomial:

• Number of cases: y = 174.6(t − 1981.2)³ + 340.
• This implies dy/dt = 3y/t (not the exponential dy/dt = y typical of other epidemics).
• The cubic growth is dramatically different and suggests a different underlying mechanism.

Don't confuse: Exponential growth (dy/dt = y) reaches a peak quickly; cubic growth (dy/dt = 3y/t) is slower and indicates something is preventing exponential spread.

💰 Applications to economics

💰 Marginal vs. average

Marginal cost/income: the derivative dy/dx, the cost or income of the next unit.

• Average: total cost y divided by quantity x, i.e., y/x.
• Marginal: the rate of change dy/dx as x changes.
• "The average describes the past, the marginal predicts the future."

Example: If the cost is y = x², then:

• Average cost per unit = x²/x = x.
• Marginal cost = dy/dx = 2x (the cost of the next unit is higher).

Why it matters: A firm decides whether to produce one more unit based on marginal cost (dy/dx), not average cost (y/x).

💰 Elasticity of demand

Elasticity E(x): the ratio of relative change in demand to relative change in price, E(x) = (dy/dx) / (y/x).

• Elasticity measures sensitivity: how much does quantity demanded change (in percentage terms) when price changes (in percentage terms)?
• Dimensionless: E is the same whether measured in liters or gallons, dollars or pesos.
• The formula is "marginal divided by average."

Elasticity	Meaning	Example
|E| < 1	Inelastic	Demand changes little with price (necessities like bread)
|E| = 1	Unit elastic	Demand y = c/x; consumers spend the same total amount xy = c at all prices
|E| > 1	Elastic	Demand changes a lot with price (luxuries like caviar)
E = 0	Perfectly inelastic	Fixed supply (e.g., wheat after harvest); y = constant
E = ∞	Perfectly elastic	Unlimited supply at fixed price; x = constant

Example: For demand y = c/x, dy/dx = −c/x² and y/x = c/x², so E = (−c/x²)/(c/x²) = −1 (unit elastic).

💰 Power functions and constant elasticity

• The demand function y = c·xⁿ has constant elasticity E = n.
• Proof: dy/dx = c·n·xⁿ⁻¹, and (dy/dx)/(y/x) = (c·n·xⁿ⁻¹)/(c·xⁿ/x) = n.
• Example: y = 20/√x = 20·x^(−1/2) has E = −1/2 (inelastic); y = x⁻³ has E = −3 (elastic).

Don't confuse: Economists often drop the minus sign in conversation (demand curves slope down, so E is negative), but the definition includes it.

💰 Income elasticity

• Income elasticity: E(I) = (dy/dI)/(y/I), where y is demand and I is income.
• Luxury: E > 1 (demand more than doubles when income doubles).
• Necessity: E < 1 (demand less than doubles).

Example: If you save $500 out of $10,000 income and E = 2, then y = c·I² with c = 5×10⁻⁶. Marginal savings dy/dI = 2c·I = 0.10 (you save 10 cents of the next dollar), even though average savings is 5%.

💰 Supply = demand vs. monopoly

• Perfect competition: Many suppliers; price is set where supply curve meets demand curve.
• Monopoly: One supplier (e.g., electricity, airport); the supplier can raise price and some demand remains.
• Calculus problem shifts: In monopoly, the goal is to maximize profit (find where marginal profit = 0), not to solve supply = demand.

🧭 Overview

🧠 One-sentence thesis

The tangent line at a point on a curve is the straight line that best approximates the curve locally, determined by matching both the point and the slope (derivative) at that point, and it emerges as the limit of secant lines as two points converge.

📌 Key points (3–5)

• What the tangent line is: the unique straight line through a point on a curve that has the same slope as the curve at that point; over a very short range, any curve looks straight and follows its tangent line.
• How to write its equation: use the point-slope form y - f(a) = m(x - a), where m = f'(a) is the derivative (slope) at x = a and f(a) is the height.
• Secant vs tangent: a secant line connects two points on the curve with slope (f(c) - f(a)) / (c - a); as the second point c approaches a, the secant slope approaches the tangent slope df/dx.
• Common confusion: the tangent line equation y - f(a) = f'(a)(x - a) is not the same as y = f'(a)x + b unless you solve for the correct intercept b = f(a) - f'(a)a.
• Why it matters: tangent lines let you approximate curved motion or change with straight-line calculations; they appear in optimization (e.g., when to step off a roller-coaster, when to sell gold) and in understanding instantaneous rates.

📐 The equation of a straight line

📐 General form and slope

A straight line has the form y = mx + b, where m is the slope and b is the y-intercept.

• Why it's straight: if two points satisfy y = mx + b, then every halfway point (and every subdivision) also satisfies it, so the graph cannot curve.
• What the slope means: moving one unit across (in x) means moving m units up (in y); slope = (distance up) / (distance across) = m / 1.
• Example: for y = 2x + 1, going from x = 0 to x = 1 raises y from 1 to 3, a rise of 2.

📌 Point-slope form

The point-slope form of a line through (a, f(a)) with slope m is:
y - f(a) = m(x - a).

• This form immediately shows that when x = a, the factor (x - a) is zero, so y = f(a) as required.
• For the tangent line, set m = f'(a) (the derivative at x = a).
• Example: the curve y = x⁴ - x² + 3 at x = 1 has height f(1) = 3 and slope dy/dx = 4x³ - 2x = 2 at x = 1, so the tangent line is y - 3 = 2(x - 1).

🔧 Finding the intercept b

• If you prefer the form y = mx + b, solve for b using the known point: f(a) = ma + b, so b = f(a) - ma.
• Example: y - 3 = 2(x - 1) expands to y = 2x + 1, where b = 1.

🎯 The tangent line to a curve

🎯 Definition and construction

The tangent line at x = a on the curve y = f(x) is the line through (a, f(a)) with slope f'(a).

• Why it works: over a very short range (zoom in with a microscope or computer), the curve looks straight and follows this line before curving away.
• The tangent line has the same slope as the curve at that one point.
• Example: for y = x³ - 2 at x = 2, height is f(2) = 6 and slope is dy/dx = 3x² = 12, so the tangent line is y - 6 = 12(x - 2) or y = 12x - 18.

🔄 The normal line

The normal line is perpendicular to the tangent line at the same point; if the tangent has slope m, the normal has slope -1/m.

• Rule: slopes of perpendicular lines multiply to give -1.
• Example: if the tangent slope is 12, the normal slope is -1/12, so the normal line is y - 6 = (-1/12)(x - 2).
• Don't confuse: the normal line is not y = (-1/12)x + b with the same intercept as the tangent; you must recalculate b from the point.
• Physical interpretation: light rays and brush fires move in the normal direction (perpendicular to the curve or fire line).

🚀 Application: stepping off a roller-coaster

Example: you are on a track y = x² + 4 and want to reach a friend at (0, 0) by stepping off and traveling in a straight line (the tangent).

• At step-off point x = a, the tangent line is y - (a² + 4) = 2a(x - a).
• For this line to pass through (0, 0), substitute: 0 - (a² + 4) = 2a(0 - a) → -(a² + 4) = -2a² → a = -2.
• So step off at x = -2.

🚗 Application: catching up to a car

Example: a red light is 72 meters away and will turn green in 4 seconds; a waiting car will then accelerate at 3 meters/sec². You want to slow down just enough to catch up without passing.

• At catchup time T, both cars have the same speed and distance (two conditions → tangent condition).
• The other car's speed at time T is 3(T - 4) (delayed by 4 seconds).
• Set your speed V = 3(T - 4) → T = (1/3)V + 4.
• Match distances: your distance VT equals the other car's distance 72 + (1/2)(3)(T - 4)².
• Solve: V = 12 meters/second (about 43 km/hr or 27 mph).

🔗 The secant line and the limit

🔗 Secant line definition

The secant line connects two points (a, f(a)) and (c, f(c)) on the curve; its slope is the average rate of change:
m = (f(c) - f(a)) / (c - a).

• Two-point form: y - f(a) = [(f(c) - f(a)) / (c - a)](x - a).
• This line automatically passes through both points.
• Example: for y = x³ - 2 from x = 2 (height 6) to x = 3 (height 25), the secant slope is (25 - 6)/(3 - 2) = 19, so the secant line is y - 6 = 19(x - 2).

🔄 From secant to tangent

• As the second point c approaches the first point a, the secant line approaches the tangent line.

• The secant slope (f(c) - f(a)) / (c - a) approaches the tangent slope f'(a).

• Fundamental idea of differential calculus:

  f'(a) = limit as c → a of (f(c) - f(a)) / (c - a).

• Equivalently, write c = a + Δx (so c - a = Δx and f(c) - f(a) = Δf):

  df/dx = limit of Δf / Δx as Δx → 0.

• Don't confuse: the limit is not 0/0 (meaningless), but a definite number (the derivative).

📊 Example: sine function

For y = sin x at x = 0:

• Starting point: (0, sin 0) = (0, 0).
• Secant to (c, sin c) has slope (sin c - 0) / (c - 0) = (sin c) / c.
• Secant equation: y = [(sin c) / c] x.
• As c → 0, the limit of (sin c) / c is 1, so the tangent line is y = 1x (slope = 1).

💰 Example: selling gold

You own gold worth √t million dollars in t years; when should you sell and buy a bond (straight interest)?

• Rate of increase (derivative): d(√t)/dt = 1/(2√t).
• Sell when this rate drops to 10% of current value: 1/(2√t) = (√t)/10 → 2t = 10 → t = 5.
• At t = 5, value is √5 ≈ 2.236 million; tangent line is y - √5 = (√5/10)(t - 5).
• At t = 25 on the tangent (bond): y - √5 = (√5/10)(25 - 5) = 2√5 → y = 3√5 ≈ 6.7 million.
• If you kept the gold: √25 = 5 million (less than the bond).

🧮 Key formulas summary

Concept	Formula	Notes
Tangent line	y - f(a) = f'(a)(x - a)	Point-slope form; slope = derivative
Normal line	y - f(a) = (-1/f'(a))(x - a)	Perpendicular to tangent
Secant line	y - f(a) = [(f(c) - f(a))/(c - a)](x - a)	Connects two points; average slope
Derivative as limit	f'(a) = lim (c→a) [(f(c) - f(a))/(c - a)]	Secant slope → tangent slope
Slope of y = mx + b	m	Constant slope; rise/run = m/1

🔍 Common pitfalls

🔍 Intercept confusion

• The tangent line y - f(a) = f'(a)(x - a) expands to y = f'(a)x + [f(a) - f'(a)a].
• The intercept is b = f(a) - f'(a)a, not just f(a).
• Example: tangent at (1, 3) with slope 2 is y = 2x + 1, not y = 2x + 3.

🔍 Normal line slope

• If tangent slope is m, normal slope is -1/m (not -m).
• Example: tangent slope 12 → normal slope -1/12.

🔍 Limit vs 0/0

• As c → a, the secant slope (f(c) - f(a))/(c - a) looks like 0/0, but the limit is the derivative f'(a), a definite number.
• The excerpt emphasizes: "Algebra stays away from 0/0, but calculus gets as close as it can."

🧭 Overview

🧠 One-sentence thesis

The derivatives of sin x and cos x reveal that these functions oscillate in a special way governed by the differential equation y″ = −y, making them fundamental for modeling all oscillatory motion.

📌 Key points (3–5)

• Core result: The derivative of sin x is cos x, and the derivative of cos x is −sin x, proven by the limit definition and trigonometric addition formulas.
• Two critical limits: (sin h)/h approaches 1 and (cos h − 1)/h approaches 0 as h → 0; these are the foundation of all sine/cosine derivatives.
• Second derivatives and oscillation: Both sin x and cos x satisfy y″ = −y, meaning acceleration equals negative distance—the signature of simple harmonic motion.
• Common confusion: A negative second derivative (y″ < 0) does not mean the function is decreasing; it means the slope (y′) is decreasing, so the curve bends downward even while rising.
• Why it matters: This differential equation y″ = −y models springs, swings, heartbeats, alternating current, and economic cycles—all real-world oscillations.

🔍 Proving the derivative of sin x

🔍 The limit definition approach

The derivative of y = sin x is dy/dx = cos x.

• Start from the standard limit:
  • dy/dx = limit as h → 0 of [sin(x + h) − sin x] / h
• Apply the addition formula: sin(x + h) = sin x · cos h + cos x · sin h
• Rearrange into two separate ratios:
  • Δy/h = sin x · [(cos h − 1)/h] + cos x · [(sin h)/h]
• As h → 0, the first ratio → 0 and the second → 1, so dy/dx = 0 + cos x = cos x.

🧮 The two critical limits

These are the heart of the proof:

Limit	Value	Why it matters
(sin h)/h as h → 0	1	Shows the sine curve starts with slope 1 at x = 0
(cos h − 1)/h as h → 0	0	Shows 1 − cos h shrinks faster than h itself

• Don't confuse: Both numerators approach 0, but the ratio behavior depends on which goes to zero more quickly.
• Example: h²/h → 0 (top faster), h/h → 1 (same speed), √h/h → ∞ (bottom faster).

📐 Why (sin h)/h approaches 1

📐 The squeeze theorem

The proof uses geometric inequalities to trap (sin h)/h between two functions that both approach 1.

Step 1: Establish inequalities

• From geometry (Figure 2.11): sin h < h < tan h for small positive h
• Divide through: (sin h)/h < 1 and (sin h)/h > cos h
• So cos h < (sin h)/h < 1

Step 2: Apply the squeeze

• As h → 0, cos h → 1 from below
• The ratio (sin h)/h is caught between cos h and 1
• Both boundaries approach 1, so the ratio must also approach 1

📏 Geometric justification

• Why sin h < h: The straight line PQ (length 2 sin h) is shorter than the circular arc (length 2h), because the shortest distance between two points is a straight line.
• Why h < tan h: Compare areas—the triangular area ½ tan h is larger than the circular sector area ½ h.
• Important: All angles must be in radians; in degrees, the derivative of sin x is not cos x but is reduced by the factor 2π/360.

Example: At h = 0.01 radians, sin(0.01) ≈ 0.01 and (sin h)/h ≈ 0.9999983, very close to 1.

🔄 Why (cos h − 1)/h approaches 0

🔄 Connecting to the sine limit

Start from the Pythagorean identity: (sin h)² + (cos h)² = 1

• Rearrange: 1 − (cos h)² = (sin h)²
• Factor: (1 − cos h)(1 + cos h) = (sin h)² < h²
• Divide by h and by (1 + cos h):
  • 0 < (1 − cos h)/h < h/(1 + cos h)
• As h → 0, the right side → 0/(1 + 1) = 0
• So (1 − cos h)/h is squeezed to 0

🔄 Approximation insight

The inequality also shows that 1 − cos h ≈ ½h² for small h.

• The factor ½ comes from 1 + cos h ≈ 2
• This means the "tangent parabola" 1 − ½h² is close to the top of the cosine curve

🌊 Derivatives and oscillation

🌊 The derivative of cos x

The derivative of y = cos x is dy/dx = −sin x.

• Use the addition formula: cos(x + h) = cos x · cos h − sin x · sin h
• Form the difference quotient:
  • Δy/h = cos x · [(cos h − 1)/h] − sin x · [(sin h)/h]
• Take limits: dy/dx = cos x · 0 − sin x · 1 = −sin x

Graphical check: Shifting the sine curve left by π/2 gives the cosine curve; the derivative also shifts by π/2, confirming the result.

🔁 Second derivatives

The second derivative is the derivative of the derivative; notation: y″ or d²y/dx².

• For y = sin t: y′ = cos t, then y″ = −sin t = −y
• For y = cos t: y′ = −sin t, then y″ = −cos t = −y
• Both satisfy the differential equation y″ = −y

What this means physically:

• y″ is acceleration (rate of change of velocity)
• y″ = −y means acceleration = −distance
• The greater the distance, the greater the restoring force pulling back

Example: Stretch a spring → restoring force pulls it back; push a swing up → gravity brings it down.

📊 Concavity and bending

Condition	Meaning	Graph behavior
y″ > 0	y′ is increasing	Curve bends upward (concave up)
y″ < 0	y′ is decreasing	Curve bends downward (concave down)

• Don't confuse: y″ < 0 does not mean y is decreasing!
  • It means the slope y′ is decreasing
  • Example: At the start of sin t, y is still increasing but y″ < 0 (the arch bends down while going up)

🎯 Simple harmonic motion

🎯 The differential equation y″ = −y

Simple harmonic motion: oscillation governed by d²y/dt² = −y.

• All solutions are combinations: y = A sin t + B cos t
• This models any system where restoring force is proportional to displacement

Real-world examples from the excerpt:

• Springs: stretch → restoring force pulls back
• Swings: push up → gravity brings down
• Hearts: fill and empty
• Balls: bounce
• Electric current: alternates
• Economy: high prices → high production → low prices → …

🎯 Why oscillation is universal

The excerpt emphasizes: "We can't live without oscillations (or differential equations)."

• Calculus models events by equations
• Equation (12), y″ = −y, models oscillation
• Sine and cosine are the only basic solutions to this equation

Example: A function with distance = 5t has velocity df/dt = 5 (constant) and acceleration d²f/dt² = 0 (no oscillation). But y = sin t has acceleration = −y, creating perpetual back-and-forth motion.

🧭 Overview

🧠 One-sentence thesis

Five fundamental differentiation rules—linearity, product, reciprocal, quotient, and power—enable us to find derivatives of complex functions built from simpler pieces whose derivatives we already know.

📌 Key points (3–5)

• What the rules do: combine known derivatives of simple functions (like x and sin x) to find derivatives of sums, products, quotients, and powers.
• Sum rule (linearity): when you add functions, you add their derivatives; when you multiply by constants, multiply the derivatives by those constants.
• Product rule: the derivative of u times v has two terms (u times dv/dx plus v times du/dx), not the product of the two derivatives.
• Common confusion: the derivative of uv is NOT (du/dx)(dv/dx); the product rule requires both u·(dv/dx) and v·(du/dx).
• Power rule generalization: the derivative of x^n is n·x^(n−1) works for all real exponents (negative, fractional, any number), and extends to [u(x)]^n with an extra factor du/dx.

➕ Sum and linearity rules

➕ The sum rule

When we add functions, we add their derivatives.

• The derivative of u(x) + v(x) is du/dx + dv/dx.
• This is the simplest rule and matches intuition.
• Example: the derivative of x + sin x is 1 + cos x.
• Physical interpretation: if you add distances, you add velocities.

🔢 The linearity rule (extended sum rule)

A linear combination au(x) + bv(x) has derivative a·(du/dx) + b·(dv/dx).

• You can add or subtract functions and multiply by constants.
• Example: the derivative of 3x − 4 sin x is 3 − 4 cos x.
• The rule works because limits can be taken separately and added.
• This is called "linearity" because it preserves addition and scalar multiplication.

✖️ Product and reciprocal rules

✖️ The product rule

The derivative of u times v is u·(dv/dx) + v·(du/dx).

• Key point: the derivative of a product has two terms, not one.
• Don't confuse: the derivative of uv is NOT (du/dx)·(dv/dx).
• Example: for x³ times x², the product rule gives x³·(2x) + x²·(3x²) = 2x⁴ + 3x⁴ = 5x⁴, which matches the derivative of x⁵.
• Example: the derivative of x sin x is x cos x + sin x (the "1" from dx/dx is implicit).
• Geometric interpretation: when a rectangle with sides u and v grows, the area change is approximately u·Δv (top strip) plus v·Δu (side strip); the corner piece Δu·Δv becomes negligible.

🔄 The reciprocal rule

The derivative of 1/v(x) is −(dv/dx)/v².

• This comes from applying the product rule to v·(1/v) = 1, whose derivative is 0.
• Example: the derivative of 1/x is −1/x².
• Example: the derivative of 1/cos x (which is sec x) is sin x/cos² x = sec x tan x.
• Dimensional check: if v is in dollars and x in hours, then (dv/dx)/v² has units (dollars/hour)/dollars² = 1/(dollars·hour), matching d(1/v)/dx.

🎯 The square rule (special case)

• When u = v in the product rule, the derivative of u² is 2u·(du/dx).
• Example: the derivative of sin² x is 2 sin x cos x.
• Example: the derivative of cos² x is −2 cos x sin x (minus sign from the derivative of cosine).
• Quick check: since sin² x + cos² x = 1, their derivatives must sum to zero.

➗ Quotient rule

➗ Dividing functions

The derivative of u(x)/v(x) is [v·(du/dx) − u·(dv/dx)]/v².

• This combines the product rule and reciprocal rule: u/v = u·(1/v).
• The v² in the denominator is familiar from the reciprocal rule.
• The numerator is v·(du/dx) − u·(dv/dx) (note the minus sign and order).
• Example: for x⁵/x³ (which is x²), the quotient rule gives [x³·(5x⁴) − x⁵·(3x²)]/x⁶ = (5x⁷ − 3x⁷)/x⁶ = 2x.

📐 Important quotient: tangent

• tan x = sin x / cos x.
• Using the quotient rule: derivative is [cos x·cos x − sin x·(−sin x)]/cos² x = (cos² x + sin² x)/cos² x = 1/cos² x = sec² x.
• Memorize: the derivative of tan x is sec² x.
• At x = 0, this slope is 1 (same as sin x and x at the origin).
• At x = π/2, the tangent curve is vertical (sec² x = ∞).

🔍 Sensitivity of quotients

• The slope generally "blows up" faster than the function because of the square in the denominator.
• Example: the slope of 1/x is −1/x², which is more sensitive.
• Example: the derivative of sin x / x is (x cos x − sin x)/x²; at x = 0 this is formally 0/0 but the true derivative is zero (the function is symmetric, so its derivative must be zero at the center).

🔋 Power rule

🔋 Powers of x

The derivative of x^n is n·x^(n−1).

• This works for all exponents: positive, negative, fractional, any real number.
• For negative powers: use the reciprocal rule on x^n to get d(x^(−n))/dx = −n·x^(−n−1).
• Example: the derivative of x^(−1) is −1·x^(−2) = −1/x².
• Example: the derivative of x^(−2) is −2·x^(−3) = −2/x³.
• Pattern: multiply by the exponent and reduce the exponent by one.

🌱 Fractional powers

• The same formula works for n = p/q (fractions).
• Proof approach: write u = x^(p/q), so u^q = x^p; differentiate both sides using the power rule; solve for du/dx.
• Example: the derivative of x^(1/3) is (1/3)·x^(−2/3); the slope is infinite at x = 0 and zero at x = ∞.
• Example: the derivative of x^(4/3) is (4/3)·x^(1/3); the slope is zero at x = 0 and infinite at x = ∞.

🎨 Power rule for composite functions

The derivative of [u(x)]^n is n·[u(x)]^(n−1)·(du/dx).

• This extends the power rule to powers of any function u(x), not just x.
• The extra factor du/dx is crucial—don't forget it.
• Example: the derivative of (sin x)^6 is 6·(sin x)^5·cos x.
• Example: the derivative of (tan x)^7 is 7·(tan x)^6·sec² x.
• Example: the derivative of (x² + 1)^8 is 8·(x² + 1)^7·(2x).
• For n = 2 this reduces to the square rule: 2u·(du/dx).
• Proof by induction: use the product rule to go from u^n to u^(n+1) = u^n·u.

📊 Summary of trigonometric derivatives

Function	Derivative
sin x	cos x
cos x	−sin x
tan x	sec² x
sec x	sec x tan x
csc x	−csc x cot x
cot x	−csc² x

🧰 Practical notes

🧰 Dimensional analysis

• Always check that dimensions (units) agree on both sides of an equation.
• Example: if v is in dollars and x in hours, dv/dx is in dollars per hour; then −(dv/dx)/v² has units (dollars/hour)/dollars² = 1/(dollars·hour), matching the units of d(1/v)/dx.
• This test ignores constants and signs but prevents major errors.
• Example: Einstein's e = mc² is dimensionally consistent (both sides have dimension mass·distance²/time²).

🧰 Three-factor products

• The derivative of uvw is u·v·(dw/dx) + u·(dv/dx)·w + (du/dx)·v·w (one derivative at a time).
• Geometric interpretation: when a box with sides u, v, w grows, three slabs are added.
• Example: the derivative of x·x·x is x·x + x·x + x·x = 3x².

🧰 What to memorize

• The five rules themselves (linearity, product, reciprocal, quotient, power).
• The derivatives of sin x, cos x, and tan x.
• The pattern: derivative of x^n is n·x^(n−1).
• Don't try to memorize every formula for sec, csc, cot—derive them as needed using the rules.

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🧭 Overview

🧠 One-sentence thesis

A rigorous definition of limits uses epsilon (ε) and delta (δ) to formalize the intuitive idea that a sequence or function can be made arbitrarily close to a target value, which is essential for defining derivatives and proving calculus rules.

📌 Key points (3–5)

• What convergence means: A sequence aₙ → L if, for any tolerance ε > 0, all terms eventually stay within ε of L (not just visit once, but stay there).
• The epsilon-delta definition: For functions, f(x) → L as x → a means: given any ε > 0, there exists δ > 0 such that |f(x) − L| < ε whenever 0 < |x − a| < δ.
• Common confusion: "Getting closer" is not enough—the sequence 1, ½, 1, ⅓, 1, ¼, ... gets arbitrarily close to zero but does not converge because it doesn't stay close.
• Necessary vs. sufficient conditions: If aₙ converges, then aₙ₊₁ − aₙ → 0 (necessary), but the converse is false—the harmonic series partial sums show that steps can shrink to zero while the sequence diverges (not sufficient).
• Why it matters: Limits underpin the definition of derivatives and enable proofs of sum, product, and quotient rules for differentiation.

🎯 What convergence really means

🎯 Four attempted definitions (only one correct)

The excerpt tests four possible definitions of aₙ → 0:

Attempt	Statement	Why it fails (or succeeds)
1	All aₙ below 10⁻¹⁰	Too weak—doesn't force approach to zero
2	Each aₙ₊₁ smaller than aₙ	1.1, 1.01, 1.001, ... is decreasing but converges to 1, not 0
3	At least one aₙ below any small number	1, ½, 1, ⅓, 1, ¼, ... satisfies this but doesn't converge
4	Eventually stay below any small number	✓ Correct: the tail end decides everything

📏 The formal definition

Convergence to zero: For any ε > 0, there exists N such that |aₙ| < ε for all n > N.

• ε (epsilon): An arbitrarily small positive tolerance chosen by a challenger (Socrates in the excerpt's metaphor).
• N: A threshold index—after this point, all terms stay within tolerance.
• The first million (or billion) terms make no difference; only the tail matters.

Example: The sequence 10⁻³, 10⁻², 10⁻⁶, 10⁻⁵, 10⁻⁹, 10⁻⁸, ... converges to zero even though it oscillates, because it eventually stays below any ε.

Counter-example: 10⁻⁴, 10⁻⁶, 10⁻⁴, 10⁻⁸, 10⁻⁴, 10⁻¹⁰, ... does not converge because it repeatedly returns to 10⁻⁴.

🎯 Convergence to any limit L

General convergence: aₙ → L means |aₙ − L| → 0.

For any ε > 0, there exists N such that |aₙ − L| < ε for all n > N.

• Visually: Only finitely many terms lie outside any strip of width 2ε centered on L.
• Negative terms are allowed; distance is measured by absolute value.

Example: The sequence 3/2, 5/4, 7/6, ... converges to L = 1 because the differences ½, ¼, ⅙, ... converge to zero.

🚫 Common pitfalls and necessary vs. sufficient conditions

🚫 The harmonic series divergence

The sequence 1, 1 + ½, 1 + ½ + ⅓, 1 + ½ + ⅓ + ¼, ... (partial sums of the harmonic series) does not converge.

• The fourth term exceeds 2 (since ⅓ + ¼ > ½).
• The eighth term exceeds 2½ (since 1/5 + 1/6 + 1/7 + 1/8 > 4 · 1/8 = ½).
• The sequence grows past any proposed limit L.

Key insight: The steps aₙ₊₁ − aₙ = 1/(n+1) → 0, yet the sequence diverges to infinity.

⚖️ Necessary vs. sufficient

The excerpt proves:

Statement: If [aₙ converges to L] then [aₙ₊₁ − aₙ converges to zero].

• Proof sketch: If |aₙ − L| < ε and |aₙ₊₁ − L| < ε, then |aₙ₊₁ − aₙ| ≤ |aₙ₊₁ − L| + |L − aₙ| < 2ε.
• Logical direction: Convergence is a sufficient condition for shrinking steps; shrinking steps are necessary for convergence.
• The converse is false: Shrinking steps do not guarantee convergence (harmonic series counter-example).

🔄 Five ways to express "A implies B"

Expression	Meaning
A ⇒ B	A implies B
If A then B	Conditional statement
A is sufficient for B	A being true is enough to guarantee B
B is necessary for A	B must be true whenever A is true
B is true if A is true	Another conditional phrasing

Don't confuse: "A ⇒ B" with "B ⇒ A" (the converse). Both require separate proofs. When both hold, write "A ⇔ B" (if and only if).

🔧 Rules for combining limits

🔧 Limit arithmetic for sequences

When aₙ → L and bₙ → M:

• Addition: aₙ + bₙ → L + M
• Subtraction: aₙ − bₙ → L − M
• Multiplication: aₙbₙ → LM
• Division: aₙ/bₙ → L/M (provided M ≠ 0)
• Scalar multiplication: caₙ → cL (constants pass through limits)

🔍 Proof sketch for multiplication rule

The key identity is:

aₙbₙ − LM = (aₙ − L)(bₙ − M) + M(aₙ − L) + L(bₙ − M)

If |aₙ − L| < ε and |bₙ − M| < ε beyond some point, the right side is less than ε² + Mε + Lε, which can be made arbitrarily small.

🔧 Corresponding rules for functions

When f(x) → L and g(x) → M as x → a, the same arithmetic rules apply:

• f(x) + g(x) → L + M
• f(x)g(x) → LM
• etc.

📐 Limits of functions: the epsilon-delta definition

📐 From sequences to functions

Instead of n → ∞ for sequences, we consider x → a for functions.

Function limit: lim_{x→a} f(x) = L means: For any ε > 0, there exists δ > 0 such that |f(x) − L| < ε whenever 0 < |x − a| < δ.

• ε (epsilon): Output tolerance—how close f(x) must be to L.
• δ (delta): Input tolerance—how close x must be to a.
• The game: Challenger picks ε; responder must find a δ that works.
• Crucial detail: We require 0 < |x − a|, meaning x ≠ a; the value f(a) is irrelevant to the limit.

📦 The box visualization

• Socrates chooses the height of a box (2ε) centered on L.
• Plato must choose the width (2δ) narrow enough that the graph exits through the sides, not the top or bottom.
• If this is possible for every ε, then the limit exists.

Example: Prove lim_{x→2} 5x = 10.

• Need |5x − 10| < ε, i.e., 5|x − 2| < ε.
• Choose δ = ε/5; then |x − 2| < δ implies |5x − 10| < ε.

🔀 One-sided limits

One-sided limit: lim_{x→a⁺} f(x) = L means x approaches a only from above (x > a).

Example: lim_{x→1⁺} √(x−1) = 0.

• For ε = 1/10, choose δ = 1/100.
• If 0 < x − 1 < 1/100, then √(x−1) < 1/10.
• The box must be extremely narrow because the slope is infinite at x = 1.

Don't confuse: Ordinary limits require x to approach from both sides; one-sided limits restrict the direction.

🗜️ The Squeeze Theorem

🗜️ Statement and proof

Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) for x near a, and both f(x) → L and h(x) → L as x → a, then g(x) → L as well.

Proof idea:

• After subtracting L, we have f(x) − L ≤ g(x) − L ≤ h(x) − L.
• If |f(x) − L| < ε and |h(x) − L| < ε, then |g(x) − L| < ε.
• For any ε, both outer inequalities hold in some region 0 < |x − a| < δ, so the middle one does too.

Why it matters: The theorem allows us to find limits of complicated functions by bounding them between simpler ones.

---

Note on practical use: The excerpt emphasizes that mathematicians rarely use the ε-δ definition directly in practice. Instead, they establish once that familiar functions (polynomials, sin x, etc.) are continuous, meaning lim_{x→a} f(x) = f(a), and then apply limit rules. The rigorous definition exists to make proofs possible, not to be invoked constantly.

🧭 Overview

🧠 One-sentence thesis

A function is continuous at a point when its limit equals its actual value there, and this property—stronger than merely having a limit but weaker than having a derivative—ensures that the function has no jumps, poles, or wild oscillations at that point.

📌 Key points (3–5)

• Three requirements for continuity: f(a) must exist, the limit of f(x) as x approaches a must exist, and that limit must equal f(a).
• Types of discontinuity: removable (wrong value assigned), jump (one-sided limits differ), pole (function blows up to infinity), and oscillation (no limit exists due to rapid fluctuation).
• Common confusion—continuity vs differentiability: having a derivative is a stronger requirement than continuity; a function can be continuous at a point but have no derivative there (e.g., absolute value at zero).
• Continuable functions: some discontinuous functions can be "fixed" by defining the right value at a problem point (like sin(x)/x at x=0), while others cannot (like 1/x at x=0).
• Two key properties on closed intervals: continuous functions on [a,b] always reach a maximum and minimum, and take on every intermediate value between them.

🔍 What continuity means

🔍 The three conditions

A function f is "continuous at x = a" if f(a) is defined and f(x) approaches f(a) as x approaches a.

The excerpt breaks this into three parts:

1. The number f(a) exists (f is defined at a)
2. The limit of f(x) exists (call it L)
3. The limit L equals f(a) (f(a) is the right value)

• Often written compactly as: f(x) → f(a) as x → a
• Intuitive test: "you can draw its graph without lifting up your pen"
• Example: All polynomials, sin(x), cos(x), and |x| are continuous functions

🎯 Continuous function vs continuous at a point

The excerpt notes a logical subtlety:

• A function like 1/x is technically called a "continuous function" because it's continuous at every point where it's defined
• But it has a discontinuity at x=0 (where it's not defined)
• This creates confusion: we speak of "a discontinuity of 1/x" while calling it a "continuous function"

🚫 Types of discontinuity

🚫 Removable discontinuity

• The limit exists but f(a) has the wrong value (or is undefined)
• Example: A function with limit 0 as x→0 but f(0)=1
• Fix: Simply redefine f(0) to match the limit
• After correction, the discontinuity disappears

📊 Jump discontinuity

• One-sided limits exist but differ
• Example: Step function jumping from 0 to 1 at x=0
  • Limit from left (x→0⁻) is 0
  • Limit from right (x→0⁺) is 1
• Another example: x/|x| jumps from -1 to +1
• Cannot be fixed: no single value makes the function continuous

♾️ Pole (infinite limit)

A function has a "pole" at x=a when the denominator goes to zero and the function goes to +∞ or -∞.

• Example: 1/x² has a pole at x=0 (approaches +∞ from both sides)
• Example: 1/x has one-sided infinite limits (−∞ from left, +∞ from right)
• Simple pole: multiplying by x makes the function continuous at that point
  • Example: x·(1/x) = 1 is continuous at x=0
• Double pole: needs multiplication by x² (not just x) to become continuous
  • Example: 1/x² needs x²·(1/x²) = 1
• Rational functions P(x)/Q(x) have poles where Q=0 (after removing common factors)

🌀 Infinite oscillation

• The function oscillates faster and faster, never settling down
• Example: sin(1/x) as x→0
  • At x=1/3, 1/4, 1/1000 it equals sin(3), sin(4), sin(1000)
  • Oscillates infinitely fast near zero
  • The sine never exceeds 1, but won't stay in any small box of height ε
• Cannot be fixed: no limit exists

🔧 Continuable functions

🔧 What "continuable" means

The excerpt proposes this distinction:

• A function is continuable if its definition can be extended to all x in a way that makes it continuous
• This clarifies the confusion about "continuous functions"

Function	Continuable?	Why
sin(x)/x	Yes	Define f(0)=1 (the limit)
√x	Yes	Already continuous where defined
1/x	No	No value at x=0 makes it continuous
tan(x)	No	Poles cannot be removed

🔧 Important examples

Example: sin(x)/x at x=0

• Limit as x→0 is 1
• Define f(0)=1 to make it continuous
• This was crucial for finding the derivative of sin(x)

Example: (1−cos(x))/x at x=0

• Limit as x→0 is 0
• Define f(0)=0 to make it continuous

Example: (1−cos(x))/x² at x=0

• Limit as x→0 is 1/2
• The excerpt shows the calculation using the identity 1−cos²(x) = sin²(x)
• Result: approaches 1/(1+cos(x)) → 1/2

Don't confuse: sin(x)/x² blows up (simple pole), but (1−cos(x))/x² has a finite limit

🔗 Continuity vs differentiability

🔗 The key distinction

Two different requirements:

Property	Requirement	Strength
Continuous at x	f(x+Δx)−f(x) → 0 as Δx→0	Weaker
Derivative at x	[f(x+Δx)−f(x)]/Δx → f'(x) as Δx→0	Stronger

Key insight from the excerpt:

Asking for a derivative is more than asking for continuity.

• For continuity: Δf goes to zero (maybe slowly)
• For derivative: Δf goes to zero as fast as Δx (because Δf/Δx has a limit)

🔗 The fundamental relationship

Rule 2I: At a point where f(x) has a derivative, the function must be continuous. But f(x) can be continuous with no derivative.

Proof sketch: The limit of Δf = (Δx)·(Δf/Δx) is (0)·(df/dx) = 0, so f(x+Δx)−f(x) → 0.

🔗 Examples of continuous but not differentiable

• |x| at x=0: continuous but slope jumps (no derivative)
• x^(1/3) at x=0: continuous but derivative (1/3)x^(−2/3) blows up
• The excerpt mentions a remarkable function (1/2)cos(3x) + (1/4)cos(9x) + ... that is continuous everywhere but has a derivative at no points

🔗 The derivative controls speed

The excerpt emphasizes:

The derivative controls the speed at which f(x) approaches the limit.

Because f(x)−f(0) is nearly f'(0) times x:

• If derivative of sin(x) is 1 → sin(x) decreases like x
• If derivative of sin²(x) is 0 → sin²(x) decreases faster than x
• If derivative of x^(1/3) is ∞ → x^(1/3) decreases more slowly than x

📐 Properties on closed intervals

📐 What's required

The excerpt specifies: continuous function on a finite closed interval [a,b]

• Closed means endpoints a and b are included
• At endpoints we require f(x) to approach f(a) and f(b)
• Contrast: open interval (a,b) leaves out the endpoints

📐 Extreme Value Property

A continuous function on the finite interval [a,b] has a maximum value M and a minimum value m.

• There are points x_max and x_min in [a,b] where it reaches those values
• f(x_max) = M ≥ f(x) ≥ f(x_min) = m for all x in [a,b]

Why the requirements matter:

• For 0 < x ≤ 1, the function f(x)=x never reaches its minimum (zero)
• If we close the interval but define f(0)=3 (discontinuous), the minimum is still not reached

📐 Intermediate Value Property

If the number F is between f(a) and f(b), there is a point c between a and b where f(c) = F.

• More generally: if F is between the minimum m and maximum M, there's a point c between x_min and x_max where f(c) = F
• Example application: cos(x) and 2x are continuous, so cos(x)=2x at some point between 0 and 1

Why continuity matters: A jump discontinuity can skip over intermediate values—the excerpt notes that if f(0)=3 (with a jump), the intermediate value F=2 is not reached.

🧭 Overview

🧠 One-sentence thesis

Linear approximation uses the tangent line at a known point to estimate function values nearby, turning complicated functions into simple straight-line predictions that are accurate for small changes.

📌 Key points (3–5)

• Core idea: Near any point a, a smooth function behaves almost like its tangent line: f(x) ≈ f(a) + f′(a)(x − a).
• Why it works: Over very short intervals, the function is "nearly constant" in slope, so the tangent line stays close to the curve.
• Error behavior: The approximation error is of order (Δx)² — linear approximation, quadratic error.
• Common confusion: Δy (true change along the curve) vs. dy (change along the tangent line); dy is the approximation to Δy.
• Practical use: Science and engineering linearize nonlinear laws (Ohm's, Hooke's, Newton's) using the derivative as a local multiplier.

📐 The tangent line formula

📐 Equation of the tangent line

At point x = a, the tangent line to y = f(x) is:

Y = f(a) + f′(a)(x − a)

• Capital Y denotes the line; lowercase y denotes the curve.
• Start at the known value f(a), then add (slope) × (horizontal distance from a).
• The slope of the curve and the slope of the line are both f′(a) at x = a.

📐 The approximation statement

The key claim is:

y ≈ Y, or equivalently f(x) ≈ f(a) + f′(a)(x − a)

• This is the "all-purpose linear approximation."
• It is accurate provided we don't move too far from a.
• Example: For y = √x near x = 100, the slope is 1/(2√x) = 1/20, so √102 ≈ 10 + (1/20)(2) = 10.1 (true value ≈ 10.0995).

📐 Alternative notation: starting from x

Instead of going from a to x, we can start at x and move by Δx:

f(x + Δx) ≈ f(x) + (slope at x) · Δx

• The letters are different but the mathematics is identical.
• This form emphasizes the change Δx from a basepoint x.

🔢 Important examples

🔢 Power function approximation

For any exponent n:

(1 + x)ⁿ ≈ 1 + n x for x near zero

• This comes from f(x) = (1 + x)ⁿ with basepoint 0; the slope at 0 is n.
• Example: (1.01)³ ≈ 1 + 3(0.01) = 1.03.
• Changing n to −n gives 1/(1 + x)ⁿ ≈ 1 − n x.

🔢 When approximation fails

The binomial expansion shows hidden terms:

(1 + x)¹⁰⁰ = 1 + 100x + (100)(99)/2 · (x)² + …

• For x = 0.01, the (x)² term ≈ 1/2 is too large to ignore.
• Linear approximation gives 2, but the true value (1.01)¹⁰⁰ ≈ 2.7 (close to e).
• Don't confuse: linear approximation is only valid when the quadratic error (1/2)(x)² f″(c) is negligible.

🔢 Square root example in detail

For y = √x near x = 100:

x	Y (line)	y (curve)	Error
100	10	10	0
102	10.1	10.0995	small
110	10.5	10.49	0.01
200	15	14.1	0.9

• Accuracy worsens as x departs from 100.
• The tangent line is above the curve because the slope is decreasing ("concave downward").
• At x = 102, squaring the approximation: (√100 + (1/20)·2)² = 100 + 2 + (1/400)·4; the last term (Δx)²/400 is the error.

🧮 Differentials notation

🧮 What differentials are

The symbols dx and dy are called differentials:

dx and dy measure changes along the tangent line.

• Until this point, dy/dx was a single symbol (the derivative), not a fraction.
• Now dx and dy become separate variables with independent meaning.
• dx measures horizontal change (same as Δx); dy measures the corresponding vertical change on the tangent line.

🧮 Differential vs. true change

Symbol	Meaning	Where measured
Δy	True change in y	Along the curve
dy	Change in Y (approximation)	Along the tangent line
Δx	Horizontal change	Same for both
dx	Horizontal change (differential)	Same as Δx

• The differential dy equals ΔY, the change along the tangent line.
• dy is the linear approximation to Δy.

🧮 Formula for differentials

dy = f′(x) dx

• This is consistent with the derivative dy/dx = f′(x), but it is a definition, not cancellation.
• Example: y = x² has dy/dx = 2x, so dy = 2x dx.
• At basepoint x = 2 with dx = 0.1: dy = 4(0.1) = 0.4; true Δy = (2.1)² − 4 = 0.41; error = (Δx)² = 0.01.

🧮 Rules for differentials

• d(xⁿ) = n xⁿ⁻¹ dx
• d(sin x) = cos x dx
• d(tan x) = sec² x dx
• d(f + g) = df + dg
• d(cf) = c df
• d(fg) = f dg + g df

🌍 Applications and measurements

🌍 Why linearization matters

Science, engineering, and virtually all applications depend on linear approximation. The true function is "linearized" using its slope v.

• Increasing time by Δt increases distance by ≈ v Δt.
• Increasing force by Δf increases deflection by ≈ v Δf.
• Increasing production by Δp increases value by ≈ v Δp.
• The multiplier v is the derivative; it gives a local prediction of change.
• Ohm's law, Hooke's law, Newton's law are linear approximations to nonlinear reality.

🌍 Three ways to measure change

Type	For f	For x
Absolute change	Δf	Δx
Relative change	Δf / f(x)	Δx / x
Percentage change	(Δf / f(x)) × 100	(Δx / x) × 100

• Relative change is often more realistic than absolute change.
• Example: Knowing the moon's distance within 3 miles (0.001%) is more impressive than knowing your height within 1 inch (1.4%).

🌍 Volume of Earth example

The radius of Earth is within 80 miles of r = 4000 miles. Volume V = (4/3)πr³.

• Derivative: dV/dr = 4πr².
• Variation in volume: dV = 4π(4000)²(80) cubic miles.
• Relative variations:
  • dr/r = 80/4000 = 2%
  • dV/V = [4π(4000)²(80)] / [(4/3)π(4000)³] = 3(80)/4000 = 6%
• A 2% relative change in radius produces a 6% relative change in volume.
• Exact calculation gives ≈6.1%; the error is small.
• Interpretation: dV = (surface area) × (change in radius) = volume of a thin shell added when radius grows by dr.

🌍 Error is quadratic

The exact error in linear approximation is:

Error ≈ (1/2)(Δx)² f″(c), where c is between x and x + Δx

• Linear approximation, quadratic error.
• Example: For y = x² at x = 2, the error is exactly (Δx)².
• When (Δx)² is not negligible, linear approximation breaks down.

🔄 Reversing the logic

🔄 From derivative back to function

The excerpt emphasizes a conceptual shift:

In Chapter 2 we struggled with y to squeeze out dy/dx. Now we use dy/dx to study y.

• Earlier: didn't know dy/dx, so used (slope) ≈ Δy/Δx.
• Now: experts at computing dy/dx, so use Δy ≈ (slope) · Δx.
• "You work with what you have."
• After computing the derivative once, the tangent line stays near the function for every nearby number.

🔄 The local-global connection

The information in dy/dx is entirely local. It tells what is happening close to the point and nowhere else.

• The problem is to connect the finite (Δy) to the infinitesimal (dy).
• The Mean Value Theorem (coming later) assures points where dy/dx equals Δy/Δx, but we cannot predict where.
• Therefore we find other ways to recover a function from its derivatives, or to estimate distance from velocity and acceleration.

🔄 The three stages of f = vt

Statement	Validity
f = vt is completely false	When v changes
f = vt is nearly true	Over very short intervals
df = v dt is exactly true	For the tangent line (differential)

• For a brief moment the function is linear and stays near its tangent line.
• This is the foundation of calculus: local linearity.

🧭 Overview

🧠 One-sentence thesis

Calculus identifies maximum and minimum values by finding where the derivative equals zero (stationary points), then comparing those values with endpoints and rough points to determine the absolute extrema.

📌 Key points (3–5)

• What the derivative tells us: positive slope means the function is increasing; negative slope means decreasing; zero slope indicates a potential maximum or minimum.
• How to find extrema: solve f'(x) = 0 for stationary points, then evaluate f(x) at all critical points (stationary points, rough points, endpoints) and compare.
• Three types of critical points: stationary points where f'(x) = 0, rough points where the derivative doesn't exist, and endpoints of the domain.
• Common confusion: a stationary point (f'(x) = 0) is not always a maximum or minimum—the function can pause and continue in the same direction if there's a double zero.
• Applied problems workflow: first choose the variable x and construct the function f(x) to optimize, then use calculus to solve f'(x) = 0, finally check all critical points.

📈 Reading the derivative

📈 Positive and negative slopes

• Positive derivative (df/dx > 0): the function f(x) is increasing between points a and b.
  • All tangent lines slope upward.
  • For any two points x < X in the interval: f(x) < f(X).
• Negative derivative (df/dx < 0): the function f(x) is decreasing.
  • For any two points x < X: f(x) > f(X).
• Don't confuse: positive slope does not mean positive function—the function itself can be negative while still increasing.

📊 Example patterns

The excerpt gives f'(x) = (x - 1)(x - 2)(x - 3)(x - 4):

• This slope is positive beyond x = 4 and up to x = 1.
• Also positive between x = 2 and x = 3.
• At x = 1, 2, 3, 4 the slope is zero and f(x) changes direction.
• The graph goes up-down-up-down-up.

🎯 Finding maxima and minima

🎯 The zero-derivative condition

Local maximum or minimum: If the maximum or minimum occurs at a point x inside an interval where f(x) and df/dx are defined, then f'(x) = 0.

Why the derivative must be zero:

• At a maximum, f(x + Δx) - f(x) ≤ 0 for any step Δx.
• If Δx < 0: the difference divided by Δx gives a ratio ≥ 0, so df/dx ≥ 0 in the limit.
• If Δx > 0: the ratio ≤ 0, so df/dx ≤ 0 in the limit.
• Both conditions together force df/dx = 0.
• The tangent line is level (horizontal).

⚠️ Exceptional cases

A function can have f'(x) = 0 without a maximum or minimum:

• Example: f(x) = 4x³ - 3x⁴ has f'(x) = 12x² - 12x³ = 0 at x = 0 and x = 1.
• At x = 0: f(0) = 0 is neither a local maximum nor minimum—the curve pauses but continues in the same direction.
• Reason: the "double zero" from the factor x² in 12x²(1 - x).
• At x = 1: f(1) = 1 is an absolute maximum (the function goes to negative infinity in both directions).

🔍 Three types of critical points

All maxima and minima occur at critical points:

Type	Description	Example
Stationary point	f'(x) = 0	Top or bottom of a smooth curve
Rough point	Derivative doesn't exist	Corner in the graph, like y = |x| at x = 0
Endpoint	Boundary of the domain	Maximum of |x| on [-3, 2] occurs at x = -3

Don't confuse: critical points (specified by x-values) vs. critical values (specified by f(x) values).

🔧 The optimization procedure

🔧 Three-step method

1. Solve f'(x) = 0 to find stationary points.
2. Compute f(x) at every critical point—stationary points, rough points, endpoints.
3. Compare those values to identify the maximum and minimum.

📐 When infinity is involved

• A function can approach but never reach a value: f(x) = 1/(1 + x²) approaches but never reaches zero as x → ∞.
• We still say "the minimum is zero" even though it's not attained.
• Exception: if f is continuous on a closed interval [a, b], then f(x) reaches its maximum and minimum (Extreme Value Theorem).

🌍 Applied problems

🌍 The real challenge: formulation

In applications, the hardest step is often choosing the variable and constructing the function:

• The problem starts with a question, not a function.
• You must decide what x represents and express the quantity to optimize as f(x).
• Only then does calculus enter: compute f'(x) and solve f'(x) = 0.

🚗 Expressway example

Problem: Where to enter an expressway to minimize driving time? Expressway speed 60 mph, ordinary speed 30 mph.

Setup:

• Distance to expressway: √(a² + x²) at 30 mph takes √(a² + x²)/30 hours.
• Distance on expressway: (b - x) at 60 mph takes (b - x)/60 hours.
• Total time: f(x) = (1/30)√(a² + x²) + (1/60)(b - x).

Solution:

• Derivative: f'(x) = (1/30) · (1/2)(a² + x²)^(-1/2) · 2x - 1/60.
• Set f'(x) = 0 and solve: 2x = √(a² + x²), so 4x² = a² + x², giving x = a/√3.
• Surprising result: the optimal entrance point x = a/√3 doesn't depend on b.
• Must also check endpoints x = 0 and x = b.
• The optimal x is the smaller of a/√3 and b.

Alternative formulation using angle:

• Let x be an angle instead of distance.
• Solving df/dx = 0 gives sin x = 1/2, so the optimal angle is 30°.

⚖️ Physics: minimum energy

Nature often chooses minimum energy:

• Spring pulled by mass: energy f(x) = (1/2)kx² - mx.
• Spring energy (1/2)kx² is positive in stretching or compression.
• Potential energy -mx decreases as mass goes down.
• Equilibrium at df/dx = kx - m = 0, giving x = m/k.
• Key insight: when f(x) is quadratic, the equilibrium equation f'(x) = 0 is linear.

💰 Economics: marginal cost and profit

Marginal cost: the derivative dC/dx, representing the cost of each additional unit.

• If cost C = 1000 + 3x dollars for x books, then dC/dx = 3 (marginal cost is $3 per book).
• Profit P(x) = income I(x) - cost C(x).
• Profit is maximized when marginal income equals marginal cost: dI/dx = dC/dx.

Advertisement example:

• Cost: C(x) = 900 + 400x - x² (setup cost, print cost, volume savings).
• Income: I(x) = 600x - 6x² (sales per ad, minus diminishing returns).
• Optimal decision: dC/dx = dI/dx gives 400 - 2x = 600 - 12x, so x = 20.
• Profit = 9600 - 8500 = 1100.

🧭 Overview

🧠 One-sentence thesis

The second derivative reveals how a curve bends—whether it curves upward or downward—and this bending information determines whether stationary points are maxima or minima, identifies inflection points where bending direction changes, and enables quadratic approximations that track curves far more accurately than tangent lines.

📌 Key points (3–5)

• What the second derivative measures: rate of change of slope (bending direction), not the slope itself or the function value.
• Concave up vs. concave down: positive second derivative means the curve bends upward (slope increasing); negative means bending downward (slope decreasing).
• How to classify stationary points: at a point where the first derivative is zero, a positive second derivative indicates a local minimum, while a negative second derivative indicates a local maximum.
• Common confusion: inflection points vs. stationary points—inflection points occur where the second derivative equals zero and changes sign (bending direction reverses), not where the first derivative is zero.
• Quadratic approximation advantage: including the second derivative term allows approximations to bend with the curve, vastly outperforming straight tangent lines.

📐 Understanding the second derivative

📊 What bending means

When the second derivative f²(x) is positive, the function is concave up (also called convex); when f²(x) is negative, the function is concave down.

• A straight line has zero second derivative because its slope never changes.
• The parabola y = x² has f' = 2x and f² = 2 (constant positive bending upward).
• The sine function has f² = -sin(x), so it alternates between bending down and bending up.

Key insight: The second derivative tells you nothing directly about whether the function is positive or negative, or even whether it's increasing or decreasing—it only reveals how the slope is changing.

👁️ Visual test for second derivative sign

• Concave up (f² > 0): tangent lines stay below the curve; linear approximation underestimates.
• Concave down (f² < 0): tangent lines stay above the curve; linear approximation overestimates.

Example: For f(x) = x², the tangent at x = 0 is the x-axis, which lies entirely below the upward-bending parabola.

⚙️ Physical interpretation

• In motion problems, the second derivative f²(t) represents acceleration (units: distance/time²).
• Acceleration is the rate of change of velocity.
• Example: sin(2t) has velocity v = 2cos(2t) (max speed 2) and acceleration a = -4sin(2t) (max acceleration 4).

🎯 Using the second derivative to classify extrema

🔍 The second derivative test

When f'(x) = 0 and f²(x) > 0, there is a local minimum at x.
When f'(x) = 0 and f²(x) < 0, there is a local maximum at x.

Why this works:

• At a minimum, the slope changes from negative (falling) to positive (rising), so the slope is increasing → f² > 0.
• At a maximum, the slope changes from positive to negative (dropping), so the slope is decreasing → f² < 0.

📝 Worked example

For f(x) = x³ - x²:

• First derivative: f'(x) = 3x² - 2x
• Stationary points: 3x² - 2x = 0 gives x = 0 and x = 2/3
• Second derivative: f²(x) = 6x - 2
• At x = 0: f²(0) = -2 < 0 → local maximum
• At x = 2/3: f²(2/3) = 2 > 0 → local minimum

⚠️ Edge case

When both f'(x) = 0 and f²(x) = 0, the test is inconclusive—anything can happen.

• Example: x³ pauses at x = 0 and continues upward (not a maximum or minimum).
• Example: x⁴ pauses at x = 0 with a very flat graph (actually a minimum).

🔄 Inflection points

🌊 Where bending direction changes

An inflection point occurs where f²(x) = 0 and f² changes sign.

• The curve switches from concave down to concave up (or vice versa).
• At an inflection point, the tangent line crosses the curve (it's below on one side, above on the other).
• For an instant at the inflection point, the graph is locally straight (f² = 0).

📐 Polynomial patterns

Polynomial degree	Inflection points
Parabola (degree 2)	None (f² is constant)
Cubic (degree 3)	Exactly one (f² is linear)
Quartic (degree 4)	Might have 0, 1, or 2 (f² is quadratic)

🌍 Real-world significance: population growth

The excerpt highlights a UN Population Fund report:

• Population is still growing (f' > 0).
• The question is whether the rate of growth is slowing (whether we've passed the inflection point where f² changes from positive to negative).
• At the inflection point, the rate of growth stops growing—this is critical for predicting whether population stabilizes at 10 billion or overshoots to 14 billion.

Don't confuse: increasing population (f > 0 and f' > 0) vs. increasing growth rate (f² > 0).

🔢 Numerical approximations with second derivatives

📏 Centered differences (better than one-sided)

The centered difference for the first derivative:

[f(x + Δx) - f(x - Δx)] / (2Δx) ≈ f'(x)

• More accurate than one-sided differences because it averages information from both sides.
• For f(x) = x², the centered difference gives exactly f' = 2x (no error), while one-sided gives 2x + Δx.
• Centered differences have error proportional to (Δx)², called "second-order accurate."

🔢 Second differences

The second difference approximates the second derivative:

[f(x + Δx) - 2f(x) + f(x - Δx)] / (Δx)² ≈ f²(x)

• This is a "difference of differences"—it measures how the slope changes.
• The notation d²f/dx² mirrors this: dx is squared, but df is not (just as in distance/time²).

📊 Accuracy comparison table

The excerpt tests f(x) = sin(x) + cos(x) at x = 0 with decreasing step sizes:

Step Δx	One-sided error	Centered error	Second difference error
1/4	0.1347	0.0104	0.0052
1/8	0.0650	0.0026	0.0013
1/16	0.0319	0.0007	0.0003

• One-sided errors are O(Δx): halving the step halves the error.
• Centered and second-difference errors are O(Δx)²: halving the step divides error by 4.

📈 Quadratic approximation

🎯 The formula

Quadratic approximation: f(x) ≈ f(a) + f'(a)(x - a) + ½f²(a)(x - a)²

This extends linear approximation (the tangent line) by adding a bending term.

✅ Why the ½ factor

• The model function f(x) = x² has f² = 2.
• To match the curve exactly, we need ½ · 2 · (x)² = (x)².
• The factor ½ cancels the 2 that comes from differentiating x² twice.

🎯 What matches at the basepoint x = a

1. Function values: f(a) = f(a) ✓
2. First derivatives: both equal f'(a) ✓
3. Second derivatives: both equal f²(a) ✓

The quadratic approximation bends with the function, unlike the tangent line which cannot bend.

📝 Examples

Example: (1 + x)ⁿ ≈ 1 + nx + ½n(n-1)x²

• First derivative at x = 0 is n.
• Second derivative at x = 0 is n(n-1).
• This is the start of the binomial expansion.

Example: 1/(1-x) ≈ 1 + x + x²

• Derivatives at x = 0: f' = 1, f² = 2.
• The ½ factor cancels the 2, giving coefficients 1, 1, 1.
• This is the beginning of the geometric series 1 + x + x² + x³ + ...

🔬 Numerical accuracy

Testing √(1+x) ≈ 1 - ½x + ⅜x²:

Approximation	Error behavior
Linear only (1 - ½x)	Error ∝ x² (divided by 4 when x halved)
Quadratic (1 - ½x + ⅜x²)	Error ∝ x³ (divided by 8 when x halved)

The quadratic approximation is dramatically more accurate—errors shrink much faster as you get closer to the basepoint.

🧭 Overview

🧠 One-sentence thesis

Understanding graphs requires combining mathematical analysis (slope, concavity, asymptotes) with computational tools, because while machines draw curves faster, calculus reveals the underlying structure that makes graphs meaningful.

📌 Key points (3–5)

• What graphs reveal: position (sign of f), slope (sign of f'), bending (sign of f''), and behavior at infinity (asymptotes).
• Why derivatives matter for graphing: finding where f' = 0 (stationary points) is far more accurate than visually locating flat spots on y = f(x); each derivative acts like an "infinite zoom."
• Centering and zoom transforms: shifting coordinates (X = x - a, Y = y - b) and rescaling (x = cX, y = dY) simplify equations and focus on key features.
• Common confusion: a flat graph near a minimum is normal, not unusual—the tangent is horizontal (dy/dx = 0), so nearby points differ by only Cx², which is tiny when x is small.
• Asymptotes come in three types: horizontal (y = b as x → ±∞), vertical (x = a where f blows up), and sloping (y = mx + b for large x).

📊 Reading real-world graphs: the electrocardiogram

💓 What an ECG shows

• An ECG records electrical impulses during heartbeats through twelve graphs (six chest leads, six limb leads).
• The fundamental pattern: P wave (atrial contraction), QRS complex (ventricular contraction), T wave (ventricular relaxation).
• Dark vertical lines are 1/5 second apart; light lines are 1/25 second apart.

🩺 Heart rate calculation

• Count dark lines between R-wave spikes (the peaks).
• Doctors memorize: 1, 2, 3, 4, 5, 6 dark lines → rates of 300, 150, 100, 75, 60, 50 beats per minute.
• Example: If beats are 3 dark lines apart (3/5 second), the rate is 5 beats per 3 seconds = 100 per minute.
• Tachycardia: rate above 100 (too fast); bradycardia: rate below 60 (slow, but normal for resting athletes).

⚠️ Rhythm and emergency signals

• Normal rhythm: regular spacing between peaks, set by the SA node pacemaker.
• Sinus arrhythmia: changing time between peaks (fairly normal variation).
• Fibrillation: irregular contractions with no normal PQRST sequence—muscles quiver independently, pumping action nearly gone; requires immediate CPR or defibrillator.
• Myocardial infarction (heart attack) shows three I's:
  • Ischemia: upside-down T wave (reduced blood supply).
  • Injury: elevated ST segment (recent attack).
  • Infarction: wide Q wave (≥ 1/25 second, occupying ~1/3 of QRS complex; indicates dead tissue).

Don't confuse: Rhythm problems (pacemaker firing irregularly) vs. blood-supply problems (blocked artery causing infarction).

🔍 Mathematical analysis of graphs

📋 Six main tests for any function f(x)

1. Sign of f(x): above or below axis; f = 0 at crossing points.
2. Sign of f'(x): increasing or decreasing; f' = 0 at stationary points.
3. Sign of f''(x): concave up or down; f'' = 0 at inflection points.
4. Behavior as x → ±∞: does f approach a limit or grow without bound?
5. Points where f → ±∞: vertical asymptotes.
6. Special properties: even/odd symmetry, periodicity, jumps, endpoints.

📐 Asymptotes defined

Horizontal asymptote y = b: if f(x) → b as x → ±∞, the line y = b is a horizontal asymptote.

Vertical asymptote x = a: if f(x) → ±∞ as x → a, the line x = a is a vertical asymptote.

Sloping asymptote y = mx + b: if f(x) - (mx + b) → 0 as x → ±∞, the line y = mx + b is a sloping asymptote.

• Vertical asymptotes usually come from zero denominators.
• For rational functions P(x)/Q(x), divide polynomials to find sloping asymptotes.

Example: f(x) = x²/(x - 1) has vertical asymptote x = 1 (denominator zero) and sloping asymptote y = x + 1 (from polynomial division: x²/(x - 1) = x + 1 + 1/(x - 1)).

🔄 Symmetry properties

• Even function: f(-x) = f(x); graph symmetric across y-axis (e.g., x² or cos x).
• Odd function: f(-x) = -f(x); graph symmetric through origin (e.g., x³ or sin x).
• Periodic function: f(x + T) = f(x) for some period T (e.g., sin x has period 2π).

💻 Computational graphing and optimization

🎯 Why graphs near minima are flat

• At a minimum, dy/dx = 0 (horizontal tangent).
• Near x*, the graph looks like y = Cx² (a parabola sitting on the tangent).
• At distance Δx = 0.01, the height is only C(0.01)² = 0.0001C—barely visible unless C is large.
• This flatness is the whole point of dy/dx = 0, not an unusual problem.

🔬 Three levels of accuracy

Method	What you use	Accuracy gain
Trace on y(x)	Function values	Read to nearest pixel
Solve f'(x) = 0	First derivative	Order of magnitude better (graph crosses axis, not just flattens)
Newton's method	Second derivative f''	Another order of magnitude (error: 0.01 → 0.0001 → 0.00000001)

• Finding where f'(x) = 0 is easier than finding where y(x) is minimum, because the derivative's graph crosses the x-axis rather than merely flattening.
• Each derivative formula represents an infinite zoom already performed by calculus.

Don't confuse: Visual zooming (stretches the picture) vs. taking derivatives (analytically captures the limit of zooming).

📦 Zoom boxes

• Draw a box around the region of interest on the graph.
• The box becomes the new viewing window (axes stretched to fit).
• Repeated zooms narrow the search: first 1 < x* < 3, then 1.5 < x* < 2, etc.
• Use a long thin box to see bending; square boxes make the graph flatter (approaching the tangent line).

🔧 Coordinate transforms

🎯 Centering transform

Centering transform: shifts the point (a, b) to the origin by setting X = x - a and Y = y - b.

• Changes y = f(x) into Y + b = f(X + a), or Y = f(X + a) - b.
• Purpose: move the interesting point (e.g., a minimum) to (0, 0) for simpler algebra.
• Example: y = 2x² - 4x has minimum at (1, -2); centering gives Y = 2X².

🔎 Zoom transform

Zoom transform: scales axes by factors c and d, setting x = cX and y = dY.

• Changes Y = F(X) into y = dF(x/c).
• Square zoom (c = d): lines keep their slope; useful for seeing overall shape.
• Rectangular zoom (c ≠ d): stretches one axis more; needed to see bending in flat graphs.

📏 Effect on derivatives

• First derivative (slope) is multiplied by d/c.
• Second derivative (bending) is multiplied by d/c².
• Square zoom (d/c = 1) preserves slopes but changes curvature.
• Zoom out (small c, d) makes the big picture more curved.

🧮 Combined transform

• First center: X = x - a, Y = y - b.
• Then zoom: x = cX, y = dY.
• Result: y = d[f(x/c + a) - b].
• The viewing window after both transforms: original [-1, 1] becomes [a - c, a + c] for x and [b - d, b + d] for y.

Example: Every upward-opening parabola can be transformed to y = x² by choosing the right a, b, c, d (center the vertex, then scale to remove the leading coefficient).

🧭 Overview

🧠 One-sentence thesis

Parabolas, ellipses, and hyperbolas are second-degree curves that connect algebraic equations to geometric shapes, forming a cornerstone of analytic geometry.

📌 Key points (3–5)

• What these curves are: second-degree curves produced by equations that include x², xy, and y² terms, in addition to 1, x, and y.
• Where they come from: all three curves (plus circles) are "conic sections"—they result from slicing a cone with a plane at different angles.
• How they differ by cutting angle: level cut → circle; moderate angle → ellipse; steep cut → hyperbola; borderline angle (matching the cone) → parabola.
• Why they matter: they bridge analysis (equations) and geometry (curves), forming the foundation of analytic geometry.
• Common confusion: straight lines use 1, x, y (first degree); these curves add x², xy, y² (second degree); going to x³ and y³ makes mathematics much more complicated.

📐 The hierarchy of important curves

📊 Ranking the top curves in mathematics

The excerpt lists four most important types of curves:

Rank	Curve type	What it uses	Purpose
1	Straight lines	1, x, y	Linear relationships
2	Sines and cosines	Trigonometric functions	Oscillation
3	Exponentials	Exponential functions	Growth and decay
4	Parabolas, ellipses, hyperbolas	1, x, y, x², xy, y²	Second-degree relationships

🏛️ Historical perspective

• The Greeks would have ranked parabolas, ellipses, and hyperbolas first.
• It is natural to progress from linear equations (first degree) to quadratic equations (second degree).
• Moving beyond second degree (to x³ and y³) makes the mathematics complicated.

🔗 Connecting equations and geometry

🧮 What analytic geometry does

Analytic geometry: the connection between analysis of equations and geometry of curves.

• Analysis focuses on the equation itself (algebraic manipulation, solving).
• Geometry focuses on the visual curve (shape, properties).
• Together they form analytic geometry, which has become central to mathematics.

📍 Numbers and points

• Numbers correspond to points in space.
• Example: "the point (5, 2)" refers to a specific location.
• This was not obvious to ancient mathematicians like Euclid.

✏️ Descartes vs. Euclid

• Where Euclid drew a 45° line through the origin (geometric description), Descartes wrote y = x (algebraic equation).
• This shift from pure geometry to equation-based representation revolutionized mathematics.

🍦 Conic sections: slicing a cone

🔪 How conic sections are created

Conic sections: curves produced by slicing a cone with a plane at different angles.

All four curves (circle, ellipse, parabola, hyperbola) come from the same geometric construction—cutting a cone with a plane.

📐 The four cuts and their curves

Cut type	Angle description	Resulting curve
Level cut	Horizontal plane	Circle
Moderate angle	Tilted, but not too steep	Ellipse
Steep cut	Sharp angle	Hyperbola (two pieces)
Borderline	Matches the cone angle	Parabola

🎯 The borderline case

• The parabola occurs at the exact angle where the slicing plane is parallel to the edge of the cone.
• This is the transition point between ellipse and hyperbola.
• Don't confuse: the parabola is not "in between" in shape, but rather the critical angle where the curve changes from closed (ellipse) to open (hyperbola).

🌟 Why this matters

• The Greeks discovered these remarkable properties geometrically, by studying cone slices.
• All these seemingly different curves share a common geometric origin.
• This unifying insight connects diverse mathematical phenomena through a single construction.

🧭 Overview

🧠 One-sentence thesis

The Fundamental Theorem of Calculus—that the area under the velocity graph equals the change in distance—can be understood through piecewise linear functions and algebra before introducing limits and curves.

📌 Key points (3–5)

• Core relationship: The velocities v are slopes of the distance function f(t), and the area under the v-graph equals f(t_end) − f(t_start).
• Piecewise approach: For piecewise linear f(t) and piecewise constant v(t), the Fundamental Theorem holds using only algebra (no limits needed yet).
• Growth patterns: Different sequences (linear 2j, quadratic j², exponential 2^j) show how velocity and distance relate in various contexts.
• Common confusion: Don't confuse marginal rates (slope at a point, like tax rate) with average rates (total divided by total, like average tax burden).
• Calculus extension: The algebraic approach with steps prepares for smooth curves, where limits become necessary to handle continuously changing velocity.

📐 The Fundamental Theorem without limits

📏 Area equals change in distance

The area under the v-graph is f(t_end) − f(t_start).

• The excerpt demonstrates this with rectangles under a staircase velocity graph.
• Each rectangle has base 1 and height equal to the velocity v at that step.
• Total area = sum of all v's = f_last − f_first.
• Example: Starting at any time and ending at t = 3.5, counting only half the last rectangle (under v = 7) gives area 1 + 3 + 5 + (1/2)(7) = 12.5, which matches f(3.5) − f(0) = 12.5 − 0.

🔗 Connection between slope and area

• The v's are slopes of f(t): velocity is the rate of change of distance.
• The area under velocity gives back the total distance change.
• This is the Fundamental Theorem of Calculus, but restricted here to piecewise linear f and piecewise constant v.
• Chapter 5 will remove this restriction and handle curved graphs using limits.

🧮 Algebraic proof example

• The excerpt shows that 1 + 3 + 5 + 7 = 4² by comparing areas.
• The triangle under a dotted line has the same area as four rectangles under the staircase.
• Triangle area = (1/2) × base × height = (1/2) × 4 × 8 = 16 = 4².
• For j rectangles: area = (1/2) × j × 2j = j².

📈 Growth patterns: linear, quadratic, exponential

📊 Comparing three types of growth

Type	Formula	Example at j=10	Velocity pattern
Linear	2j	20	Constant differences
Quadratic	j²	100	Linearly increasing differences
Exponential	2^j	1024	Exponentially increasing differences

• Exponential 2^j grows much faster than the others.
• Don't confuse: 2j (linear), j² (quadratic), and 2^j (exponential) are fundamentally different growth rates.

🚀 Exponential velocity and distance

• Start with f = 1, 2, 4, 8, 16 (powers of 2).
• These are 2^0, 2^1, 2^2, 2^3, 2^4.
• The differences (velocities) are v = 1, 2, 4, 8—also powers of 2.
• Formula: f_j = 2^j and v_j = 2^(j−1).
• Key property: v(t) is proportional to f(t)—the velocity is proportional to the distance.
• This pattern appears in population growth, bank interest, and national debt.
• The smooth curve version is f(t) = 2^t with slope v(t) = c·2^t, where c ≈ 0.693 (the natural logarithm of 2).

🌊 Oscillating velocity and distance

• Example: f = 0, 1, 1, 0, −1, −1, 0 (returns to zero after 6 steps).
• Velocities: v = 1, 0, −1, −1, 0, 1 (also returns to zero).
• The sum of v's is zero, agreeing with f_last − f_first = 0 − 0 = 0.
• The f-graph roughly resembles a sine curve; the v-graph roughly resembles a cosine curve.
• This is a periodic motion: the pattern repeats every 6 steps (the period is 6).
• Smooth versions (sine and cosine) repeat every 2π radians (or 360 degrees).
• Don't confuse: These digitized step functions are like analog waveforms but go in jumps, not smoothly.

🚗 Burst of speed and step functions

⚡ Short burst example

• A car travels at speed V until distance reaches f = 1, then stops suddenly.
• Velocity: v(t) = V up to time T, then v(t) = 0 after T.
• Distance: f(t) = V·t up to time T, then f(t) = 1 after T.
• Area under v-graph = V × T = 1, so T = 1/V.
• Example: Three cars (Jeep, Corvette, Maserati) with different speeds V but all reach distance 1, so all have area 1 under their velocity graphs; faster cars have taller, narrower rectangles.

📍 Step function and infinite slope

• Extreme case: speed becomes infinitely fast, so the car jumps instantly from 0 to 1.
• This is the unit step function U(t): U = 0 for t < 0, then U = 1 for t > 0.
• Slope is zero everywhere except at the jump (t = 0), where the slope is infinite.
• The velocity is an impulse (a spike), often denoted by δ (delta function).
• Area under the infinite spike is still 1.
• Don't confuse: Calculus is about curves, not jumps; delta functions are an advanced topic beyond ordinary calculus.

💰 Income tax: marginal vs average rates

📋 Tax brackets and rates (1991 example)

• Taxable income x determines tax f(x) through three brackets:
  • Bracket 1: 0 ≤ x ≤ $20,350, tax = 0.15 × x (rate 15%).
  • Bracket 2: $20,350 ≤ x ≤ $49,300, tax = $3,052.50 + 0.28 × (x − $20,350) (rate 28%).
  • Bracket 3: x ≥ $49,300, tax = $11,158.50 + 0.31 × (x − $49,300) (rate 31%).
• The tax function f(x) is piecewise linear with slopes 0.15, 0.28, 0.31.

🔍 Marginal rate vs average rate

Marginal rate: the tax on each additional dollar of income; it is the slope at point x.

Average rate: total tax divided by total income.

• The rates 0.15, 0.28, 0.31 are marginal rates, not average rates.
• Only the income in a bracket is taxed at that bracket's rate.
• Example: Someone earning $50,000 pays 31% only on the amount over $49,300, not on the entire $50,000.
• Tax is like area or distance—it adds up over all brackets.
• Tax rate is like slope or velocity—it depends on where you are (your income level).
• Don't confuse: News media often conflate marginal and average rates, leading to misunderstanding.

📐 Equation for the top bracket

• The top bracket begins at x = $49,300 with tax f(x) = $11,158.50.
• The slope (marginal rate) is 0.31.
• Equation: f(x) = $11,158.50 + 0.31 × (x − $49,300) for x ≥ $49,300.

🧭 Overview

🧠 One-sentence thesis

Newton's method provides a powerful iterative technique for solving equations by following tangent lines, but even simple equations can lead to unpredictable chaotic behavior that reveals fundamental limits on our ability to compute long-term outcomes.

📌 Key points (3–5)

• Core mechanism: Newton's method replaces a function with its tangent line at each step, using both the function value f(x₀) and slope f'(x₀) to find where the tangent crosses the axis.
• Typical convergence: When starting near a solution, the error is squared at each step (superconvergence), doubling the number of correct decimals per iteration.
• Chaos from simple rules: Even innocent-looking iterations like solving x² + 1 = 0 produce chaotic sequences that are extremely sensitive to starting values and unpredictable over long times.
• Common confusion: Don't confuse the function f(x) whose root we seek with the iteration function F(x); convergence depends on the slope F'(x*) at the fixed point, not just on f(x*) = 0.
• Practical limits: Simple deterministic rules can generate chaos, meaning no measurement can ever be accurate enough to predict long-term behavior—this applies to weather, planetary orbits, and mathematical iterations alike.

🎯 The Newton iteration formula

🎯 Building the method from tangent lines

Newton's method: x_{n+1} = x_n - f(x_n)/f'(x_n)

• What we know at x₀: the graph height f(x₀) and the slope f'(x₀).
• What we don't know: whether the curve bends (we lack f'').
• The strategy: follow the tangent line, which uses all available information.
• The tangent line equation is f(x) ≈ f(x₀) + f'(x₀)(x - x₀).
• Set the right side to zero to find where the tangent crosses the axis: f(x₀) + f'(x₀)(x₁ - x₀) = 0.
• Solving for x₁ gives the Newton formula.

🔄 The iteration process

At each new point x_n:

1. Compute the height f(x_n) and slope f'(x_n).
2. Draw the tangent line from that point.
3. Find where it crosses the axis to get x_{n+1}.
4. Repeat until convergence (or chaos).

Example: The tangent line from x_n crosses the axis at x_{n+1}, while the actual curve crosses at the true solution x*.

🔗 Connection to linear approximation

The excerpt emphasizes that calculus uses three related calculations:

1. Estimate slope f'(x) from Δf/Δx (differentiation).
2. Estimate change Δf from f'(x)Δx (linear approximation).
3. Estimate change Δx from Δf/f'(x) (Newton's method).

Newton's method is exactly Δx = -f(x_n)/f'(x_n), where the desired Δf is -f(x_n).

🚀 Convergence and superconvergence

🚀 Error squaring behavior

Superconvergence: the error is squared at each Newton step when F'(x*) = 0.

• For the iteration x_{n+1} = F(x_n), the slope of F at the solution determines convergence speed.
• Newton achieves F'(x*) = 0 when f(x*) = 0, which is optimal.
• Why F' = 0: The slope of F(x) = x - f(x)/f'(x) equals f(x)f''(x)/(f'(x))², which is zero when f(x) = 0.
• This means the multiplier m = 0, far better than the usual convergence test |F'(x*)| < 1.

📈 Doubling correct decimals

Example (square roots): To find √4, iterate x_{n+1} = ½(x_n + 4/x_n) starting from x₀ = 1:

• x₁ = 2.5
• x₂ = 2.05
• x₃ = 2.0006
• x₄ = 2.000000009

The wrong decimal is twice as far out at each step—the number of correct decimals doubles.

⚠️ When convergence fails

• Starting at a horizontal tangent: If f'(x₀) = 0, the tangent line is horizontal and never crosses the axis (division by zero).
• Wrong basin of attraction: Starting at x₀ = -1 for √4 converges to -√4 instead of +√4.
• Divergence: If |1 - 2x₀| > 1 in certain iterations, the sequence diverges rather than converges.

Don't confuse: Fast convergence near x* does not guarantee convergence from far away; basins of attraction can be complicated.

🧮 Classic examples

🧮 Square roots (Example 1)

To solve f(x) = x² - b = 0 (finding x* = √b):

• The slope is f'(x_n) = 2x_n.
• Newton's formula becomes x_{n+1} = x_n - (x_n² - b)/(2x_n).
• This simplifies to x_{n+1} = ½(x_n + b/x_n).
• Interpretation: Guess the square root, divide into b, and average the two numbers.
• The Babylonians used this same method without knowing calculus.
• The error equation is x_{n+1} - 2 = (1/(2x_n))(x_n - 2)², showing error squaring.

🧮 Division without dividing (Example 2)

To solve 1/x - a = 0 (finding x* = 1/a without division):

• Here f(x) = 1/x - a and f'(x) = -1/x².
• Newton gives x_{n+1} = x_n - (1/x_n - a)/(-1/x_n²) = x_n + x_n - ax_n² = 2x_n - 2x_n².
• For a = 2 (aiming for x* = 1/2), the error equation is x_{n+1} - 1/2 = -2(x_n - 1/2)².
• Convergence condition: Fast convergence if 0 < x₀ < 1; divergence if x₀ < 0 or x₀ > 1.
• The algebra confirms: (1 - 2x_{n+1}) = (1 - 2x_n)², so convergence requires |1 - 2x₀| < 1.

🧮 Imaginary roots and chaos (Example 3)

To solve f(x) = x² + 1 = 0 (no real solutions, only x* = i and x* = -i):

• Newton's method becomes x_{n+1} = ½(x_n - 1/x_n).
• The x's cannot approach i or -i (nothing is imaginary).
• What happens: The sequence bounces around unpredictably.
• Starting from x₀ = 1 gives x₁ = 0, then x₂ divides by zero and blows up.
• Large x_n produces x_{n+1} about half as large, but eventually a small number appears, then division by that small number sends the sequence far out again.
• This is chaos: no convergence, strong sensitivity to initial conditions.

🌀 The cotangent formula and chaos

🌀 Trigonometric identity reveals the pattern

The key identity: ½(cot θ - 1/cot θ) = cot 2θ.

For x² + 1 = 0, if x₀ = cot θ, then:

• x₁ = cot 2θ
• x₂ = cot 4θ
• x_n = cot(2ⁿθ)

Our points are on the cotangent curve, and every iteration doubles the angle.

🌀 Three behaviors

Starting angle	Sequence behavior	Explanation
θ = π/4	x₀ = 1, x₁ = 0, x₂ = ∞	Blows up due to division by zero
θ = π/3	x₀ = 1/√3, x₂ = 1/√3, ...	Cycles forever with period 2
Small θ (large x₀)	Chaotic bouncing	Angles 4θ, 8θ, ... eventually hit large cotangents

🌀 Extreme sensitivity

• After 10 steps, θ is multiplied by 2¹⁰ = 1024.
• Starting angles 60° and 61° look close, but after 10 steps they differ by 1024°.
• The x₁₀ values are 0.6 and 14—completely different.
• This is chaos: small errors snowball, making long-term prediction impossible.

Don't confuse: Deterministic rules (like cot 2ⁿθ) with predictable outcomes; chaos means the formula is "absolutely hopeless after 100 steps."

🌪️ Quadratic iteration and period doubling

🌪️ The parabolic family

Changing variables from x to z = 1/(1 + x²) transforms Newton's iteration into:

• z_{n+1} = 4z_n - 4z_n²

This is the most famous quadratic iteration, with z_n = (sin 2ⁿθ)².

The general family is z_{n+1} = az_n - az_n² for 0 < a ≤ 4.

🌪️ Behavior as parameter a increases

Range of a	Behavior	Fixed point
0 < a < 1	Converge to z* = 0	Single limit
1 < a < 3	Converge to z* = (a-1)/a	Single limit
a ≈ 3.4	2-cycle: alternates between two values	Period 2
a ≈ 3.5	4-cycle: repeats after four steps	Period 4
a ≈ 3.55	8-cycle	Period 8
3.57 < a < 4	Chaos, period-3 windows, fractals	Complex behavior

Convergence test: |F'(z*)| < 1 at the fixed point z* = F(z*).

🌪️ Period doubling cascade

• Stable cycles of length 2, 4, 8, 16, 32, 64, ... appear as a increases.
• Each cycle becomes unstable and doubles in period.
• The stability windows shrink by the Feigenbaum factor 4.6692...
• All cycles are unstable before a = 3.57.
• Between 3.57 and 4: chaos, with occasional windows of stable periods (like period 3).

Example: At a = 3.5, starting from any random z₀, after 100 steps the sequence settles into the 4-cycle: 0.875 → 0.383 → 0.827 → 0.501 → 0.875.

🌪️ What happens at a = 4

Three possibilities:

1. The z_n's cycle with some long period.
2. The z_n's come close to every point between 0 and 1.
3. The z_n's approach a very thin limit set (like a Cantor set).

The behavior is chaotic and statistical tests find no pattern—the numbers are effectively random.

🔬 Fractals and the Cantor set

🔬 Constructing the Cantor set

Cantor set: divide [0,1] into three pieces and remove the open middle third (1/3, 2/3); repeat on each remaining piece.

• At each step, take out the middle thirds.
• What remains: all endpoints 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, ... plus other points like 3/4.
• The removed intervals have lengths summing to 1, so the Cantor set has "measure zero."
• Self-similarity: Between 0 and 1/3, you see the same Cantor set scaled down by 3; from 0 to 1/9, scaled down by 9.

🔬 Fractals and dimension

Fractal: a set with self-similarity at infinitely many scales.

• Fractional dimension: The Cantor set has dimension between 0 and 1; a fractal snowflake boundary has dimension between 1 and 2.
• Covering a line segment with circles of radius r takes c/r circles; for fractals it takes c/r^D circles, where D is the dimension.
• Mathematical snowflake: Start with a triangle, add a bump in the middle of each side; repeat. The boundary lengthens by 4/3 at each step, becoming infinitely long.

🔬 Chaos in nature

• Weather forecasting: Small errors snowball, destroying forecasts after 6 days; a plane's flight can change everything.
• Pluto's orbit: Chaotic despite obeying gravity; motion is unpredictable over long times.
• Revolutionary idea: Simple rules can lead to answers too sensitive to compute; we are not accustomed to innocent formulas that are "absolutely hopeless after 100 steps."

Don't confuse: Complicated formulas with unpredictability; even simple deterministic rules (like x_{n+1} = 4x_n - 4x_n²) can produce chaos.

🛠️ Alternative methods

🛠️ Secant method

Secant method: x_{n+1} = x_n - f(x_n)/(Δf/Δx)n, where (Δf/Δx)n = (f(x_n) - f(x{n-1}))/(x_n - x{n-1}).

• Key difference from Newton: Uses two previous points to approximate the slope, avoiding the need to compute f'(x).
• The secant line connects the two latest points on the graph; set y = 0 to find where it crosses the axis.
• Prediction: Three secant steps ≈ two Newton steps; both give four times as many correct decimals.
• Probably also chaotic for x² + 1 = 0.

🛠️ Bisection method

If f(x) changes sign between x₀ and x₁:

1. Find the sign at the midpoint x₂ = ½(x₀ + x₁).
2. Decide whether f(x) changes sign between x₀ and x₂, or x₂ and x₁.
3. Repeat on that half-length interval.
4. Switch to a faster method when the interval is small enough.

Example: f(x) = x² - 4 is negative at x = 1, positive at x = 2.5, negative at midpoint x = 1.75, so x* lies in [1.75, 2.5].

Three bisection steps reduce the interval by a factor of 8 (= 2³).

🛠️ Comparison table

Method	Slope information	Convergence speed	Advantages
Newton	Uses f'(x_n)	Error squared (superfast)	Fewest steps when close to x*
Secant	Approximates f' from two points	Between linear and quadratic	No derivative needed
Bisection	None (only sign changes)	Linear (halves interval)	Always converges if f changes sign; robust

Don't confuse: The function f(x) we're solving with the iteration function F(x); Newton's F has F'(x*) = 0, which is why it converges so fast.

🧭 Overview

🧠 One-sentence thesis

The Mean Value Theorem bridges local behavior (instantaneous slope at a point) to global behavior (average slope across an interval), and this connection enables l'Hôpital's Rule to resolve indeterminate forms like 0/0 by comparing derivatives.

📌 Key points (3–5)

• What the MVT connects: relates the derivative df/dx at some interior point c to the average change Δf/Δx over an interval [a, b].
• Rolle's theorem as special case: when f(a) = f(b) = 0, the derivative must equal zero at some interior point c.
• l'Hôpital's Rule for 0/0: when both f(x) and g(x) approach zero, their ratio f/g has the same limit as the ratio of their derivatives f'/g'.
• Common confusion: l'Hôpital is NOT the quotient rule—derivatives are taken separately for numerator and denominator, and it applies only when you have 0/0 or ∞/∞.
• Why it matters: the MVT proves that continuous change cannot "jump over" intermediate values, and l'Hôpital provides a practical tool for evaluating limits that initially appear meaningless.

🔗 The Mean Value Theorem core idea

🎯 What the theorem states

Mean Value Theorem: If f(x) is continuous on the closed interval [a, b] and has a derivative everywhere in the open interval (a, b), then [f(b) - f(a)] / (b - a) = f'(c) at some point a < c < b.

• The left side is the average slope over the entire interval.
• The right side is the instantaneous slope at one unknown interior point c.
• The theorem guarantees c exists but does not tell you its exact value.

Example: If average velocity over 2 hours is 75 mph, then instantaneous velocity must equal 75 at some moment during those 2 hours (assuming velocity exists at all interior points).

📐 Closed vs open intervals

Notation	Meaning	MVT requirement
[a, b]	Closed interval, includes endpoints	f must be continuous here
(a, b)	Open interval, excludes endpoints	f' must exist here

• The derivative f' is required only in the open interval (a, b).
• Continuity at the endpoints a and b is required, but f'(a) and f'(b) need not exist.

🔄 Rolle's theorem as the foundation

Rolle's theorem: If f(a) = f(b) = 0 (zero at both ends), then f'(c) = 0 at some point a < c < b.

• This is the special case when the function starts and returns to zero.
• Average slope is zero, so the derivative must be zero somewhere inside.
• Proof sketch: A continuous function on [a, b] reaches its maximum and minimum (Extreme Value Theorem). If the max or min occurs at an interior point c, then df/dx = 0 there. If both extremes are at the endpoints, then f(x) ≡ 0 in between, so f' = 0 everywhere.

Historical note: Rolle himself did not believe in the logic behind calculus and fought against it, yet his special case leads directly to the general MVT.

🛠️ Proving and applying the MVT

🧮 Proof strategy

The proof tilts the graph back to Rolle's case by subtracting the secant line:

• Define F(x) = f(x) - [f(a) + (Δf/Δx)(x - a)].
• This measures the vertical distance between the curve and the secant line.
• At both endpoints, F(a) = F(b) = 0.
• Apply Rolle's theorem to F(x): there exists c where F'(c) = 0.
• Taking the derivative: 0 = f'(c) - (Δf/Δx), which rearranges to the MVT.

📏 Exact prediction formula

f(x) = f(a) + f'(c)(x - a), where a < c < x.

• This replaces the approximate linear prediction f(x) ≈ f(a) + f'(a)(x - a).
• The approximation uses the slope at a; the exact formula uses the slope at an unknown c between a and x.

Example: sin x = (cos c) · x for some c between 0 and x. Since cos c < 1 for c > 0, this proves sin x < x for positive x.

⚙️ Key consequence: constant functions

If f'(c) = 0 at all points in an interval, then f(x) is constant.

• Proof: The MVT gives Δf = f'(c) · Δx = 0 · Δx = 0 for every pair of points.
• Therefore f(b) = f(a) for all a and b; the graph is a horizontal line.
• This simple case is essential for the Fundamental Theorem of Calculus.

🎲 l'Hôpital's Rule for indeterminate forms

🔍 The 0/0 problem

When f(x) and g(x) both approach zero, the ratio f(x)/g(x) is indeterminate:

• Examples: x²/x, (sin x)/x, (x - sin x)/(1 - cos x) all become 0/0 at x = 0.
• You cannot work with f and g separately; it is a "race toward zero."
• The limit of the ratio might be any number, or ±∞, or might not exist.

📜 l'Hôpital's Rule statement

l'Hôpital's Rule: Suppose f(x) and g(x) both approach zero as x → a. Then f(x)/g(x) approaches the same limit as f'(x)/g'(x), if that second limit exists. Normally this limit is f'(a)/g'(a).

Critical warning: This is NOT the quotient rule! Derivatives of f and g are taken separately, not as a quotient.

✅ When to use l'Hôpital

Situation	Use l'Hôpital?	Why
f(x)/g(x) → 0/0	Yes	Both numerator and denominator → 0
f(x)/g(x) → ∞/∞	Yes	Extended version applies
f(x)/g(x) → a/(a-1) where a ≠ 0	No	Ordinary limit: divide limits separately

Don't confuse: If f(x) → a (nonzero) and g(x) → b (nonzero), just compute a/b. l'Hôpital enters only for 0/0 or ∞/∞.

🧪 Worked examples

Example (old friend): lim[x→0] (1 - cos x)/x

• Both numerator and denominator → 0.
• Apply l'Hôpital: derivatives are sin x (numerator) and 1 (denominator).
• Limit = sin 0 / 1 = 0.

Example (repeated application): lim[x→0] (x - sin x)/(1 - cos x)

• First application: f'/g' = (1 - cos x)/(sin x), still 0/0 at x = 0.
• Apply again: f''/g'' = (sin x)/(cos x) → 0/1 = 0.

Example (geometric insight): lim[x→0] (tan x)/(sin x)

• Derivatives: (sec² x)/(cos x).
• At x = 0: 1/1 = 1.

🧩 Why l'Hôpital works

The algebra behind the rule:

• f(x)/g(x) = [f(x) - f(a)]/(x - a) ÷ [g(x) - g(a)]/(x - a).
• Both f(a) and g(a) are zero by assumption, so this is exact.
• As x → a, the numerator approaches f'(a) and the denominator approaches g'(a).
• Therefore the ratio approaches f'(a)/g'(a).

The more general proof uses the Generalized Mean Value Theorem (Cauchy's extension).

🔬 Advanced topics and error analysis

🎯 Generalized MVT (Cauchy)

If f(x) and g(x) are continuous on [a, b] and differentiable on (a, b), there exists a < c < b where [f(b) - f(a)]g'(c) = [g(b) - g(a)]f'(c).

• When g(x) = x, this reduces to the ordinary MVT.
• This form proves l'Hôpital's Rule rigorously.
• Proof: construct F(x) = [f(b) - f(a)]g(x) - [g(b) - g(a)]f(x), which satisfies F(a) = F(b) = 0, then apply Rolle.

📊 Error in linear approximation

The error in the tangent-line approximation grows quadratically:

Error = (1/2) f''(c)(x - a)² for some a < c < x.

• Linear approximation: f(x) ≈ f(a) + f'(a)(x - a).
• Exact formula: f(x) = f(a) + f'(a)(x - a) + (1/2)f''(c)(x - a)².
• The error term involves the second derivative and the square of the distance.

Example: For f(x) = √x near a = 100, approximating √102:

• √102 ≈ 10 + (1/20)·2 + (1/2)·(-1/4000)·2² = 10.1 - 0.0005.
• Predicted error ≈ -0.0005; actual error ≈ -0.000496.

⚠️ Other indeterminate forms

Beyond 0/0, l'Hôpital extends to:

• ∞/∞: Apply the rule directly to f'(x)/g'(x).
• 0·∞: Rewrite as f(x)/(1/g(x)) to get 0/0.
• ∞ - ∞, 0⁰, 1^∞, ∞⁰: Take logarithms to reduce to 0/0 or 0·∞.

Don't confuse: When f(x) → 0 and g(x) → ∞, the product f·g is indeterminate, but the ratio f/g → 0 (no l'Hôpital needed).

🧭 Overview

🧠 One-sentence thesis

The chain rule enables us to find the derivative of composite functions by multiplying the derivative of the outside function (evaluated at the inside function) by the derivative of the inside function.

📌 Key points (3–5)

• What composition means: A composite function f(g(x)) applies g first to get y, then applies f to y to get the final output z.
• The chain rule formula: dz/dx = (dz/dy) × (dy/dx), meaning we multiply the derivatives of the outer and inner functions.
• Common confusion: The derivative of sin(x²) is NOT (cos x)(2x); it is (cos x²)(2x)—the cosine must be evaluated at x², not at x.
• Order matters: f(g(x)) is usually different from g(f(x)); applying functions in different orders produces different results.
• Recognition skill: A major part of using the chain rule is identifying which function is "inside" and which is "outside."

🔗 Understanding composite functions

🔗 What composition means

Composite function z = f(g(x)): Start with x, compute y = g(x) (the inside function), then compute z = f(y) (the outside function).

• The inside function g(x) produces an intermediate value y.
• The outside function f(y) takes that y as input and produces the final output z.
• Notation: Written as f ∘ g or more commonly f(g(x)).
• Example: For sin(x²), the inside function is g(x) = x² and the outside function is f(y) = sin y.

🔄 Order matters in composition

• f(g(x)) is usually different from g(f(x)).
• Example with f(x) = sin x and g(x) = x²:
  • f(g(x)) = sin(x²): square first, then take sine
  • g(f(x)) = (sin x)² (often written sin²x): take sine first, then square
• These produce completely different functions and graphs.
• Don't confuse: sin x² means sin(x²), never (sin x)².

🧮 Calculator analogy

• On a calculator: input x, push the "g" button, then push the "f" button.
• This performs the operations in sequence: x → g(x) → f(g(x)).
• The squaring and sine functions are used in that order to create sin(x²).

🎯 The chain rule mechanics

🎯 The fundamental formula

Chain Rule: If z = f(g(x)), then dz/dx = (dz/dy) × (dy/dx) = f'(g(x)) × g'(x).

• The derivative of the composite is the product of two derivatives.
• First factor: derivative of the outside function f, evaluated at y = g(x).
• Second factor: derivative of the inside function g, evaluated at x.
• Key insight: Δz/Δx = (Δz/Δy) × (Δy/Δx) becomes exact in the limit.

⚠️ Critical evaluation point

• The most common mistake: evaluating the outer derivative at the wrong place.
• The derivative of sin(x²) is NOT (cos x)(2x).
• Correct: (cos x²)(2x)—the cosine is evaluated at x², not at x.
• General pattern: derivative of f(g(x)) requires f'(y) at y = g(x), not f'(x).

🔢 The power rule as a special case

• For z = [g(x)]ⁿ, the chain rule gives: dz/dx = n[g(x)]^(n-1) × g'(x).
• Example: derivative of (x² - 1)^(1/2) is (1/2)(x² - 1)^(-1/2) × (2x) = x / √(x² - 1).
• This extends the simple power rule to any function raised to a power.

🔍 Recognizing and applying chains

🔍 Identifying inside and outside functions

• Look at the function for a moment to see the structure.
• Example: (x³ + 1)⁵ is u⁵ where u = x³ + 1 (inside function).
• Example: cos(2x + 1) is cos u where u = 2x + 1.
• Example: sin √(1 - x) involves three functions: z = sin y, y = √u, u = 1 - x.

📝 Two calculation approaches

Careful way: Write down all functions explicitly:

• z = cos u, u = 2x + 1
• dz/dx = (-sin u)(2) = -2 sin(2x + 1)

Quick way: Keep "the derivative of what's inside" in mind:

• Derivative of cos(2x + 1) is -sin(2x + 1), times 2 from the chain rule.

🎲 Common examples with solutions

Function z	Inside y or u	Outside f	Derivative dz/dx
sin(x²)	x²	sin y	(cos x²)(2x)
(sin x)³	sin x	y³	3(sin x)² cos x
sin(3x)	3x	sin y	3 cos(3x)
(1 - x)²	1 - x	y²	2(1 - x)(-1)

⚡ Speed interpretation

• Example: z = sin(3t) oscillates three times as fast as sin t.
• The function sin(3t) completes a full wave at time 2π/3 instead of 2π.
• The velocity naturally contains the extra factor 3 from the chain rule.

🧩 Special cases and extensions

🧩 Triple chains

• Some functions involve three compositions: z = f(y), y = g(u), u = h(x).
• Example: sin √(1 - x) chains z = sin y, y = √u, u = 1 - x.
• Derivative: (cos √(1 - x)) × (1/(2√u)) × (-1) evaluated properly.
• Multiply all three derivatives in sequence.

🔁 Iteration and self-composition

• Newton's method composes F(x) with itself: x_(n+1) = F(x_n).
• Example: F(x) = (1/2)x + 4 gives F(F(x)) = (1/4)x + 6.
• Derivative of F(x) is 1/2; derivative of F(F(x)) is 1/4 = (1/2) × (1/2).

📐 Second derivatives

• The chain rule gives dz/dx as a product, so the second derivative needs the product rule.
• Formula: d²z/dx² = (dz/dy)(d²y/dx²) + (d/dx)[dz/dy] × (dy/dx).
• The last term requires the chain rule again: becomes (d²z/dy²) × (dy/dx)².
• Example: derivative of sin(x²) is 2x cos(x²); second derivative is 2 cos(x²) - 4x² sin(x²).

🎯 Practical word problem

• A Buick uses 1/20 gallon per mile; you drive at 60 miles per hour.
• Gallons per hour = (gallons/mile) × (miles/hour) = (1/20) × 60 = 3.
• This is the chain rule: (dy/dt) = (dy/dx) × (dx/dt).

🚫 Common pitfalls

🚫 Wrong evaluation points

• Don't confuse: The derivative of sin(x²) is NOT (cos x)(2x).
• The cosine must be evaluated at the inside function x², not at x.
• Always evaluate the outer function's derivative at y = g(x).

🚫 Forgetting the inside derivative

• The derivative of (sin x)³ is 3(sin x)², times cos x.
• That extra factor cos x (the derivative of sin x) is easy to forget.
• The derivative of (1 - x)² is 2(1 - x), times (-1).

🚫 Confusing composition with products

• f(g(x)) is composition; f(x)g(x) is multiplication—completely different.
• Example: f(x) = x⁴, g(x) = x³ gives f(g(x)) = x¹² but f(x)g(x) = x⁷.
• Use the chain rule for composition, the product rule for multiplication.

🚫 Identity and inverse special cases

• Identity function: f(x) = x and g(x) = x gives f(g(x)) = x (derivative is 1 × 1 = 1).
• Inverse functions: If g adds 5 and f subtracts 5, then f(g(x)) = x (output equals input).

🧭 Overview

🧠 One-sentence thesis

Implicit differentiation allows us to find derivatives of functions that cannot be solved explicitly for y, and related rates problems use the chain rule to connect the rates of change of different quantities through their relationships.

📌 Key points (3–5)

• Implicit differentiation (ID): differentiate equations directly without solving for y, using the chain rule to include dy/dx wherever y appears.
• When to use ID: equations like y⁵ + xy = 3 or sin y + sin x = 1 cannot be solved explicitly for y, but ID still finds dy/dx at any point.
• Related rates setup: given dg/dt, find df/dt by identifying the relationship between f and g, then apply the chain rule: df/dt = (df/dg)(dg/dt).
• Common confusion: in related rates, substitute known values after differentiating, not before—differentiating a constant gives zero and loses information.
• Why it matters: ID reveals slopes and tangent lines for curves defined implicitly; related rates solve real-world problems where one changing quantity affects another.

🔍 Core concept: Implicit differentiation

🔍 What implicit differentiation means

Implicit differentiation: differentiating an equation F(x, y) = 0 directly by the chain rule, without solving for y in terms of x.

• Many equations cannot be solved for y explicitly (e.g., y⁵ + xy = 3—Galois proved no solution formula exists for fifth-degree equations).
• The function y(x) is defined implicitly by the equation, not given as an explicit formula.
• ID works by differentiating every term with respect to x, including dy/dx from the chain rule wherever y appears.

🧮 How to perform ID

Step-by-step process:

1. Differentiate every term in the equation with respect to x.
2. When differentiating terms containing y, include dy/dx (from the chain rule).
3. Collect all dy/dx terms on one side.
4. Solve algebraically for dy/dx.
5. Substitute the specific point (x, y) if needed.

Example: For y⁵ + xy = 3:

• Differentiate: 5y⁴(dy/dx) + x(dy/dx) + y = 0
• At point (2, 1): 5(dy/dx) + 2(dy/dx) + 1 = 0
• Solve: dy/dx = -1/7

🔄 Verification approaches

The excerpt shows two ways to check ID results:

• Method 1: If possible, solve for y explicitly and differentiate normally—should match ID result.
• Method 2: Solve for x instead (e.g., x = 3/y - y⁴), differentiate to get 1 = ... dy/dx, then solve for dy/dx.

Example: For xy = 2, ID gives x(dy/dx) + y = 0, so dy/dx = -y/x. Solving explicitly gives y = 2/x, so dy/dx = -2/x² = -y/x ✓

🔁 Second derivatives and geometric applications

🔁 Finding second derivatives implicitly

After finding dy/dx by ID, differentiate again to get d²y/dx²:

• Use the quotient rule or product rule on the dy/dx expression.
• Remember dy/dx itself contains y, so apply the chain rule again.

Example: Circle x² + y² = 25

• First derivative: 2x + 2y(dy/dx) = 0 → dy/dx = -x/y
• Second derivative: d²y/dx² = -(y - x(dy/dx))/y² = -(y² + x²)/y³

📐 Geometric interpretation: perpendicular lines

For the circle x² + y² = 25:

• Radius slope: y/x (goes across x, up y)
• Tangent slope: -x/y (goes across -y, up x)
• Product: (-x/y)(y/x) = -1, confirming perpendicularity

The second derivative -(y² + x²)/y³ is negative at the top of the circle, confirming it is concave down.

⏱️ Related rates problems

⏱️ What related rates means

Related rates: problems where you are given the rate of change of one quantity (dg/dt) and must find the rate of change of another quantity (df/dt) using a relationship between them.

• The chain rule connects the rates: df/dt = (df/dg)(dg/dt)
• The variable is typically t (time) because these are real-world applications.
• You must identify or derive the relationship between f and g from the problem setup.

🎯 Three-step solution method

Standard approach emphasized in the excerpt:

1. Write down a relation from the figure or problem description.
2. Take its derivative with respect to t (using implicit differentiation).
3. Substitute known information after differentiating.

Critical warning: Don't confuse: substitute values after differentiating, not before. If you substitute y = 50 into tan θ = y/100 to get tan θ = 1/2, then differentiate, you get zero (useless).

🌐 Example: Circle circumference

Simple case showing the pattern:

• Given: radius growing at dr/dt = 7
• Relation: C = 2πr
• Differentiate: dC/dt = 2π(dr/dt)
• Substitute: dC/dt = 2π(7) = 14π

Surprising implication: To put a rope around Earth that any 7-footer can walk under (adding 7 feet to radius), you only need 14π ≈ 44 feet more rope, regardless of Earth's size.

📊 Complex related rates examples

📊 Rectangle with changing sides

Problem: Rectangle sides change so dz/dt = 1 and dx/dt = 3(dy/dt). At x = 4, y = 3, find dx/dt.

Solution:

• Relation: x² + y² = z² (diagonal)
• Differentiate: 2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)
• Substitute x = 4, y = 3, z = 5, dz/dt = 1: 8(dx/dt) + 6(dy/dt) = 10
• Use dx/dt = 3(dy/dt): 6(dy/dt) = 2(dx/dt), so 10(dx/dt) = 10
• Answer: dx/dt = 1

🚶 Shadow lengthening problem

Problem: Person 2 meters tall walks from streetlight 8 meters high. Shadow lengthens at ds/dt = 4/9 meters/second. How fast is the person walking?

Solution:

• Draw a figure to identify variables (this was emphasized as hard—took three tries).
• Relation from similar triangles: x/6 = s/2
• Differentiate: dx/dt = (6/2)(ds/dt) = 3(4/9) = 4/3 meters/second
• Note: Never needed to know actual values of x, s, or the angle.

🎈 Balloon rising problem

Problem: Balloon rises at dy/dt = 3 m/s from point C, observer at A is 100 meters from C.

Three parts:

• (a) Rate z changes when y = 50: z² = y² + 100² → 2z(dz/dt) = 2y(dy/dt) → dz/dt = (2·50·3)/(2·50√5) = 3√5/5
• (b) Rate area changes: A = (1/2)(100)(y) = 50y → dA/dt = 50(dy/dt) = 150
• (c) Rate angle θ changes: tan θ = y/100 → sec²θ(dθ/dt) = (1/100)(dy/dt) → dθ/dt = 3/125

Key insight: Substitute y = 50 after differentiating tan θ = y/100. Substituting first gives tan θ = 1/2 (constant), whose derivative is zero.

⚡ Speed-of-light paradox

⚡ The paradox setup

Lighthouse problem: Light at A turns once per second (dθ/dt = 2π radians/second). How fast does point B move up shoreline?

• Relation: y = 100 tan θ
• Speed: dy/dt = 100 sec²θ (dθ/dt) = 200π sec²θ

Paradox: As θ approaches 90°, sec θ → ∞, so dy/dt → ∞. Point B moves faster than light c, contradicting relativity!

⚡ Resolution: light travel time

The error: we forgot light takes time to reach B.

• Arrival time: t = θ/(2π) + z/c (rotation time + travel time)
• Differentiate: dθ/dt = 2π(1 - (dz/dt)/c)
• Combine with z' = y' sin θ and y = 100 tan θ
• Result: y' = 200πc/(c cos²θ + 200π sin θ)

As θ → 90°, this approaches 200πc (finite), not infinity. The shadow can appear to move faster than light, but no physical object or information actually does.

⚠️ Small paradox about y and z

Apparent contradiction: From y = z sin θ, it looks like dy/dt = (dz/dt) sin θ. But the exact opposite is true: dz/dt = (dy/dt) sin θ.

Explanation: The angle θ is also changing with time. Correct differentiation: z² = y² + 100² → z(dz/dt) = y(dy/dt) → dz/dt = (y/z)(dy/dt) = (dy/dt) sin θ.

Don't confuse: you cannot treat θ as constant when it is actually changing—this is why we must differentiate the full relationship, not just substitute.

🧭 Overview

🧠 One-sentence thesis

Inverse functions undo each other's operations, and their derivatives obey the fundamental rule that (dx/dy)(dy/dx) = 1, which follows directly from the chain rule applied to the identity f(g(x)) = x.

📌 Key points (3–5)

• What inverse functions do: If g(x) = y, then the inverse g⁻¹(y) = x; one function undoes what the other does.
• The derivative rule: The slope of an inverse function is the reciprocal of the original function's slope: dx/dy = 1/(dy/dx).
• Domain and range swap: The domain of g matches the range of g⁻¹, and vice versa; inputs to one are outputs from the other.
• Common confusion: Not all functions have inverses—only functions that are steadily increasing or decreasing (one x for each y) can be inverted.
• How to find inverses: Solve the equation y = g(x) for x to get x = g⁻¹(y); inverting a chain h(g(x)) requires reversing the order: g⁻¹(h⁻¹(z)).

🔄 What inverse functions are

🔄 The core relationship

Inverse functions: If y = g(x) then x = g⁻¹(y); if x = g⁻¹(y) then y = g(x).

• The defining equation is f(g(x)) = x.
• Start with any input x, compute y = g(x), then compute f(y), and you must get back x.
• What one function does, the inverse undoes.
• Example: If g(x) = x - 2 and f(y) = y + 2, then starting from x = 5 gives y = 3, and f(3) = 5.
• The notation g⁻¹ is pronounced "g inverse" and is not the same as 1/g(x).

🔁 Symmetry in both directions

• The relationship is completely symmetric: if f is the inverse of g, then g is the inverse of f.
• Both compositions return to the start: g⁻¹(g(x)) = x and g(g⁻¹(y)) = y.
• Example: For y = √x and x = y², the square root of y² is y, and the square of √x is x.

🎯 Domain and range requirements

🎯 The matching rule

• The domain of a function matches the range of its inverse.
• The inputs to g⁻¹ are the outputs from g; the inputs to g are the outputs from g⁻¹.
• Example: For y = √x, the domain is x ≥ 0 (nonnegative inputs), which matches the range of x = y² when y ≥ 0.

⚠️ When inverses don't exist

• Not all functions have inverses.
• For each y, the equation g(x) = y must produce only one x.
• If two points x₁ and x₂ give g(x₁) = g(x₂), then g has no inverse—because g⁻¹(y) cannot equal both x₁ and x₂.
• Example: sin(x) = 1/2 has many solutions on the interval 0 ≤ x ≤ π, so sine is not invertible over that full interval.
• To be invertible over an interval, g must be steadily increasing or steadily decreasing.

🧪 Testing for invertibility

• Horizontal line test: If no horizontal line touches the graph twice, then f(x) is invertible (one x for each y).
• Don't confuse with the vertical line test (which checks if something is a function at all).

📐 The derivative of inverse functions

📐 The fundamental rule

From f(g(x)) = x, the chain rule gives:

(dx/dy)(dy/dx) = 1 or equivalently dx/dy = 1/(dy/dx)

• The slope of x = g⁻¹(y) times the slope of y = g(x) equals one.
• This comes from differentiating both sides of f(g(x)) = x, which gives 1.
• Example: For y = x³ and x = y^(1/3), we have dy/dx = 3x² and dx/dy = (1/3)y^(-2/3) = 1/(3x²).

🔍 Why the rule works

• The chain rule applied to f(g(x)) = x gives f'(g(x)) · g'(x) = 1.
• Writing y = g(x) and x = f(y), this becomes (dx/dy)(dy/dx) = 1.
• This is not ordinary algebra with fractions, but it is true—the derivatives are limits of fractions Δx/Δy and Δy/Δx.
• Example: For y = 3x, the slopes dy/dx = 3 and dx/dy = 1/3 multiply to give 1.

📊 Computing derivatives two ways

You can find dx/dy either:

• Directly: differentiate x = g⁻¹(y) with respect to y.
• Indirectly: compute 1/(dy/dx) using the original function.

Example: For y = x³ and x = y^(1/3):

• Direct: dx/dy = (1/3)y^(-2/3)
• Indirect: dx/dy = 1/(dy/dx) = 1/(3x²) = 1/(3y^(2/3))

Both give the same answer.

📈 Graphs of inverse functions

📈 The mirror-image property

• The graph of x = g⁻¹(y) is the mirror image of the graph of y = g(x).
• The reflection is across the 45° line (the line y = x).
• If point (2, 6) is on the graph of y = g(x), then point (6, 2) is on the graph of x = g⁻¹(y).
• The same pairs (x, y) appear on both graphs, but with roles swapped.

🔄 Why the reflection works

• The graph of y = g(x) shows x across (horizontal) and y up (vertical).
• The graph of x = g⁻¹(y) would naturally show y across and x up—which is the same picture turned across the 45° line.
• Don't confuse: some books graph y = g⁻¹(x), but the excerpt emphasizes x = g⁻¹(y) as the correct inverse notation.

🔗 Inverting chains of functions

🔗 The reversal rule

The inverse of z = h(g(x)) is a chain of inverses in the opposite order: x = g⁻¹(h⁻¹(z)).

• h⁻¹ is applied first because h was applied last.
• The key equation: g⁻¹(h⁻¹(h(g(x)))) = x.
• In the middle, h⁻¹ and h cancel, leaving g⁻¹(g(x)) = x.

🧩 Why order matters

• Example: For z = 3(x - 2), we have g(x) = x - 2 and h(y) = 3y.
• To invert: first divide by 3 (apply h⁻¹), then add 2 (apply g⁻¹).
• The inverse is x = (z/3) + 2, which is g⁻¹(h⁻¹(z)).
• The inverse of h ∘ g is g⁻¹ ∘ h⁻¹ (opposite order).

🔢 Practical example

For z = √(x - 2):

• Write z² = x - 2, then x = z² + 2.
• The inverse adds 2 and takes the square—but not in that order.
• Correct: z² + 2 (square first, then add).
• Wrong: (z + 2)² (this would add first, then square).

🌟 Important examples

🌟 Temperature conversion

• Fahrenheit to Celsius: y = (5/9)(x - 32)
• Celsius to Fahrenheit: x = (9/5)y + 32
• These are inverse functions: (5/9) subtracts 32 first; (9/5)y + 32 adds 32 last.
• The slopes dy/dx = 5/9 and dx/dy = 9/5 multiply to give 1.

🌟 Exponentials and logarithms (preview)

• If y = 2^x then x = log₂(y).
• What the exponential does, the logarithm undoes.
• The logarithm of 2^x is the exponent x.
• The excerpt mentions these are "overwhelmingly important" and will get a full chapter later.
• Natural logarithms use base e: y = e^x is the inverse of x = ln(y).

🧭 Overview

🧠 One-sentence thesis

Inverting trigonometric functions requires restricting their domains to intervals where they are one-to-one, and the chain rule reveals that the derivative of each inverse function is the reciprocal of the original function's slope expressed in terms of the output variable.

📌 Key points (3–5)

• Why domain restriction matters: Sine, cosine, and tangent repeat infinitely, so we must choose one piece of the curve where each output comes from exactly one input to make the function invertible.
• How to find derivatives of inverse trig functions: Use the chain rule: the slope of the inverse function equals 1 divided by the slope of the original function, then rewrite in terms of the new variable.
• The three core derivative formulas: The derivatives of inverse sine, inverse tangent, and inverse secant are the building blocks; the other three (inverse cosine, inverse cotangent, inverse cosecant) differ only by a minus sign.
• Common confusion—cofunctions: Inverse sine and inverse cosine are "cofunctions" that always add to π/2 (a right angle), so their derivatives add to zero; the same pattern holds for the other pairs.
• Why these matter in calculus: These derivatives provide new velocity–distance pairs (e.g., velocity 1/√(1 − t²) corresponds to distance sin⁻¹ t) and answer the question "What function has derivative 1/(1 + x²)?" (answer: tan⁻¹ x).

🔄 Why we restrict domains

🔄 The problem with unrestricted trig functions

• Trigonometric functions like sine go up and down infinitely often.
• If we allow the whole sine curve, infinitely many angles would satisfy sin x = 0.
• The sine function could not have an inverse because one output y would correspond to many inputs x.

✂️ Choosing the right interval

Domain restriction: To make a trigonometric function invertible, we select an interval where the function is steadily increasing (or steadily decreasing), so each output comes from exactly one input.

Function	Restricted domain for x	Range for y	Behavior on interval
sin x	−π/2 ≤ x ≤ π/2	−1 ≤ y ≤ 1	Increasing
cos x	0 ≤ x ≤ π	−1 ≤ y ≤ 1	Decreasing
tan x	−π/2 < x < π/2 (open interval)	All y	Increasing
sec x	0 ≤ x ≤ π, x ≠ π/2	|y| ≥ 1	Increasing (with gap)

• Don't confuse: The restricted domain is for the original function y = sin x; the inverse function x = sin⁻¹ y swaps the roles, so y becomes the input (between −1 and 1) and x becomes the output (between −π/2 and π/2).

📐 Inverse sine and inverse cosine

📐 Definition and notation

x = sin⁻¹ y means that y = sin x and |x| ≤ π/2.

• The inverse sine starts with a number y between −1 and 1 and produces an angle x (the angle whose sine is y).
• Historically called "arc sine" (arcsin in computing); the notation sin⁻¹ has nothing to do with 1/sin x.
• Example: The 30° angle x = π/6 has sine y = 1/2, so sin⁻¹(1/2) = π/6.

🔁 Round-trip identities

• sin⁻¹(sin x) = x for −π/2 ≤ x ≤ π/2
• sin(sin⁻¹ y) = y for −1 ≤ y ≤ 1
• Similarly for cosine: cos⁻¹(cos x) = x and cos(cos⁻¹ y) = y.

🧮 Key formula: cosine in terms of sine

If sin x = y, then cos x = √(1 − y²).

• This comes from cos² x = 1 − sin² x.
• Example: If sin x = y, then cos x = cos(sin⁻¹ y) = √(1 − y²).
• Why it matters: This formula is crucial for computing derivatives.

📉 Derivative of inverse sine

Using the chain rule:

• y = sin x gives dy/dx = cos x
• So dx/dy = 1/cos x = 1/√(1 − y²)

Derivative of sin⁻¹ y: dx/dy = 1/√(1 − y²)

• This gives a new velocity–distance pair: velocity v(t) = 1/√(1 − t²) corresponds to distance f(t) = sin⁻¹ t.
• Extreme case: At y = 1, the slope is infinite (1/0) because the graph of y = sin x is horizontal there (slope zero), so its mirror image is vertical.

🔻 Derivative of inverse cosine

The inverse cosine and inverse sine are cofunctions:

Cofunction identity: cos⁻¹ y + sin⁻¹ y = π/2

• The sum is constant (π/2), so its derivative is zero.
• Therefore the derivatives of cos⁻¹ y and sin⁻¹ y must add to zero (opposite signs).

Derivative of cos⁻¹ y: dx/dy = −1/√(1 − y²)

• Don't confuse: Two functions can have the same derivative if they differ by a constant; here sin⁻¹ y = −cos⁻¹ y + C with C = π/2.

📊 Inverse tangent and its derivative

📊 Definition and domain

x = tan⁻¹ y means that y = tan x and −π/2 < x < π/2 (open interval).

• The tangent can be any number (all real y), but the inverse tangent produces an angle between −π/2 and π/2.
• The interval is "open" because the endpoints are not included (the tangents of ±π/2 are not defined).
• Not a ratio: The inverse tangent is not sin⁻¹ y / cos⁻¹ y; it is the angle whose tangent is y.

🧮 Derivative of inverse tangent

The slope of y = tan x is dy/dx = sec² x.

• By the chain rule: dx/dy = 1/sec² x = 1/(1 + tan² x) = 1/(1 + y²)

Derivative of tan⁻¹ y: df/dy = 1/(1 + y²)

• Important application: What function has derivative 1/(1 + x²)? Answer: f(x) = tan⁻¹ x (just change letters).
• Example: The tangent of x = π/4 is y = 1. On the inverse tangent curve, dx/dy = 1/(1 + 1) = 1/2. On the tangent curve, dy/dx = sec²(π/4) = 2. The slopes multiply to give 1, as required by the chain rule.

🔢 The remaining three inverse functions

🔢 Inverse cotangent, secant, and cosecant

The idea is to use 1/(dy/dx) for y = cot x, y = sec x, and y = csc x.

Inverse function	Derivative	Notes
cot⁻¹ y	−1/(1 + y²)	Same as tan⁻¹ y but with minus sign
sec⁻¹ y	1/(|y|√(y² − 1))	Requires |y| ≥ 1; absolute value for positive slope
csc⁻¹ y	−1/(|y|√(y² − 1))	Same as sec⁻¹ y but with minus sign

🔗 Cofunction pairs and minus signs

Each inverse function and its "cofunction" add to π/2, so their derivatives add to zero:

• sin⁻¹ y + cos⁻¹ y = π/2 → derivatives differ by minus sign
• tan⁻¹ y + cot⁻¹ y = π/2 → derivatives differ by minus sign
• sec⁻¹ y + csc⁻¹ y = π/2 → derivatives differ by minus sign

Only three derivatives to learn: The other three just have minus signs.

📝 Note on inverse secant

• When y is negative, there is a choice for x = sec⁻¹ y.
• The excerpt selects the angle in the second quadrant (between π/2 and π) so that sec⁻¹ y = cos⁻¹(1/y), matching sec x = 1/cos x.
• This choice makes sec⁻¹ y an increasing function (cos⁻¹ y is decreasing), requiring the absolute value |y| in the derivative.
• Don't confuse: Some tables make a different choice (third quadrant), which omits the absolute value.

🚫 Domain gaps

• For sec⁻¹ y and csc⁻¹ y, the input must satisfy |y| ≥ 1 (the graph misses all points −1 < y < 1).
• The graph of sec⁻¹ y misses x = π/2 (where cosine is zero, so secant would be 1/0).
• The graph of csc⁻¹ y misses x = 0 (because csc 0 would be 1/sin 0 = 1/0).

📋 Summary table

📋 Quick reference for all six inverse functions

Function f(y)	Inputs y	Outputs x	Slope dx/dy
sin⁻¹ y, cos⁻¹ y	|y| ≤ 1	[−π/2, π/2], [0, π]	±1/√(1 − y²)
tan⁻¹ y, cot⁻¹ y	All y	(−π/2, π/2), (0, π)	±1/(1 + y²)
sec⁻¹ y, csc⁻¹ y	|y| ≥ 1	[0, π], [−π/2, π/2]	±1/(|y|√(y² − 1))

*Asterisks indicate that certain points (x = π/2 for sec⁻¹, x = 0 for csc⁻¹) are removed.

• The column of derivatives is what we need and use in calculus.
• Pattern: Each pair (sin⁻¹ and cos⁻¹, tan⁻¹ and cot⁻¹, sec⁻¹ and csc⁻¹) has the same formula except for sign.

🧭 Overview

🧠 One-sentence thesis

Integration solves the problem of adding infinitely many infinitesimal quantities by working backward from derivatives—finding a function f(x) whose derivative is the given v(x)—rather than performing the addition directly.

📌 Key points (3–5)

• Core idea: Integration is the reverse of differentiation; if v(x) is the derivative of f(x), then f(x) is the integral (antiderivative) of v(x).
• The fundamental insight: Taking sums reverses taking differences in algebra; taking integrals reverses taking derivatives in calculus.
• Geometric meaning: The integral represents the area under a curve, computed as the limit of sums of rectangle areas as rectangles become infinitely thin.
• Common confusion: Don't try to add infinitely many infinitesimals directly—the "whole point of calculus is to offer a better way" by finding the antiderivative.
• From discrete to continuous: The transition from algebra (finite sums) to calculus (integrals) happens by taking limits as the number of terms approaches infinity and each term becomes infinitesimal.

🔄 The Fundamental Connection Between Sums and Differences

➕ How sums reverse differences (algebra version)

Fundamental Theorem (before limits): If each v_j = f_j − f_(j−1), then v_1 + v_2 + ⋯ + v_n = f_n − f_0.

• Start with two sets of n numbers: v's (suggesting velocity) and f's (suggesting distance).
• Going from f to v: Take differences between consecutive f's to get each v.
• Going from v to f: Add up all the v's to recover the final f (minus the starting f_0).
• Example from the excerpt: Given f = 1, 3, 6, 10, the differences are v = 1, 2, 3, 4; conversely, summing v = 1, 2, 3, 4 gives back f_4 = 10 (when f_0 = 0).

🔭 The telescoping sum

The sum "telescopes" because intermediate terms cancel:

• v_1 + v_2 + v_3 + ⋯ = (f_1 − f_0) + (f_2 − f_1) + (f_3 − f_2) + ⋯
• Every f_1 is canceled by −f_1, every f_2 by −f_2, etc.
• Only the last term f_n and the starting term −f_0 survive.
• Why this matters: We can compute a sum by finding the right f's, rather than adding term by term.

🎯 Example: adding odd numbers

• Question: How to add 1 + 3 + 5 + ⋯ + 99?
• The odd numbers are differences between squares: 0, 1, 4, 9, 16, …
• There are 50 odd numbers from 1 to 99.
• By the Fundamental Theorem, the sum equals (50)² = 2500.
• The tricky part: Discovering that the f's are squares; in calculus, the tricky part is finding the right f(x).

📐 From Rectangles to Areas Under Curves

💰 The income example (discrete case)

The excerpt uses a company earning √x million dollars per year after x years.

• Bar graph approach: In the first 4 years, income rates are √1, √2, √3, √4 million/year.
• Each rectangle has base = 1 year and height = income rate.
• Total income = sum of rectangle areas = √1 + √2 + √3 + √4 ≈ 6.15 million.
• Two perspectives: (1) arithmetic sum of rates × time; (2) geometric sum of rectangle areas.

🔬 Refining the estimate

• Problem: The bar graph overstates income because the rate changes continuously, not in yearly jumps.
• Divide time into quarters (16 rectangles): total area ≈ 5.56 million, closer to truth.
• Divide into weeks (208 rectangles), then days (1460 rectangles), then hours…
• Each refinement gives thinner rectangles with heights √(Δx), √(2Δx), …, √4.
• The limit: As Δx → 0, the number of rectangles → ∞, and the sum approaches the true area under the √x curve.

🎨 The integral as a limit

The area under the curve is the limit of the sum as Δx → 0.

• This limiting area is called the integral.
• The discrete sum v_1 + v_2 + ⋯ + v_n becomes a continuous integral.
• Don't confuse: The integral is not an infinite sum you compute by adding; it's a limit that you find by discovering the antiderivative f(x).

🔑 The Key Idea: Reversing the Derivative

🧮 Algebra vs. calculus comparison

Algebra (finite)	Calculus (limit)
Compute v_1 + ⋯ + v_n	Compute limit of Δx[v(x) + v(2Δx) + ⋯]
Find f's such that v_j = f_j − f_(j−1)	Find f(x) such that v(x) = df/dx
Sum = f_n − f_0	Integral = area under v(x) curve

🔍 What integration means

• Integration is finding an antiderivative: If v(x) is the derivative of some function f(x), then f(x) is the integral of v(x).
• Example from the excerpt: The integral of v = cos x is f = sin x; the integral of v = x is f = (1/2)x².
• Why this works: Instead of adding infinitely many infinitesimals, we find f and use the Fundamental Theorem (in its calculus form).

⚠️ The starting point issue

• In the discrete case, we need f_0 as a starting value.
• In calculus, this becomes the "constant of integration" C.
• Don't confuse: Any constant can be added to f(x) because the derivative of a constant is zero; the differences between f's remain unchanged.

🎓 The Transition to Calculus

📊 Optimist vs. pessimist estimates

• Optimist: Uses the income rate at the end of each period (heights √(Δx), √(2Δx), …, √4).
• Pessimist: Uses the rate at the beginning of each period (heights 0, √(Δx), …, √(4−Δx)).
• The optimist always overestimates; the pessimist always underestimates.
• As time periods shrink, both estimates converge to the same limit: the true area under the √x curve.
• Key insight: The difference between them is the area of the last rectangle, which shrinks to zero.

🌊 The limiting process

• Years → weeks → days → hours → seconds → continuous time.
• The number of rectangles grows: 4 → 16 → 208 → 1460 → …
• Each rectangle's base Δx shrinks toward zero.
• The sum Δx · [v(x) + v(2Δx) + ⋯] approaches a definite number: the integral.
• Why limits are essential: This is "the essential difference between algebra and calculus"—we must take the limit "to add up infinitely many infinitesimals."

🧩 The derivative connection

From Problem 15 in the excerpt:

• Let f(x) = area under the √x curve from 0 to x.
• The area from 0 to x + Δx is f(x + Δx).
• The extra area Δf is almost a rectangle with base Δx and height √x.
• So Δf/Δx is close to √x.
• As Δx → 0, we suspect df/dx = √x.
• This confirms: If f(x) is the integral of v(x), then v(x) is the derivative of f(x).

🧭 Overview

🧠 One-sentence thesis

The antiderivative reverses differentiation to find the area under a curve, enabling exact calculation of integrals through the Fundamental Theorem of Calculus.

📌 Key points (3–5)

• What an antiderivative is: a function f(x) whose derivative equals the given function v(x); finding it means working backward from derivatives.
• The integral symbol and its meaning: ∫ (stretched S from Latin "sum") represents the limit of rectangular areas approaching the curved area under a graph.
• Definite vs indefinite integrals: indefinite integrals are functions f(x) + C (with arbitrary constant C); definite integrals are numbers f(b) − f(a) between limits a and b.
• Common confusion: the constant C appears in indefinite integrals but cancels when computing definite integrals (the difference f(b) − f(a)).
• The core technique: to find the antiderivative of x^n, raise the exponent to n+1 and divide by n+1, then adjust to cancel unwanted factors from differentiation.

🔄 Working backward from derivatives

🔄 The antiderivative concept

Antiderivative: a function f(x) such that df/dx = v(x); it reverses the derivative operation.

• Instead of finding the derivative of a given function, we find a function that produces a given derivative.
• The excerpt calls this "the opposite of Chapters 2–4" and requires working backwards.
• Example: since the derivative of x^5 is 5x^4, the antiderivative of x^4 is x^5/5 (divide by 5 to cancel the factor from differentiation).

🧮 The power rule in reverse

General formula: the antiderivative of x^n is x^(n+1)/(n+1), provided n+1 ≠ 0.

Why it works:

• The derivative lowers the exponent (x^n becomes nx^(n−1)).
• The antiderivative raises it back (x^n becomes x^(n+1)).
• Dividing by (n+1) cancels the unwanted factor that differentiation would produce.

Example from the excerpt: for v(x) = √x = x^(1/2):

• Raise exponent: x^(3/2)
• The derivative would be (3/2)x^(1/2), so divide by 3/2
• Result: f(x) = (2/3)x^(3/2)

📐 From rectangles to exact areas

📐 The limiting process

The excerpt describes approximating curved areas with rectangles:

• Rectangles have base Δx and heights v(x) at sample points.
• As Δx → 0, the sum v₁Δx + v₂Δx + ⋯ approaches the integral ∫v(x)dx.
• The "dx" in the integral notation indicates that Δx approaches zero.

Two requirements for valid approximation:

1. The largest width Δx_max must approach zero.
2. The top of each rectangle must touch or cross the curve.

🎯 Computing exact areas

Once the antiderivative f(x) is known, the area from x = 0 to x = 4 is simply f(4) − f(0).

Example: for v(x) = √x with f(x) = (2/3)x^(3/2):

• At x = 4: f(4) = (2/3)(8) = 16/3
• At x = 0: f(0) = 0
• Total area = 16/3 = 5⅓ million dollars (in the income context)

The excerpt notes this is the exact answer that thousands of rectangles were slowly approaching.

🔀 Definite vs indefinite integrals

🔀 Indefinite integrals (functions)

Indefinite integral: the most general antiderivative, written ∫v(x)dx = f(x) + C, where C is an arbitrary constant.

• It is a function, not a number.
• Contains the variable x.
• Includes constant C because the derivative of any constant is zero.
• Example: ∫(4 − x)dx = 4x − (1/2)x² + C

All these are antiderivatives of the same function:

• f(x) = 4x − (1/2)x²
• f(x) = 4x − (1/2)x² + 1
• f(x) = 4x − (1/2)x² − 9
• f(x) = 4x − (1/2)x² + C (general form)

🔢 Definite integrals (numbers)

Definite integral: the area under v(x) between limits a and b, written ∫ₐᵇv(x)dx = f(b) − f(a).

• It is a number, not a function.
• Contains no variable x and no arbitrary constant C.
• Determined by the function v(x) and the endpoints (limits of integration).

Why C cancels:

• At x = 3: f(3) = 7½ + C
• At x = 1: f(1) = 3½ + C
• Difference: f(3) − f(1) = (7½ + C) − (3½ + C) = 4

The constant C appears in both terms and cancels in the subtraction.

📊 Comparison table

Feature	Indefinite integral	Definite integral
Result type	Function f(x)	Number
Contains x?	Yes	No
Contains C?	Yes (arbitrary constant)	No (cancels)
Notation	∫v(x)dx	∫ₐᵇv(x)dx
Example	4x − (1/2)x² + C	8 (from 0 to 4)

🎓 The Fundamental Theorem preview

🎓 Connecting sums and integrals

The excerpt draws a parallel between discrete and continuous cases:

In algebra (discrete):

• Difference: fⱼ − fⱼ₋₁ = vⱼ
• Sum: v₁ + ⋯ + vₙ = fₙ − f₀

In calculus (continuous):

• Derivative: df/dx = v(x)
• Integral: ∫ₐᵇv(x)dx = f(b) − f(a)

🎓 The Fundamental Theorem statement

The excerpt states (equation 7):

∫ₐᵇv(x)dx = ∫ₐᵇ(df/dx)dx = f(b) − f(a)

What this means:

• The integral of v(x) equals the difference in its antiderivative f(x) at the endpoints.
• Integration and differentiation are inverse operations.
• The excerpt notes the proof comes later but gives the answer by following the analogy.

Don't confuse: the excerpt emphasizes this is a preview; the full proof is "the Fundamental Theorem of Calculus" covered later.

🧭 Overview

🧠 One-sentence thesis

Summation formulas for discrete sums (like 1 + 2 + ... + n) approach integral formulas for continuous areas as the number of rectangles increases and their width shrinks to zero, with correction terms vanishing in the limit.

📌 Key points (3–5)

• Sigma notation provides a compact way to write sums: Σ from j=1 to n of v_j means v₁ + v₂ + ... + vₙ
• Gauss's formula for the sum 1 + 2 + ... + n equals (1/2)n(n+1), and the sum of squares 1² + 2² + ... + n² equals (1/3)n³ plus correction terms
• Correction terms disappear in limits: discrete sums like (1/3)n³ + (1/2)n² + (1/6)n approach the clean integral result (1/3)n³ as rectangles become infinitely thin
• Common confusion: summation (discrete) vs integration (continuous)—sums have correction terms that depend on n, but integrals give exact answers without corrections
• The Fundamental Theorem connection: if f_n - f_(n-1) = v_n for sums, then df/dx = v(x) for integrals; both relate accumulation to rates of change

📝 Sigma notation and dummy variables

📝 What sigma notation means

Summation notation uses the capital Greek letter Σ (sigma) to express sums compactly.

• Structure: Σ from j=1 to n of v_j means add v₁ + v₂ + ... + vₙ
• The number below Σ is the lower limit (where to start)
• The number above Σ is the upper limit (where to stop)
• Example: Σ from j=1 to 4 of j² = 1² + 2² + 3² + 4² = 30

🔄 Dummy variables have no meaning

• The letter used for the index (j, k, i, etc.) is a dummy variable
• It appears only on the Σ side, not in the final numerical answer
• Σ from j=1 to 4 of j² = Σ from k=1 to 4 of k² = 30 (same result, different letter)
• Don't confuse: the upper limit n appears on both sides and affects the sum; the dummy variable does not

🔀 Changing limits and variables

When you shift the index, you must change three things consistently:

• The lower limit
• The upper limit
• The expression being summed

Example: Σ from j=101 to 200 of j equals Σ from k=1 to 100 of (k+100)

• Here j = k + 100, so when j starts at 101, k starts at 1
• When j ends at 200, k ends at 100
• The sum 101 + 102 + ... + 200 equals (1+100) + (2+100) + ... + (100+100)

🧮 Special summation formulas

🧮 Gauss's sum of integers

The sum 1 + 2 + 3 + ... + n equals (1/2)n(n+1).

Gauss's insight: pair first and last terms

• 1 + 100 = 101
• 2 + 99 = 101
• 3 + 98 = 101
• There are 50 such pairs, so the sum is 50 × 101 = 5050

General formula: (1/2)n(n+1) = (1/2)n² + (1/2)n

• The leading term is (1/2)n²
• The correction term is (1/2)n

Example: The sum from 101 to 200 equals (sum to 200) - (sum to 100) = 15,050

🔲 Sum of squares

The sum 1² + 2² + 3² + ... + n² equals (1/3)n³ + (1/2)n² + (1/6)n.

How to find it: guess and verify using mathematical induction

• Try f_n = (1/3)n³ first (copying from integrals)
• Check if f_n - f_(n-1) = n²
• Result: (1/3)n³ - (1/3)(n-1)³ = n² - n + 1/3 (not quite right)
• Add correction terms (1/2)n² + (1/6)n to fix the difference

Important: the leading term (1/3)n³ dominates for large n

• For n = 100: exact sum is 338,350
• Leading term alone: (1/3)(100)³ ≈ 333,333
• Correction terms add about 5,017

🔢 Mathematical induction principle

To prove a formula f_n is correct for all n:

1. Check the base case: verify f₁ is correct
2. Check the change: verify f_n - f_(n-1) equals v_n

If both checks pass, the formula is proven for all n.

🏗️ From rectangles to integrals

🏗️ Rectangular approximation of area

To find the area under v = x² from 0 to 100:

• Divide into n rectangles, each of width Δx = 100/n
• The j-th rectangle has height (j·Δx)²
• Total area = Σ from j=1 to n of (j·Δx)² · Δx

Factor out (Δx)³: area = (Δx)³ · [1² + 2² + ... + n²]

• Substitute the sum of squares formula
• Result: (Δx)³ · [(1/3)n³ + (1/2)n² + (1/6)n]
• Since n = 100/Δx, this becomes: (1/3)(100)³ + (1/2)(100)²·Δx + (1/6)(100)·(Δx)²

✨ Correction terms vanish

As rectangles get thinner (Δx → 0):

• The leading term (1/3)(100)³ stays constant
• The term (1/2)(100)²·Δx → 0
• The term (1/6)(100)·(Δx)² → 0 even faster

Why they vanish: correction terms account for the small corners of rectangles above the curve; as rectangles thin out, these corners disappear.

Example with 100, 1000, 10000 rectangles:

• 100 rectangles: area ≈ 338,350
• 1000 rectangles: area ≈ 333,833.5
• Limit: area = exactly 333,333.333... = (1/3)(100)³

🎯 The clean integral answer

The integral of v = x² from 0 to n is exactly (1/3)n³ with no correction terms.

Contrast with summation:

Method	Formula	Correction terms?
Summation	(1/3)n³ + (1/2)n² + (1/6)n	Yes, depend on n
Integration	(1/3)n³	No, exact answer

The antiderivative of v(x) = x² is f(x) = (1/3)x³, so the area from 0 to 100 is f(100) - f(0) = (1/3)(100)³.

⚡ Convergence speed matters

⚡ Slow convergence is unacceptable

For v = x from 0 to 4 with n rectangles:

• Exact area = 8
• Rectangular area = 8 + 2Δx (where Δx = 4/n)
• Error is proportional to Δx = 4/n

The problem: to get error of 10⁻⁶ requires 8 million rectangles

• Each time you double the rectangles, error only halves
• This is "much too slow" for practical computation

🎯 Better methods exist

Endpoint rule (rectangles touch curve at right edge):

• Error proportional to Δx
• Convergence: error ∝ 1/n

Midpoint rule (rectangles cross curve at center):

• Much better: error proportional to (Δx)²
• Convergence: error ∝ 1/n²
• Only 1000 rectangles needed for 10⁻⁶ accuracy

Don't confuse: unbalanced rectangles (one side too high) vs balanced rectangles (crossing the curve in the middle)

📊 General pattern for p-th powers

The sum 1^p + 2^p + ... + n^p equals [1/(p+1)]n^(p+1) plus correction terms.

The integral of v = x^p from 0 to n is exactly [1/(p+1)]n^(p+1).

Key insight: calculus doesn't care if n or p are integers; the formula works for any p where p+1 > 0.

🔗 The Fundamental Theorem connection

🔗 Parallel structure

For discrete sums:

• If f₁ + f₂ + ... + fₙ is the sum, then fₙ - f_(n-1) is the last term added
• Formula: if Σ v_j = f_n, then v_n = f_n - f_(n-1)

For continuous integrals:

• If ∫v(x)dx = f(x), then df/dx = v(x)
• The reverse of slope (derivative) is area (integral)

🌉 The limit transition

As Δx → 0 and n → ∞:

• Σ becomes ∫ (sum becomes integral)
• v_j becomes v(x) (discrete values become continuous function)
• Δx becomes dx (finite width becomes infinitesimal)
• Correction terms → 0 (discrete errors vanish)

Not yet fully proved: the text acknowledges this requires careful handling of multiple limits simultaneously, deferred to a later section.

🧭 Overview

🧠 One-sentence thesis

Substitution is the most powerful technique for finding antiderivatives by reversing the chain rule, allowing us to transform complicated integrals into simpler forms that we already know how to integrate.

📌 Key points (3–5)

• What indefinite integrals are: finding a function f(x) whose derivative is the given v(x), rather than computing areas as definite integrals.
• Linearity makes combinations easy: constants factor out, and integrals of sums equal sums of integrals.
• Substitution reverses the chain rule: choose an inside function u whose derivative du/dx appears in the integral, transform to u, integrate, then substitute back to x.
• Common confusion: you cannot substitute u for any expression—the derivative du/dx must be present (possibly after adjusting constants).
• Why some integrals fail: not every function has an elementary antiderivative; substitution requires the right structure.

🔍 Two approaches to integration

🔍 Indefinite vs definite integrals

• Definite integral: computes a number by summing rectangular areas as delta-x approaches zero.
• Indefinite integral: finds a function f(x) whose derivative is v(x).
• The excerpt emphasizes these give the same answer but work differently.
• Computers can always compute definite integrals numerically; symbolic codes like MACSYMA or Mathematica are needed to find antiderivative formulas.

📋 Known antiderivative pairs

The excerpt provides a reference table of basic pairs where df/dx = v(x):

Type	v(x)	f(x)
Powers	x to the n	x to the (n+1) divided by (n+1), plus C
Trig	cos x	sin x plus C
Trig	sin x	negative cos x plus C
Inverse trig	1 divided by square root of (1 minus x squared)	inverse sine of x plus C
Inverse trig	1 divided by (1 plus x squared)	inverse tangent of x plus C

• Every integration formula comes directly from reversing a differentiation formula.
• The constant C appears because derivatives of constants are zero.

➕ Linearity rules

➕ Sum and constant rules

Sum rule: the antiderivative of v(x) plus w(x) is the sum of their separate antiderivatives.

Constant rule: the antiderivative of c times v(x) is c times the antiderivative of v(x).

• These combine into full linearity: the antiderivative of a times v plus b times w equals a times the antiderivative of v plus b times the antiderivative of w.
• Proof: derivatives are linear, so (af plus bg) prime equals af prime plus bg prime.
• Practical tip: factor out constants immediately to simplify the integral.

🔢 The plus C convention

• All antiderivatives of the same function differ only by a constant.
• For combinations like av(x) plus bw(x), the constants from each part combine into a single C.
• Write "plus C" once at the end to give all possible antiderivatives.

Example: The antiderivative of x squared plus x to the negative 2 is (x cubed divided by 3) plus (x to the negative 1 divided by negative 1) plus C.

🔄 Substitution method

🔄 The core idea

Substitution reverses the chain rule for derivatives:

• The chain rule says: the derivative of f(g(x)) is f prime of g(x) times dg/dx.
• Reversing: if you see v(u(x)) times du/dx, the antiderivative is f(u(x)) plus C.

Example: sin of (x squared) has derivative (cos of x squared) times 2x. Therefore, the integral of x times cos of (x squared) is (1/2) times sin of (x squared) plus C.

🎯 Choosing the inside function u

Two critical points:

1. Constants are fixable: you can multiply and divide by constants like 2 or 15 to get the exact du/dx you need.
2. The derivative du/dx must be present: you cannot substitute u for any expression unless its derivative appears (possibly after constant adjustment).

Success cases:

• Integral of 2x times cos of (x squared): works because u = x squared gives du/dx = 2x, which is there.
• Integral of x squared times (x cubed plus 1) to the fourth: works because u = x cubed plus 1 gives du/dx = 3x squared, and we can fix the factor of 3.

Failure cases:

• Integral of cos of (x squared): fails because du/dx = 2x is missing.
• Integral of x squared times cos of (x squared): fails because du/dx = 2x is wrong (we have x squared instead).

📝 Four-step substitution process

1. Choose u(x) and compute du/dx.
2. Locate v(u) times du/dx times dx, or equivalently v(u) times du.
3. Integrate the integral of v(u) du to find f(u) plus C.
4. Substitute back: replace u with u(x) in the antiderivative.

Example: Integral of (cos of square root of x) times dx divided by (2 times square root of x):

• Step 1: u = square root of x, so du/dx = 1 divided by (2 times square root of x).
• Step 2: This is integral of cos u times du.
• Step 3: Integral is sin u plus C.
• Step 4: Answer is sin of (square root of x) plus C.

⚙️ Special cases: shifts and rescaling

Shifts: u = x plus c has du/dx = 1 (automatic).

• Integral of (x plus 2) cubed dx equals (1/4) times (x plus 2) to the fourth plus C.

Rescaling: u = cx has du/dx = c.

• Integral of cos of (2x) dx equals (1/2) times sin of (2x) plus C.
• The factor c from the chain rule cancels the 1/c you introduce.

General rules:

• Integral of v(x plus c) dx = f(x plus c)
• Integral of v(cx) dx = (1/c) times f(cx)

Combined: Integral of cos of (3x plus 7) dx equals (1/3) times sin of (3x plus 7) plus C.

⚠️ Common mistakes

Don't confuse dx with du: The factor du/dx from the chain rule is absolutely needed.

Nonexample: The integral of (x squared plus 1) squared dx does NOT equal (1/3) times (x squared plus 1) cubed. Why? Because if u = x squared plus 1, then du/dx = 2x is missing from the integral.

When substitution fails: Some functions like cos of (x squared) or 1 divided by square root of (a to the fourth minus sine squared of x) have no elementary antiderivative. They can be computed numerically but not expressed in closed form.

🧭 Overview

🧠 One-sentence thesis

The definite integral assigns a specific numerical value to the area under a curve by choosing a starting constant and can be rigorously defined as the limit of rectangular approximations, even when no simple antiderivative formula exists.

📌 Key points (3–5)

• From indefinite to definite: The indefinite integral f(x) + C becomes definite by setting C = -f(a) so that area starts at zero, yielding the formula ∫ from a to b of v(x) dx = f(b) - f(a).
• Two viewpoints: When f is known, the formula computes area; when f is unknown, the area itself defines the integral f(x).
• Riemann sums as definition: The integral is the common limit of lower sums s (using minimum heights) and upper sums S (using maximum heights) as the mesh width approaches zero.
• Common confusion: Not every function is Riemann integrable—the lower and upper sums must converge to the same limit; continuous functions always satisfy this, but some discontinuous functions (like the rational/irrational indicator) do not.
• Substitution changes limits: When substituting u = u(x), the limits change from [a, b] on x to [u(a), u(b)] on u.

🎯 Choosing the constant of integration

🎯 Setting the starting area to zero

The constant C in f(x) + C is chosen so that f(a) + C = 0, giving C = -f(a).

• The indefinite integral contains an arbitrary constant because any vertical shift has the same derivative.
• For a definite integral (a specific number), we want area = 0 at the starting point x = a.
• This yields the area from a to x as: ∫ from a to x of v(t) dt = f(x) - f(a).
• The area from a to b is: ∫ from a to b of v(x) dx = f(b) - f(a).

📐 The fundamental calculation pattern

The excerpt emphasizes a two-step process:

1. Find f(x) such that df/dx = v(x).
2. Substitute limits: Compute f(b) - f(a), written as f(x) evaluated from a to b (using brackets or a vertical bar).

Example: For v(x) = 5(x + 1)⁴, the antiderivative is f(x) = (x + 1)⁵, so ∫ from a to b equals (b + 1)⁵ - (a + 1)⁵.

Don't confuse: Adding any constant to f(x) (like writing (x + 1)⁵ - 1) gives an equally valid antiderivative, but f(b) - f(a) remains unchanged because the constant cancels.

🔄 Substitution with definite integrals

🔄 Changing the variable changes the limits

When substituting u = u(x):

• The new limits on u are u(a) and u(b).
• The formula becomes: ∫ from a to b of v(u(x)) · (du/dx) dx = ∫ from u(a) to u(b) of v(u) du.

Example: For ∫ from 0 to 1 of (x² + 5)³ · x dx, set u = x² + 5, so du/dx = 2x. The integral becomes ∫ from 5 to 6 of u³ · (du/2) = (1/8)u⁴ evaluated from 5 to 6.

⚠️ When no elementary formula exists

The excerpt notes that ∫ sin(x²) dx has no simple algebraic antiderivative.

• Trying f(x) = cos(x²) fails because the chain rule produces an extra 2x that cannot be adjusted away.
• Key insight: Every continuous v(x) still has an antiderivative f(x), even if we cannot write it in closed form.
• The area under the graph provides the definition of f(x) through Riemann sums.

📏 Riemann sums and the definition of the integral

📏 The setup: dividing the interval

Problem: Integrate continuous v(x) over [a, b].

Step 1: Split [a, b] into n subintervals using meshpoints x₁, x₂, ..., with x₀ = a and xₙ = b.

• Each subinterval k has length Δxₖ = xₖ - xₖ₋₁.
• In subinterval k, let mₖ = minimum of v(x) and Mₖ = maximum of v(x).

📊 Lower and upper sums

Lower sum s: The total area of rectangles with height mₖ (minimum) in each subinterval.
Upper sum S: The total area of rectangles with height Mₖ (maximum) in each subinterval.

• s = m₁Δx₁ + m₂Δx₂ + ... + mₙΔxₙ
• S = M₁Δx₁ + M₂Δx₂ + ... + MₙΔxₙ
• The true area under v(x) satisfies: s ≤ area ≤ S.

Key behavior: As new dividing points are added, the lower sum s increases (finer rectangles capture more area) and the upper sum S decreases (finer rectangles exclude more excess).

🎯 The definition via limits

Definition: The area A is the common limit of s and S as the maximum mesh width approaches zero: s → A and S → A as Δx_max → 0.

• If this common limit exists, A is the Riemann integral of v(x) from a to b.
• The limit exists for all continuous functions and some discontinuous ones.

🔢 Riemann sums (intermediate rectangles)

Between s and S, we can use any height v(xₖ) where xₖ is any point in subinterval k.

• Riemann sum S*: v(x₁)Δx₁ + v(x₂)Δx₂ + ... + v(x*ₙ)Δxₙ.
• The midpoint rule chooses x*ₖ at the center of each subinterval and is often more accurate than s or S.
• All such sums approach the same limit A if v is continuous.

🧪 Which functions are integrable?

✅ Continuous functions are always integrable

The excerpt states (with proof sketch):

• Every continuous function on a closed interval [a, b] is Riemann integrable.
• The proof relies on uniform continuity: for any ε, there exists a δ (independent of position) such that v(x) varies by less than 2ε within any interval of width 2δ.
• As ε → 0, the gap S - s shrinks to zero, so s and S converge to the same number A.

❌ Some discontinuous functions are not integrable

Example (Remark 1): Define V(x) = 1 at every fraction (rational number) and V(x) = 0 at every irrational number.

• Every subinterval contains both rationals and irrationals.
• Therefore mₖ = 0 and Mₖ = 1 in every subinterval.
• The lower sum is always s = 0; the upper sum is always S = b - a.
• The gap never closes, so V(x) is not Riemann integrable.

Don't confuse: A step function (with finitely many jumps) is integrable because only the intervals containing jumps have mₖ ≠ Mₖ, and their total width shrinks to zero.

🔬 Advanced note: Lebesgue integration

The excerpt mentions (Remark 5) that modern mathematics uses Lebesgue's approach to integrate functions like V(x).

• Lebesgue allows infinitely many subintervals (of shrinking width) to cover all rationals with total width ε.
• Since irrationals are "uncountable" and V(x) = 0 there, the Lebesgue integral of V(x) is zero.
• This extends integration beyond Riemann's definition but is not needed for continuous functions.

🔑 Key properties and notation

🔑 Standard notation

• f(x) evaluated from a to b is written with brackets [f(x)] from a to b or a vertical bar f(x)|ᵇₐ.
• This always means f(b) - f(a): the upper limit gives +f(b), the lower limit gives -f(a).

🔑 Why the definition matters

The excerpt emphasizes two perspectives:

1. Computational: When we know f(x), the formula ∫ from a to b of v(x) dx = f(b) - f(a) computes area.
2. Theoretical: When we don't know f(x), the area (defined via Riemann sums) constructs the antiderivative.

This completes the circle: the integral (area) leads back to the derivative through the Fundamental Theorem of Calculus.

🧭 Overview

🧠 One-sentence thesis

The integral obeys seven fundamental properties—including addition over neighboring intervals, sign reversal when going backward, and the Mean Value Theorem—that enable its most important application: computing the average value of a continuous function over an interval, which extends the discrete average to a continuum and forms the foundation of continuous probability.

📌 Key points (3–5)

• Seven basic properties: integrals add over neighboring intervals, equal zero from a point to itself, reverse sign when limits are swapped, cancel for odd functions, add for even functions, preserve inequalities, and guarantee an average value at some point c.
• Average value formula: the average of v(x) from a to b is (1/(b − a)) times the integral from a to b, extending the discrete average (sum divided by n) to the continuous case.
• Mean Value Theorem for integrals: if v(x) is continuous, there exists a point c where v(c) equals the average value—visualized as a rectangle with the same area as the region under the curve.
• Common confusion: "expected value" does not mean the outcome you literally expect; it is the probability-weighted average, which is predictable even when individual outcomes are random.
• From discrete to continuous probability: as intervals shrink, sums of (outcome × probability) become integrals of x p(x) dx, where p(x) is the probability density.

📐 The Seven Properties

📐 Property 1: Addition over neighboring intervals

If v(x) is integrated from a to b and then from b to c, the sum equals the integral from a to c.

• Formula: integral from a to b plus integral from b to c equals integral from a to c.
• Why it works: rectangular areas obey this rule, and their limits (the integrals) inherit it.
• Example: integrating velocity from time 0 to 5 and then 5 to 10 gives the same total distance as integrating from 0 to 10.

📐 Property 2: Integral from a point to itself is zero

• Formula: integral from b to b equals 0.
• Reason: the area over a single point has no width, so it contributes zero area.
• This follows from Property 1 when c = b: the two identical integrals on the left must cancel.

📐 Property 3: Reversing limits reverses the sign

Going backward from b to a produces the negative of the integral from a to b.

• Formula: integral from a to b equals − (integral from b to a).
• Why: when the "lower limit" is larger than the "upper limit," the steps Δx are negative, so rectangular areas are negative.
• Example: integral from 0 to x of t² dt = x³/3; integral from x to 0 equals −x³/3.
• Don't confuse: this is not about the function being negative; it is about the direction of integration.

📐 Property 4: Odd and even functions

Odd functions:

If v(−x) = −v(x), then the integral from −a to a equals 0.

• Reason: areas on opposite sides of the origin cancel.
• Example: integral from −a to a of 6x⁵ dx = x⁶ evaluated from −a to a = a⁶ − (−a)⁶ = 0.
• Curious fact: if v(x) is odd, its antiderivative f(x) is even.

Even functions:

If v(−x) = +v(x), then the integral from −a to a equals 2 times the integral from 0 to a.

• Reason: areas on both sides add.
• Example: integral from −a to a of cos x dx = sin a − sin(−a) = 2 sin a.

📐 Property 5: Positive functions have positive integrals

• If v(x) > 0 for a ≤ x ≤ b, then the integral from a to b is positive.
• Proof: lower sums s are positive and increase toward the integral, so the integral must be positive.
• Interpretation: positive velocity means positive distance; a positive function lies above a positive area.

📐 Property 6: Inequalities are preserved

If l(x) ≤ v(x) ≤ u(x) for all x in [a, b], then the integral of l ≤ integral of v ≤ integral of u.

• Why: rectangles for v stay between rectangles for l and u, and this holds in the limit.
• Example 1: cos t ≤ 1, so integrating from 0 to x gives sin x ≤ x.
• Example 2: 1 ≤ sec² t, so integrating from 0 to x gives x ≤ tan x (for x > 0).
• Example 3: 1/(1 + x²) ≤ 1, so tan⁻¹ x ≤ x.

📐 Property 7: Mean Value Theorem for integrals

If v(x) is continuous, there exists a point c in [a, b] where v(c) equals the average value of v(x).

• Formula: v(c) = (1/(b − a)) times the integral from a to b of v(x) dx.
• This is the same as the ordinary Mean Value Theorem for derivatives: f′(c) = (f(b) − f(a))/(b − a).
• Direct proof: imagine a rectangle across [a, b]; raise its height from v_min to v_max until its area equals the area under the curve—at that height v(c), the function equals its average.
• Relies on the intermediate value theorem: a continuous function takes on every height between its minimum and maximum.

📊 Average Value of a Function

📊 Definition and formula

The average value of v(x) from a to b is (1/(b − a)) times the integral from a to b of v(x) dx.

• Parallel to discrete average: (1/n) times (v₁ + v₂ + ⋯ + v_n).
• Integration is the continuous analog of summation.
• Example: the average of 1/n, 2/n, 3/n, …, n/n is (n + 1)/(2n); as n → ∞, this approaches 1/2, which equals the integral from 0 to 1 of x dx.

📊 Examples of average values

Example: Average of x from 0 to 1

• Integral from 0 to 1 of x dx = 1/2.
• The average value of all numbers between 0 and 1 is 1/2.

Example: Average of x² from −1 to 1

• (1/2) times integral from −1 to 1 of x² dx = x³/6 evaluated from −1 to 1 = 1/6 − (−1/6) = 1/3.
• Where does x² equal 1/3? At c = ±1/√3, the Gauss points (important for numerical integration).

Example: Average of sin² x from 0 to π

• (1/π) times integral from 0 to π of sin² x dx = (x − sin x cos x)/2 evaluated from 0 to π = 1/2.
• The function sin² x = 1/2 − (1/2) cos 2x oscillates around its average value 1/2.
• The point c is π/4 or 3π/4, where sin² c = 1/2.

Example: Average of an odd function

• The average of x from −1 to 1 is 0 (the center point c = 0).

📊 Geometric interpretation

• The average value v_ave is the height of a rectangle over [a, b] with the same area as the region under v(x).
• Equal areas lie above and below the line y = v_ave.

🎲 Expected Value and Continuous Probability

🎲 Discrete expected value

Expected value: multiply each outcome by its probability and add.

Example: Rolling two dice

• Outcomes range from 2 to 12.
• Probability of 2 is (1/6)(1/6) = 1/36 (two ones).
• Probabilities: 2 → 1/36, 3 → 2/36, 4 → 3/36, …, 7 → 6/36, …, 12 → 1/36.
• All probabilities sum to 1 (all possibilities covered).
• Expected value: 2·(1/36) + 3·(2/36) + 4·(3/36) + ⋯ + 12·(1/36) = 7.
• Key insight: one roll is unpredictable, but the average of a million rolls is almost completely predictable.

🎲 Continuous probability

Transition from discrete to continuous:

• In the continuous case, the probability of hitting any particular number x is zero.
• Instead, an interval has nonzero probability: the probability of an outcome between x and x + Δx is p(x) Δx, where p(x) is the probability density.
• The integral of p(x) over all outcomes equals 1 (some outcome must happen).

Example: Uniform distribution from 2 to 12

• All numbers between 2 and 12 are equally probable, so p(x) = 1/10.
• Discrete approximation with Δx = 1: average = 2·(1/10) + 3·(1/10) + ⋯ + 11·(1/10) = 6.5.
• Continuous limit: expected value E(x) = integral from 2 to 12 of x p(x) dx = integral from 2 to 12 of x (dx/10) = x²/20 evaluated from 2 to 12 = 7.

Example: Uniform distribution from 0 to 1

• p(x) = 1 (since the interval has length 1).
• E(x^n) = integral from 0 to 1 of x^n dx = 1/(n + 1).
• E(1/√x) = integral from 0 to 1 of dx/√x = 2.
• E(1/x) = integral from 0 to 1 of dx/x = ∞ (does not exist).

🎲 Common confusion: "expected" does not mean "what you expect"

• Don't confuse: the expected value is not the outcome you literally expect to see.
• It is the probability-weighted average, which is predictable even when individual outcomes are random.
• Example: if you pick a random number between 0 and 1, its expected value is 1/2, but you don't expect to pick exactly 1/2.

🎲 The class-size paradox

Setup:

• 95 classes have 20 students; 5 classes have 200 students.
• Total: 100 classes, 2900 students.

Two perspectives:

Viewpoint	Calculation	Result
Random professor	(95·20 + 5·200)/100	29 students
Random student	20·(1900/2900) + 200·(1000/2900)	82 students

• Why the difference: a random student is more likely to be in a large class because large classes contain more students.
• Probability for the student: 1900/2900 of being in a small class, 1000/2900 of being in a large class.
• Don't confuse: (a) the percentage of cars with one person vs. (b) the percentage of people alone in a car—these are different.

🧭 Overview

🧠 One-sentence thesis

The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse operations, enabling us to compute integrals by finding antiderivatives and to understand that integration applies far beyond simple rectangular areas.

📌 Key points (3–5)

• Part 1 says: The derivative of an integral (area function from a to x) equals the original function v(x).
• Part 2 says: The integral of a derivative df/dx recovers the original function f(x) plus a constant; definite integrals equal f(b) − f(a).
• Variable limits: When both endpoints of an integral depend on x, use the chain rule: derivative = v(b(x))·(db/dx) − v(a(x))·(da/dx).
• Common confusion: Integration is not limited to vertical rectangles under curves—thin rings, shells, horizontal strips, and triangles can also be integrated.
• Why it matters: The theorem connects local properties (derivatives at points) to global properties (integrals over intervals) and extends to volumes, probabilities, and other applications.

🔄 Part 1: Derivative of an integral

🔄 The main statement

Fundamental Theorem, Part 1: If v(x) is continuous and f(x) = integral from a to x of v(t) dt, then df/dx = v(x).

• The integral from a fixed point a to a variable point x defines a function f(x) (the accumulated area).
• Even without a formula for f(x), we can find its derivative.
• The derivative of this area function equals the height of the original curve.

📐 How the proof works

• Consider a small change Δx: the new area minus the old area is the thin strip from x to x + Δx.
• That strip has area ≈ v(x) · Δx (a thin rectangle).
• Dividing by Δx gives Δf/Δx ≈ v(x).
• As Δx → 0, the approximation becomes exact: df/dx = v(x).
• Key tool: The Mean Value Theorem for integrals says the average value on a short interval equals v(c) for some point c in that interval; as the interval shrinks, c is squeezed toward x.

🖼️ Graphical meaning

• The f-graph gives cumulative area under the v-graph.
• A thin vertical strip has area Δf ≈ height v(x) × base Δx.
• Dividing by the base gives the height: Δf/Δx → v(x).

Example: If f(x) = integral from 0 to x of t² dt, then df/dx = x².

🔀 Variable endpoints

🔀 Lower limit moving

If g(x) = integral from x to b of v(t) dt, then dg/dx = −v(x).

• When the lower limit moves forward, area is removed from the left.
• Hence the minus sign.
• Quick proof: reverse the limits (integral from b to x = − integral from x to b), then apply Part 1.

🔀 Both limits moving

If A = integral from a(x) to b(x) of v(t) dt, then dA/dx = v(b(x))·(db/dx) − v(a(x))·(da/dx).

• Two thin strips: one added at the upper limit, one subtracted at the lower limit.
• Each strip's area ≈ height × thickness.
• This is the chain rule in action.

Example: A = integral from 2x to x³ of cos t dt has dA/dx = (cos x³)·(3x²) − (cos 2x)·(2).

⚠️ Don't confuse

• When a = 0 and b = x, we get da/dx = 0 and db/dx = 1, so the derivative is just v(x) (Part 1).
• When a = x and b = constant, the derivative is −v(x) (lower limit case).
• The general formula covers both.

🔁 Part 2: Integral of a derivative

🔁 The main statement

Fundamental Theorem, Part 2: If v(x) = df/dx, then integral from a to b of v(x) dx = f(b) − f(a).

• To integrate v(x), find any antiderivative f(x) (a function whose derivative is v).
• Evaluate f at the endpoints and subtract.
• This is the tool we use most, because integrals are harder to compute than derivatives.

🔑 The key lemma

• Special case: If df/dx = 0 everywhere in an interval, then f(x) = constant.
• Proof uses the Mean Value Theorem: if f(a) ≠ f(b), then f′(c) = [f(b) − f(a)]/(b − a) ≠ 0 for some c, contradicting df/dx = 0.
• General case: If dA/dx = df/dx, then A(x) = f(x) + C.
• The derivative of A(x) − f(x) is zero, so A(x) − f(x) must be constant.

🔗 Connecting Part 1 and Part 2

• Part 1 says A(x) = integral from a to x of v(t) dt has dA/dx = v(x).
• If f(x) is any other antiderivative of v(x), then A(x) = f(x) + C.
• At the lower limit, A(a) = 0, so f(a) + C = 0, hence C = −f(a).
• At the upper limit, A(b) = f(b) + C = f(b) − f(a).
• Therefore the definite integral equals f(b) − f(a).

🧩 Why this matters

• We can compute integrals by finding antiderivatives (often easier than summing rectangles).
• The theorem connects local information (derivative at each point) to global information (total change over an interval).

🌀 Beyond rectangles: New applications

🌀 Thin rings for circles

• Question: Why does the area A of a circle satisfy dA/dr = C (circumference)?
• The area is πr², and its derivative 2πr is the circumference.
• Geometric reason: Divide the circle into thin concentric rings.
• A ring of thickness Δr has area ≈ circumference × Δr = 2πr · Δr.
• Integrating from 0 to r: A = integral of 2πr dr = πr².
• The ring "unwinds" into a thin strip of width Δr and length ≈ C.

Example: This is the first step away from rectangles—we add up rings instead.

🌀 Thin shells for spheres

• Question: Why does the volume V of a sphere satisfy dV/dr = A (surface area)?
• A thin spherical shell from radius r to r + Δr has volume ≈ surface area × thickness = 4πr² · Δr.
• Integrating from 0 to r: V = integral of 4πr² dr = (4/3)πr³.
• The derivative of volume is surface area.

Main point: Integration is not restricted to rectangles; shells, rings, and other shapes work too.

🔲 Puzzle: The square

• A square with side s has area A = s².
• The derivative dA/ds = 2s is only half the perimeter (4s).
• Why? The excerpt leaves this as an exercise (the figure is misleading).

📏 Horizontal rectangles

• Problem: Find the area under v(x) = arccos x from x = 0 to x = 1.
• We don't have an easy antiderivative for arccos x.
• Solution: Use horizontal strips instead of vertical ones.
• A horizontal strip at height v has length x = cos v and thickness Δv.
• Integrate upward: area = integral from 0 to π/2 of cos v dv = sin v evaluated from 0 to π/2 = 1.

Don't confuse: The limits are now on v, not on x; we integrate in the direction perpendicular to the usual one.

🧮 Summary of techniques

Situation	Formula	Key idea
Fixed lower limit a, variable upper x	df/dx = v(x)	Part 1: derivative of area = height
Variable lower x, fixed upper b	dg/dx = −v(x)	Area removed from left
Both limits a(x) and b(x) variable	dA/dx = v(b)·(db/dx) − v(a)·(da/dx)	Chain rule on both ends
Antiderivative known	integral v dx = f(b) − f(a)	Part 2: evaluate at endpoints
Non-rectangular regions	Use rings, shells, horizontal strips, etc.	Match the geometry to the problem

🧭 Overview

🧠 One-sentence thesis

Numerical integration methods improve accuracy by fitting higher-degree polynomials to the function being integrated, with the quality of a formula determined by how many powers of x it integrates exactly.

📌 Key points (3–5)

• Goal: compute the definite integral I = ∫ from a to b of y(x) dx accurately and quickly when an antiderivative is unavailable or rectangle methods are too crude.
• Order of accuracy: a method has order p if it integrates 1, x, x², ..., x^(p−1) exactly but fails on x^p; higher p means the error involves (Δx)^p, so smaller errors.
• Common confusion: the trapezoidal rule (second-order) and midpoint rule (second-order) both beat rectangles (first-order), but Simpson's rule (fourth-order) is much better because it unexpectedly integrates x³ exactly, not just x².
• Why it matters: each evaluation of y(x) can be expensive (a subroutine call), so minimizing the number of sample points while staying below error tolerance is critical.
• Practical insight: second-order rules need ~1000 values for 10^(−6) tolerance; fourth-order methods (Simpson, Gauss) are far more efficient.

📏 Rectangle rules and first-order accuracy

📏 Right and left rectangle rules

Right rectangle rule R_n: Δx · (y₁ + y₂ + ⋯ + y_n)
Left rectangle rule L_n: Δx · (y₀ + y₁ + ⋯ + y_(n−1))

• Divide [a, b] into n equal intervals of length Δx = (b − a)/n.
• R_n uses the function value at the right endpoint of each interval; L_n uses the left endpoint.
• Simple to compute and visualize, but very inaccurate.

🔺 Error in rectangle rules

• When y(x) is a straight line, the errors form triangles with base Δx and height (y_(j+1) − y_j).
• The total error telescopes:
  • R_n − I = (1/2) Δx · (y_n − y₀) = (1/2) Δx · [y(b) − y(a)]
  • L_n − I = −(1/2) Δx · [y(b) − y(a)]
• First-order accuracy: error is proportional to Δx (the first power).
• Example: integrating √x from 0 to 1 with n = 1000 gives error ≈ 0.0005 ≈ (1/2) Δx.

⚠️ Why rectangles are slow

• The greater the slope of y(x), the greater the error—rectangles have zero slope and cannot follow a slant.
• To achieve error 1/1,000,000 for y = x from 0 to 1 requires 500,000 rectangles.
• The error prediction (1/2) Δx [y(b) − y(a)] is "asymptotically correct": as Δx → 0, the ratio of predicted to actual error approaches 1.

🔄 Special cases

• When y(b) = y(a): the leading error term vanishes; the next term involves (Δx)² and the error drops much faster.
• Periodic functions over a complete period: y(0) = y(1) and all derivatives match at endpoints, so errors go to zero exponentially fast (e.g., 1/(10 + cos 2x) integrated from 0 to 1 shows practically zero error even with few rectangles).

🔷 Second-order methods: trapezoidal and midpoint

🔷 Trapezoidal rule T_n

Trapezoidal rule: T_n = (1/2) R_n + (1/2) L_n = Δx · [(1/2) y₀ + y₁ + ⋯ + y_(n−1) + (1/2) y_n]

• Average of the right and left rectangle rules.
• Geometrically: the area of a trapezoid with base Δx and heights y_(j−1) and y_j is (1/2) Δx · (y_(j−1) + y_j).
• Exact for y = x: trapezoids fit under a sloping line, so the error for straight lines is zero.
• Second-order accuracy: error is proportional to (Δx)².

📐 Midpoint rule M_n

Midpoint rule: M_n = Δx · (y_(1/2) + y_(3/2) + ⋯ + y_(n−1/2))

• Evaluate y(x) at the midpoint of each interval: x = (1/2) Δx, (3/2) Δx, (5/2) Δx, …
• A rectangle with height at the midpoint has the same area as a trapezoid when the graph is a straight line.
• Also exact for y = x, so also second-order.

📊 Error comparison for y = x²

n	Trapezoidal error T_n − I	Midpoint error M_n − I
1	1/6	−1/12
10	1/600	−1/1200
100	1/60000	−1/120000

• Errors fall by 100 when n is multiplied by 10 (confirming (Δx)² behavior).
• Midpoint is twice as accurate as trapezoidal: the leading error for M is −(1/24) (Δx)² [y′(b) − y′(a)], versus (1/12) (Δx)² [y′(b) − y′(a)] for T.
• Don't confuse: both are second-order, but the constant in front of (Δx)² differs.

🧮 Error formulas

• Trapezoidal: T_n − I ≈ (1/12) (Δx)² · [y′(b) − y′(a)]
• Midpoint: M_n − I ≈ −(1/24) (Δx)² · [y′(b) − y′(a)]
• The exact formulas replace [y′(b) − y′(a)] with (b − a) y″(c) for some unknown point c (like the Mean Value Theorem).
• In practice, the unknown c limits usefulness; the key takeaway is the (Δx)² factor.

🎯 Fourth-order method: Simpson's rule

🎯 Simpson's rule S_n

Simpson's rule: S_n = (1/3) T_n + (2/3) M_n = (Δx/6) · [y₀ + 4y_(1/2) + 2y₁ + 4y_(3/2) + 2y₂ + ⋯ + 4y_(n−1/2) + y_n]

• Combines trapezoidal and midpoint with weights 1/3 and 2/3 to cancel more error.
• The famous 1–4–2–4–2–⋯–4–1 pattern: endpoints get weight 1, midpoints get weight 4, interior full-step points get weight 2, all multiplied by Δx/6.
• Parabolic approximation: fits a parabola through three consecutive points (y₀, y_(1/2), y₁) and integrates the parabola exactly.

🚀 Why Simpson is fourth-order

• By construction, Simpson integrates 1, x, and x² exactly (it was designed for parabolas).
• Surprise: it also integrates x³ exactly, even though it was not designed to do so.
• Reason: over the symmetric interval [−1, 1], x³ is an odd function with zero integral, and Simpson's symmetric weights give zero by symmetry.
• Fourth-order accuracy: error is proportional to (Δx)⁴, so errors drop by 10,000 when n is multiplied by 10.

📉 Error behavior for powers of x

y(x)	n = 1 error	n = 10 error	n = 100 error
x²	0	0	0
x³	0	0	0
x⁴	8.33×10⁻³	8.33×10⁻⁷	8.33×10⁻¹¹

• Exact for x² and x³; first error appears at x⁴.
• The error for x⁴ confirms (Δx)⁴ scaling: multiplying n by 10 divides error by 10⁴.

🔑 Why Simpson is popular

• Efficiency: achieves high accuracy with far fewer function evaluations than second-order methods.
• Example: to reach 10⁻⁶ tolerance, second-order methods need ~1000 samples; Simpson needs far fewer.
• Each evaluation of y(x) can be expensive (a subroutine call), so minimizing the number of samples is crucial.

🌟 Gauss rule (optional advanced method)

🌟 Gauss two-point rule

Gauss rule over [−1, 1]: ∫ from −1 to 1 of y(x) dx ≈ y(−1/√3) + y(1/√3)

• Uses only two points per interval, yet achieves fourth-order accuracy.
• The "Gauss points" x = ±1/√3 are chosen so that 1, x, x², and x³ are all integrated exactly.
• By symmetry, odd powers (x, x³) integrate to zero; the key is x²: the integral of x² from −1 to 1 is 2/3, and (−1/√3)² + (1/√3)² = 1/3 + 1/3 = 2/3.

⚖️ Gauss vs Simpson

• Gauss: two points per interval, fourth-order; best for thousands of integrations over one interval.
• Simpson: also uses two new y-values per interval (midpoint and next full step), fourth-order; better when intervals go back-to-back because sample points align.
• For y = x⁴ from 0 to 1:
  • Simpson error (n = 1): 8.33×10⁻³
  • Gauss error (n = 1): −5.56×10⁻³ (slightly better constant, same (Δx)⁴ scaling)

🔧 When to use Gauss

• Gauss points are not as convenient as equally spaced points (hand calculators prefer Simpson).
• Gauss shines when the same interval is integrated many times with different functions.
• Don't confuse: both are fourth-order, but Gauss uses unequal spacing within each interval.

💻 Practical numerical integration on calculators

💻 How calculators actually work

• Points are not equally spaced: machines may internally change variables (e.g., replace x by 3u² − 2u³) to avoid singularities at endpoints.
• Example: ∫ from 0 to 1 of dx/√x has an infinite value at x = 0; the substitution x = 3u² − 2u³ gives dx = 6(u − u²) du, which vanishes at u = 0, removing the singularity.
• The differential 6(u − u²) du was chosen to be zero at both u = 0 and u = 1, so the machine does not need y(x) at the endpoints (where infinity is most common).

⚠️ Difficult cases

• Singularities inside [a, b]: break the interval into two pieces.
• Integrals to infinity: chop off the tail where the integrand is negligible (e.g., stop e^(−x²) at x = 10).
• Rapid oscillations: the answer depends on cancellation of highs and lows; the calculator requires many integration points.
• Aliasing danger: if sin(8x) is sampled with Δx = 1/8, it is always zero—a high frequency 8 is confused with frequency 0. Unequal spacing prevents this.

🎛️ User interface

• Input: the function y(x), endpoints a and b, and the desired accuracy (e.g., 0.00001).
• Output: the integral I and an estimated error bound.
• The machine samples y(x) at points x₁, …, x_k and estimates accuracy based on how answers converge as more points are added.
• Formulas using 1, 3, 7, 15, … sample points are common; each new formula reuses samples from the previous one, stopping when answers are close.

🛡️ How any method can be deceived

• Ask for the integral of y = 0 and note the sample points x₁, …, x_k.
• Then integrate Y(x) = (x − x₁)² ⋯ (x − x_k)²; this also returns zero (now wrong) because the calculator follows the same steps and Y is zero at all sample points.
• Lesson: no method is foolproof; the function must be reasonably smooth and not adversarially constructed.

🧭 Overview

🧠 One-sentence thesis

Logarithms and exponentials are inverse operations—logarithms convert multiplication into addition by expressing numbers as exponents of a base—and understanding them is essential for modeling natural phenomena through differential equations.

📌 Key points (3–5)

• What logarithms do: they turn multiplication into addition by expressing numbers as exponents (powers) of a base.
• Logarithm as exponent: the logarithm of a number is simply the exponent needed to produce that number from the base.
• Why exponentials matter: the function e^x and the differential equation dy/dx = y are central to modeling change in nature, science, economics, and engineering.
• Common confusion: logarithms are "mirror images" (inverses) of exponentials—they undo each other.
• Three key applications: understanding logarithms as exponents, drawing graphs on different scales (ordinary, semilog, log-log), and finding derivatives of exponential functions.

🔢 What logarithms are

🔢 Logarithm as exponent

The logarithm of a number is the exponent needed to produce that number from a given base.

• Logarithms express numbers as powers of a base.
• Example: In base 10, the numbers 10, 100, 1000 are written as 10¹, 10², 10³; their logarithms are 1, 2, 3 respectively.
• The logarithm of 1 is always 0, regardless of base, because any base raised to the zeroth power equals 1 (b⁰ = 1).

➕ Turning multiplication into addition

• The fundamental property: b^(m+n) = (b^m)(b^n).
• This rule means that adding exponents corresponds to multiplying the original numbers.
• Why it matters: addition is simpler than multiplication, making logarithms useful for computation and analysis.

🪞 Mirror images of exponentials

• Logarithms and exponentials are inverse operations.
• If you take the exponential of a logarithm (or vice versa), you get back the original number.
• Don't confuse: exponentials grow from exponents; logarithms extract exponents from numbers.

🌍 Why exponentials and logarithms matter

🧬 Differential equations and natural phenomena

• The chapter emphasizes that dy/dx = y is a central differential equation in mathematics.
• The exponential function e^x is the solution to this equation.
• Applications span life sciences, physical sciences, economics, and engineering—anywhere change depends on the current state.

📐 Three key uses in Section 6.1

Application	What it does
Understanding logarithms	Viewing them as exponents of a base
Graphing	Drawing on ordinary, semilog, and log-log paper
Finding derivatives	Using the property b^(x+Δx) = (b^x)(b^Δx) to find slopes

🎯 The special role of e^x

• Among all exponential functions (2^x, 10^x, etc.), e^x is the most important.
• The chapter is devoted to understanding, differentiating, integrating, solving equations with, and inverting exponentials.
• The placement of this chapter early (before other integration techniques) reflects the priority: differential equations and natural laws come before computational techniques.

📊 Bases and notation

🔟 Base 10 logarithms

• Powers of 10 are fundamental to the decimal system: 10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1000.
• The logarithms "to base 10" of these numbers are their exponents: 0, 1, 2, 3.

🔤 General base b

• The same principle applies to any base b.
• The logarithm of 1 in any base is always 0, because b⁰ = 1 for any base.
• The fundamental multiplication rule b^(m+n) = (b^m)(b^n) holds for any base and drives all three applications in the section.

🧭 Overview

🧠 One-sentence thesis

The excerpt provided contains no substantive content about the exponential function e^x; it consists entirely of trigonometry exercises, a discussion of sine wave patterns in discrete graphs, and an introduction to the derivative definition.

📌 Key points (3–5)

• The excerpt does not cover the exponential function e^x despite the title "6.2 The Exponential e^x."
• The material includes trigonometry problems (sine, cosine, tangent functions and their properties).
• A lengthy section discusses visual patterns ("A Thousand Points of Light") when plotting discrete sine values.
• The excerpt ends with the formal definition of the derivative from Chapter 2.
• Common confusion: The title suggests exponential content, but the actual text is unrelated.

📭 Content mismatch

📭 What the excerpt contains

The source text does not discuss the exponential function e^x at all. Instead, it includes:

• Trigonometry exercises: problems about sine, cosine, tangent, secant, cosecant, cotangent, their periods, addition formulas, and identities.
• Discrete sine graphs: a detailed exploration of visual patterns when plotting y = sin n (discrete integer values) rather than y = sin x (continuous curve).
• Moiré patterns: an explanation of hexagonal visual artifacts arising from interference between periodic patterns.
• Derivative definition: the formal limit definition of the derivative from Chapter 2.

🚫 Missing exponential content

No information is provided about:

• The definition or properties of e^x
• The base e (Euler's number)
• Exponential growth or decay
• The derivative of e^x
• Applications of exponential functions

🔢 Trigonometry exercises (partial content)

🔢 Types of problems

The excerpt lists 32 numbered exercises covering:

• Evaluating trigonometric functions at specific angles (e.g., theta = 3π/2)
• Verifying trigonometric identities (e.g., cos²θ + sin²θ)
• Graphing functions and determining periods
• Addition formulas for sine and cosine
• Solving trigonometric equations
• Geometric applications (law of cosines, triangle problems)

🔢 Key formulas mentioned

The exercises reference but do not fully explain:

• Addition formulas for cos(s + t) and sin(s + t)
• Complementary angle relationships: sin θ = cos(π/2 - θ)
• Period properties of tan θ and cot θ
• Identities like sec²θ - tan²θ and csc²θ - cot²θ

📊 Discrete sine patterns

📊 The main observation

The excerpt describes plotting y = sin n for integer values n = 1, 2, 3, ..., 10,000.

• Continuous vs discrete: The graph of sin x is one continuous curve; the graph of sin n picks discrete points from that curve.
• Visual illusion: When 10,000 points are plotted, they appear to lie on more than 40 separate sine curves.
• Scale effect: The same 1,000 points look like sine curves in one scale and hexagons in another scale.

📊 Why points cluster on curves

The excerpt explains that certain integer values n produce sin n values close to zero:

• sin 22 ≈ 0 because 22/7 is close to π, so 22 is close to 7π (whose sine is zero).
• sin 44 ≈ 0 because 44 is close to 14π.
• Pattern: Points at n = 0, 44, 88, 132, ... (multiples of 44) lie on the "middle sine curve."

The excerpt states there are 44 curves in total, starting near the heights sin 0, sin 1, ..., sin 43.

📊 Moiré patterns and hexagons

When the same points are plotted at a different scale, hexagonal patterns appear.

Moiré pattern: visual interference between periodic patterns.

• The hexagons arise from the way human eyes perceive overlapping periodic structures.
• The excerpt notes this is "a problem with your eyes," not the mathematics itself.
• Applications mentioned: engineering, optics; problems: printing misalignment, TV vertical lines, dizziness in cloth manufacturing.

🧮 Derivative definition

🧮 Formal definition

The excerpt ends with the beginning of Chapter 2, introducing the derivative:

At time t, the derivative f'(t) or df/dt or v(t) is f'(t) = lim (Δt → 0) [f(t + Δt) - f(t)] / Δt

• Interpretation: The ratio [f(t + Δt) - f(t)] / Δt is the average velocity (or average rate of change) over a short time interval Δt.
• Limit process: The derivative is the limit of this ratio as Δt approaches zero.
• Notation: The derivative can be written as f'(t), df/dt, or v(t) (when representing velocity).

🧮 Examples mentioned

The excerpt briefly recalls two examples from Chapter 1:

• When distance is t², the velocity (derivative) is 2t.
• When f(t) = sin t, the derivative is v(t) = cos t.

No further explanation or derivation is provided in the excerpt.

---

Note: This excerpt does not contain the expected content on the exponential function e^x. To study that topic, a different source or section is required.

🧭 Overview

🧠 One-sentence thesis

The differential equation dy/dt = cy (and its extension dy/dt = cy + s with a source term) governs exponential growth and decay across science and economics, determining population dynamics, radioactive dating, financial investments, and steady-state behaviors.

📌 Key points (3–5)

• Core equation: dy/dt = cy has the solution y = y₀e^(ct), where c is the growth rate (c > 0) or decay rate (c < 0).
• Three-quantity relationship: Given any two of {y₀, c, t} plus one additional fact, the third can be determined—leading to problems about finding doubling time, decay constants, or initial values.
• Source term extension: dy/dt = cy + s (with continuous deposits/withdrawals s) has solution y = (y₀ + s/c)e^(ct) - s/c, combining exponential growth from y₀ with contributions from the source.
• Common confusion: Don't confuse annual rates with continuous rates—continuous compounding at 5% yields an effective annual rate of about 5.13% (e^0.05 ≈ 1.0513).
• Steady state vs transient: When c < 0 (decay), the transient term y₀e^(ct) → 0 and the system approaches steady state y∞ = -s/c, independent of initial conditions.

📐 The fundamental exponential equation

📐 What dy/dt = cy means

The differential equation dy/dt = cy states that the rate of change is proportional to the current amount.

• This is fundamentally different from dy/dt = x (which asks for an antiderivative).
• The equation dy/dt = y can be rewritten as dy/y = dx, leading to ln y = x + C, hence y = e^x · e^C.
• Geometric interpretation: A field of tangent arrows where slopes grow steeper as y grows (not as x grows).
• The solution curve must stay tangent to these arrows at every point.

🔑 Standard solution form

The solution to dy/dt = cy starting from y = y₀ at t = 0 is:

y = y₀e^(ct)

• The constant c is the "growth rate" (c > 0) or "decay rate" (c < 0).
• Time variable t replaces x in most applications.
• The exponential e^(ct) is the growth/decay factor; y₀ is the initial condition.

🔢 Three types of problems

🔢 Type 1: Finding doubling time T (given c)

When does y₀e^(cT) = 2y₀?

• This gives e^(cT) = 2, so cT = ln 2.
• Doubling time: T = (ln 2)/c ≈ 0.7/c.
• Example: At c = 0.1 (10% continuous growth), doubling takes about 7 time units.
• More generally: time to multiply by factor k is (ln k)/c.

🔢 Type 2: Finding decay constant c (given half-life T)

For carbon-14 with half-life T = 5568 years, when does y = (1/2)y₀?

• This gives e^(cT) = 1/2, so cT = ln(1/2) = -ln 2.
• Decay constant: c = -ln(2)/T (negative for decay).
• The ratio y(T)/y(0) = e^(cT) determines c via logarithms.

🔢 Type 3: Finding initial value y₀ (given y at time t)

If y(1) = 5 and c = 2, what was y₀?

• From y(t) = y₀e^(ct), we get y₀ = y(t)e^(-ct).
• This "runs the process backward"—growth forward becomes decay backward.
• Example: y₀ = 5e^(-2).

⏱️ Key time property

For any step Δt, the multiplier is always e^(cΔt):

• y(T + t) = y₀e^(c(T+t)) = (y₀e^(cT))e^(ct).
• Each time interval multiplies by the same factor, whether at the start or later.

🌍 Applications in science

🌍 Population growth

Model: dy/dt = by - dy = cy, where b = birth rate, d = death rate, c = b - d (net rate).

• Solution: y = y₀e^(ct).
• Example: Earth's population with c = 0.02/year doubles in T ≈ 0.7/0.02 = 35 years.
• Limitation: This model cannot hold for very large t (populations can't grow exponentially forever).
• Dimensions: b, c, d have units of "1/time" (per person per unit time).

☢️ Radioactive dating

Living organisms maintain radiocarbon balance; dead material decays.

• Carbon-14 dating: Ratio of disintegrations = e^(ct), where c is known from half-life.
• Example: Charcoal giving 0.97 disintegrations vs 6.68 in living wood:
  • e^(ct) = 0.97/6.68
  • t = (1/c) ln(0.97/6.68) ≈ 14,400 years (Lascaux cave paintings).
• Uranium dating: Comparing U-238 and U-235 with different half-lives determines when uranium was created (about 6 billion years ago).

🌡️ Newton's Law of Cooling

Model: dy/dt = c(y - y∞), where y∞ is ambient temperature.

• The rate is proportional to the temperature difference.
• Solution: (y - y∞) = (y₀ - y∞)e^(ct).
• Example: Body found at 90° in 70° room, later 80°—work backward to find time of death.
• Don't confuse: y∞ is the steady-state temperature (not zero).

💰 Financial applications

💰 Continuous vs annual rates

At continuous rate c, the growth over time dt is dy = cy·dt.

• After one year: e^c ≈ 1 + (annual rate).
• Example: 5% continuous = e^0.05 ≈ 1.0513 = 5.13% annual (effective rate).
• Banks quote both: the "effective rate" accounts for compounding.

💰 Present and future value

Concept	Formula	Meaning
Future value	y₀e^(ct)	What y₀ now becomes in t years
Present value	ye^(-ct)	What y in t years is worth now
Doubling time	(ln 2)/c	Time for money to double

Example: $24 in 1626 at 8% continuous interest → $24 · e^(29.2) ≈ 115 trillion after 365 years.

🔄 Adding a source term

🔄 The extended equation

dy/dt = cy + s (constant source s, starting from y₀)

• s represents continuous deposits (s > 0) or withdrawals (s < 0).
• Units: s has dimensions "amount/time" to match dy/dt.
• Example: Initial investment y₀ = $8000 with deposits s = $1000/year.

🔄 Solution methods

Method 1 (fast): Assume y = Ae^(ct) + B

• Substitute into equation: cAe^(ct) = c(Ae^(ct) + B) + s.
• This gives cB + s = 0, so B = -s/c.
• From y(0) = y₀: A = y₀ + s/c.
• Key formula: y = (y₀ + s/c)e^(ct) - s/c = y₀e^(ct) + (s/c)(e^(ct) - 1).

Method 2 (Duhamel's principle): Add up all outputs

• Initial deposit y₀ grows to y₀e^(ct).
• Small deposit s·dT at time T grows by factor e^(c(t-T)) to reach time t.
• Total: y(t) = y₀e^(ct) + ∫₀ᵗ e^(c(t-T)) s dT.
• Integration yields the same formula.

Method 3 (for steady state): Look at difference y - y∞

• If y∞ = -s/c is the steady state, then d/dt(y - y∞) = c(y - y∞).
• This is a pure exponential: (y - y∞) = (y₀ - y∞)e^(ct).

💵 Six financial questions

💵 Questions 1–2: No source (s = 0)

1. Future value: y₀ → y₀e^(ct) after time t.
2. Present value: To get y later, deposit ye^(-ct) now.

💵 Questions 3–4: No initial deposit (y₀ = 0)

3. Deposits to target: Depositing s = $1000/year at 5% for 20 years yields y = (1000/0.05)(e - 1) ≈ $34,400.
4. Required deposit rate: To reach $20,000 in 20 years requires s = 1000/(e - 1) ≈ $582/year.

💵 Questions 5–6: End with nothing (y = 0)

5. Annuity: To withdraw $1000/year for 20 years, deposit y₀ = (s/c)(1 - e^(-ct)) ≈ $12,640.
6. Loan repayment: To clear $20,000 loan in 20 years requires payments s = -cy₀e^(ct)/(e^(ct) - 1) ≈ $1582/year.

Puzzle solution: The difference $1582 - $582 = $1000 is exactly the interest on $20,000 at 5%—you can pay $1000/year to stay even with interest, and the extra $582 builds to $20,000 to repay principal.

🎯 Steady state behavior

🎯 When decay dominates (c < 0)

The solution y = (y₀ + s/c)e^(ct) - s/c has two parts:

• Transient term: (y₀ + s/c)e^(ct) → 0 as t → ∞ (dies out).
• Steady state: y∞ = -s/c (remains constant).

At steady state: dy/dt = 0, meaning cy + s = 0.

• The source s exactly balances the decay cy.
• y∞ depends on s and c, but not on y₀.

🎯 Population example

Bermuda with birth rate b = 0.02, death rate d = 0.03 (net c = -0.01), immigration s = 1200/year:

• Steady state: y∞ = -1200/(-0.01) = 120,000.
• Population approaches 120,000 regardless of whether it starts at 5,000 or 5,000,000.
• If y₀ < 120,000, population grows; if y₀ > 120,000, population decays.

🎯 Don't confuse

• Transient vs steady: y₀e^(ct) depends on initial conditions; y∞ = -s/c does not.
• Growth vs decay: c > 0 means no steady state (exponential growth); c < 0 means convergence to y∞.

🧭 Overview

🧠 One-sentence thesis

The natural logarithm ln x, defined as the integral of 1/x from 1 to x, provides the theoretical foundation for exponential functions and yields powerful techniques for differentiation and integration through its properties and the chain rule.

📌 Key points (3–5)

• Definition as integral: ln x is defined as the area under the curve y = 1/x from 1 to x, making it the "missing integral" of the −1 power.
• Two fundamental properties: ln(ab) = ln a + ln b (product becomes sum) and ln(b^n) = n ln b (power becomes multiple), both provable from the integral definition.
• Logarithmic differentiation (LD): for complicated products or powers, differentiate ln p instead of p directly, then multiply back by p to recover the derivative.
• Common confusion: ln x is only defined for x > 0; for integrals that may cross negative values, use ln |u| to handle both positive and negative u (but never u = 0).
• Integration yields logarithms: the integral of du/u equals ln |u|, which extends via substitution to many rational forms like 1/(cx + 7) or tan x.

📐 Definition and foundational properties

📐 The natural logarithm as area

ln x = integral from 1 to x of (1/t) dt

• This is the direct definition: logarithm is the area under the hyperbola y = 1/x, measured from the starting point x = 1.
• At x = 1, the area is zero, so ln 1 = 0.
• The integral does not extend to x ≤ 0; ln x is defined only for x > 0.
• As x → 0⁺, the area becomes infinite in the negative direction: ln 0 = −∞.
• As x → ∞, the area grows without bound: ln ∞ = +∞.

Why this definition matters: Earlier chapters integrated all powers x^n except the −1 power. The logarithm fills that gap and avoids circular reasoning (we cannot define e^x as its own integral).

🔗 Property 1: Logarithm of a product

ln(ab) = ln a + ln b

Proof from the integral definition:

• The area from 1 to a is ln a.
• The area from a to ab must equal ln b.
• Use substitution u = x/a in the integral from a to ab of (1/x) dx.
• Then x = au, dx = a du, so dx/x = du/u.
• The limits become u = 1 to u = b, yielding ln b.
• Neighboring areas combine: integral from 1 to a plus integral from a to ab equals integral from 1 to ab.

Example: ln 6 = ln(2·3) = ln 2 + ln 3.

🔢 Property 2: Logarithm of a power

ln(b^n) = n ln b

Proof from the integral definition:

• Use substitution x = u^n in the integral from 1 to b^n of (1/x) dx.
• Then dx = n u^(n−1) du, so dx/x = n du/u.
• The limits become u = 1 to u = b, yielding n times the integral from 1 to b of (1/u) du = n ln b.

Example: ln 8 = ln(2³) = 3 ln 2.

Don't confuse: ln(b^n) pulls the exponent down as a multiplier; ln(a + b) has no simplification.

🔄 Inverse relationship with e^x

🔄 Defining e and e^x via logarithms

• e is defined as the unique number whose logarithm equals 1: ln e = 1, so the area from 1 to e under 1/x is exactly 1.
• e^π is defined as the unique number whose logarithm equals π: ln(e^π) = π.
• This approach constructs the exponential function as the inverse of the logarithm.

Why the area reaches every value:

• The area from 1 to 2 is more than 1/2 (because 1/x > 1/2 on that interval of length 1).
• The combined area from 1 to 4 is more than 1, so we reach area = 1 before x = 4 (actually at e ≈ 2.718).
• Since 1/x is always positive, the area is strictly increasing and never returns to any value.

📈 Growth rate comparison

• ln x grows very slowly: ln x → ∞ as x → ∞, but (ln x)/x → 0.
• ln x is passed by every root: (ln x)/x^(1/n) → 0 for any n, because e^x eventually surpasses every power x^n (and they are inverses).
• Example: at x = 10, ln 10 ≈ 2.3 versus √10 ≈ 3.2; but at x = e^10, ln(e^10) = 10 versus √(e^10) = e^5 ≈ 148, and ln x loses badly.

Key insight: To double the area under 1/x, you must square the distance (because ln(x²) = 2 ln x).

🧮 Approximation near x = 1

🧮 Linear and quadratic approximations

For x near zero:

• Linear: ln(1 + x) ≈ x
• Quadratic: ln(1 + x) ≈ x − (1/2)x²
• Also: e^x ≈ 1 + x (linear) and e^x ≈ 1 + x + (1/2)x² (quadratic)

Why the linear approximation works:

• Between 1 and 1 + x, the area under 1/x is nearly a rectangle with base x and height 1.
• The curved area ln(1 + x) is close to the rectangular area x.
• A small triangle is chopped off at the top, with area approximately (1/2)x².

Example: ln 1.01 = 0.0099503 (actual) versus 0.01 (linear approximation); the difference 0.0000497 is predicted almost exactly by −(1/2)(0.01)² = −0.00005.

Don't confuse: Two wrongs make a right here: ln(e^x) ≈ ln(1 + x) ≈ x, and ln(e^x) = x exactly.

🔍 Differentiation with logarithms

🔍 Basic derivative and chain rule

The derivative of ln x is 1/x. The derivative of ln u(x) is (1/u)(du/dx).

• By the Fundamental Theorem of Calculus, since ln x is defined as the integral of 1/x, its derivative must be 1/x.
• For ln u(x), apply the chain rule: inside function is u, outside is ln.

Examples:

• d/dx ln(3x) = (1/(3x))·3 = 1/x (the 3 cancels!)
• d/dx ln(x³) = 3x²/x³ = 3/x
• d/dx ln(x² + 1) = 2x/(x² + 1)
• d/dx ln(cos x) = −sin x/cos x = −tan x
• d/dx ln(ln x) = (1/ln x)·(1/x)

Common confusion: The slope of ln(3x) is not 3/x; it equals 1/x because ln(3x) = ln 3 + ln x, and the constant ln 3 disappears when differentiating.

🪵 Logarithmic differentiation (LD)

For products and powers, differentiate ln p instead of p directly:

Method:

1. Take ln of both sides: ln p = (sum or simpler expression)
2. Differentiate: (1/p)(dp/dx) = (derivative of right side)
3. Multiply by p: dp/dx = p·(derivative of ln p)

Example (product): If p(x) = x·√(x−1), then

• ln p = ln x + (1/2)ln(x−1)
• (1/p)(dp/dx) = 1/x + 1/(2(x−1))
• dp/dx = p·[1/x + 1/(2(x−1))]

Example (power): If p = x^(1/x), then

• ln p = (1/x)ln x
• (1/p)(dp/dx) = (1/x²)(−ln x + 1)
• dp/dx = x^(1/x)·[(1 − ln x)/x²]

The catch: You must multiply back by p at the end, which can complicate the answer—but for complex products or variable exponents, LD is often simpler than the product or power rule.

∫ Integration producing logarithms

∫ The fundamental integral pattern

Integral of (du/dx)/u(x) dx = ln |u(x)| + C Or equivalently: integral of du/u = ln |u| + C

Strategy: Try to identify u(x) so that the integrand contains (du/dx) divided by u.

Examples:

• Integral of dx/(x + 7) = ln |x + 7| + C
• Integral of dx/(cx + 7) = (1/c)ln |cx + 7| + C
• Integral of x dx/(x² + 7) = (1/2)ln(x² + 7) + C (here u = x² + 7, du/dx = 2x)
• Integral of dx/(x ln x) = ln |ln x| + C (here u = ln x, du/dx = 1/x)

∫ Trigonometric integrals

• Tangent: integral of tan x dx = integral of (sin x)/(cos x) dx = −ln |cos x| + C (u = cos x, du = −sin x dx)
• Cotangent: integral of cot x dx = integral of (cos x)/(sin x) dx = ln |sin x| + C (u = sin x, du = cos x dx)
• Secant (requires a trick): integral of sec x dx = ln |sec x + tan x| + C
  • Multiply numerator and denominator by (sec x + tan x)
  • Then u = sec x + tan x, and du/dx = sec x tan x + sec² x appears in the numerator
• Cosecant: integral of csc x dx = ln |csc x + cot x| + C (similar trick)

∫ Absolute value and domain

Why write ln |u|:

• When u > 0, the integral of 1/u is ln u.
• When u < 0, ln u is undefined, but we can switch to −u: integral of (du/dx)/u dx = integral of (−du/dx)/(−u) dx = ln(−u).
• Combining both cases: ln |u| works for u ≠ 0.

Forbidden case: u = 0 gives infinite area on both sides; the integral cannot cross zero.

Example: Integral from −1 to −e of dx/x = [ln |x|] from −1 to −e = ln e − ln 1 = 1 − 0 = 1 (the area is below the x-axis, but we measure magnitude).

Don't confuse: We do not have logarithms of negative numbers (in real calculus); we use absolute value to handle negative u. Never integrate across u = 0.

∫ Special integrals

• Integral of ln x: integral of ln x dx = x ln x − x + C (derived by recognizing that d/dx(x ln x) = ln x + 1, so subtract x to remove the extra 1)
• Definite integral examples:
  • Integral from x to 3x of dt/t = ln(3x) − ln x = ln 3
  • Integral from 0.1 to 1 of dx/x = ln 1 − ln 0.1 = 0 − ln(1/10) = ln 10

Application: The area integral from a to b of dx/(ln x) (no elementary formula) is approximately the number of primes between a and b—this is the prime number theorem. Near e^1000, about 1/1000 of integers are prime.

🧭 Overview

🧠 One-sentence thesis

Separable differential equations allow us to solve important nonlinear models like the logistic equation by splitting variables into separate integrals, revealing how populations grow rapidly at first and then level off into an S-curve as competition limits further expansion.

📌 Key points (3–5)

• What separation means: rearranging dy/dt = u(y)v(t) so all y terms are on one side and all t terms on the other, then integrating both sides independently.
• The logistic equation models realistic growth: dy/dt = cy - by² captures both natural growth (cy) and competition that slows growth (-by²), producing an S-shaped curve.
• Steady state at c/b: when competition term by² balances growth term cy, the population stops changing and reaches equilibrium y = c/b.
• Common confusion—linear vs nonlinear: the equation dy/dt = cy is separable and linear (exponential growth forever), but dy/dt = y + t is not separable; dy/dt = cy - by² is separable but nonlinear (growth stops).
• Why it matters: these methods solve real models in biology (population dynamics), chemistry (reaction rates), and ecology (resource competition).

🔧 The separation method

🔧 Core idea of separation

To solve dy/dt = u(y)v(t), separate dy/u(y) from v(t)dt and integrate both sides.

• The goal: isolate all y-related terms on the left, all t-related terms on the right.
• Then integrate: ∫ dy/u(y) = ∫ v(t)dt + C.
• After integration, substitute the initial condition (t = 0, y = y₀) to find the constant C.
• Finally, solve for y(t) explicitly if possible.

📝 Example: dy/dt = y²

• Separate: dy/y² = dt.
• Integrate: -1/y = t + C.
• At t = 0, y = y₀ gives C = -1/y₀.
• Solution: y = y₀/(1 - ty₀).
• Warning: this solution blows up (goes to infinity) when t reaches 1/y₀.

📝 Example: dy/dt = ty

• Separate: dy/y = t dt.
• Integrate: ln y = (1/2)t² + C.
• At t = 0, y = y₀ gives C = ln y₀.
• Solution: y = y₀ exp(t²/2).
• The interest rate c = t (time-dependent) produces exponent t²/2.

❌ When separation fails

• The equation dy/dt = y + t is not separable—you cannot isolate y from t.
• Other methods (like assuming y = Ae^t + B + Dt) may still work.

🌱 The logistic equation

🌱 Why the simple model fails

Problem: dy/dt = cy predicts exponential growth forever, but real populations face limits.

• Constant growth rate c = (birth rate - death rate) is unrealistic long-term.
• Competition for food and space must slow growth as population y gets large.
• The true rate c depends on population size: c(y), not a constant.

🌱 The logistic model

dy/dt = cy - by²

• Growth term cy: natural increase when population is small.
• Competition term -by²: interactions between individuals slow growth; the number of interactions is proportional to y times y.
• Typically b is very small compared to c, so competition only matters when y is large.
• This is the basic model of "growth versus competition."

🎯 Steady state and equilibrium

• When dy/dt = 0, growth stops: cy - by² = 0.
• Solving: y = c/b (the steady state or carrying capacity).
• Example (world population): c ≈ 0.029/year, b ≈ 3×10⁻¹²/year gives y∞ = 10 billion people.
• At equilibrium, loss from competition exactly balances gain from growth.

📈 The inflection point

• Halfway to steady state, at y = c/(2b), the S-curve bends from accelerating to decelerating.
• The second derivative d²y/dt² = 0 here; the slope dy/dt is maximum.
• Found by differentiating: y'' = (c - 2by)y' = 0 when y = c/(2b).

📐 Solving the logistic equation

📐 Separation and integration

• Start: dy/dt = cy - by².
• Separate: dy/(cy - by²) = dt.
• Integrate both sides: ∫ dy/(cy - by²) = ∫ dt.
• The y-integral requires "partial fractions" (covered in Section 7.4).
• Result: ln[y/(c - by)] = ct + C.

📐 Finding the S-curve formula

• At t = 0, y = y₀ gives C = ln[y₀/(c - by₀)].
• Take exponentials: y/(c - by) = e^(ct) · y₀/(c - by₀).
• Key insight: it is y/(c - by) that grows exponentially, not y itself.
• Final formula: y = c/[b + de^(-ct)], where d depends on initial conditions.
• As t → ∞, e^(-ct) → 0, so y → c/b (the steady state).

🔄 Alternative: linear equation for z = 1/y

• Surprising fact: if you set z = 1/y, then z satisfies a linear equation.
• By calculus: z' = -y'/y² = -(cy - by²)/y² = -cz + b.
• This is linear: z' = -cz + b.
• Solution: z = Ae^(-ct) + b/c.
• Turn upside down: y = 1/z gives the same S-curve.

🧪 Applications in biology and chemistry

🧪 The Law of Mass Action

The reaction rate is proportional to the product of concentrations.

• For reaction mA + nB → pC, the rate is proportional to [A] times [B].
• Example: d[A]/dt = -r[A][B] and d[C]/dt = +k[A][B].
• This produces quadratic terms (like y²) in the differential equations.
• Why it matters: interactions between two populations of size y and z give rate proportional to yz; one population competing with itself gives y².

🧪 The Michaelis-Menten (MM) equation

dy/dt = -cy/(y + K)

• Models enzyme-catalyzed reactions in biochemistry.
• Enzyme: a catalyst that speeds up reactions and emerges unchanged.
• The Michaelis constant K depends on reaction rates.
• Behavior:
  • When y is large: dy/dt ≈ -c (maximum rate).
  • When y is small: dy/dt ≈ -cy/K (proportional to y).
• Separation gives: ∫(y + K)/y dy = -∫c dt, leading to y + K ln y = -ct + C.
• No simple formula for y(t), but the solution can be graphed by computer.

🧪 Examples of catalysts

• Platinum in catalytic converters (reacts with pollutants).
• Spray propellants (destroy ozone layer).
• Enzymes in blood clotting (Factor VIII missing in hemophilia).
• Yeast (makes bread rise).
• Meat tenderizer (predigests protein).

🎨 The y-line method

🎨 Understanding dy/dt = f(y)

• Draw a horizontal "y line" (the y-axis).
• Add arrows showing the sign of f(y):
  • When f(y) > 0, y increases (arrow points right).
  • When f(y) < 0, y decreases (arrow points left).
  • When f(y) = 0, y is stationary (steady state).

🎨 Stable vs unstable steady states

• Stable: arrows point toward the steady state from both sides.
• Unstable: arrows point away from the steady state.
• The y-line shows which direction y moves and where it stops.
• Example: for logistic equation, y = 0 is unstable, y = c/b is stable.