Calculus Bible Chapter 3: Applications of Differentiation

This post summarizes Chapter 3 of James Stewart's Calculus.

3.1 Maximum and Minimum Values#

Optimization problems ultimately reduce to finding the maximum or minimum value of a function. (p.150)

Let $c$ be a number in the domain $D$ of $f$. Then for all $x$ in $D$:

• If $f(c) \ge f(x)$, then $f(c)$ is the absolute maximum$^{absolute\ maximum}$ of $f$ on $D$.
• If $f(c) \le f(x)$, then $f(c)$ is the absolute minimum$^{absolute\ minimum}$ of $f$ on $D$.

The maximum and minimum values of $f$ are called the extreme values$^{extreme\ values}$ of $f$.

When $x$ is near $c$:

• If $f(c) \ge f(x)$, then $f(c)$ is a local maximum$^{local\ maximum}$ of $f$.
• If $f(c) \le f(x)$, then $f(c)$ is a local minimum$^{local\ minimum}$ of $f$.

Fermat's Theorem$^{Fermat's\ Theorem}$ (p.152)

If $f$ has a local maximum or minimum at $c$ and $f'(c)$ exists, then $f'(c) = 0$. (The converse does not hold.)

Critical Numbers (p.154)

A critical number$^{critical\ number}$ of a function $f$ is a number $c$ in the domain of $f$ such that either $f'(c) = 0$ or $f'(c)$ does not exist.

3.2 The Mean Value Theorem#

Rolle's Theorem$^{Rolle's\ Theorem}$ (p.157)

Suppose $f$ satisfies the following three conditions:

• $f$ is continuous on the closed interval $[a, b]$.
• $f$ is differentiable on the open interval $(a, b)$.
• $f(a) = f(b)$

Then there exists a number $c$ in $(a, b)$ such that $f'(c) = 0$.

The Mean Value Theorem (p.159)

Suppose $f$ satisfies the following conditions:

1. $f$ is continuous on the closed interval $[a, b]$.
2. $f$ is differentiable on the open interval $(a, b)$.

Then there exists a number $c$ in $(a, b)$ such that:

$f'(c)=\frac{f(b)-f(a)}{b-a}$

or equivalently:

$f(b)-f(a)= f'(c)(b-a)$

In plain terms: "There is always a point where the instantaneous rate of change equals the average rate of change over the interval."

The Mean Value Theorem can be used to prove several fundamental facts in calculus. (See corollaries on p.162.)

3.3 How Derivatives Affect the Shape of a Graph#

Most applications of calculus depend on the ability to deduce information about a function $f$ from information about its derivatives. (p.164)

What $f'$ tells us about $f$

Increasing/Decreasing Test

(a) If $f'(x) > 0$ on an interval, then $f$ is increasing on that interval. (b) If $f'(x) < 0$ on an interval, then $f$ is decreasing on that interval.

The First Derivative Test

Let $c$ be a critical number of a continuous function $f$. (a) If $f'$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$. (b) If $f'$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$. (c) If $f'$ does not change sign at $c$, then $f$ has neither a local maximum nor a local minimum at $c$.

What $f''$ tells us about $f$ (p.167)

If the graph of $f$ lies above all of its tangents on an interval $I$, then $f$ is concave upward$^{Concave\ Upward}$ on $I$. If the graph of $f$ lies below all of its tangents on an interval $I$, then $f$ is concave downward$^{Concave\ Downward}$ on $I$.

Inflection Points (p.168)

A point $P$ on the curve $y = f(x)$ is called an inflection point$^{inflection\ point}$ if $f$ is continuous at $P$ and the curve changes concavity (from upward to downward, or from downward to upward) at $P$.

Concavity Test (p.168)

(a) If $f''(x) > 0$ for all $x$ in $I$, then the graph of $f$ is concave upward on $I$. (b) If $f''(x) < 0$ for all $x$ in $I$, then the graph of $f$ is concave downward on $I$.

The Second Derivative Test (p.169)

Suppose $f''$ is continuous near $c$. (a) If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $c$. (b) If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $c$.

3.4 Curve Sketching#

SKIP (p.172)

3.5 Optimization Problems#

We can apply the methods for finding extreme values to solve optimization problems. (p.180) These include problems that maximize area, volume, or profit, and problems that minimize distance, time, or cost.

Steps for Solving Optimization Problems

1. Understand the problem. Read it carefully until you understand it. Identify: what you want to find, what is given, and what conditions are imposed.
2. Draw a diagram.
3. Introduce notation. Assign a symbol to the quantity to be maximized or minimized (call it $Q$). Also assign symbols to other unknown quantities and label them on the diagram (e.g., area $A$, height $h$, time $t$, etc.).
4. Express $Q$ in terms of the symbols from Step 3.
5. If $Q$ is expressed as a function of more than one variable, use the given information to find equations relating those variables. Eliminate variables to express $Q$ as a function of a single variable: $Q = f(x)$. Determine the domain of this function.
6. Find the absolute maximum or minimum of $f$ using the methods from Sections 3.1 and 3.3.

See worked examples on p.181 and beyond.

3.6 Newton's Method#

There is no general formula for the exact roots of polynomial equations of degree five or higher, or for transcendental equations involving trigonometric functions. (p.190)

For example, suppose we want to solve:

$48x(1+x)^{60}-(1+x)^{60}+1=0$

Numerical roots are most commonly found using the Newton-Raphson method$^{Newton\text{-}Raphson\ method}$, or simply Newton's method$^{Newton's\ method}$:

$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

If the sequence $x_n$ converges to $r$ as $n$ grows large, we say the sequence converges to $r$ and write:

$\lim_{n \to \infty} x_n = r$

Note, however, that Newton's method does not work for every function. For some functions the successive approximations may diverge, or the method may converge very slowly.

3.7 Antiderivatives#

A function $F$ is called an antiderivative$^{antiderivative}$ of $f$ on an interval $I$ if $F'(x) = f(x)$ for all $x$ in $I$. (p.196)

If $F$ is an antiderivative of $f$ on an interval $I$, then the most general antiderivative of $f$ on $I$ is $F(x) + C$, where $C$ is an arbitrary constant.

Summary#

• Absolute extrema: extreme values over the entire domain — candidates are found at critical numbers
• Fermat's Theorem: if an extreme value exists, then $f'(c) = 0$ (converse fails)
• Rolle's Theorem: if $f(a) = f(b)$, there must exist $c$ with $f'(c) = 0$
• Mean Value Theorem: there always exists a point where the instantaneous rate equals the average rate of change
• Increasing/Decreasing: $f' > 0$ means increasing; $f' < 0$ means decreasing
• Concavity: $f'' > 0$ means concave up; $f'' < 0$ means concave down
• Optimization: define variables → build objective function → find extrema via derivatives
• Newton's Method: numerically approximates roots of equations using tangent lines
• Antiderivatives: $F(x)$ satisfying $F'(x) = f(x)$ — general solution is $F(x) + C$