Calculus

Laura Taalman, Peter Kohn

Chapter 3

Applications of the Derivative - all with Video Answers

Educators

Section 1

The Mean Value Theorem

Problem 1

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: Rolle's Theorem is a special case of the Mean Value Theorem.
(b) True or False: The Mean Value Theorem is so named because it concerns the average (or "mean") rate of change of a function on an interval.
(c) True or False: If $f$ is differentiable on $\mathbb{R}$ and has an extremum at $x=-2,$ then $f^{\prime}(-2)=0$
(d) True or False: If $f$ has a critical point at $x=1$, then $f$ has a local minimum or maximum at $x=1$.
(e) True or False: If $f$ is any function with $f(2)=0$ and $f(8)=0,$ then there is some $c$ in the interval (2,8) such that $f^{\prime}(c)=0$
(f) True or False: If $f$ is continuous and differentiable on [-2,2] with $f(-2)=4$ and $f(2)=0,$ then there is some $c \in(-2,2)$ with $f^{\prime}(c)=-1$
(g) True or False: If $f$ is continuous and differentiable on [0,10] with $f^{\prime}(5)=0,$ then $f$ has a local maximum or minimum at $x=5$
(h) True or False: If $f$ is continuous and differentiable on [0,10] with $f^{\prime}(5)=0,$ then there are some values $a$ and $b$ in (0,10) for which $f(a)=0$ and $f(b)=0$

Raj Bala

Raj Bala

Numerade Educator

Problem 2

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A function with a local minimum at $x=3$ that is continuous but not differentiable at $x=3$. (b) A function with a local maximum at $x=-2$ that is not differentiable at $x=-2$ because of a removable discontinuity.
(c) A function with a local minimum at $x=1$ that is not differentiable at $x=1$ because of a jump discontinuity.

Raj Bala

Numerade Educator

Problem 3

If $f$ has a local maximum at $x=1$, then what can you say about $f^{\prime}(1) ?$ What if you also know that $f$ is differentiable at $x=1 ?$

Raj Bala

Numerade Educator

Problem 4

If $f$ has a local minimum at $x=0$ and a local maximum at $x=2,$ what can you say about $f^{\prime}(0)$ and $f^{\prime}(2)$ ? Is there anything else you can say about $f^{\prime} ?$

Raj Bala

Numerade Educator

Problem 5

Suppose that $f$ is defined on $(-\infty, \infty)$ and differentiable everywhere except at $x=-2$ and $x=4,$ and that $f^{\prime}(x)=0$ only at $x=0$ and $x=5$. List all the critical points of $f$ and sketch a possible graph of $f$.

Raj Bala

Numerade Educator

Problem 6

Suppose that $f$ is defined for $x \neq 0$ and differentiable everywhere except at $x=0$ and $x=1,$ and that $f^{\prime}(x)=0$ only at $x=\pm 2$. List all the critical points of $f$ and sketch a possible graph of $f$.

Raj Bala

Numerade Educator

Problem 7

If a continuous, differentiable function $f$ has zeroes at $x=-4, x=1,$ and $x=2,$ what can you say about $f^{\prime}$ on [-4,2]$?$

Raj Bala

Numerade Educator

Problem 8

If a continuous, differentiable function $f$ is equal to 2 at $x=3$ and at $x=5$, what can you say about $f^{\prime}$ on [3,5]$?$

Raj Bala

Numerade Educator

Problem 9

If a continuous, differentiable function $f$ has values $f(-2)=3$ and $f(4)=1$, what can you say about $f^{\prime}$ on [-2,4]$?$

Raj Bala

Numerade Educator

Problem 10

Restate Theorem 3.3 so that its conclusion has to do with tangent lines.

Raj Bala

Numerade Educator

Problem 11

Restate Rolle's Theorem so that its conclusion has to do with tangent lines.

Raj Bala

Numerade Educator

Problem 12

Restate the Mean Value Theorem so that its conclusion has to do with tangent lines. In Exercises $13-22,$ sketch the graph of a function that satisfies the given description. Label or annotate your graph so that it is clear that it satisfies each part of the description.

Raj Bala

Numerade Educator

Problem 13

A function that satisfies the hypothesis, and therefore the conclusion, of Rolle's Theorem on [2,6]

Raj Bala

Numerade Educator

Problem 14

A function that satisfies the hypothesis, and therefore the conclusion, of the Mean Value Theorem.

Raj Bala

Numerade Educator

Problem 15

A function $f$ that satisfies the hypotheses of Rolle's Theorem on [-2,2] and for which there are exactly three values $c \in(-2,2)$ that satisfy the conclusion of the theorem.

Raj Bala

Numerade Educator

Problem 16

A function $f$ that satisfies the hypothesis of the Mean Value Theorem on [0,4] and for which there are exactly three values $c \in(0,4)$ that satisfy the conclusion of the theorem.

Raj Bala

Numerade Educator

Problem 17

A function $f$ that is defined on [-2,2] with $f(-2)=f(2)=$ 0 such that $f$ is continuous everywhere, differentiable everywhere except at $x=-1,$ and fails the conclusion of Rolle's Theorem.

Raj Bala

Numerade Educator

Problem 18

A function $f$ defined on [1,5] with $f(1)=f(5)=0$ such that $f$ is continuous everywhere except for $x=2$, differentiable everywhere except at $x=2$, and fails the conclusion of Rolle's Theorem.

Raj Bala

Numerade Educator

Problem 19

A function $f$ defined on [-3,-1] with $f(-3)=f(-1)=0$ such that $f$ is continuous everywhere except at $x=-1$ and differentiable everywhere except at $x=-1,$ and fails the conclusion of Rolle's Theorem.

Raj Bala

Numerade Educator

Problem 20

A function $f$ defined on [0,4] such that $f$ is continuous everywhere, differentiable everywhere except at $x=2$, and fails the conclusion of the Mean Value Theorem with $a=0$ and $b=4$

Raj Bala

Numerade Educator

Problem 21

A function $f$ defined on [-3,3] such that $f$ is continuous everywhere except at $x=1$, differentiable everywhere except at $x=1$, and fails the conclusion of the Mean Value Theorem with $a=-3$ and $b=3$.

Raj Bala

Numerade Educator

Problem 22

A function $f$ defined on [-2,0] such that $f$ is continuous everywhere except at $x=-2,$ differentiable everywhere except at $x=-2,$ and fails the conclusion of the Mean Value Theorem with $a=-2$ and $b=0$.

Raj Bala

Numerade Educator

Problem 23

Approximate all the values $x \in(0,4)$ for which the derivative of $f$ is zero or does not exist. Indicate whether $f$ has a local maximum, minimum, or neither at each of these critical points.

Raj Bala

Numerade Educator

Problem 24

Approximate all the values $x \in(0,4)$ for which the derivative of $f$ is zero or does not exist. Indicate whether $f$ has a local maximum, minimum, or neither at each of these critical points.

Raj Bala

Numerade Educator

Problem 25

Approximate all the values $x \in(0,4)$ for which the derivative of $f$ is zero or does not exist. Indicate whether $f$ has a local maximum, minimum, or neither at each of these critical points.

Raj Bala

Numerade Educator

Problem 26

Approximate all the values $x \in(0,4)$ for which the derivative of $f$ is zero or does not exist. Indicate whether $f$ has a local maximum, minimum, or neither at each of these critical points.

Raj Bala

Numerade Educator

Problem 27

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=(x-1.7)(x+3)$

Raj Bala

Numerade Educator

Problem 28

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=x^{3}+x^{2}+1$

Raj Bala

Numerade Educator

Problem 29

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=3 x^{4}+8 x^{3}-18 x^{2}$

Raj Bala

Numerade Educator

Problem 30

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=(2 x-1)^{5}$

Raj Bala

Numerade Educator

Problem 31

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=3 x-2 e^{x}$

Raj Bala

Numerade Educator

Problem 32

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=3^{x}-2^{x}$

Raj Bala

Numerade Educator

Problem 33

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=\frac{\ln 2 x}{x}$

Raj Bala

Numerade Educator

Problem 34

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=2^{1-\ln x}$

Raj Bala

Numerade Educator

Problem 35

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=\cos x$

Raj Bala

Numerade Educator

Problem 36

Find the critical points of each function. Then use a graphing utility to determine whether $f$ has a local minimum, a local maximum, or neither at each of these critical points.
$f(x)=\sec x$

Raj Bala

Numerade Educator

Problem 37

Explain why $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b]$. Then approximate any values $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.

Raj Bala

Numerade Educator

Problem 38

Explain why $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b]$. Then approximate any values $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.

Raj Bala

Numerade Educator

Problem 39

Explain why $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b]$. Then approximate any values $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.

Raj Bala

Numerade Educator

Problem 40

Explain why $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b]$. Then approximate any values $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.

Raj Bala

Numerade Educator

Problem 41

Determine whether or not each function $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=x^{3}-4 x^{2}+3 x,[a, b]=[0,3]$

Raj Bala

Numerade Educator

Problem 42

Determine whether or not each function $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=x^{3}-4 x^{2}+3 x,[a, b]=[1,3]$

Raj Bala

Numerade Educator

Problem 43

Determine whether or not each function $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=x^{4}-3.24 x^{2}-3.04,[a, b]=[-2,2]$

Raj Bala

Numerade Educator

Problem 44

Determine whether or not each function $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=\frac{x^{2}-4 x}{x^{2}-4 x+3},[a, b]=[0,4]$

Raj Bala

Numerade Educator

Problem 45

Determine whether or not each function $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=\cos x,[a, b]=\left[-\frac{\pi}{2}, \frac{3 \pi}{2}\right]$

Raj Bala

Numerade Educator

Problem 46

Determine whether or not each function $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=\sin 2 x,[a, b]=[0,2 \pi]$

Raj Bala

Numerade Educator

Problem 47

Determine whether or not each function $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=e^{x}\left(x^{2}-2 x\right),[a, b]=[0,2]$

Raj Bala

Numerade Educator

Problem 48

Determine whether or not each function $f$ satisfies the hypotheses of Rolle's Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=\ln \left|x^{2}-1\right|,[a, b]=[-\sqrt{2}, \sqrt{2}]$

Raj Bala

Numerade Educator

Problem 49

Explain why $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b]$ and approximate any values $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.

Raj Bala

Numerade Educator

Problem 50

Explain why $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b]$ and approximate any values $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.

Raj Bala

Numerade Educator

Problem 51

Explain why $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b]$ and approximate any values $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.

Raj Bala

Numerade Educator

Problem 52

Explain why $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b]$ and approximate any values $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.

Raj Bala

Numerade Educator

Problem 53

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=x^{2}+\frac{1}{x},[a, b]=[-3,2]$

Raj Bala

Numerade Educator

Problem 54

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=x^{3}-2 x+1,[a, b]=[0,6]$

Raj Bala

Numerade Educator

Problem 55

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=-x^{3}+3 x^{2}-7,[a, b]=[-2,3]$

Raj Bala

Numerade Educator

Problem 56

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=\left(x^{2}-1\right)\left(x^{2}-4\right),[a, b]=[-3,3]$

Raj Bala

Numerade Educator

Problem 57

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=\ln \left(x^{2}+1\right),[a, b]=[0,1]$

Raj Bala

Numerade Educator

Problem 58

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=2^{x},[a, b]=[0,3]$

Raj Bala

Numerade Educator

Problem 59

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=\sin x,[a, b]=\left[0, \frac{\pi}{2}\right]$

Raj Bala

Numerade Educator

Problem 60

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b] .$ For those that do, use derivatives and algebra to find the exact values of all $c \in(a, b)$ that satisfy the conclusion of the Mean Value Theorem.
$f(x)=\tan x,[a, b]=[-\pi, \pi]$

Raj Bala

Numerade Educator

Problem 61

The cost of manufacturing a container for frozen orange juice is $C(h)=h^{2}-7.4 h+13.7$ cents, where $h$ is the height of the container in inches. Your boss claims that the containers will be cheapest to make if they are 4 inches tall. Use Theorem 3.3 to quickly show that he is wrong.

Melissa Munoz

Numerade Educator

Problem 62

Last night at 6 P.M., Linda got up from her blue easy chair. She did not return to her easy chair until she sat down again at 8 p.m. Let $s(t)$ be the distance between Linda and her easy chair $t$ minutes after 6 P.M. last night.
(a) Sketch a possible graph of $s(t),$ and describe what Linda did between 6 P.M. and 8 p.M. according to your graph. (Questions to think about: Will Linda necessarily move in a continuous and differentiable way? What are good ranges for $t$ and $s$ ?)
(b) Use Rolle's Theorem to show that at some point between 6 P.M. and 8 . P.M., Linda's velocity $v(t)$ with respect to the easy chair was zero. Find such a place on the graph of $s(t)$.

Raj Bala

Numerade Educator

Problem 63

It took Alina half an hour to drive to the grocery store that is 20 miles from her house.
(a) Use the Mean Value Theorem to show that, at some point during her trip, Alina must have been travelling exactly 40 miles per hour.
(b) Why does what you have shown in part
(a) make sense in real-world terms?

Raj Bala

Numerade Educator

Problem 64

Prove the part of Theorem 3.3 that was not proved in the reading: If a function $f$ has a local minimum at $x=c$, then either $f^{\prime}(c)$ does not exist or $f^{\prime}(c)=0$.

Raj Bala

Numerade Educator

Problem 65

Prove Rolle's Theorem: If $f$ is continuous on $[a, b]$ and differentiable on $(a, b),$ and if $f(a)=f(b)=0,$ then there is some value $c \in(a, b)$ with $f^{\prime}(c)=0$.

Raj Bala

Numerade Educator

Problem 66

Prove the Mean Value Theorem: If $f$ is continuous on $[a, b]$ and differentiable on $(a, b),$ then there is some value $c \in(a, b)$ with $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} .$

Raj Bala

Numerade Educator

Problem 67

Use Rolle's Theorem to prove that if $f$ is continuous and differentiable everywhere and has three roots, then its derivative $f^{\prime}$ has at least two roots.

Raj Bala

Numerade Educator

Problem 68

Follow the method of proof that we used for Rolle's Theorem to prove the following slightly more general theorem: If $f$ is continuous on $[a, b]$ and differentiable on $(a, b),$ and if $f(a)=f(b),$ then there is some value $c \in(a, b)$ with $f^{\prime}(c)=0$

Raj Bala

Numerade Educator

Problem 69

Use Rolle's Theorem to prove the slightly more general theorem from Exercise 68 : If $f$ is continuous on $[a, b]$ and differentiable on $(a, b),$ and if $f(a)=f(b),$ then there is some value $c \in(a, b)$ with $f^{\prime}(c)=0 .$ (Hint: Apply Rolle's Theorem to the function $g(x)=f(x)-f(a) .)$

Raj Bala

Numerade Educator