Calculus: Single Variable

Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason

Chapter 4

Using the Derivative - all with Video Answers

Educators

Section 1

Using First and Second Derivatives

Problem 1

Indicate all critical points on the graph of $f$ in Figure 4.12 and determine which correspond to local maxima of $f$ which to local minima, and which to neither.
(FIGURE CAN'T COPY)

Lauren Shelton

Lauren Shelton

Numerade Educator

Problem 2

Graph a function which has exactly one critical point, at $x=2,$ and exactly one inflection point, at $x=4$

Kian Manafi

Numerade Educator

Problem 3

Graph a function with exactly two critical points, one of which is a local minimum and the other is neither a local maximum nor a local minimum.

Kian Manafi

Numerade Educator

Problem 4

Use derivatives to find the critical points and inflection points.
$$f(x)=x^{3}-9 x^{2}+24 x+5$$

Kian Manafi

Numerade Educator

Problem 5

Use derivatives to find the critical points and inflection points.
$$f(x)=x^{5}-10 x^{3}-8$$

Kian Manafi

Numerade Educator

Problem 6

Use derivatives to find the critical points and inflection points.
$$f(x)=x^{5}+15 x^{4}+25$$

Kian Manafi

Numerade Educator

Problem 7

Use derivatives to find the critical points and inflection points.
$$f(x)=5 x-3 \ln x$$

Kian Manafi

Numerade Educator

Problem 8

Use derivatives to find the critical points and inflection points.
$$f(x)=4 x e^{3 x}$$

Kian Manafi

Numerade Educator

Problem 9

Find all critical points and then use the first derivative test to determine local maxima and minima. Check your answer by graphing.
$$f(x)=3 x^{4}-4 x^{3}+6$$

Joanie Morris

Numerade Educator

Problem 10

Find all critical points and then use the first derivative test to determine local maxima and minima. Check your answer by graphing.
$$f(x)=\left(x^{2}-4\right)^{7}$$

Joanie Morris

Numerade Educator

Problem 11

Find all critical points and then use the first derivative test to determine local maxima and minima. Check your answer by graphing.
$$f(x)=\left(x^{3}-8\right)^{4}$$

Joanie Morris

Numerade Educator

Problem 12

Find all critical points and then use the first derivative test to determine local maxima and minima. Check your answer by graphing.
$$f(x)=\frac{x}{x^{2}+1}$$

Joanie Morris

Numerade Educator

Problem 13

Find the critical points of the function and classify them as local maxima or local minima or neither.
$$g(x)=x e^{-3 x}$$

Joanie Morris

Numerade Educator

Problem 14

Find the critical points of the function and classify them as local maxima or local minima or neither.
$$h(x)=x+1 / x$$

Joanie Morris

Numerade Educator

Problem 15

(a) Use a graph to estimate the $x$ -values of any critical points and inflection points of $f(x)=e^{-x^{2}}$
(b) Use derivatives to find the $x$ -values of any critical points and inflection points exactly.

Bryan Lynn

Numerade Educator

Problem 16

The function $f$ is defined for all $x$. Use the graph of $f^{\prime}$ to decide:
(a) Over what intervals is $f$ increasing? Decreasing?(b) Does $f$ have local maxima or minima? If so, which, and where?
(GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 17

The function $f$ is defined for all $x$. Use the graph of $f^{\prime}$ to decide:
(a) Over what intervals is $f$ increasing? Decreasing?(b) Does $f$ have local maxima or minima? If so, which, and where?
(GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 18

The function $f$ is defined for all $x$. Use the graph of $f^{\prime}$ to decide:
(a) Over what intervals is $f$ increasing? Decreasing?(b) Does $f$ have local maxima or minima? If so, which, and where?
(GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 19

The function $f$ is defined for all $x$. Use the graph of $f^{\prime}$ to decide:
(a) Over what intervals is $f$ increasing? Decreasing?(b) Does $f$ have local maxima or minima? If so, which, and where?
(GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 20

(a) Show that if $a$ is a positive constant, then $x=0$ is the only critical point of $f(x)=x+a \sqrt{x}$
(b) Use derivatives to show that $f$ is increasing and its graph is concave down for all $x > 0$

Carson Merrill

Numerade Educator

Problem 21

(a) If $b$ is a positive constant and $x>0,$ find all critical points of $f(x)=x-b \ln x$
(b) Use the second-derivative test to determine whether the function has a local maximum or local minimum at each critical point.

Kian Manafi

Numerade Educator

Problem 22

(a) If $a$ is a nonzero constant, find all critical points of
$$
f(x)=\frac{a}{x^{2}}+x
$$
(b) Use the second-derivative test to show that if $a$ is positive then the graph has a local minimum, and if $a$ is negative then the graph has a local maximum.

Kian Manafi

Numerade Educator

Problem 23

If $U$ and $V$ are positive constants, find all critical points of
$$
F(t)=U e^{t}+V e^{-t}
$$

Zachary Watson

Numerade Educator

Problem 24

Indicate on the graph of the derivative function $f^{\prime}$ in Figure 4.13 the $x$ -values that are critical points of the function $f$ itself. At which critical points does $f$ have local maxima, local minima, or neither?
(FIGURE CAN'T COPY)

Joanie Morris

Numerade Educator

Problem 25

Indicate on the graph of the derivative $f^{\prime}$ in Figure 4.14 the $x$ -values that are inflection points of the function $f$
(FIGURE CAN'T COPY)

Lauren Shelton

Numerade Educator

Problem 26

Indicate on the graph of the second derivative $f^{\prime \prime}$ in Figure 4.15 the $x$ -values that are inflection points of the function $f$
(FIGURE CAN'T COPY)

Noah Mekonnen

Numerade Educator

Problem 27

Sketch a possible graph of $y=f(x)$ using the given information about the derivatives $y^{\prime}=f^{\prime}(x)$ and $y^{\prime \prime}=f^{\prime \prime}(x) .$ Assume that the function is defined and continuous for all real $x$.
(EQUATION CAN'T COPY)

Noah Mekonnen

Numerade Educator

Problem 28

Sketch a possible graph of $y=f(x)$ using the given information about the derivatives $y^{\prime}=f^{\prime}(x)$ and $y^{\prime \prime}=f^{\prime \prime}(x) .$ Assume that the function is defined and continuous for all real $x$.
(EQUATION CAN'T COPY)

Noah Mekonnen

Numerade Educator

Problem 29

Sketch a possible graph of $y=f(x)$ using the given information about the derivatives $y^{\prime}=f^{\prime}(x)$ and $y^{\prime \prime}=f^{\prime \prime}(x) .$ Assume that the function is defined and continuous for all real $x$.
(EQUATION CAN'T COPY)

Noah Mekonnen

Numerade Educator

Problem 30

Sketch a possible graph of $y=f(x)$ using the given information about the derivatives $y^{\prime}=f^{\prime}(x)$ and $y^{\prime \prime}=f^{\prime \prime}(x) .$ Assume that the function is defined and continuous for all real $x$.
(EQUATION CAN'T COPY)

Noah Mekonnen

Numerade Educator

Problem 31

Suppose $f$ has a continuous derivative whose values are given in the following table.
(a) Estimate the $x$ -coordinates of critical points of $f$ for $0 \leq x \leq 10$
(b) For each critical point, indicate if it is a local maximum of $f,$ local minimum, or neither.
$$\begin{array}{c|c|c|c|r|r|r|r|r|r|r|r}
\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline f^{\prime}(x) & 5 & 2 & 1 & -2 & -5 & -3 & -1 & 2 & 3 & 1 & -1 \\
\hline
\end{array}$$

Joanie Morris

Numerade Educator

Problem 32

(a) The following table gives values of the differentiable function $y=f(x) .$ Estimate the $x$ -values of critical points of $f(x)$ on the interval $0 < x < 10 .$ Classify each critical point as a local maximum, local minimum, or neither.
(b) Now assume that the table gives values of the continuous function $\left.y=f^{\prime}(x) \text { (instead of } f(x)\right) .$ Estimate and classify critical points of the function $f(x)$
$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}
\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline y & 1 & 2 & 1 & -2 & -5 & -3 & -1 & 2 & 3 & 1 & -1 \\
\hline
\end{array}$$

Carson Merrill

Numerade Educator

Problem 33

If water is flowing at a constant rate (i.e., constant volume per unit time) into the vase in Figure $4.16,$ sketch a graph of the depth of the water against time. Mark on the graph the time at which the water reaches the corner of the vase.
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 34

If water is flowing at a constant rate (i.e., constant volume per unit time) into the Grecian urn in Figure 4.17 , sketch a graph of the depth of the water against time. Mark on the graph the time at which the water reaches the widest point of the urn.
(FIGURE CAN'T COPY)

Bryan Lynn

Numerade Educator

Problem 35

Find and classify the critical points of $f(x)=x^{3}(1-x)^{4}$ as local maxima and minima.

Joanie Morris

Numerade Educator

Problem 36

If $m, n \geq 2$ are integers, find and classify the critical points of $f(x)=x^{m}(1-x)^{n}$

Carson Merrill

Numerade Educator

Problem 37

The rabbit population on a small Pacific island is approximated by
$$
P=\frac{2000}{1+e^{5.3-0.4 t}}
$$
with $t$ measured in years since $1774,$ when Captain James Cook left 10 rabbits on the island.
(a) Graph $P .$ Does the population level off?
(b) Estimate when the rabbit population grew most rapidly. How large was the population at that time?
(c) What natural causes could lead to the shape of the graph of $P ?$

Carson Merrill

Numerade Educator

Problem 38

Find values of $a$ and $b$ so that the function $f(x)=$ $x^{2}+a x+b$ has a local minimum at the point (6,-5)

Carson Merrill

Numerade Educator

Problem 39

Find the value of $a$ so that the function $f(x)=x e^{a x}$ has a critical point at $x=3$

Joanie Morris

Numerade Educator

Problem 40

Find constants $a$ and $b$ in the function $f(x)=a x e^{b x}$ such that $f\left(\frac{1}{3}\right)=1$ and the function has a local maximum at $x=\frac{1}{3}$

Joanie Morris

Numerade Educator

Problem 41

Graph $f(x)=x+\sin x,$ and determine where $f$ is increasing most rapidly and least rapidly.

Carson Merrill

Numerade Educator

Problem 42

You might think the graph of $f(x)=x^{2}+\cos x$ should look like a parabola with some waves on it. Sketch the actual graph of $f(x)$ using a calculator or computer. Explain what you see using $f^{\prime \prime}(x)$

Carson Merrill

Numerade Educator

Problem 43

Show graphs of the three functions $f, f^{\prime}, f^{\prime \prime}$ Identify which is which.
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 44

Show graphs of the three functions $f, f^{\prime}, f^{\prime \prime}$ Identify which is which.
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 45

Show graphs of $f, f^{\prime}, f^{\prime \prime} .$ Each of these three functions is either odd or even. Decide which functions are odd and which are even. Use this information to identify which graph corresponds to $f,$ which to $f^{\prime},$ and which to $f^{\prime \prime}$.
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 46

Show graphs of $f, f^{\prime}, f^{\prime \prime} .$ Each of these three functions is either odd or even. Decide which functions are odd and which are even. Use this information to identify which graph corresponds to $f,$ which to $f^{\prime},$ and which to $f^{\prime \prime}$.
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 47

Use the derivative formulas and algebra to find the intervals where $f(x)=(x+50) /\left(x^{2}+525\right)$ is increasing and the intervals where it is decreasing. It is possible, but difficult, to solve this problem by graphing $f ;$ describe the difficulty.

Carson Merrill

Numerade Educator

Problem 48

Let $f$ be a function with $f(x)>0$ for all $x .$ Set $g=1 / f$
(a) If $f$ is increasing in an interval around $x_{0},$ what about $g ?$
(b) If $f$ has a local maximum at $x_{1},$ what about $g ?$
(c) If $f$ is concave down at $x_{2},$ what about $g ?$

Carson Merrill

Numerade Educator

Problem 49

The differentiable function $f$ has $x=1$ as its only zero and $x=2$ as its only critical point. For the given functions, find all
(a) Zeros
(b) Critical points.
$$y=f\left(x^{2}-3\right)$$

Nick Derr

Numerade Educator

Problem 50

The differentiable function $f$ has $x=1$ as its only zero and $x=2$ as its only critical point. For the given functions, find all
(a) Zeros
(b) Critical points.
$$y=(f(x))^{2}+3$$

Nick Derr

Numerade Educator

Problem 51

The graph of $f$ lies entirely above the $x$ axis and $f^{\prime}(x)<0$ for all $x$.
(a) Give the critical point(s) of the function, or explain how you know there are none.
(b) Say where the function increases and where it decreases.
$$y=(f(x))^{2}$$

Nick Derr

Numerade Educator

Problem 52

The graph of $f$ lies entirely above the $x$ axis and $f^{\prime}(x)<0$ for all $x$.
(a) Give the critical point(s) of the function, or explain how you know there are none.
(b) Say where the function increases and where it decreases.
$$y=f\left(x^{2}\right)$$

Nick Derr

Numerade Educator

Problem 53

Explain what is wrong with the statement.
An increasing function has no inflection points.

Kian Manafi

Numerade Educator

Problem 54

Explain what is wrong with the statement.
For any function $f,$ if $f^{\prime \prime}(0)=0,$ there is an inflection point at $x=0$

Kian Manafi

Numerade Educator

Problem 55

Give an example of:
A function which has no critical points on the interval between 0 and 1

Kian Manafi

Numerade Educator

Problem 56

Give an example of:
A function, $f,$ which has a critical point at $x=1$ but for which $f^{\prime}(1) \neq 0$

Kian Manafi

Numerade Educator

Problem 57

Give an example of:
A function with local maxima and minima at an infinite number of points.

Kian Manafi

Numerade Educator

Problem 58

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
A local minimum of $f$ occurs at a critical point of $f$.

Nick Derr

Numerade Educator

Problem 59

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
If $x=p$ is not a critical point of $f,$ then $x=p$ is not a local maximum of $f$.

Nick Derr

Numerade Educator

Problem 60

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
A local maximum of $f$ occurs at a point where
$$
f^{\prime}(x)=0.
$$

Nick Derr

Numerade Educator

Problem 61

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
If $x=p$ is not a local maximum of $f,$ then $x=p$ is not a critical point of $f$.

Nick Derr

Numerade Educator

Problem 62

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
If $f^{\prime}(p)=0,$ then $f(x)$ has a local minimum or local maximum at $x=p$.

Nick Derr

Numerade Educator

Problem 63

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
If $f^{\prime}(x)$ is continuous and $f(x)$ has no critical points, then $f$ is everywhere increasing or everywhere decreasing.

Nick Derr

Numerade Educator

Problem 64

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
If $f^{\prime \prime}(x)$ is continuous and the graph of $f$ has an inflection point at $x=p,$ then $f^{\prime \prime}(p)=0$.

Nick Derr

Numerade Educator

Problem 65

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
A critical point of $f$ must be a local maximum or minimum of $f$.

Nick Derr

Numerade Educator

Problem 66

Are the statements in Problems true or false for a function $f$ whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample.
Every cubic polynomial has an inflection point.

Nick Derr

Numerade Educator

Problem 67

Give an example of a function $f$ that makes the statement true, or say why such an example is impossible. Assume that $f^{\prime \prime}$ exists everywhere.
$f$ is concave up and $f(x)$ is positive for all $x$.

Carson Merrill

Numerade Educator

Problem 68

Give an example of a function $f$ that makes the statement true, or say why such an example is impossible. Assume that $f^{\prime \prime}$ exists everywhere.
$f$ is concave down and $f(x)$ is positive for all $x$.

Kian Manafi

Numerade Educator

Problem 69

Give an example of a function $f$ that makes the statement true, or say why such an example is impossible. Assume that $f^{\prime \prime}$ exists everywhere.
$f$ is concave down and $f(x)$ is negative for all $x$.

Carson Merrill

Numerade Educator

Problem 70

Give an example of a function $f$ that makes the statement true, or say why such an example is impossible. Assume that $f^{\prime \prime}$ exists everywhere.
$f$ is concave up and $f(x)$ is negative for all $x$.

Kian Manafi

Numerade Educator

Problem 71

Give an example of a function $f$ that makes the statement true, or say why such an example is impossible. Assume that $f^{\prime \prime}$ exists everywhere.
$f(x) f^{\prime \prime}(x)<0$ for all $x$.

Carson Merrill

Numerade Educator

Problem 72

Give an example of a function $f$ that makes the statement true, or say why such an example is impossible. Assume that $f^{\prime \prime}$ exists everywhere.
$f(x) f^{\prime}(x) f^{\prime \prime}(x) f^{\prime \prime \prime}(x) < 0$ for all $x$.

Carson Merrill

Numerade Educator

Problem 73

Given that $f^{\prime}(x)$ is continuous everywhere and changes from negative to positive at $x=a,$ which of the following statements must be true?
(a) $a$ is a critical point of $f(x)$
(b) $f(a)$ is a local maximum
(c) $f(a)$ is a local minimum
(d) $f^{\prime}(a)$ is a local maximum
(e) $f^{\prime}(a)$ is a local minimum

Nick Derr

Numerade Educator