Applied Calculus

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

Chapter 4

Using the Derivative - all with Video Answers

Educators

WM

Section 1

Local Maxima and Minima

Problem 1

Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. (GRAPH CAN'T COPY)

Joanie Morris

Joanie Morris

Numerade Educator

Problem 2

Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. (GRAPH CAN'T COPY)

Joanie Morris

Numerade Educator

Problem 3

Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. (GRAPH CAN'T COPY)

Joanie Morris

Numerade Educator

Problem 4

Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. (GRAPH CAN'T COPY)

Joanie Morris

Numerade Educator

Problem 5

(a) Graph a function with two local minima and one local maximum.
(b) Graph a function with two critical points. One of these critical points should be a local minimum, and the other should be neither a local maximum nor a local minimum.

Joanie Morris

Numerade Educator

Problem 6

Graph two continuous functions $f$ and $g,$ each of which has exactly five critical points, the points $A-E$ in Figure $4.13,$ and that satisfy the following conditions: (GRAPH CAN'T COPY)
(a) $f(x) \rightarrow \infty$ as $x \rightarrow-\infty \quad$ and
$f(x) \rightarrow \infty$ as $x \rightarrow \infty$
(b) $g(x) \rightarrow-\infty$ as $x \rightarrow-\infty \quad$ and
$g(x) \rightarrow 0$ as $x \rightarrow \infty$

Joanie Morris

Numerade Educator

Problem 7

During an illness a person ran a fever. His temperature rose steadily for eighteen hours, then went steadily down for twenty hours. When was there a critical point for his temperature as a function of time?

Joanie Morris

Numerade Educator

Problem 8

Using a calculator or computer, graph the functions in Problems $8-13 .$ Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph.
$$f(x)=x^{3}+6 x+1$$

Zachary Watson

Numerade Educator

Problem 9

Using a calculator or computer, graph the functions in Problems $8-13 .$ Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph.
$$f(x)=x^{3}-6 x+1$$

Joanie Morris

Numerade Educator

Problem 10

Using a calculator or computer, graph the functions in Problems $8-13 .$ Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph.
$$f(x)=3 x^{5}-5 x^{3}$$

Joanie Morris

Numerade Educator

Problem 11

Using a calculator or computer, graph the functions in Problems $8-13 .$ Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph.
$$f(x)=e^{x}-10 x$$

Joanie Morris

Numerade Educator

Problem 12

Using a calculator or computer, graph the functions in Problems $8-13 .$ Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph.
$$f(x)=x \ln x, \quad x>0$$

Joanie Morris

Numerade Educator

Problem 13

Using a calculator or computer, graph the functions in Problems $8-13 .$ Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph.
$$f(x)=x+2 \sin x$$

Joanie Morris

Numerade Educator

Problem 14

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

Joanie Morris

Numerade Educator

Problem 15

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 16

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 17

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 18

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 19

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 20

The function $f(x)=x^{4}-4 x^{3}+8 x$ has a critical point at $x=1 .$ Use the second-derivative test to identify it as a local maximum or local minimum.

Joanie Morris

Numerade Educator

Problem 21

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

Joanie Morris

Numerade Educator

Problem 22

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 23

Indicate on the graph of the derivative function $f^{\prime}$ in Figure 4.14 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 24

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

Joanie Morris

Numerade Educator

Problem 25

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

Joanie Morris

Numerade Educator

Problem 26

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

Joanie Morris

Numerade Educator

Problem 27

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

Joanie Morris

Numerade Educator

Problem 28

Figure 4.15 is a graph of $f^{\prime} .$ For what values of $x$ does $f$ have a local maximum? A local minimum? (FIGURE CAN'T COPY)

Joanie Morris

Numerade Educator

Problem 29

Consumer demand for a product is changing over time, and the rate of change of demand, $f^{\prime}(t),$ in units/week, is given, in week $t,$ for $0 \leq t \leq 10,$ in the following table. $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline f^{\prime}(t) & 12 & 10 & 4 & -2 & -3 & -1 & 3 & 7 & 11 & 15 & 10\\\hline\end{array}$$
(a) When is the demand for this product increasing? When is it decreasing?
(b) Approximately when is demand at a local maximum? A local minimum?

Joanie Morris

Numerade Educator

Problem 30

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.
$$\begin{array}{c|c|c|c|r|r|r|r|r|r|r|c}\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline f^{\prime}(x) & 5 & 2 & 1 & -2 & -5 & -3 & -1 & 2 & 3 & 1 & -1 \\\hline\end{array}$$

Joanie Morris

Numerade Educator

Problem 31

The derivative of $f(t)$ is given by $f^{\prime}(t)=t^{3}-6 t^{2}+8 t$ for $0 \leq t \leq 5 .$ Graph $f^{\prime}(t),$ and describe how the function $f(t)$ changes over the interval $t=0$ to $t=5 .$ When is $f(t)$ increasing and when is it decreasing? Where does $f(t)$ have a local maximum and where does it have a local minimum?

Joanie Morris

Numerade Educator

Problem 32

Find constants $a$ and $b$ so that the minimum for the parabola $f(x)=x^{2}+a x+b$ is at the given point.
$$(3,5)$$

Joanie Morris

Numerade Educator

Problem 33

Find constants $a$ and $b$ so that the minimum for the parabola $f(x)=x^{2}+a x+b$ is at the given point.
$$(-2,-3)$$

Joanie Morris

Numerade Educator

Problem 34

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

Khushbu Rani

Numerade Educator

Problem 35

For what values of $a$ and $b$ does $f(x)=a(x-b \ln x)$ have a local minimum at the point (2,5)$?$ Figure 4.16 shows a graph of $f(x)$ with $a=1$ and $b=1$ (GRAPH CAN'T COPY)

Joanie Morris

Numerade Educator

Problem 36

Sketch several members of the family $y=x^{3}-a x^{2}$ on the same axes. Discuss the effect of the parameter $a$ on the graph. Find all critical points for this function.

Zachary Watson

Numerade Educator

Problem 37

(a) For $a$ a positive constant, find all critical points of $f(x)=x-a \sqrt{x}$
(b) What value of $a$ gives a critical point at $x=5 ?$ Does $f(x)$ have a local maximum or a local minimum at this critical point?

Joanie Morris

Numerade Educator

Problem 38

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 39

investigate the one-parameter family of functions. Assume that $a$ is positive.
(a) Graph $f(x)$ using three different values for $a$
(b) Using your graph in part (a), describe the critical points of $f$ and how they appear to move as $a$ increases.
(c) Find a formula for the $x$ -coordinates of the critical $\operatorname{point}(s)$ of $f$ in terms of $a$
$$f(x)=(x-a)^{2}$$

Zachary Watson

Numerade Educator

Problem 40

investigate the one-parameter family of functions. Assume that $a$ is positive.
(a) Graph $f(x)$ using three different values for $a$
(b) Using your graph in part (a), describe the critical points of $f$ and how they appear to move as $a$ increases.
(c) Find a formula for the $x$ -coordinates of the critical $\operatorname{point}(s)$ of $f$ in terms of $a$
$$f(x)=x^{3}-a x$$

Zachary Watson

Numerade Educator

Problem 41

investigate the one-parameter family of functions. Assume that $a$ is positive.
(a) Graph $f(x)$ using three different values for $a$
(b) Using your graph in part (a), describe the critical points of $f$ and how they appear to move as $a$ increases.
(c) Find a formula for the $x$ -coordinates of the critical $\operatorname{point}(s)$ of $f$ in terms of $a$
$$f(x)=x^{2} e^{-a x}$$

Zachary Watson

Numerade Educator

Problem 42

investigate the one-parameter family of functions. Assume that $a$ is positive.
(a) Graph $f(x)$ using three different values for $a$
(b) Using your graph in part (a), describe the critical points of $f$ and how they appear to move as $a$ increases.
(c) Find a formula for the $x$ -coordinates of the critical $\operatorname{point}(s)$ of $f$ in terms of $a$
If $m, n \geq 2$ are integers, find and classify the critical points of $f(x)=x^{m}(1-x)^{n}$

Zachary Watson

Numerade Educator