Calculus: Graphical, Numerical, Algebraic

Ross L. Finney, Franklin D. Demana, Bet K. Waits, Daniel Kennedy

Chapter 3

Derivatives - all with Video Answers

Educators

+ 6 more educators

Section 1

Derivative of a Function

Problem 1

In Exercises 1-4, use the definition $f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ to find the derivative of the given function at the indicated point. $f(x)=1 / x, a=2$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 2

In Exercises 1-4, use the definition $f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ to find the derivative of the given function at the indicated point.
$$f(x)=x^{2}+4, a=1$$

Carson Merrill

Numerade Educator

Problem 3

In Exercises 1-4, use the definition $f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ to find the derivative of the given function at the indicated point.
$$f(x)=3-x^{2}, a=-1$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 4

In Exercises 1-4, use the definition $f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ to find the derivative of the given function at the indicated point.
$$f(x)=x^{3}+x, a=0$$

Linh Vu

Numerade Educator

Problem 5

In Exercises $5-8,$ use the definition $f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$ to find the derivative of the given function at the indicated point.
$$f(x)=1 / x, a=2$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 6

In Exercises $5-8,$ use the definition $f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$ to find the derivative of the given function at the indicated point.
$$f(x)=x^{2}+4, a=1$$

Linh Vu

Numerade Educator

Problem 7

In Exercises $5-8,$ use the definition $f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$ to find the derivative of the given function at the indicated point.
$$f(x)=\sqrt{x+1}, a=3$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 8

In Exercises $5-8,$ use the definition $f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$ to find the derivative of the given function at the indicated point.
$$f(x)=2 x+3, a=-1$$

Linh Vu

Numerade Educator

Problem 9

In Exercises $5-8,$ use the definition $f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$ to find the derivative of the given function at the indicated point.
$$f^{\prime}(x) \text { if } f(x)=3 x-12$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 10

Find $d y / d x \text { if } y=7 x$

Linh Vu

Numerade Educator

Problem 11

Find $\frac{d}{d x}\left(x^{2}\right)$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 12

Find $\frac{d}{d x} f(x) \text { if } f(x)=3 x^{2}$

Linh Vu

Numerade Educator

Problem 13

In Exercises $13-16,$ match the graph of the function with the graph of
the derivative shown here:

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 14

In Exercises $13-16,$ match the graph of the function with the graph of
the derivative shown here:

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 15

In Exercises $13-16,$ match the graph of the function with the graph of
the derivative shown here:

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 16

In Exercises $13-16,$ match the graph of the function with the graph of
the derivative shown here:

Linh Vu

Numerade Educator

Problem 17

If $f(2)=3$ and $f^{\prime}(2)=5,$ find an equation of (a) the tangent line, and (b) the normal line to the graph of $y=f(x)$ at the point where $x=2$ .

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 18

Find the derivative of the function $y=2 x^{2}-13 x+5$ and use it to find an equation of the line tangent to the curve at $x=3 .$

Alex Arteaga

Numerade Educator

Problem 19

Find the lines that are (a) tangent and (b) normal to the curve
$y=x^{3}$ at the point $(1,1)$ .

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 20

Find the lines that are (a) tangent and (b) normal to the curve
$y=\sqrt{x}$ at $x=4 .$

Linh Vu

Numerade Educator

Problem 21

The viewing window below shows the number of hours of daylight in Fairbanks, Alaska, on each day for a typical 365 -day period from January 1 to December 31 . Answer the following questions by estimating slopes on the graph in hours per day. For the purposes of estimation, assume that each month has 30 days.
(a) On about what date is the amount of daylight increasing at the fastest rate? What is that rate?
(b) Do there appear to be days on which the rate of change in the amount of daylight is zero? If so, which ones?
(c) On what dates is the rate of change in the number of daylight hours positive? negative?

Stephen Hobbs

Numerade Educator

Problem 22

Graphing $f^{\prime}$ from $f$ Given the graph of the function $f$ below,
sketch a graph of the derivative of $f .$

Linh Vu

Numerade Educator

Problem 23

The graphs in Figure 3.10 a show the numbers of rabbits and foxes in a small arctic population. They are plotted as functions of time for 200 days. The number of rabbits increases at first, as the rabbits reproduce. But the foxes prey on the rabbits and, as
the number of foxes increases, the rabbit population levels off and then drops. Figure 3.10 $\mathrm{b}$ shows the graph of the derivative of the rabbit population. We made it by plotting slopes, as in Example $3 .$ (a) What is the value of the derivative of the rabbit population in Figure 3.10 when the number of rabbits is largest? smallest? (b) What is the size of the rabbit population in Figure 3.10 when its derivative is largest? smallest?

Carson Merrill

Numerade Educator

Problem 24

Shown below is the graph of $f(x)=x \ln x-x .$ From what you know about the graphs of functions (i) through (v), pick out the one that is the derivative of $f$ for $x>0$ . $\begin{array}{ll}{\text { i. } y=\sin x} & {\text { ii. } y=\ln x} \\ {\text { iv. } y=x^{2}} & {\text { v. } y=3 x-1}\end{array}$

Linh Vu

Numerade Educator

Problem 25

From what you know about the graphs of functions (i) through (v), pick out the one that is its own derivative. i. $y=\sin x \quad$ ii. $y=x \quad$ iii. $y=\sqrt{x}$ iv. $y=e^{x} \quad$ v. $y=x^{2}$

Willis James

Numerade Educator

Problem 26

The graph of the function $y=f(x)$ shown here is made of line segments joined end to end. (a) Graph the function's derivative.
(b) At what values of $x$ between $x=-4$ and $x=6$ is the
function not differentiable?

Linda Hand

Numerade Educator

Problem 27

Graphing $f$ from $f^{\prime}$ Sketch the graph of a continuous function $f$ with $f(0)=-1$ and $f^{\prime}(x)=\left\{\begin{array}{ll}{1,} & {x<-1} \\ {-2,} & {x>-1}\end{array}\right.$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 28

Graphing $f$ from $f^{\prime}$ Sketch the graph of a continuous function $f$ with $f(0)=1$ and $f^{\prime}(x)=\left\{\begin{array}{ll}{2,} & {x<2} \\ {-1,} & {x>2}\end{array}\right.$

Linh Vu

Numerade Educator

Problem 29

In Exercises 29 and $30,$ use the data to answer the questions. A Downhill Skier Table 3.3 gives the approximate distance traveled by a downhill skier after $t$ seconds for $0 \leq t \leq 10$ . Use the method of Example 5 to sketch a graph of the derivative; then answer the following questions:
(a) What does the derivative represent?
(b) In what units would the derivative be measured?
(c) Can you guess an equation of the derivative by considering
its graph?

Nick Johnson

Numerade Educator

Problem 30

A Whitewater River Bear Creek, a Georgia river known to kayaking enthusiasts, drops more than 770 feet over one stretch of 3.24 miles. By reading a contour map, one can estimate the elevations $(y)$ at various distances $(x)$ downriver from the start
of the kayaking route (Table $3.4 ) .$
(a) Sketch a graph of elevation $(y)$ as a function of distance downriver $(x) .$
(b) Use the technique of Example 5 to get an approximate graph of the derivative, $d y / d x$ .
(c) The average change in elevation over a given distance is
called a gradient. In this problem, what units of measure would
be appropriate for a gradient?
(d) In this problem, what units of measure would be appropriate
for the derivative?
(e) How would you identify the most dangerous section of the
river (ignoring rocks) by analyzing the graph in $($ a)? Explain.
(f) How would you identify the most dangerous section of the
river by analyzing the graph in (b)? Explain.

AG

Ankit Gupta

Numerade Educator

Problem 31

Using one-sided derivatives, show that the function $f(x)=\left\{\begin{array}{c}{x^{2}+x,} & {x \leq 1} \\ {3 x-2,} & {x>1}\end{array}\right.$ does not have a derivative at $x=1$

Sanchit Jain

Numerade Educator

Problem 32

Using one-sided derivatives, show that the function $f(x)=\left\{\begin{array}{ll}{x^{3},} & {x \leq 1} \\ {3 x,} & {x>1}\end{array}\right.$ does not have a derivative at $x=1$

Linh Vu

Numerade Educator

Problem 33

Writing to Learn Graph $y=\sin x$ and $y=\cos x$ in the same viewing window. Which function could be the derivative of the other? Defend your answer in terms of the behavior of the graphs.

Linh Vu

Numerade Educator

Problem 34

In Example 2 of this section we showed that the derivative of $y=\sqrt{x}$ is a function with domain $(0, \infty) .$ However, the function $y=\sqrt{x}$ itself has domain $[0, \infty),$ so it could have a right-hand derivative at $x=0 .$ Prove that it does not.

Linh Vu

Numerade Educator

Problem 35

Writing to Learn Use the concept of the derivative to define what it might mean for two parabolas to be parallel. Construct equations for two such parallel parabolas and graph them. Are the parabolas "everywhere equidistant," and if so, in what sense?

Alison Rodriguez

Alison Rodriguez

Numerade Educator

Problem 36

True or False If $f(x)=x^{2}+x,$ then $f^{\prime}(x)$ exists for every real
number $x .$ Justify your answer. True. $f^{\prime}(x)=2 x+1$

Linh Vu

Numerade Educator

Problem 37

True or False If the left-hand derivative and the right-hand derivative of $f$ exist at $x=a,$ then $f^{\prime}(a)$ exists. Justify your answer.

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 38

Multiple Choice Let $f(x)=4-3 x$ . Which of the following is equal to $f^{\prime}(-1) ?
$(\mathbf{A})-7 \quad(\mathbf{B}) 7 \quad(\mathbf{C})-3 \quad(\mathbf{D}) 3 \quad(\mathbf{E})$ does not exist

Linh Vu

Numerade Educator

Problem 39

Multiple Choice Let $f(x)=4-3 x .$ Which of the following is equal to $f^{\prime}(-1) ? $(\mathbf{A})-6 \quad(\mathbf{B})-5 \quad(\mathbf{C}) 5 \quad(\mathbf{D}) 6 \quad(\mathbf{E})$ does not exist

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 40

In Exercises 40 and $41,$ let $f(x)=\left\{\begin{array}{ll}{x^{2}-1,} & {x<0} \\ {2 x-1,} & {x \geq 0}\end{array}\right.$
Multiple Choice Which of the following is equal to the left-hand derivative of $f$ at $x=0 ?$
$(\mathbf{A})-2 \quad(\mathbf{B}) 0 \quad(\mathbf{C}) 2 \quad(\mathbf{D}) \propto(\mathbf{E})-\infty$

Madi Sousa

Numerade Educator

Problem 41

Multiple Choice Which of the following is equal to the right-hand derivative of $f$ at $x=0 ?$
$(\mathbf{A})-2 \quad(\mathbf{B}) 0 \quad(\mathbf{C}) 2 \quad(\mathbf{D}) \infty \quad(\mathbf{E})-\infty$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 42

Explorations
Let $f(x)=\left\{\begin{array}{ll}{x^{2},} & {x \leq 1} \\ {2 x,} & {x>1}\end{array}\right.$
\begin{array}{ll}{\text { (a) Find } f^{\prime}(x) \text { for } x<1 .} & {\text { (b) Find } f^{\prime}(x) \text { for } x>1.2} \\ {\text { (c) Find } \lim _{x \rightarrow 1}-f^{\prime}(x) .2} &{\text { (d) Find } \lim _{x \rightarrow 1^{+}} f^{\prime}(x)}\end{array}
\begin{array}{l}{\text { (e) Does } \lim _{x \rightarrow 1} f^{\prime}(x) \text { exist? Explain. }} \\ {\text { (f) Use the definition to find the left-hand derivative of } f^ {}} \\ {\text { at } x=1 \text { if it exists. } } \\ {\text { (g) Use the definition to find the right-hand derivative of } f} \\ {\text { at } x=1 \text { if it exists.}} \\ {\text { (h) Does $f^{\prime}(1)$} \text{exist?} \text{Explain.}} \end{array}

Vinnu M

Numerade Educator

Problem 43

Group Activity Using graphing calculators, have each person in your group do the following:
(a) pick two numbers $a$ and $b$ between 1 and $10 ;$
(b) graph the function $y=(x-a)(x+b)$ ;
(c) graph the derivative of your function (it will be a line with
slope 2$)$
(d) find the $y$ -intercept of your derivative a simple way to predict
the $y$ -intercept, given the values of $a$ and $b$ . Test your result.

Vinnu M

Numerade Educator

Problem 44

Extending the ldeas
Find the unique value of $k$ that makes the function
$f(x)=\left\{\begin{array}{ll}{x^{3},} & {x \leq 1} \\ {3 x+k,} & {x>1}\end{array}\right.$
differentiable at $x=1 .$

Linh Vu

Numerade Educator

Problem 45

Generating the Birthday Probabilities Example 5 of this section concerns the probability that, in a group of $n$ people, at least two people will share a common birthday. You can
generate these probabilities on your calculator for values of $n$ from 1 to $365 .$
Step 1: Set the values of $N$ and $P$ to zero:
Step $2 :$ Type in this single, multi-step command:
Now each time you press the ENTER key, the command will
print a new value of $N($ the number of people in the room)
alongside $P$ (the probability that at least two of them share a
common birthday):
If you have some experience with probability, try to answer the
following questions without looking at the table:
(a) If there are three people in the room, what is the probability
that they all have different birthdays? (Assume that there are 365
possible birthdays, all of them equally likely.)
(b) If there are three people in the room, what is the probability
that at least two of them share a common birthday?
(c) Explain how you can use the answer in part (b) to find the
probability of a shared birthday when there are four people
in the room. (This is how the calculator statement in Step 2
generates the probabilities.)
(d) Is it reasonable to assume that all calendar dates are equally
likely birthdays? Explain your answer.

AG

Ankit Gupta

Numerade Educator