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Calculus: Graphical, Numerical, Algebraic

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

Chapter 3

Derivatives - all with Video Answers

Educators

Section 1

Derivative of a Function

02:03

Problem 1

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 given value of $a$.
$f(x)=1 / x, a=2$

Kian Manafi
Kian Manafi
Numerade Educator
03:12

Problem 2

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 given value of $a$.

Rachel Mcbride
Rachel Mcbride
Numerade Educator
01:42

Problem 3

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 given value of $a$. .
$f(x)=3-x^{2}, a=-1$

Kian Manafi
Kian Manafi
Numerade Educator
02:57

Problem 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 given value of $a$.
$f(x)=x^{3}+x, a=0$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:03

Problem 5

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 given value of $a$.
f(x)=1 / x, a=2

Kian Manafi
Kian Manafi
Numerade Educator
01:23

Problem 6

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 given value of $a$.
$f(x)=x^{2}+4, a=1$

Kian Manafi
Kian Manafi
Numerade Educator
01:47

Problem 7

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 given value of $a$.
$f(x)=\sqrt{x}+1, a=3$

Kian Manafi
Kian Manafi
Numerade Educator
01:20

Problem 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 given value of $a$.
$f(x)=2 x+3, a=-1$

Kian Manafi
Kian Manafi
Numerade Educator
01:30

Problem 9

Find $f^{\prime}(x)$ if $f(x)=3 x-12$

Kian Manafi
Kian Manafi
Numerade Educator

Problem 10

Find $d y / d x$ if $y=7 x$

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Problem 11

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

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00:34

Problem 12

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

Willis James
Willis James
Numerade Educator

Problem 13

Match the graph of the function with the graph of the derivative shown here:

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Problem 14

Match the graph of the function with the graph of the derivative shown here:

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Problem 15

Match the graph of the function with the graph of the derivative shown here:

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01:15

Problem 16

Match the graph of the function with the graph of the derivative shown here:

Rachel Mcbride
Rachel Mcbride
Numerade Educator
02:43

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 $x=2$ [Hint: Recall that the normal line is perpendicular to the tangent line.]

Sheryl Ezze
Sheryl Ezze
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 $x=3$.

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Problem 19

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

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Problem 20

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

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03:12

Problem 21

Daylight in Fairbanks 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?

Rachel Mcbride
Rachel Mcbride
Numerade Educator

Problem 22

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

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Problem 23

Lynxes and Hares In some northern parts of its range, the snowshoe hare is known to have a population that varies dramatically over time. The variation is cyclic and is tied to the population cycle of the Canada lynx, which preys almost exclusively on snowshoe hares. A simplified version of the "predatorprey population model" is shown in Figure $3.10 .$ Figure $3.10 \mathrm{a}$ shows populations of hares (blue) and lynxes (red) varying over a 9-year period in the $1800^{\circ} \mathrm{s},$ while Figure $3.10 \mathrm{~b}$ shows the graph of the derivative of the hare population, determined by the slopes of the blue curve in Figure $3.10 \mathrm{a}$.
(a) What is the derivative of the hare population in Figure 3.10 when the number of hares is the largest? smallest?
(b) What is the size of the hare population in Figure 3.10 when the population of hares is largest? smallest?

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00:53

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 $x>0$.
i. $y=e^{x-1}$
ii. $y=\ln x$
iii. $y=-\ln (x)$
iv. $y=\ln (x)-1$
v. $y=3 x-1$

Gio Maya
Gio Maya
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. $y=\sin x$
ii. $y=x$
iii. $y=\sqrt{x}$
iv. $y=e^{x}$
v. $y=x^{2}$

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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?

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Problem 27

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

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Problem 28

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

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Problem 29

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?

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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 ).
$$
\begin{array}{cc}
\text { TABLE 3.4 Elevations Along Bear Creek } \\
\begin{array}{c}
\text { Distance Downriver } \\
\text { (miles) }
\end{array} & \begin{array}{c}
\text { River Elevation } \\
\text { (feet) }
\end{array} \\
\hline 0.00 & 1577 \\
\hline 0.56 & 1512 \\
\hline 0.92 & 1448 \\
\hline 1.19 & 1384 \\
\hline 1.30 & 1319 \\
\hline 1.39 & 1255 \\
\hline 1.57 & 1191 \\
\hline 1.74 & 1126 \\
\hline 1.98 & 1062 \\
\hline 2.18 & 998 \\
\hline 2.41 & 933 \\
\hline 2.64 & 869 \\
\hline 3.24 & 805 \\
\hline
\end{array}
$$
(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.

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Problem 31

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

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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 $x=1$.

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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.

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01:03

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 $r i g h t$ hand derivative at $x=0 .$ Prove that it does not.

Gio Maya
Gio Maya
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?

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01:36

Problem 36

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

Kian Manafi
Kian Manafi
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.

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Problem 38

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

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01:42

Problem 39

Multiple Choice Let $f(x)=1-3 x^{2}$. Which of the following is equal to $f^{\prime}(1) ?$
(A) -6
(B) -5
(C) 5
(D) 6
(E) does not exist

Kian Manafi
Kian Manafi
Numerade Educator
01:10

Problem 40

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 lefthand derivative of $f$ at $x=0 ?$
(A) -2
(B) 0
(C) 2
(D) $\infty$
(E) $-\infty$

Kian Manafi
Kian Manafi
Numerade Educator
01:09

Problem 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 righthand derivative of $f$ at $x=0 ?$
$(\mathbf{A})-2$
(B) 0
(C) 2
(D) $\infty$
(E) $-\infty$

Kian Manafi
Kian Manafi
Numerade Educator
02:26

Problem 42

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

Rachel Mcbride
Rachel Mcbride
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 graph.
(e) Compare your answers and determine a simple way to predict the $y$ -intercept, given the values of $a$ and $b .$ Test your result.

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01:44

Problem 44

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 $x=1$

Kian Manafi
Kian Manafi
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 I: Set the values of $\mathrm{N}$ and $\mathrm{P}$ 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 $\mathrm{N}$ (the number of people in the room) along-side $\mathrm{P}$ (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 $\overline{2}$ generates the probabilities.)
(d) Is it reasonable to assume that all calendar dates are equally likely birthdays? Explain your answer.

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Problem 46

Algebraic Challenge Construct polynomial functions that fit the graphs of $f$ and $f^{\prime}$ that are shown in Example 3 of this section.

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