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Chapter 2

DERIVATIVES

Educators


Problem 1

(a) Find the slope of the tangent line to the parabola
$y=4 x-x^{2}$ at the point $(1,3)$
(i) using Definition 1
(b) Find an equation of the tangent line in part (a).
(c) Graph the park, zoom in toward the point $(1,3)$ until the
parabola and the tangent line are indistinguishable.

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Problem 2

(a) Find the slope of the tangent line to the curve
$y=x-x^{3}$ at the point $(1,0)$
(i) using Definition 1
(b) Find an equation of the tangent line in part (a).
(c) Graph the curve and the tangent line in successively
smaller viewing rectangles centered at $(1,0)$ until the
curve and the line appear to coincide.

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Problem 3

Find an equation of the tangent line to the curve at the
given point.
$$
y=4 x-3 x^{2}, \quad(2,-4)
$$

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Problem 4

Find an equation of the tangent line to the curve at the
given point.
$$
y=x^{3}-3 x+1, \quad(2,3)
$$

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Problem 5

Find an equation of the tangent line to the curve at the
given point.
$$
y=\sqrt{x}, \quad(1,1)
$$

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Problem 6

Find an equation of the tangent line to the curve at the
given point.
$$
y=\frac{2 x+1}{x+2}, \quad(1,1)
$$

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Problem 7

(a) Find the slope of the tangent to the curve
$y=3+4 x^{2}-2 x^{3}$ at the point where $x=a$ .
(b) Find equations of the tangent lines at the points $(1,5)$
and $(2,3) .$
(c) Graph the curve and both tangents on a common screen.

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Problem 8

(a) Find the slope of the tangent to the curve $y=1 / \sqrt{x}$ at
the point where $x=a$ .
(b) Find equations of the tangent lines at the points $(1,1)$
and $\left(4, \frac{1}{2}\right) .$
(c) Graph the curve and both tangents on a common screen.

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Problem 9

The graph shows the position function of a car. Use the
shape of the graph to explain your answers to the following questions.
(a) What was the initial velocity of the car?
(b) Was the car going faster at $B$ or at $C ?$
(c) Was the car slowing down or speeding up at $A, B$ ,
and $C ?$
(d) What happened between $D$ and $E ?$

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Problem 10

Shown are graphs of the position functions of two runners,
A and B, who run a 100 -m race and finish in a tie.
(a) Describe and compare how the runners run the race.
(b) At what time is the distance between the runners the
greatest?
(c) At what time do they have the same velocity?

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Problem 11

If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s},$ its
height (in feet) after $t$ seconds is given by $y=40 t-16 t^{2}$ .
Find the velocity when $t=2$

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Problem 12

If an arrow is shot upward on the moon with a velocity of
$58 \mathrm{m} / \mathrm{s},$ its height (in meters) after $t$ seconds is given by
$H=58 t-0.83 t^{2}$
(a) Find the velocity of the arrow after one second.
(b) Find the velocity of the arrow when $t=a$ .
(c) When will the arrow hit the moon?
(d) With what velocity will the arrow hit the moon?

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Problem 13

The displacement (in meters) of a particle moving in a
straight line is given by the equation of motion $s=1 / t^{2}$ ,
where $t$ is measured in seconds. Find the velocity of the
particle at times $t=a, t=1, t=2,$ and $t=3 .$

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Problem 14

The displacement (in meters) of a particle moving in a
straight line is given by $s=t^{2}-8 t+18,$ where $t$ is measured in seconds.
(a) Find the average velocity over each time interval:
$\begin{array}{ll}{\text { (i) }[3,4]} & {\text { (ii) }[3.5,4]} \\ {\text { (iii) }[4,5]} & {\text { (iv) }[4,4.5]}\end{array}$
(b) Find the instantancous velocity when $t=4$ .
(c) Draw the graph of $s$ as a function of $t$ and draw the
secant lines whose slopes are the average velocities in
part (a) and the tangent line whose slope is the instantaneous velocity in part (b).

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Problem 15

For the function $g$ whose graph is given, arrange the
following numbers in increasing order and explain your
reasoning:
$$
0 \quad g^{\prime}(-2) \quad g^{\prime}(0) \quad g^{\prime}(2) \quad g^{\prime}(4)
$$

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Problem 16

Find an equation of the tangent line to the graph of
$y=g(x)$ at $x=5$ if $g(5)=-3$ and $g^{\prime}(5)=4$ .

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Problem 17

If an equation of the tangent line to the curve $y=f(x)$ at
the point where $a=2$ is $y=4 x-5,$ find $f(2)$ and $f^{\prime}(2)$

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Problem 18

If the tangent line to $y=f(x)$ at $(4,3)$ passes through the
point $(0,2),$ find $f(4)$ and $f^{\prime}(4)$

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Problem 19

Sketch the graph of a function $f$ for which $f(0)=0$
$f^{\prime}(0)=3, f^{\prime}(1)=0,$ and $f^{\prime}(2)=-1$

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Problem 20

Sketch the graph of a function $g$ for which
$$
\begin{array}{l}{g(0)=g(2)=g(4)=0, g^{\prime}(1)=g^{\prime}(3)=0} \\ {g^{\prime}(0)=g^{\prime}(4)=1, g^{\prime}(2)=-1, \lim _{x \rightarrow \infty} g(x)=\infty, \text { and }} \\ {\lim _{x \rightarrow-\infty} g(x)=-\infty}\end{array}
$$

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Problem 21

If $f(x)=3 x^{2}-x^{3},$ find $f^{\prime}(1)$ and use it to find an equation
of the tangent line to the curve $y=3 x^{2}-x^{3}$ at the point
$(1,2)$ .

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Problem 22

If $g(x)=x^{4}-2,$ find $g^{\prime}(1)$ and use it to find an cquation
of the tangent line to the curve $y=x^{4}-2$ at the point
$(1,-1)$ .

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Problem 23

(a) If $F(x)=5 x /\left(1+x^{2}\right),$ find $F^{\prime}(2)$ and use it to find an
equation of the tangent line to the curve
$$
y=\frac{5 x}{1+x^{2}}
$$
at the point $(2,2) .$
(b) Illustrate part (a) by graphing the curve and the tangent
line on the same screen.

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Problem 24

(a) If $G(x)=4 x^{2}-x^{3}$ , find $G^{\prime}(a)$ and use it to find equations of the tangent lines to the curve $y=4 x^{2}-x^{3}$ at
the points $(2,8)$ and $(3,9)$ .
(b) Illustrate part (a) by graphing the curve and the tangent
lines on the same screen.

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Problem 25

Find $$f^{\prime}(a)$$
$$
f(x)=3 x^{2}-4 x+1
$$

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Problem 26

Find $$f^{\prime}(a)$$
$$
f(t)=2 t^{3}+t
$$

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Problem 27

Find $$f^{\prime}(a)$$
$$
f(t)=\frac{2 t+1}{t+3}
$$

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Problem 28

Find $$f^{\prime}(a)$$
$$
f(x)=x^{-2}
$$

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Problem 29

Find $$f^{\prime}(a)$$
$$
f(x)=\sqrt{1-2 x}
$$

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Problem 30

Find $$f^{\prime}(a)$$
$$
f(x)=\frac{4}{\sqrt{1-x}}
$$

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Problem 31

Each limit represents the derivative of some function
$f$ at some number $a .$ State such an $f$ and $a$ in each case.
$$
\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}
$$

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Problem 32

Each limit represents the derivative of some function
$f$ at some number $a .$ State such an $f$ and $a$ in each case.
$$
\lim _{n \rightarrow 0} \frac{\sqrt[4]{16+h}-2}{h}
$$

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Problem 33

Each limit represents the derivative of some function
$f$ at some number $a .$ State such an $f$ and $a$ in each case.
$$
\lim _{x \rightarrow 5} \frac{2^{x}-32}{x-5}
$$

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Problem 34

Each limit represents the derivative of some function
$f$ at some number $a .$ State such an $f$ and $a$ in each case.
$$
\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}
$$

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Problem 35

Each limit represents the derivative of some function
$f$ at some number $a .$ State such an $f$ and $a$ in each case.
$$
\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}
$$

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Problem 36

Each limit represents the derivative of some function
$f$ at some number $a .$ State such an $f$ and $a$ in each case.
$$
\lim _{r \rightarrow 1} \frac{t^{4}+t-2}{t-1}
$$

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Problem 37

A warm can of soda is placed in a cold refrigerator. Sketch
the graph of the temperature of the soda as a function of
time. Is the initial rate of change of temperature greater or
less than the rate of change after an hour?

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Problem 38

A roast turkey is taken from an oven when its temperature
has reached $185^{\circ} \mathrm{F}$ and is placed on a table in a room
where the temperature is $75^{\circ} \mathrm{F}$ . The graph shows how the
temperature of the turkey decreases and eventually
approaches room temperature. By measuring the slope of
the tangent, estimate the rate of change of the temperature
after an hour.

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Problem 39

The number $N$ of US cellular phone subscribers (in
millions) is shown in the table. (Midyear estimates are
given.)
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline t & {1996} & {1998} & {2000} & {2002} & {2004} & {2006} \\ \hline N & {44} & {69} & {109} & {141} & {182} & {233} \\ \hline\end{array}
$$
(a) Find the average rate of cell phone growth
(i) from 2002 to 2006
(iii) from 2000 to 2002
In each case, include the units.
(b) Estimate the instantaneous rate of growth in 2002 by
taking the average of two average rates of change.
What are its units?
(c) Estimate the instantaneous rate of growth in 2002 by
measuring the slope of a tangent.

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Problem 40

The number $N$ of locations of a popular coffeehouse chain
is given in the table. (The numbers of locations as of October 1 are given.)
$$
\begin{array}{|c|c|c|c|c|c|}\hline \text { Year } & {2004} & {2005} & {2006} & {2007} & {2008} \\ \hline N & {8569} & {10,241} & {12,440} & {15,011} & {16,680} \\ \hline\end{array}
$$
(a) Find the average rate of growth
(i) from 2006 to 2008 (ii) from 2006 to 2007
(iii) from 2005 to 2006
In each case, include the units.
(b) Estimate the instantaneous rate of growth in 2006 by
taking the average of two average rates of change.
What are its units?
(c) Estimate the instantaneous rate of growth in 2006 by
measuring the slope of a tangent.
(d) Estimate the intantaneous rate of growth in 2007 and
compare it with the growth rate in $2006 .$ What do you
conclude?

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Problem 41

The cost (in dollars) of producing $x$ units of a certain commodity is $C(x)=5000+10 x+0.05 x^{2}$
(a) Find the avcrage rate of change of $C$ with respect to $x$
when the production level is changed
(i) from $x=100$ to $x=105$
(ii) from $x=100$ to $x=101$
(b) Find the instantaneous rate of change of $C$ with respect
to $x$ when $x=100$ . (This is called the marginal cost. Its
significance will be explained in Section $2.3 . )$

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Problem 42

If a cylindrical tank holds $100,000$ gallons of water, which
can be drained from the bottom of the tank in an hour, then
Torricelli's Law gives the volume $V$ of water remaining in
the tank after $t$ minutes as
$$
V(t)=100,000\left(1-\frac{1}{\infty} t\right)^{2} \quad 0 \leqq t \leqq 60
$$
Find the rate at which the water is flowing out of the tank
(the instantaneous rate of change of $V$ with respect to $t )$ as a
function of $t$ . What are its units? For times $t=0,10,20,30$ ,
$40,50,$ and 60 min, find the flow rate and the amount of
water remaining in the tank. Summarize your findings in a
sentence or two. At what time is the flow rate the greatest?
The least?

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Problem 43

The cost of producing $x$ ounces of gold from a new gold
mine is $C=f(x)$ dollars.
(a) What is the meaning of the derivative $f^{\prime}(x) ?$ What are
its units?
(b) What does the statement $f^{\prime}(800)=17$ mean?
(c) Do you think the values of $f^{\prime}(x)$ will increase or
decrease in the short term? What about the long term?
Explain.

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Problem 44

The number of bacteria after $t$ hours in a controlled laboratory experiment is $n=f(t)$ .
(a) What is the meaning of the derivative $f^{\prime}(5) ?$ What are
its units?
(b) Suppose there is an unlimited amount of space and
nutrients for the bacteria. Which do you think is larger,
$f^{\prime}(5)$ or $f^{\prime}(10) ?$ If the supply of nutrients is limited,
would that affect your conclusion? Explain.

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Problem 45

Let $T(t)$ be the temperature $\left(\mathrm{in}^{\circ} \mathrm{F}\right)$ in Phoenix $t$ hours after
midnight on September $10,2008 .$ The table shows values of
this function recorded every two hours. What is the meaning
of $T^{\prime}(8) ?$ Estimate its value.
$$
\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline t & {0} & {2} & {4} & {6} & {8} & {10} & {12} & {14} \\ \hline T & {82} & {75} & {74} & {75} & {84} & {90} & {93} & {94} \\ \hline\end{array}
$$

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Problem 46

The quantity (in pounds) of a gourmet ground coffee that is
sold by a coffee company at a price of $p$ dollars per pound
is $Q=f(p) .$
(a) What is the meaning of the derivative $f^{\prime}(8) ?$ What are
its units?
(b) Is $f^{\prime}(8)$ positive or negative? Explain.

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Problem 47

The quantity of oxygen that can dissolve in water depends
on the temperature of the water. (So thermal pollution influences the oxygen content of water.) The graph shows how
oxygen solubility $S$ varies as a function of the water temperature $T .$
(a) What is the meaning of the derivative $S^{\prime}(T) ?$ What are
its units?
(b) Estimate the value of $S^{\prime}(16)$ and interpret it.

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Problem 48

The graph shows the influence of the temperature $T$ on the
maximum sustainable swimming speed $S$ of Coho salmon.
(a) What is the meaning of the derivative $S^{\prime}(T) ?$ What are
its units?
(b) Estimate the values of $S^{\prime}(15)$ and $S^{\prime}(25)$ and interpret
them.

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Problem 49

Determine whether $f^{\prime}(0)$ exists.
$$
f(x)=\left\{\begin{array}{ll}{x \sin \frac{1}{x}} & {\text { if } x \neq 0} \\ {0} & {\text { if } x=0}\end{array}\right.
$$

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Problem 50

Determine whether $f^{\prime}(0)$ exists.
$$
f(x)=\left\{\begin{array}{ll}{x^{2} \sin \frac{1}{x}} & {\text { if } x \neq 0} \\ {0} & {\text { if } x=0}\end{array}\right.
$$

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