(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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(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.

Benjamin S.

Numerade Educator

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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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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Find an equation of the tangent line to the curve at the

given point.

$$

y=\sqrt{x}, \quad(1,1)

$$

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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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(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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(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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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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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?

Benjamin S.

Numerade Educator

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 $t=2$

Andrea L.

Numerade Educator

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?

Benjamin S.

Numerade Educator

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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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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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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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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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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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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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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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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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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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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(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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(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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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}

$$

Christopher S.

Numerade Educator

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}

$$

Benjamin S.

Numerade Educator

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}

$$

Benjamin S.

Numerade Educator

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}

$$

Benjamin S.

Numerade Educator

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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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}

$$

Benjamin S.

Numerade Educator

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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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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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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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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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 $2.3 . )$

Benjamin S.

Numerade Educator

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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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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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.

Benjamin S.

Numerade Educator

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}

$$

Benjamin S.

Numerade Educator

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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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.

Samantha K.

Numerade Educator

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.

Benjamin S.

Numerade Educator

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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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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