Find the average rate of change of the function over the given interval or intervals.

$$

f(x)=x^{3}+1

$$

$$

\text { a. }[2,3] \quad \text { b. }[-1,1]

$$

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Find the average rate of change of the function over the given interval or intervals.

$$

g(x)=x^{2}

$$

$$

\text { a. }[-1,1] \quad \text { b. }[-2,0]

$$

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Find the average rate of change of the function over the given interval or intervals.

$$

h(t)=\cot t

$$

$$

\text { a. }[\pi / 4,3 \pi / 4] \quad \text { b. }[\pi / 6, \pi / 2]

$$

Lu H.

Numerade Educator

Find the average rate of change of the function over the given interval or intervals.

$$

g(t)=2+\cos t

$$

$$

\text { a. }[0, \pi] \quad \text { b. }[-\pi, \pi]

$$

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Find the average rate of change of the function over the given interval or intervals.

$$

R(\theta)=\sqrt{4 \theta+1} ; \quad[0,2]

$$

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Find the average rate of change of the function over the given interval or intervals.

$$

P(\theta)=\theta^{3}-4 \theta^{2}+5 \theta ; \quad[1,2]

$$

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Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

y=x^{2}-3, \quad P(2,1)

$$

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Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

y=5-x^{2}, \quad P(1,4)

$$

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Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

y=x^{2}-2 x-3, \quad P(2,-3)

$$

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Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

y=x^{2}-4 x, \quad P(1,-3)

$$

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Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

y=x^{3}, \quad P(2,8)

$$

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Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

y=2-x^{3}, \quad P(1,1)

$$

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Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

y=x^{3}-12 x, \quad P(1,-11)

$$

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Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

y=x^{3}-3 x^{2}+4, \quad P(2,0)

$$

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Speed of a car The accompanying figure shows the time-to-distance graph for a sports car accelerating from a standstill.

a. Estimate the slopes of secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ , arranging them in order in a table like the one in Figure 2.6 . What are the appropriate units for these slopes?

b. Then estimate the car's speed at time $t=20 \mathrm{sec}$.

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The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 $\mathrm{m}$ to the surface of the moon.

a. Estimate the slopes of the secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in a table like the one in Figure 2.6 .

b. About how fast was the object going when it hit the surface?

Ahmed A.

Numerade Educator

The profits of a small company for each of the first five years of its operation are given in the following table:

a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can.

b. What is the average rate of increase of the profits between 2002 and 2004$?$

c. Use your graph to estimate the rate at which the profits were changing in $2002 .$

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Make a table of values for the function $F(x)=(x+2) /(x-2)$ at the points $x=1.2, x=11 / 10, x=101 / 100, x=1001 / 1000$ $x=10001 / 10000,$ and $x=1 .$

a. Find the average rate of change of $F(x)$ over the intervals $[1, x]$

b. Extending the table if necessary, try to determine the rate of change of $F(x)$ at $x=1 .$

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Let $g(x)=\sqrt{x}$ for $x \geq 0$

a. Find the average rate of change of $g(x)$ with respect to $x$ owerthe intervals $[1,2],[1,1.5]$ and $[1,1+h]$.

b. Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $[1,1+h]$ for some values of $h$ approaching zero, say $h=0.1,0.01,0.001,0.0001,0.00001$ and $0.000001 .$

c. What does your table indicate is the rate of change of $g(x)$ with respect to $x$ at $x=1 ?$

d. Calculate the limit as $h$ approaches zero of the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,1+h]$ .

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Let $f(t)=1 / t$ for $t \neq 0$

a. Find the average rate of change of $f$ with respect to $t$ over the intervals (i) from $t=2$ to $t=3,$ and (ii) from $t=2$ to $t=T$ .

b. Make a table of values of the average rate of change of $f$ with respect to $t$ over the interval $[2, T]$ , for some values of $T$ approaching $2,$ say $T=2.1,2.01,2.001,2.0001,2.00001,$ and $2.000001 .$

c. What does your table indicate is the rate of change of $f$ with respect to $t$ at $t=2 ?$

d. Calculate the limit as $T$ approaches 2 of the average rate of change of $f$ with respect to $t$ over the interval from 2 to $T$ . You will have to do some algebra before you can substitute $T=2$ .

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The accompanying graph shows the total distance s traveled by a bicyclist after $t$ hours.

a. Estimate the bicyclist's average speed over the time intervals $[0,1],[1,2.5],$ and $[2.5,3.5] .$

b. Estimate the bicyclist's instantaneous speed at the times $t=\frac{1}{2}$ , $t=2,$ and $t=3 .$

c. Estimate the bicyclist's maximum speed and the specific time at which it occurs.

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The accompanying graph shows the total amount of gasoline $A$ in the gas tank of an automobile after being driven for $t$ days.

a. Estimate the average rate of gasoline consumption over the time intervals $[0,3],[0,5],$ and $[7,10]$ .

b. Estimate the instantancous rate of gasoline consumption at the times $t=1, t=4,$ and $t=8$ .

c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

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