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Thomas Calculus 12

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 2

Limits and Continuity

Educators


Problem 1

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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Problem 2

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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Problem 3

Find the average rate of change of the function over the given interval or intervals.
$$
h(t)=\cot t
$$
$$
\text { a. }[\pi / 4,3 \pi / 4] \quad \text { b. }[\pi / 6, \pi / 2]
$$

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Problem 4

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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Problem 5

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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Problem 6

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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Problem 7

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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Problem 8

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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Problem 9

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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Problem 10

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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Problem 11

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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Problem 12

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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Problem 13

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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Problem 14

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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Problem 15

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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Problem 16

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

Problem 17

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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Problem 18

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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Problem 19

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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Problem 20

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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Problem 21

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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Problem 22

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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