University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 2

Limits and Continuity

Educators


Problem 1

Find the average rate of change of the function over the given interval or intervals.
$f(x)=x^{3}+1$
a. [2,3]
b. [-1,1]

Rebecca P.
Numerade Educator

Problem 2

Find the average rate of change of the function over the given interval or intervals.
$g(x)=x^{2}-2 x$
a. [1,3]
b. [-2,4]

Rebecca P.
Numerade Educator

Problem 3

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

Rebecca P.
Numerade Educator

Problem 4

Find the average rate of change of the function over the given interval or intervals.
$g(t)=2+\cos t$
a. $[0, \pi]$
b. $[-\pi, \pi]$

Rebecca P.
Numerade Educator

Problem 5

Find the average rate of change of the function over the given interval or intervals.
$$R(\theta)=\sqrt{4 \theta+1} ; \quad[0,2]$$

Rebecca P.
Numerade Educator

Problem 6

Find the average rate of change of the function over the given interval or intervals.
$$P(\theta)=\theta^{3}-4 \theta^{2}+5 \theta ; \quad[1,2]$$

Rebecca P.
Numerade Educator

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

Sajin S.
Numerade Educator

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 $P$
$$y=7-x^{2}, \quad P(2,3)$$

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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

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 $P$
$$y=\frac{1}{x}, \quad P(-2,-1 / 2)$$

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Problem 16

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 $P$
$$y=\frac{x}{2-x}, \quad P(4,-2)$$

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Problem 17

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 $P$
$$y=\sqrt{x}, \quad P(4,2)$$

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Problem 18

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 $P$
$$y=\sqrt{7-x}, \quad P(-2,3)$$

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Problem 19

The accompanying figure shows the time-todistance graph for a sports car accelerating from a standstill. (FIGURE CAN'T COPY)
a. Estimate the slopes of secant lines $P Q_{1}, P Q_{2}, P Q_{3},$ 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 $t=20 \mathrm{sec}$

Sajin S.
Numerade Educator

Problem 20

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. (FIGURE CAN'T COPY)
a. Estimate the slopes of the secant lines $P Q_{1}, P Q_{2}, P Q_{3},$ 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?

Sajin S.
Numerade Educator

Problem 21

The profits of a small company for each of the first five years of its operation are given in the following table: $$\begin{array}{lc} \hline \text { Year } & \text { Profit in } \$ 1000 \mathrm{s} \\\hline 2010 & 6 \\2011 & 27 \\2012 & 62 \\2013 & 111 \\2014 & 174 \\\hline\end{array}$$
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 2012 and $2014 ?$
c. Use your graph to estimate the rate at which the profits were changing in 2012

Rebecca P.
Numerade Educator

Problem 22

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 $[1, x]$ for each $x \neq 1$ in your table.
b. Extending the table if necessary, try to determine the rate of change of $F(x)$ at $x=1$

Rebecca P.
Numerade Educator

Problem 23

Let $g(x)=\sqrt{x}$ for $x \geq 0$
a. Find the average rate of change of $g(x)$ with respect to $x$ over the 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 $x=1 ?$

Rebecca P.
Numerade Educator

Problem 24

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 $t=2 ?$

Sajin S.
Numerade Educator

Problem 25

The accompanying graph shows the total distance $s$ traveled by a bicyclist after $t$ hours. (FIGURE CAN'T COPY)
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.

Sajin S.
Numerade Educator

Problem 26

The accompanying graph shows the total amount of gasoline $A$ in the gas tank of an automobile after it has been driven for $t$ days. (FIGURE CAN'T COPY)
a. Estimate the average rate of gasoline consumption over the time intervals $[0,3],[0,5],$ and [7,10]
b. Estimate the instantaneous 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.

Sajin S.
Numerade Educator