Thomas' Calculus: Early Transcendentals in SI Units

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

Chapter 2

Limits and Continuity - all with Video Answers

Educators

Section 1

Rates of Change and Tangents to Curves

Problem 1

In Exercises $1-6,$ find the average rate of change of the function over the given interval or intervals.
$$
\begin{aligned}
&f(x)=x^{3}+1\\
&\text { a. }[2,3]
\end{aligned}
$$

Carson Merrill

Carson Merrill

Numerade Educator

Problem 2

In Exercises $1-6,$ 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]

Carson Merrill

Numerade Educator

Problem 3

In Exercises $1-6,$ 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. $[\pi / 6, \pi / 2]$

Carson Merrill

Numerade Educator

Problem 4

In Exercises $1-6,$ find the average rate of change of the function over the given interval or intervals.
$g(t)=2+\cos t$
a. $[0, \pi]$
b. $[-\pi, \pi]$

Carson Merrill

Numerade Educator

Problem 5

In Exercises $1-6,$ find the average rate of change of the function over the given interval or intervals.
$$
R(\theta)=\sqrt{4 \theta+1} ; \quad[0,2]
$$

Carson Merrill

Numerade Educator

Problem 6

In Exercises $1-6,$ find the average rate of change of the function over the given interval or intervals.
$$
P(\theta)=\theta^{3}-4 \theta^{2}+5 \theta ; \quad[1,2]
$$

Carson Merrill

Numerade Educator

Problem 7

In Exercises $7-14,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathbf{b})$ an equation of the tangent line at $P .$
$$
y=x^{2}-5, \quad P(2,-1)
$$

Willis James

Numerade Educator

Problem 8

In Exercises $7-14,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathbf{b})$ an equation of the tangent line at $P .$
$$
y=7-x^{2}, \quad P(2,3)
$$

Carson Merrill

Numerade Educator

Problem 9

In Exercises $7-14,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathbf{b})$ an equation of the tangent line at $P .$
$$
y=x^{2}-2 x-3, \quad P(2,-3)
$$

Carson Merrill

Numerade Educator

Problem 10

In Exercises $7-14,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathbf{b})$ an equation of the tangent line at $P .$
$$
y=x^{2}-4 x, \quad P(1,-3)
$$

Carson Merrill

Numerade Educator

Problem 11

In Exercises $7-14,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathbf{b})$ an equation of the tangent line at $P .$
$$
y=x^{3}, \quad P(2,8)
$$

Carson Merrill

Numerade Educator

Problem 12

In Exercises $7-14,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathbf{b})$ an equation of the tangent line at $P .$
$$
y=2-x^{3}, \quad P(1,1)
$$

Carson Merrill

Numerade Educator

Problem 13

In Exercises $7-14,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathbf{b})$ an equation of the tangent line at $P .$
$$
y=x^{3}-12 x, \quad P(1,-11)
$$

Carson Merrill

Numerade Educator

Problem 14

In Exercises $7-14,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathbf{b})$ an equation of the tangent line at $P .$
$$
y=x^{3}-3 x^{2}+4, \quad P(2,0)
$$

Carson Merrill

Numerade Educator

Problem 15

Speed of a car The accompanying figure shows the time-todistance 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 $t=20 \mathrm{~s}$.

Carson Merrill

Numerade Educator

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 Abualsaud

Ahmed Abualsaud

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 2012 and $2014 ?$
c. Use your graph to estimate the rate at which the profits were changing in 2012 .

Carson Merrill

Numerade Educator

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 $[1, x]$ 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 Polasek

Rebecca Polasek

Numerade Educator

Problem 19

Let $g(x)=\sqrt{x}$ for $x \geq 0$
a. Find the average rate of change of $g(x)$ with respect to $x$ 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 $x=1 ?$
d. Calculate the limit as $h$ approaches zero of the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,1+h]$.

Mj Santos

Numerade Educator

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 $T=2$.

Carson Merrill

Numerade Educator

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.

Ef

Esraa Fathi

Numerade Educator

Problem 22

The accompanying graph shows the total amount of gasoline $A$ in the gas tank of a motorcycle 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 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.

Carson Merrill

Numerade Educator