Calculus Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillett

Chapter 2

Limits - all with Video Answers

Educators

Section 1

The idea of Limits

Problem 1

Suppose $s(t)$ is the position of an object moving along a line at time $t \geq 0 .$ What is the average velocity between the times $t=a$ and $t=b ?$

George Stanisic

George Stanisic

Numerade Educator

Problem 2

Suppose $s(t)$ is the position of an object moving along a line at time $t \geq 0 .$ Describe a process for finding the instantaneous velocity at $t=a$.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 3

What is the slope of the secant line that passes through the points $(a, f(a))$ and $(b, f(b))$ on the graph of $f ?$

George Stanisic

George Stanisic

Numerade Educator

Problem 4

Describe a process for finding the slope of the line tangent to the graph of $f$ at $(a, f(a).$

George Stanisic

George Stanisic

Numerade Educator

Problem 5

Describe the parallels between finding the instantaneous velocity of an object at a point in time and finding the slope of the line tangent to the graph of a function at a point on the graph.

George Stanisic

George Stanisic

Numerade Educator

Problem 6

Graph the parabola $f(x)=x^{2} .$ Explain why the secant lines between the points $(-a, f(-a))$ and $(a, f(a))$ have zero slope. What is the slope of the tangent line at $x=0 ?$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 7

Average velocity The function $s(t)$ represents the position of an object at time $t$ moving along a line. Suppose $s(2)=136$ and $s(3)=156 .$ Find the average velocity of the object over the interval of time $[2,3].$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 8

The function $s(t)$ represents the position of an object at time $t$ moving along a line. Suppose $s(1)=84$ and $s(4)=144 .$ Find the average velocity of the object over the interval of time $[1,4]$.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 9

The position of an object moving vertically along a line is given by the function $s(t)=-16 t^{2}+128 t .$ Find the average velocity of the object over the following intervals.
a. [1,4]
b. [1,3]
c. [1,2]
d. $[1,1+h],$ where $h>0$ is a real number

George Stanisic

George Stanisic

Numerade Educator

Problem 10

Average velocity The position of an object moving vertically along a line is given by the function $s(t)=-4.9 t^{2}+30 t+20$ Find the average velocity of the object over the following intervals.
a. [0,3]
b. [0,2]
c. [0,1]
d. $[0, h],$ where $h>0$ is a real number.

George Stanisic

George Stanisic

Numerade Educator

Problem 11

The table gives the position $s(t)$ of an object moving along a line at time $t,$ over a two-second interval. Find the average velocity of the object over the following intervals.
a. [0,2]
b. [0,1.5]
c. [0,1]
d. [0,0.5]

Carson Merrill

Numerade Educator

Problem 12

The table gives the position $s(t)$ of an object moving along a line at time $t,$ over a two-second interval. Find the average velocity of the object over the following intervals.
a. [0,2]
b. [0,1.5]
c. [0,1]
d. [0,0.5]

Carson Merrill

Numerade Educator

Problem 13

Consider the position function $s(t)=-16 t^{2}+100 t$ representing the position of an object moving vertically along a line. Sketch a graph of $s$ with the secant line passing through $(0.5, s(0.5))$ and $(2, s(2)) .$ Determine the slope of the secant line and explain its relationship to the moving object.

Carson Merrill

Numerade Educator

Problem 14

Consider the position function $s(t)=\sin \pi t$ representing the position of an object moving along a line on the end of a spring. Sketch a graph of $s$ together with a secant line passing through $(0, s(0))$ and $(0.5, s(0.5)) .$ Determine the slope of the secant line and explain its relationship to the moving object.

Carson Merrill

Numerade Educator

Problem 15

Consider the position function $s(t)=-16 t^{2}+128 t$ (Exercise 9). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=1$
$$\begin{array}{|l|l|l|l|l|l|}\hline \begin{array}{l}\text { Time } \\\text { interval }\end{array} & {[1,2]} & {[1,1.5]} & {[1,1.1]} & {[1,1.01]} & {[1,1.001]} \\\hline \begin{array}{l}\text { Average } \\\text { velocity }\end{array} & & & & & \\\hline\end{array}$$

Collin Parker

Numerade Educator

Problem 16

Consider the position function $s(t)=-4.9 t^{2}+30 t+20$ (Exercise 10). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=2$.
$$\begin{array}{|l|l|l|l|l|l|}\hline \begin{array}{l}\text { Time } \\\text { interval }\end{array} & {[2,3]} & {[2,2.5]} & {[2,2.1]} & {[2,2.01]} & {[2,2.001]} \\\hline \begin{array}{l}
\text { Average } \\\text { velocity }\end{array} & & & & & \\\hline\end{array}$$

Carson Merrill

Numerade Educator

Problem 17

The following table gives the position $s(t)$ of an object moving along a line at time $t .$ Determine the average velocities over the time intervals $[1,1.01],[1,1.001]$ and [1,1.0001] . Then make a conjecture about the value of the instantaneous velocity at $t=1.$
$$\begin{array}{|l|l|l|l|l|}\hline t & 1 & 1.0001 & 1.001 & 1.01 \\\hline s(t) & 64 & 64.00479984 & 64.047984 & 64.4784 \\\hline\end{array}$$

George Stanisic

George Stanisic

Numerade Educator

Problem 18

The following table gives the position $s(t)$ of an object moving along a line at time $t .$ Determine the average velocities over the time intervals [2,2.01],[2,2.001] and $[2,2.0001] .$ Then make a conjecture about the value of the instantaneous velocity at $t=2$
$$\begin{array}{|l|c|c|c|c|}\hline t & 2 & 2.0001 & 2.001 & 2.01 \\\hline s(t) & 56 & 55.99959984 & 55.995984 & 55.9584 \\\hline\end{array}$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 19

Consider the position function $s(t)=-16 t^{2}+100 t .$ Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=3$
$$\begin{array}{|l|l|}\hline \text { Time interval } & \text { Average velocity } \\\hline[2,3] & \\
\hline[2.9,3] & \\\hline[2.99,3] & \\\hline[2.999,3] & \\\hline[2.9999,3] & \\\hline\end{array}$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 20

Consider the position function $s(t)=3 \sin t$ that describes a block bouncing vertically on a spring. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=\pi / 2.$
$$\begin{array}{|l|l|}\hline {\text { Time interval }} & \text { Average velocity } \\\hline[\pi / 2, \pi] & \\\hline[\pi / 2, \pi / 2+0.1] & \\\hline[\pi / 2, \pi / 2+0.01] & \\\hline[\pi / 2, \pi / 2+0.001] & \\\hline[\pi / 2, \pi / 2+0.0001] & \\\hline\end{array}$$

Carson Merrill

Numerade Educator

Problem 21

For the following position functions, make a table of average velocities similar to those in Exercises $19-20$ and make a conjecture about the instantaneous velocity at the indicated time.
$$s(t)=-16 t^{2}+80 t+60 \quad \text { at } t=3$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 22

For the following position functions, make a table of average velocities similar to those in Exercises $19-20$ and make a conjecture about the instantaneous velocity at the indicated time.
$$s(t)=20 \cos t \quad \text { at } t=\pi / 2$$

Carson Merrill

Numerade Educator

Problem 23

For the following position functions, make a table of average velocities similar to those in Exercises $19-20$ and make a conjecture about the instantaneous velocity at the indicated time.
$$s(t)=40 \sin 2 t \quad \text { at } t=0$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 24

For the following position functions, make a table of average velocities similar to those in Exercises $19-20$ and make a conjecture about the instantaneous velocity at the indicated time.
$$s(t)=20 /(t+1) \quad \text { at } t=0$$

Carson Merrill

Numerade Educator

Problem 25

For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.
$$f(x)=2 x^{2} \quad \text { at } x=2$$

Carson Merrill

Numerade Educator

Problem 26

For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.
$$f(x)=3 \cos x \quad \text { at } x=\pi / 2$$

George Stanisic

George Stanisic

Numerade Educator

Problem 27

For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.
$$f(x)=e^{x} \quad \text { at } x=0$$

Carson Merrill

Numerade Educator

Problem 28

For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point.
$$f(x)=x^{3}-x \quad \text { at } x=1$$

Carson Merrill

Numerade Educator

Problem 29

Tangent lines with zero slope
a. Graph the function $f(x)=x^{2}-4 x+3$
b. Identify the point $(a, f(a))$ at which the function has a tangent line with zero slope.
c. Confirm your answer to part (b) by making a table of slopes of secant lines to approximate the slope of the tangent line at this point.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 30

Tangent lines with zero slope
a. Graph the function $f(x)=4-x^{2}$
b. Identify the point $(a, f(a))$ at which the function has a tangent line with zero slope.
c. Consider the point $(a, f(a))$ found in part (b). Is it true that the secant line between $(a-h, f(a-h))$ and $(a+h, f(a+h))$ has slope zero for any value of $h \neq 0 ?$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 31

A projectile is fired vertically upward and has a position given by $s(t)=-16 t^{2}+128 t+192,$ for $0 \leq t \leq .$
a. Graph the position function, for $0 \leq t \leq 9$
b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time $t=a$
c. Confirm your answer to part (b) by making a table of average velocities to approximate the instantaneous velocity at $t=a$
d. For what values of $t$ on the interval [0,9] is the instantaneous velocity positive (the projectile moves upward)?
e. For what values of $t$ on the interval [0,9] is the instantaneous velocity negative (the projectile moves downward)?

Carson Merrill

Numerade Educator

Problem 32

A rock is dropped off the edge of a cliff, and its distance $s$ (in feet) from the top of the cliff after $t$ seconds is $s(t)=16 t^{2} .$ Assume the distance from the top of the cliff to the ground is $96 \mathrm{ft}$
a. When will the rock strike the ground?
b. Make a table of average velocities and approximate the velocity at which the rock strikes the ground.

Carson Merrill

Numerade Educator

Problem 33

Given the function $f(x)=1-\cos x$ and the points $A(\pi / 2, f(\pi / 2)), B(\pi / 2+0.05, f(\pi / 2+0.05)).$ $C(\pi / 2+0.5, f(\pi / 2+0.5)),$ and $D(\pi, f(\pi))$ (see figure), find the slopes of the secant lines through $A$ and $D, A$ and $C,$ and $A$ and $B .$ Then use your calculations to make a conjecture about the slope of the line tangent to the graph of $f$ at $x=\pi / 2.$

Carson Merrill

Numerade Educator