Calculus: Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillet

Chapter 2

Limits - all with Video Answers

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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 $t$ as $0 .$ Describe a process for finding the instantaneous velocity at $t=a$

Bobby Barnes

University of North Texas

Problem 3

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]

George Stanisic

George Stanisic

Numerade Educator

Problem 4

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]

Juan Pablo Olloqui

Juan Pablo Olloqui

Numerade Educator

Problem 4

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]

George Stanisic

George Stanisic

Numerade Educator

Problem 5

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]
$$\begin{array}{|l|l|l|l|l|l|} \hline t & 0 & 0.5 & 1 & 1.5 & 2 \\ \hline s(t) & 0 & 30 & 52 & 66 & 72 \\ \hline \end{array}$$

George Stanisic

George Stanisic

Numerade Educator

Problem 6

The graph gives the position $s(t)$ of an object moving along a line at time $t,$ over a 2.5 -second interval. Find the average velocity of the object over the following intervals.
$\begin{array}{rlr}\text { a. }[0.5,2.5] & \text { b. }[0.5,2] & \text { c. }[0.5,1.5]\end{array}$ d. [0.5,1]
(graph cannot copy)

George Stanisic

George Stanisic

Numerade Educator

Problem 7

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 8

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 instantancous velocity at $t=2$
$$\begin{array}{|l|l|l|l|l|} \hline t & 2 & 2.0001 & 2.001 & 2.01 \\ \hline s(t) & 56 & 55.99959984 & 55.995984 & 55.9584 \\ \hline \end{array}$$

George Stanisic

George Stanisic

Numerade Educator

Problem 9

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

Nick Johnson

Numerade Educator

Problem 10

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

Khushbu Rani

Numerade Educator

Problem 11

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 12

Given the function $f(x)=-16 x^{2}+64 x,$ complete the following.
a. Find the slopes of the secant lines that pass though the points $(x, f(x))$ and $(2, f(2)),$ for $x=1.5,1.9,1.99,1.999,$ and
$1.9999(\text { see figure })$
b. Make a conjecture about the value of the limit of the slopes of the secant lines that pass through $(x, f(x))$ and $(2, f(2))$ ) as
$x$ approaches 2
c. What is the relationship between your answer to part (b) and the slope of the line tangent to the curve at $x=2$ (see figure)?
(FIGURE CANNOT COPY)

George Stanisic

George Stanisic

Numerade Educator

Problem 13

Average velocity 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

Nick Johnson

Numerade Educator

Problem 14

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

Nick Johnson

Numerade Educator

Problem 15

Average velocity 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.

George Stanisic

George Stanisic

Numerade Educator

Problem 16

Average velocity 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 the 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.

George Stanisic

George Stanisic

Numerade Educator

Problem 17

Instantaneous velocity Consider the position function $\left.s(t)=-16 t^{2}+128 t \text { (Exercise } 13\right) .$ Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=1$

George Stanisic

George Stanisic

Numerade Educator

Problem 18

Instantancous velocity Consider the position function $\left.s(t)=-4.9 t^{2}+30 t+20 \text { (Exercise } 14\right) .$ Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=2$
$$\begin{array}{|l|l|l|l|l|l|} \hline \begin{array}{l} \text { Time } \\ \text { interval } \end{array} & [2,3] & [2,2.5] & [2,2.1] & [2,2.01] & [2,2.001] \\ \hline \begin{array}{l} \text { Average } \\ \text { velocity } \end{array} & & & & \\ \hline \end{array}$$

George Stanisic

George Stanisic

Numerade Educator

Problem 19

Instantancous velocity 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 instantancous velocity at $t=3$
$$\begin{array}{|l|l|} \hline \text { Time interval } & \text { Areraze velocity } \\ \hline[2,3] & \\ \hline[2.9,3] & \\ \hline[2.99,3] & \\ \hline[2.999,3] & \\ \hline[2.9999,3] & \\ \hline \end{array}$$

George Stanisic

George Stanisic

Numerade Educator

Problem 20

Instantancous velocity 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 $t=\pi / 2$
$$\begin{array}{|l|l|}
\hline \text { Time interval } & \text { Average velocity } \\
\hline[\pi / 2, \pi] & \\
\hline[\pi / 2, \pi / 2+0.1] & \\
\hline[\pi / 2, \pi / 2+0.01] & \\
\hline[\pi / 2, \pi / 2+0.001] & \\
\hline[\pi / 2, \pi / 2+0.0001] & \\
\hline
\end{array}$$

George Stanisic

George Stanisic

Numerade Educator

Problem 21

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

George Stanisic

George Stanisic

Numerade Educator

Problem 22

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

George Stanisic

George Stanisic

Numerade Educator

Problem 23

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

Drew Scalzo

Numerade Educator

Problem 24

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

George Stanisic

George Stanisic

Numerade Educator

Problem 25

Slopes of tangent lines 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.
$$f(x)=2 x^{2} \text { at } x=2$$

George Stanisic

George Stanisic

Numerade Educator

Problem 26

Slopes of tangent lines 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.
$$f(x)=3 \cos x \text { at } x=\pi / 2$$

George Stanisic

George Stanisic

Numerade Educator

Problem 27

Slopes of tangent lines 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.
$$f(x)=e^{x} \text { at } x=0$$

George Stanisic

George Stanisic

Numerade Educator

Problem 28

Slopes of tangent lines 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.
$$f(x)=x^{3}-x \text { at } x=1$$

George Stanisic

George Stanisic

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.

George Stanisic

George Stanisic

Numerade Educator

Problem 30

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 $h \neq 0$ ?

George Stanisic

George Stanisic

Numerade Educator

Problem 31

Zero velocity A projectile is fired vertically upward and has a position given by $s(t)=-16 t^{2}+128 t+192,$ for $0 \leq t \leq 9$
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 instantancous 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 upwand)?
e. For what values of $t$ on the interval [0,9] is the instantancous velocity negative (the projectile moves downward)?

George Stanisic

George Stanisic

Numerade Educator

Problem 32

Impact speed 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.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 33

Slope of tangent line 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. Use your calculations to make a conjecture about the slope of the line tangent to the graph of $f$ at $x=\pi / 2$
(graph cannot copy)

George Stanisic

George Stanisic

Numerade Educator