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Calculus for Scientists and Engineers: Early Transcendental

William Briggs, Lyle Cochran, Bernard Gillett

Chapter 2

Limits - all with Video Answers

Educators

Section 1

The Idea of limits

01:01

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 $t=b ?$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
02:28

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
01:07

Problem 3

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

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
02:00

Problem 4

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

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
02:05

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.

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
02:42

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
01:03

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
01:08

Problem 8

Average velocity 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
04:03

Problem 9

Average velocity The position of an object moving 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.
c. [1,2]
a. [1,4]
b. [1,3]
d. $[1,1+h],$ where $h>0$ is a real number

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
03:44

Problem 10

Average velocity The position of an object moving 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

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:42

Problem 11

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

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:51

Problem 12

Average velocity 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.
a. [0.5,2.5]
b. [0.5,2]
c. [0.5,1.5]
d. [0.5,1]
(FIGURE CANNOT COPY)

Nick Johnson
Nick Johnson
Numerade Educator
05:42

Problem 13

Average velocity Consider the position function $s(t)=-16 t^{2}+100 t$ representing the position of an object moving 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.

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
04:28

Problem 14

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 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.

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
03:06

Problem 15

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

Carson Merrill
Carson Merrill
Numerade Educator
03:27

Problem 16

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
04:41

Problem 17

Instantaneous velocity 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 $t=1.$
$$\begin{array}{|l|c|c|c|c|}
\hline t & 1 & 1.0001 & 1.001 & 1.01 \\
\hline s(t) & 64 & 64.00479984 & 64.047984 & 64.4784 \\
\hline
\end{array}$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:24

Problem 18

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

Carson Merrill
Carson Merrill
Numerade Educator
04:01

Problem 19

Instantaneous 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 instantaneous velocity at $t=3.$
$$\begin{array}{|l|l|}
\hline \text { Time interval } & \text { Average velocity } \\
\hline[2,3] & \\
\hline[2.9,3] & \\
\hline[2.99,3] & \\
\hline[2.999,3] & \\
\hline[2.9999,3] & \\
\hline
\end{array}$$

William Semus
William Semus
Numerade Educator
03:04

Problem 20

Instantaneous 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 $t=\pi / 2.$
$$\begin{array}{|l|l|}
\hline \multicolumn{1}{|c|} {\text { Time interval }} & \text { Average velocity } \\
\hline[\pi / 2, \pi] & \\
\hline[\pi / 2, \pi / 2+0.1] & \\
\hline[\pi / 2, \pi / 2+0.01] & \\
\hline[\pi / 2, \pi / 2+0.001] & \\
\hline[\pi / 2, \pi / 2+0.0001] & \\
\hline
\end{array}$$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
05:21

Problem 21

Instantaneous velocity 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
06:23

Problem 22

Instantaneous velocity 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.
$$s(t)=20 \cos t \quad \text { at } t=\pi / 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
04:09

Problem 23

Instantaneous velocity 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
04:45

Problem 24

Instantaneous velocity 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.
$$s(t)=20 /(t+1) \quad \text { at } t=0$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
05:09

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.
$$f(x)=2 x^{2} \quad \text { at } x=2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
05:22

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.
$$f(x)=3 \cos x \quad \text { at } x=\pi / 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
04:09

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.
$$f(x)=e^{x} \quad \text { at } x=0$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
03:49

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.
$$f(x)=x^{3}-x \quad \text { at } x=1$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
06:02

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
04:59

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
04:52

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 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)?

George Stanisic
George Stanisic
Numerade Educator
04:38

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
04:10

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 $D(\pi, f(\pi))$ (sce 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 CAN'T COPY)

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator