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Calculus Early Transcendentals 2

James Stewart

Chapter 2

Limits and Derivatives - all with Video Answers

Educators

Section 1

The Tangent and Velocity Problems

01:05

Problem 1

A tank holds 1000 gallons of water, which drains from the
bottom of the tank in half an hour. The values in the table
show the volume $V$ of water remaining in the tank (in gallons)
after $t$ minutes.
(Table cant copy)
(a) If $P$ is the point $(15,250)$ on the graph of $V$, find the
slopes of the secant lines $P Q$ when $Q$ is the point on the
graph with $t=5,10,20,25,$ and $30 .$
(b) Estimate the slope of the tangent line at $P$ by averaging
the slopes of two secant lines.
(c) Use a graph of the function to estimate the slope of the
tangent line at $P$. (This slope represents the rate at which
the water is flowing from the tank after 15 minutes.)

Carson Merrill
Carson Merrill
Numerade Educator
04:57

Problem 2

A cardiac monitor is used to measure the heart rate of a patient
after surgery. It compiles the number of heartbeats after $t$ minutes. When the data in the table are graphed, the slope of the
tangent line represents the heart rate in beats per minute.

(Table cant copy)

The monitor estimates this value by calculating the slope of
a secant line. Use the data to estimate the patient's heart rate
after 42 minutes using the secant line between the points with
the given values of $t .$
$$
\text { (a) } t=36 \text { and } t=42 \quad \text { (b) } t=38 \quad \text { and } \quad t=42
$$
$$
\text { (c) } t=40 \text { and } t=42 \quad \text { (d) } t=42 \text { and } t=44
$$
What are your conclusions?

Bobby Barnes
Bobby Barnes
University of North Texas
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Problem 3

The point $P(2,-1)$ lies on the curve $y=1 /(1-x)$
(a) If $Q$ is the point $(x, 1 /(1-x))$, use your calculator to find
the slope of the secant line $P Q$ (correct to six decimal
places) for the following values of $x$ :
$$
\begin{array}{llll}{\text { (i) } 1.5} & {\text { (ii) } 1.9} & {\text { (iii) } 1.99} & {\text { (iv) } 1.999}\end{array}
$$
$$
\begin{array}{llll}{\text { (v) } 2.5} & {\text { (vi) } 2.1} & {\text { (vii) } 2.01} & {\text { (viii) } 2.001}\end{array}
$$
(b) Using the results of part (a), guess the value of the slope
of the tangent line to the curve at $P(2,-1) .$
(c) Using the slope from part (b), find an equation of the
tangent line to the curve at $P(2,-1) .$

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 4

The point $P(0.5,0)$ lies on the curve $y=\cos \pi x .$
(a) If $Q$ is the point $(x, \cos \pi x),$ use your calculator to find the
slope of the secant line $P Q$ (correct to six decimal places)
for the following values of $x$ :
$$
\begin{array}{llll}{\text { (i) } 0} & {\text { (ii) } 0.4} & {\text { (iii) } 0.49}\end{array}
$$
$$
\begin{array}{llll}{\text { (iv) } 0.499} & {\text { (v) } 1} & {\text { (vi) } 0.6}\end{array}
$$
$$
\text { (vii) } 0.51 \quad \text { (viii) } 0.501
$$
(b) Using the results of part (a), guess the value of the
slope of the tangent line to the curve at $P(0.5,0) .$
(c) Using the slope from part (b), find an equation of the
tangent line to the curve at $P(0.5,0) .$
(d) Sketch the curve, two of the secant lines, and the
tangent line.

Carson Merrill
Carson Merrill
Numerade Educator
05:40

Problem 5

If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its
height in feet $t$ seconds later is given by $y=40 t-16 t^{2}$.
(a) Find the average velocity for the time period beginning
when $t=2$ and lasting
$$
\begin{array}{ll}{\text { (i) } 0.5 \text { seconds }} & {\text { (ii) } 0.1 \text { seconds }} \\ {\text { (iii) } 0.05 \text { seconds }} & {\text { (iv) } 0.01 \text { seconds }}\end{array}
$$
(b) Estimate the instantaneous velocity when $t=2$

Wesley Hines
Wesley Hines
Numerade Educator
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Problem 6

If a rock is thrown upward on the planet Mars with a
velocity of $10 \mathrm{m} / \mathrm{s}$, its height in meters $t$ seconds later is
given by $y=10 t-1.86 t^{2}$
(a) Find the average velocity over the given time intervals:
$$
\begin{array}{ll}{\text { (i) }[1,2]} & {\text { (ii) }[1,1.5]} \\ {\text { (iii) }[1,1.1]} & {\text { (iv) }[1,1.01]} \\ {\text { (v) }[1,1.001]}\end{array}
$$
(b) Estimate the instantaneous velocity when $t=1$

Carson Merrill
Carson Merrill
Numerade Educator
05:56

Problem 7

The table shows the position of a motorcyclist after accelerating from rest.
(Table cant copy)
(a) Find the average velocity for each time period:
$$
\begin{array}{llll}{\text { (i) }[2,4]} & {\text { (ii) }[3,4]} & {\text { (iii) }[4,5]} & {\text { (iv) }[4,6]}\end{array}
$$
(b) Use the graph of $s$ as a function of $t$ to estimate the
instantaneous velocity when $t=3$

Mary Wakumoto
Mary Wakumoto
Numerade Educator
01:12

Problem 8

The displacement (in centimeters) of a particle moving
back and forth along a straight line is given by the equation
of motion $s=2 \sin \pi t+3 \cos \pi t,$ where $t$ is measured in
seconds.
(a) Find the average velocity during each time period:
$$
\begin{array}{ll}{\text { (i) }[1,2]} & {\text { (ii) }[1,1,1]} \\ {\text { (iii) }[1,1,01]} & {\text { (iv) }[1,1,001]}\end{array}
$$
(b) Estimate the instantaneous velocity of the particle
when $t=1$

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 9

The point $P(1,0)$ lies on the curve $y=\sin (10 \pi / x)$
(a) If $Q$ is the point $(x, \sin (10 \pi / x))$, find the slope of the
secant line $P Q$ (correct to four decimal places) for
$x=2,1.5,1.4,1.3,1.2,1.1,0.5,0.6,0.7,0.8,$ and 0.9
Do the slopes appear to be approaching a limit?
(b) Use a graph of the curve to explain why the slopes of
the secant lines in part (a) are not close to the slope of
the tangent line at $P .$
(c) By choosing appropriate secant lines, estimate the slope
of the tangent line at $P .$

Carson Merrill
Carson Merrill
Numerade Educator