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Single Variable Calculus: Early Transcendentals

James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

Chapter 2

Limits and Derivatives - all with Video Answers

Educators

Section 1

The Tangent and Velocity Problems

10:40

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 can't 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 $V$ 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.)

Brian Beasley
Brian Beasley
Numerade Educator
02:10

Problem 2

A student bought a smartwatch that tracks the number of steps she walks throughout the day. The table shows the number of steps recorded $t$ minutes after 3:00 PM on the first day she wore the watch.
Table can't copy

(b) Estimate the student's walking pace, in steps per minute, at 3:20 PM by averaging the slopes of two secant lines.

James Kiss
James Kiss
Numerade Educator
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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 $(x, 1 /(1-x))$, find the slope of the secant line $P Q$ (correct to six decimal places) for the following values of $x$ :
(i) 1.5
(ii) 1.9
(iii) 1.99
(iv) 1.999
(v) 2.5
(vi) 2.1
(vii) 2.01
(viii) 2.001
(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 $(x, \cos \pi x)$, find the slope of the secant line $P Q$ (correct to six decimal places) for the following values of $x$ :
(i) 0
(ii) 0.4
(iii) 0.49
(iv) 0.499
(v) 1
(vi) 0.6
(vii) 0.51
(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
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Problem 5

The deck of a bridge is suspended 275 feet above a river. If a pebble falls off the side of the bridge, the height, in feet, of the pebble above the water surface after $t$ seconds is given by $y=275-16 t^2$.
(a) Find the average velocity of the pebble for the time period beginning when $t=4$ and lasting
(i) 0.1 seconds
(ii) 0.05 seconds
(iii) 0.01 seconds
(b) Estimate the instantaneous velocity of the pebble after 4 seconds.

Carson Merrill
Carson Merrill
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:
(i) $[1,2]$
(ii) $[1,1.5]$
(iii) $[1,1.1]$
(iv) $[1,1.01]$
(v) $[1,1.001]$
(b) Estimate the instantaneous velocity when $t=1$.

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 7

The table shows the position of a motorcyclist after accelerating from rest.
Table can't copy

(a) Find the average velocity for each time period:
(i) $[2,4]$
(ii) $[3,4]$
(iii) $[4,5]$
(iv) $[4,6]$
(b) Use the graph of $s$ as a function of $t$ to estimate the instantaneous velocity when $t=3$.

Carson Merrill
Carson Merrill
Numerade Educator
01:17

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:
(i) $[1,2]$
(ii) $[1,1.1]$
(iii) $[1,1.01]$
(iv) $[1,1.001]$
(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