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Math
CalculusSingle Variable Calculus: Early Transcendentals, Volume I8th Edition

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Chapter 2.1, Problem 5E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet t seconds later is given by y = 40t – 16t2.

(a) Find the average velocity for the time period beginning when t = 2 and lasting

(i) 0.5 seconds

(ii) 0.1 seconds

(iii) 0.05 seconds

(iv) 0.0 I seconds

(b) Estimate the instantaneous velocity when t = 2.

(a)

To determine

To find: The average velocity for the time period beginning when t=2 and given lasting time.

Explanation

Given:

The velocity of the ball is 40 ft/s.

The height (in feet) of the ball after t seconds is y=40t−16t2, which is also the position function.

Calculation:

Obtain the average velocity over the time interval [t, t + h].

vavg=Change in positiontime elapsed=y(t+h)−y(t)(t+h)−t=y(t+h)−y(t)h

Obtain average velocity of the position function over the interval [2, 2+h].

vavg=y(2+h)−y(2)h=[40(2+h)−16(2+h)2]−[40(2)−16(2)2]h=[80+40h−16(4+h2+4h)]−[80−64]h=80+40h−64−16h2−64h−80+64h

Simplify further and obtain the average velocity as follows:

vavg=−16h2−24hh=h(−16h−24)h=−16h−24

Thus, the average velocity of the position function y=40t−16t2 over the interval

[2, 2+h] is,

vavg=−16h−24 (1)

Section-(i):

Obtain the average velocity for the time period beginning when t=2 and lasting 0

(b).

To determine

To estimate: The instantaneous velocity when t=2.

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