BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2.1, Problem 7E

(a)

To determine

To find: The average velocity for given time periods.

Expert Solution

Answer to Problem 7E

(i) The average velocity over the time interval [2, 4] is 29.3 ft/s.

(ii) The average velocity over the time interval [3, 4] is 32.7 ft/s.

(iii) The average velocity over the time interval [4, 5] is 45.6 ft/s.

(iv) The average velocity over the time interval [4, 6] is 48.75 ft/s.

Explanation of Solution

Formula used:

The average velocity over the time interval [t,t+h] is,

vavg=s(t+h)s(t)(t+h)t (1)

Calculation:

Section-(i)

Obtain the average velocity over the time interval [2, 4].

Substitute t=2 and t+h=4 in equation (1),

vavg=s(t+h)s(t)(t+h)t=s(4)s(2)42=s(4)s(2)2

From the given table, it is observed that,

(i) When t=2 seconds, the value of s(2)=20.6 ft.

(ii) When t=4 seconds, the value of s(4)=79.2 ft.

Thus, the average velocity over the time interval [2, 4] is computed as follows.

vavg=s(4)s(2)2=79.220.62=58.62=29.3

Therefore, the average velocity over the time interval [2, 4] is 29.3 ft/s.

Section-(ii)

Obtain the average velocity over the time interval [3, 4].

Substitute t=3 and t+h=4 in equation (1),

vavg=s(t+h)s(t)(t+h)t=s(4)s(3)43=s(4)s(3)1

From the given table, it is observed that,

(i) When t=3 seconds, the value of s(3)=46.5 ft.

(ii) When t=4 seconds, the value of s(4)=79.2.

Thus, the average velocity over the time interval [3, 4] is computed as follows.

vavg=s(4)s(3)=79.246.5=32.7

Therefore, the average velocity over the time interval [3, 4] is 32.7 ft/s.

Section-(iii)

Obtain the average velocity over the time interval [4, 5].

Substitute t=4 and t+h=5 in equation (1),

vavg=s(t+h)s(t)(t+h)t=s(5)s(4)54=s(5)s(4)1

From the table, it is observed that,

(i) When t=4 seconds, the value of s(4)=79.2 ft.

(ii) When t=5 seconds, the value of s(5)=124.8.

Thus, the average velocity over the time interval [4, 5] is computed as follows.

vavg=s(5)s(4)=124.879.2=45.6

Therefore, the average velocity over the time interval [4, 5] is 45.6 ft/s.

Section-(iv)

Obtain the average velocity over the time interval [4, 6].

Substitute t=4 and t+h=6 in equation (1),

vavg=s(t+h)s(t)(t+h)t=s(6)s(4)64=s(6)s(4)2

From the table, it is observed that,

(i) When t=4 seconds, the value of s(4)=79.2 ft.

(ii) When t=6 seconds, the value of s(6)=176.7.

Thus, the average velocity over the time interval [4, 6] is computed as follows.

vavg=s(6)s(4)2=176.779.22=97.52=48.75

Therefore, average velocity over the time interval [4, 6] is 48.75 ft/s .

(b)

To determine

To estimate: The instantaneous velocity when t=3.

Expert Solution

Answer to Problem 7E

The estimated instantaneous velocity when t=3 is 29.67 ft/s.

Explanation of Solution

Plot a curve using the points (0, 0), (1, 4.9), (2, 20.6), (3, 46.5), (4, 79.2), (5, 124.8) and (6, 176.7) as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.1, Problem 7E , additional homework tip  1

Draw the slope of the tangent line at t=3 seconds as shown below in Figure 2.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.1, Problem 7E , additional homework tip  2

The instantaneous velocity at t=3 is equal to the slope of the tangent line at (3, 46.5).

From Figure 2, the slope of the tangent line at (3, 46.5) is obtained below.

mΔRSΔST105165289329.67

The estimated instantaneous velocity when t=3 is 29.67 ft/s.

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