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 2.6, Problem 16E

(a)

To determine

To find: The average velocity of the particle at the given time intervals.

Expert Solution

Answer to Problem 16E

The average velocity of the particle,

(i) At the interval [4, 8] is 0 ft/s_.

(ii) At the interval [6, 8] is 1 ft/s_.

(iii) At the interval [8, 10] is 3 ft/s_.

(iv) At the interval [8, 12] is 4 ft/s_.

Explanation of Solution

Given:

The displacement (in feet) of a particle moving in a straight line is s=12t26t+23, where t is measured in seconds.

Formula used:

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

f(a+h)f(a)h (1)

Calculation:

Obtain the average velocity of the particle over the time interval [a,a+h].

Take the position function s(t)=f(t) and use equation (1) to compute the average velocity over the time interval [a,a+h] as follows.

Average velocity=f(a+h)f(a)h=(12(a+h)26(a+h)+23)(12(a)26(a)+23)h=(12(a2+h2+2ha)6a6h+23)(12a26a+23)h=12a2+12h2+ha6a6h+2312a2+6a23h

Simplify further and obtain the value of average velocity as follows.

Average velocity=12h2+ha6hh=h(12h+a6)h=(a+12h6)

Thus, the average velocity of the particle over the time interval [a,a+h] is (a+12h6) m/s_.

Section (i)

Obtain the average velocity of the particle over the time interval [4,8].

Consider a=4 and a+h=8.

h=8a=8(4)=4

Substitute a=4 and h=4 in the average velocity,

Average velocity=a+12h6=(4)+12(4)6=2+2=0

Thus, the average velocity of the particle over the time interval [4,8] is 0 ft/s.

Section (ii)

Obtain the average velocity of the particle over the time interval [6,8].

Consider a=6 and a+h=8.

h=8a=8(6)=2

Substitute a=6 and h=2 in the average velocity,

Average velocity=a+12h6=(6)+12(2)6=1

Thus, the average velocity of the particle over the time interval [6,8] is 1 ft/s.

Section (iii)

Obtain the average velocity of the particle over the time interval [8,10].

Consider a=8 and a+h=10.

h=10a=10(8)=2

Substitute a=8 and h=2 in the average velocity,

Average velocity=a+12h6=(8)+12(2)6=2+1=3

Thus, the average velocity of the particle over the time interval [8,10] is 3 ft/s  .

Section (iv)

Obtain the average velocity of the particle over the time interval [8,12].

Take a=8 and a+h=12, so that

h=12a=12(8)=4

Substitute a=8 and h=4 in the average velocity,

Average velocity=a+12h6=(8)+12(4)6=2+2=4

Thus, the average velocity of the particle over the time interval [8,12] is 4 ft/s.

(b)

To determine

To find: The instantaneous velocity at t=8 seconds.

Expert Solution

Answer to Problem 16E

The instantaneous velocity when t=8 is 2 ft/s_.

Explanation of Solution

Formula used:

The derivative v(a) is the instantaneous velocity of the particle at time t=a.

v(a)=limh0f(a+h)f(a)h (2)

Calculation:

From part (a), f(a+h)f(a)h=a+12h6.

Use equation (2) to obtain the instantaneous velocity,

v(a)=limh0f(a+h)f(a)h=limh0(a+12h6)=a+12(0)6=a6

Therefore, the instantaneous velocity at t=a is v(a)=a6.

Substitute a=8 in v(a)=a6,

v(8)=86=2

Thus, the instantaneous velocity at t=8 is 2 ft/s.

(c)

To determine

To sketch: The graph of the function s(t), secant lines and the tangent line.

Expert Solution

Explanation of Solution

Given:

The equation of the position function is s=12t26t+23.

The secant lines whose slopes are the average velocities in part (a).

The tangent line whose slope is the instantaneous velocity in part (b).

Graph:

Use the online graphing calculator to draw the graph of the function s(t), secant lines and tangent line as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.6, Problem 16E

From the Figure 1, it is observed that the tangent line touches the curve at t=8. The secant lines passes through the curve at the respected given time intervals.

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!