The average velocity of the particle at the given time intervals.

Single Variable Calculus: Concepts...

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

Single Variable Calculus: Concepts...

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

Solutions

Chapter 2.6, Problem 16E

(a)

To determine

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

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

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.

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.

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