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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.6, Problem 16E

(a)

To determine

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

Expert Solution

The average velocity of the particle,

(i) At the interval [4, 8] is

(ii) At the interval [6, 8] is

(iii) At the interval [8, 10] is

(iv) At the interval [8, 12] is

**Given:**

The displacement (in feet) of a particle moving in a straight line is *t* is measured in seconds.

**Formula used:**

The average velocity over the time interval

**Calculation:**

Obtain the average velocity of the particle over the time interval

Take the position function

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

Thus, the average velocity of the particle over the time interval

**Section (i)**

Obtain the average velocity of the particle over the time interval

Consider

Substitute

Thus, the average velocity of the particle over the time interval

**Section (ii)**

Obtain the average velocity of the particle over the time interval

Consider

Substitute

Thus, the average velocity of the particle over the time interval

**Section (iii)**

Obtain the average velocity of the particle over the time interval

Consider

Substitute

Thus, the average velocity of the particle over the time interval

**Section (iv)**

Obtain the average velocity of the particle over the time interval

Take

Substitute

Thus, the average velocity of the particle over the time interval

**(b)**

To determine

**To find:** The instantaneous velocity at

Expert Solution

The instantaneous velocity when

**Formula used:**

The derivative

**Calculation:**

From part (a),

Use equation (2) to obtain the instantaneous velocity,

Therefore, the instantaneous velocity at

Substitute

Thus, the instantaneous velocity at

**(c)**

To determine

**To sketch:** The graph of the function

Expert Solution

**Given:**

The equation of the position function is

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

From the Figure 1, it is observed that the tangent line touches the curve at