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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.7, Problem 18E

(a)

To determine

**To estimate:** The value of
*f.*

Expert Solution

The value of

**Given:**

The function is

**Estimation:**

Obtain

Use the online graphing calculator to zoom toward the point

The calculation of

From Figure 1, the tangent to the curve at
*x*-axis.

So,

Thus,

Obtain

Use the online graphing calculator to zoom toward the point

The calculation of

From Figure 2,the slope of the tangent is as follows,

Thus,

Obtain

Use the online graphing calculator to zoom toward the point

The calculation of

From Figure 3, the slope of tangent is,

Thus,

Obtain

Use the online graphing calculator to zoom toward the point

The calculation of

From Figure 4,the slope of tangent is

Thus,

Obtain

Use the online graphing calculator to zoom toward the point

The calculation of

From Figure 5, the slope of tangent is

Thus,

(b)

To determine

**To deduce:** The values of

Expert Solution

The values of

**Result Used:**

For any odd and even function,

**Calculation:**

The function

Thus, the function

From part (a),

Using the symmetry,

Thus,

Using the symmetry,

Thus,

Using the symmetry,

Thus,

Using the symmetry,

Thus,

(c)

To determine

**To sketch:** The graph of

Expert Solution

**Graph:**

Use the information from part a and b and sketch the graph of

From Figure 6, it is observed that the function

(d)

To determine

**To guess:** The formula of

Expert Solution

The formula of

From part a and part b,

It is observed from the above calculations that the derivative is thrice the square of the input value.

Thus, the formula for the function is,

(e)

To determine

**To prove:** The guess in part d by using the definition of derivative.

Expert Solution

**Definition used:**

The derivation of a function is given by the formula,

**Proof:**

Consider the function,

Use the definition of derivative to obtain the derivative of

Simplify the terms in numerator,

Since the limit *h* approaches zero but not equal to zero, cancel the common term *h* from both the numerator and the denominator,

So,

Thus, the guess in part d is correct.