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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.7, Problem 17E

(a)

To determine

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

Expert Solution

The value of

**Given:**

The function is

**Estimation:**

Obtain the value of

Use the online graphing calculator to zoom toward the point

The calculation of

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

Thus,

Obtain the value of

Use the online graphing calculator to zoom toward the point

The calculation of

From Figure 2, the slope of the tangent line to the curve is computed as follows.

Thus,

Obtain the value of

Use the online graphing calculator to zoom toward the point

The calculation of

From Figure 3, the slope of the tangent line to the curve is computed as follows.

Thus,

Obtain

Use the online graphing calculator to zoom toward the point

The calculation of

From Figure 4, the slope of the tangent line is computed as follows.

Thus,

(b)

To determine

**To deduce:** The values of

Expert Solution

The values of

**Result Used:**

For an odd function *f*,

For an even function *f*,

**Calculation:**

The function

Thus, the function

From part (a),

Using the symmetry,

Thus,

Using the symmetry,

Thus,

Using the symmetry,

Thus,

Therefore, the values of

(c)

To determine

**To guess:** The formula for

Expert Solution

The formula of

From part (a) and part (b),

From the above calculations, it is observed that the derivative of the function is twice the input value.

Thus,

Thus, the formula of

(d)

To determine

**To prove:** The answer in part (c) is correct 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 is not equal to zero, cancel the common term *h* from both the numerator and the denominator.

So,

Thus, the guess in part (c) is correct.