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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.7, Problem 49E

(a)

To determine

**To find**: The derivative of the function

Expert Solution

The value of

**Given**:

The given function is,

**Result Used**:

The derivative of a function *f* at *,* denoted by

Difference of cube formula:

**Calculation**:

Obtain the derivative of the function

Compute

Apply the difference of cube formula in the numerator as follows,

Simplify the denominator,

Thus, the value of

**(b)**

To determine

**To Show:** The function

Expert Solution

**Result used:**

The derivative of a function *f ,* denoted by

**Proof:**

Consider the function **.**

Compute

Here, the function *h* tends to zero. That is,

Therefore, the derivative of the function does not exist at

Thus, the required proof is obtained.

**(c)**

To determine

**To show:** The

Expert Solution

**Result Used:**

A curve has a vertical tangent line at *f* is continuous at

**Proof:**

Consider the equation

Substitute

Thus

The limit of the function

Therefore,

From part (a),

Take the limit of the function *x* approaches zero.

Since the function

By result, the curve

Thus, the curve

**Graph:**

Use the online graphing calculator to draw the graph of the function

From Figure 1, it is clear that the *y*-axis is the vertical tangent to the curve