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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.4, Problem 46E

(a)

To determine

**To find:** The equation of the tangent line to the curve at the point.

Expert Solution

The equation of the tangent line to the curve

**Given:**

The function is

**Result used:**

**The Power Rule combined with the Chain Rule:**

If *n* is any real number and

**Quotient Rule:**

If

**Formula used:**

The equation of the tangent line at

where, *m* is the slope of the tangent line at

**Calculation:**

For *,*

The derivative of

Apply the quotient rule as shown in equation (2),

Apply the power rule combined with the chain rule as shown in equation (1),

On further simplification, the derivative of the function becomes,

Therefore, the derivative of

The slope of the tangent line at

Thus, the slope of the tangent line is

Substitute *m* in equation (1),

Therefore, the equation of the tangent line is

(b)

To determine

**To sketch:** The graph of the curve and the tangent line.

Expert Solution

**Given:**

The equation of the curve is

The equation of the tangent line is

**Graph:**

Use the online graphing calculator to draw the graph of the functions as shown below in Figure 1.

From Figure 1, it is observed that the equation of the tangent line touches on the curve