Start your trial now! First week only $4.99!*arrow_forward*

BuyFind*launch*

4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.4, Problem 50E

To determine

**To find:** The point on which the curve is the tangent line perpendicular to the given line.

Expert Solution

The point on which the curve is the tangent line perpendicular to the given line is (4, 3).

**Given:**

The equation of the curve

The equation of the line

**Result used: Chain Rule**

If *h* is differentiable at *x* and *g* is differentiable at *x* and

**Derivative rules:**

(1) Constant Multiple Rule:

(2) Power Rule:

**Calculation:**

The derivative of

Let

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

The derivative of

Apply the power rule (2),

Substitute

Thus, the derivative

The derivative of

Apply the constant multiple rule (1),

Thus, the derivative

Substitute

Therefore, the derivative of the curve *y* is

Obtain the slope of the line

Rewrite the line equation as slope intercept form.

Therefore, the slope of the line is

Obtain the point on the curve is if slope of tangent line is perpendicular to slope of the line

Note that, if two lines are perpendicular with slopes are

The required slope

That is,

Since slope of the tangent line is

Take square on both sides,

Substitute

Thus, the required point is

Therefore, the point on which the curve is the tangent line perpendicular to the given line is (4, 3).