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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.1, Problem 58E

(a)

To determine

**To find:** The equation of the tangent lines to the parabola through the points.

Expert Solution

The tangent line of curve at

The tangent line of curve at

**Given:**

The parabola is

The parabola through the point is

**Derivative rules:**

(1) Power Rule:

(2) Sum Rule:

**Formula used:**

The equation of tangent line at

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

**Calculation:**

The derivative of

Apply the sum rule (2).

Apply the power rule (1) and simplify the expression.

Therefore, the derivative of the parabola

Given parabola is

Obtain the equation of the tangent line to the parabola

The slope of the tangent line at

Thus, the slope of tangent line at

Note that, the slope of the line passing through the point

The tangent line passes through the point

Therefore, the slope

Compare the equations (2) and (3),

Cross multiply the equation and simplify the terms,

Solve the above equation and obtain

Substitute the values

Since the slope of the tangent line at

Substitute *m* in equation (1),

The tangent line to the curve at

Substitute *m* in equation,

The tangent line of curve at

**The graph:**

The graph of the curve and tangent line as shown below in Figure 1.

From Figure 1, it is observed that the lines

(b)

To determine

**To show:** There is no line through the point

Expert Solution

The given parabola is

Note that, the slope of the line passing through the point

The tangent line passing through the point

Therefore, the slope of the tangent line

From part (a), the slope of tangent line at

Then, the equation becomes,

Note that, the quadratic equation

Thus, the equation

Therefore, it is concluded that there is no line such that the tangent line passing through the point

**Graph:**

The graph of the parabola

From Figure 2, it is observed that there is no tangent line to the parabola through the point