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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.1, Problem 74E

To determine

**To sketch:** The parabolas

Expert Solution

**Derivative rules:**

(1) Power Rule:

(2) Constant multiple rule:

(3) Sum rule:

(4) Difference rule:

**Result used:**

The equation of the tangent line at

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

**Graph:**

The graph of two parabolas

From Figure 1, it is observed that there may be a line that is tangent to both the parabolas.

It is required to find the equation of the tangent line to the parabolas.

**Calculation:**

Consider the parabolas

Choose the point *P**Q*

Suppose the slope of the required tangent line passes through the points *P**Q*

The derivative of parabola

Apply the power rule (1) and simplify the terms,

Thus, the derivative of *x*.

Therefore, the slope of the tangent to *a*. (3)

The derivative of parabola

Apply the derivative rules (1), (2), (3) and (4),

Thus, the derivative of

Therefore, the slope of the tangent to

Since the required equation of the tangent is linear from

From equations (2) and (3),

From equations (3) and (4),

Substitute

Add 2 on both sides and obtain the value of *b.*

Substitute the value

For *a* is 1 and the point *P*

Substitute *m* in equation (1),

Therefore, the equation of the tangent line to the parabolas is