Chapter 1.4, Problem 22E

a.

To determine

To find: The slope of the curve $y=9-x^2$ at $x=a$ .

Answer to Problem 22E

  $-2a$

Explanation of Solution

Given information:

The function is $y=9-x^2$ at $x=a$ .

Concept used:

For any function f , slope at any point $x=a$ can be calculated as: $\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$ .

Calculation:

Let $f\left(x\right)=y$

For $y=9-x^2$ at $x=a$ the slope will be:

  $\begin{array}{c}\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}=\underset{h\to 0}{\lim }\frac{9-{\left(a-h\right)}^2-9+a^2}{h}\\ =\underset{h\to 0}{\lim }\frac{-a^2-h^2-2ah+a^2}{h}\\ =\underset{h\to 0}{\lim }\frac{-h^2-2ah}{h}\\ =\underset{h\to 0}{\lim }\left(-h-2a\right)\\ =-2a\end{array}$

The slope of the function is $-2a$ .

b.

To determine

To find: what happens to the slope if the value of a changes.

Answer to Problem 22E

The slope of the tangent steadily decreases as a increases.

Explanation of Solution

Given information:

The function is $y=9-x^2$ at $x=a$ .

Calculation:

From part (a) it was found that the slope of the tangent is $-2a$ .

The behavior of the slope depends on the value of a .

In an expression like $-2a$ , if a increases, $-2a$ decrease and if a decreases, $-2a$ will increase.

Conclusion:

The slope of the tangent steadily decreases as a increases.