Chapter 11.3, Problem 23E

To determine

To find: the slopes at a given points

Answer to Problem 23E

The slopes at a given points are -4 and 8

Explanation of Solution

Given:

  $f\left(x\right)=4-x^2$

  $\left(2,0\right),\left(-4,-12\right)$

Calculation:

Let the given equation be $f(x)=-x^2+4$

Formula for the slope of the graph is,

  $\begin{array}{l}\frac{-{(x+h)}^2+4-\left(x^2+4\right)}{h}\\ =\frac{-x^2-2xh-h^2-4+x^2-4}{h}\\ =\frac{2xh-h^2}{h}=\frac{A(-2x-h)}{\pi }\\ =-2x-h\end{array}$

When $h=0$

  $(-2x-h)=-2x$

Substitute the x values from the given points

  $x=2\Rightarrow m=-2(2)=-4$

  $x=-4\Rightarrow m=-2(-4)=8$

Conclusion:

Therefore, the slopes at a given points are -4 and 8