Chapter 12.3, Problem 17E

(a)

To determine

To find: The slope of the graph of the function at the specified point.

Answer to Problem 17E

The slope of the graph of the function at the specified point is $0$ .

Explanation of Solution

Given information:

The given function is,

  $f\left(x\right)=4-x^2$

The given point is $\left(0,4\right)$ .

Concept used:

The slope of the tangent line to the graph at $\left(x,f\left(x\right)\right)$ is equal to the slope of its tangent line at $\left(x,f\left(x\right)\right)$ and is given by,

  $m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

The given function is,

  $f\left(x\right)=4-x^2$

The slope of the graph of the function at the specified point is calculated as,

  $\begin{array}{c}m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[4-{\left(x+h\right)}^2\right]-\left[4-x^2\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[4-\left(x^2+h^2+2xh\right)\right]-\left[4-x^2\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{-h^2-2xh}{h}\end{array}$

Simplify further,

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\left(-h-2x\right)\\ =-3\left(0\right)-6x\\ =-6x\end{array}$

The slope at the point $\left(0,4\right)$ is calculated as,

  $\begin{array}{c}m=-6x\\ =-6\left(0\right)\\ =0\end{array}$

Therefore, the slope of the graph of the function at the specified pointis $0$ .

(b)

To determine

To find: The slope of the graph of the function at the specified point.

Answer to Problem 17E

The slope of the graph of the function at the specified point is $6$ .

Explanation of Solution

Given information:

The given function is,

  $f\left(x\right)=4-x^2$

The given point is $\left(-1,3\right)$ .

Concept used:

The slope of the tangent line to the graph at $\left(x,f\left(x\right)\right)$ is equal to the slope of its tangent line at $\left(x,f\left(x\right)\right)$ and is given by,

  $m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

The given function is,

  $f\left(x\right)=4-x^2$

The slope of the graph of the function at the specified point is calculated as,

  $\begin{array}{c}m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[4-{\left(x+h\right)}^2\right]-\left[4-x^2\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[4-\left(x^2+h^2+2xh\right)\right]-\left[4-x^2\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{-h^2-2xh}{h}\end{array}$

Simplify further,

  $\begin{array}{c}m=\underset{h\to 0}{\lim }\left(-h-2x\right)\\ =-3\left(0\right)-6x\\ =-6x\end{array}$

The slope at the point $\left(-1,3\right)$ is calculated as,

  $\begin{array}{c}m=-6x\\ =-6\left(-1\right)\\ =6\end{array}$

Therefore, the slope of the graph of the function at the specified point is $6$ .