Chapter 11.3, Problem 16E

To determine

To find: the slope of the function at the given point

Answer to Problem 16E

The slope of the graph at $x=4$ is $-\frac{1}{4}$

Explanation of Solution

Given:

  $g\left(x\right)=\frac{1}{x-2}$ and $\left(4,\frac{1}{2}\right)$

Calculation:

In order to find the slope of the graph at the point $(c,f(c)),$ use the formula

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

Therefore, using the given function $g(x)$ and plugging in $x=4,$ expand the terms in the difference quotient, and then simplify, divide out, and evaluate the limit by direct substitution to obtain the slope:

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{\frac{1}{4+h-2}-\frac{1}{4-2}}{h}\\ =\underset{h\to 0}{\lim }\frac{\frac{1}{2+h}-\frac{1}{2}}{h}\times \frac{2(2+h)}{2(2+h)}\\ =\underset{h\to 0}{\lim }\frac{2-(2+h)}{2h(2+h)}\\ =\underset{h\to 0}{\lim }\frac{-h}{2h(2+h)}\\ =\underset{h\to 0}{\lim }\frac{-1}{4+2h}\\ =-\frac{1}{4}\text{ (By direct substitution}\text{.)}\end{array}$

Your graphing calculator most likely has a "Trace" feature with which you can calculate $\frac{dy}{dx}$ at a given point, which will give the slope of the graph at that point.

The slope of the graph at $x=4$ is $-\frac{1}{4}$

Conclusion:

Therefore, the slope of the graph at $x=4$ is $-\frac{1}{4}$