Chapter 11, Problem 61CR

To determine

To evaluate: the formula for the slope of the graph of f at the point ( x , f ( x )) and then use it to find the slopes at the two specified points.

Answer to Problem 61CR

The slope at the (9, 4/3) is 4/9 and the slope at the (8, 2) is 1.

Explanation of Solution

Given:

  $f\left(x\right)=\frac{4}{x-6}$

To find the slope of the graph, using limit definition of the derivative,

  $f\text{'}\left(x\right)=\frac{d}{dx}\left(f\left(x\right)\right)$

So,

  $\begin{array}{c}f\text{'}\left(x\right)=\frac{d}{dy}\left(\frac{4}{x-6}\right)\\ =\frac{4}{{\left(x-6\right)}^2}\end{array}$

At (9, 4/3), the slope,

  $\begin{array}{l}f\text{'}\left(x\right)=\frac{4}{{\left(x-6\right)}^2}\\ f\text{'}\left(9\right)=\frac{4}{{\left(9-6\right)}^2}\\ f\text{'}\left(9\right)=\frac{4}{9}\end{array}$

At (8, 2), the slope,

  $\begin{array}{l}f\text{'}\left(x\right)=\frac{4}{{\left(x-6\right)}^2}\\ f\text{'}\left(8\right)=\frac{4}{{\left(8-6\right)}^2}\\ f\text{'}\left(8\right)=\frac{4}{4}\\ f\text{'}\left(8\right)=1\end{array}$

The slope at the (9, 4/3) is 4/9 and the slope at the (8, 2) is 1.