Chapter 12, Problem 61RE

a.

To determine

To find the slope of the graph of the given function at the given point.

Answer to Problem 61RE

  $m=-4$

Explanation of Solution

Given:

  $f\left(x\right)=\frac{4}{x-6},\left(7,4\right)$

Concept used:

The formula for the slope at any point on the graph of a function $f$ is given by,

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

Calculation:

Given,

  $f\left(x\right)=\frac{4}{x-6},\left(7,4\right)$

Now the formula for the slope of the function $f$ is,

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{\left[\frac{4}{x+h-6}\right]-\left[\frac{4}{x-6}\right]}{h}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{\left[\frac{4\left(x-6\right)-4\left(x+h-6\right)}{\left(x+h-6\right)\left(x-6\right)}\right]}{h}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{4x-24-4x-4h+24}{h\left(x+h-6\right)\left(x-6\right)}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{-4h}{h\left(x+h-6\right)\left(x-6\right)}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{-4}{\left(x+h-6\right)\left(x-6\right)}\\ \Rightarrow m=\frac{-4}{\left(x+0-6\right)\left(x-6\right)}\\ \Rightarrow m=\frac{-4}{{\left(x-6\right)}^2}\end{array}$

Now substituting the point $\left(7,4\right)$ in the formula for slope, we get

  $\begin{array}{l}m=\frac{-4}{{\left(7-6\right)}^2}\\ \Rightarrow m=-4\end{array}$

b.

To determine

To find the slope of the graph of the given function at the given point.

Answer to Problem 61RE

  $m=-1$

Explanation of Solution

Given:

  $f\left(x\right)=\frac{4}{x-6},\left(8,2\right)$

Calculation:

From subpart (a), we have the formula for the slope as,

  $m=\frac{-4}{{\left(x-6\right)}^2}$

Now substituting the point $\left(8,2\right)$ in the formula for slope, we get

  $\begin{array}{l}m=\frac{-4}{{\left(8-6\right)}^2}\\ \Rightarrow m=\frac{-4}{4}\\ \Rightarrow m=-1\end{array}$