Chapter 11.3, Problem 15E

To determine

To calculate:

Slope of the following function and use the graphing utility to confirm the result:

  $g\left(x\right)=\frac{3}{x},\left(2,\frac{3}{2}\right)$

Answer to Problem 15E

Slope of the function $g\left(x\right)=\frac{3}{x}$ is $-\frac{3}{4}$ .

Explanation of Solution

Given information:

  $\begin{array}{l}g\left(x\right)=\frac{3}{x}\\ \left(2,\frac{3}{2}\right)\end{array}$

Formula used:

The slope of graph function is given by the following expression:

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

Calculation:

The given function is:

  $h\left(x\right)=5x+9$

The slope of graph function $h\left(x\right)=5x+9$ is given by the following expression:

  $\begin{array}{l}m=\frac{\frac{3}{\left(x+h\right)}-\frac{3}{x}}{h}\\ m=\frac{3x-3x-3h}{xh\left(x+h\right)}\\ m=\frac{-3h}{xh\left(x+h\right)}\\ m=\frac{-3}{x\left(x+h\right)}\end{array}$

Take the limit of slope of graph function:

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{h\left(x+h\right)-h\left(x\right)}{h}\\ m=\underset{h\to 0}{\lim }\frac{-3}{x\left(x+h\right)}\\ m=\frac{-3}{x\left(x+0\right)}\left[\because h=0\right]\\ m=\frac{-3}{x^2}\end{array}$

Hence the slope of graph at the point $\left(2,\frac{3}{2}\right)$ is $m=\frac{-3}{x^2}$ is calculated as:

  $\begin{array}{l}m=\frac{-3}{x^2}\\ m=\frac{-3}{{\left(2\right)}^2}\\ m=-\frac{3}{4}\end{array}$

Now, verified the slope result by using the following graph of function: $g\left(x\right)=\frac{3}{x}$