Chapter 11.3, Problem 14E

To determine

To calculate:

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

  $h\left(x\right)=5x+9,\left(-1,4\right)$

Answer to Problem 14E

The slope of graph function $h\left(x\right)=5x+9$ is $m=5$

Explanation of Solution

Given information:

  $\begin{array}{l}h\left(x\right)=5x+9\\ \left(-1,4\right)\end{array}$

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:

  $\begin{array}{l}m=\frac{h\left(x+h\right)-h\left(x\right)}{h}\\ m=\frac{5\left(x+h\right)+9-5x-9}{h}\\ m=\frac{5x+5h+9-5x-9}{h}\\ m=\frac{5h}{h}\\ m=5\end{array}$

Take the limit of slope of graph function:

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{h\left(x+h\right)-h\left(x\right)}{h}\\ m=\underset{h\to 0}{\lim }\left(5\right)\\ m=5\end{array}$

Hence the slope of graph at the point $\left(-1,4\right)$ is $m=5$ .

Now, verified the slope result by using the following graph of function: $h\left(x\right)=5x+9$