Chapter 12.3, Problem 12E

To determine

To find: The slope of the graph of the function at the specified point.

Answer to Problem 12E

The slope of the graph of the function at the specified point is $2$ .

Explanation of Solution

Given information:

The given function is,

  $h\left(x\right)=2x+5$

The given point is $\left(-1,3\right)$ .

Concept used:

The slope of the tangent line to the graph at $\left(x,f\left(x\right)\right)$ is equal to the slope of its tangent line at $\left(x,f\left(x\right)\right)$ and is given by,

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

The given function is,

  $h\left(x\right)=2x+5$

The slope of the graph of the function at the specified point is calculated as,

  $\begin{array}{c}m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[2\left(x+h\right)+5\right]-\left[2x+5\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[\left(2x+2h\right)+5\right]-\left[2x+5\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{2h}{h}\end{array}$

Simplify further,

  $m=2$

Consider the graph of the given function as,

  

Therefore, the slope of the graph of the function at the specified point is $2$ .