Problems 81– 88 require the following discussion of a secant line. The slope of the secant line containing the two points $(x,f(x\left)\right)$and $(x+h,f(x+h\left)\right)$on the graph of a function $y=f\left(x\right)$may be given as localid="1647347070725" $m_{sec}=\frac{f(x+h)-f\left(x\right)}{h}$, where localid="1647347075908" $h\ne 0$.

(a) Express the slope of the secant line of each function in terms of localid="1647347079379" $x$and localid="1647347084201" $h$. Be sure to simplify your answer.

(b) Find $m_{sec}$for $h=0.5,0.1,$and $0.01$at $x=1$. What value does localid="1647347087975" $m_{sec}$approach as $h$approaches $0$?

(c) Find the equation for the secant line at $x=1$with $h=0.01$.

(d) Use a graphing utility to graph localid="1647347093532" $f$and the secant line found in part (c) on the same viewing window.

$f\left(x\right)=2x+5$

Consider the function.

$f\left(x\right)=2x+5$

The slope of secant line of the given function is as follows:

$\begin{array}{rcl}{m}_{sec}& =& \frac{f(x+h)-f\left(x\right)}{h}\\ & =& \frac{2(x+h)+5-(2x+5)}{h}\\ & =& \frac{2x+2h+5-2x-5}{h}\\ & =& \frac{2h}{h}\\ & =& 2\end{array}$

From part (a), the value is $\begin{array}{rcl}{m}_{sec}& =& 2\end{array}$for all values of $h$since $\begin{array}{rcl}& & m_{sec}\end{array}$is constant in $h$.

Therefore, the value of $m_{sec}$for role="math" localid="1647175804973" $h=0.5,0.1,$and $0.01$at $x=1$are $2,2,$and $2$.

Thus, conclude that as $h$approaches to $0$, $m_{sec}$approaches $2$.

From part (b), $m_{sec}=2$at $x=1$and $h=0.01$.

The equation of a straight line with point $(x_1,y_1)$and the slope$m$is given by localid="1647245016311" $y-y_1=m(x-x_1)$.

$\begin{array}{rcl}(1,f(1\left)\right)& =& (1,2(1)+5)\\ & =& (1,2+5)\\ & =& (1,7)\end{array}$

Thus, the equation of a line with point $(1,f(1\left)\right)$and the slope $m=2$is as follows:

localid="1647175466966" $\begin{array}{rcl}y-7& =& 2(x-1)\\ y-7& =& 2x-2\\ y& =& 2x-2+7\\ y& =& 2x+5\end{array}$

The function is $f\left(x\right)=2x+5$and the secant line is $y=2x+5$.

The graph of $f$and the secant line is as follows: