Chapter 11.3, Problem 12E

To determine

To find: the slope of the function at the given point

Answer to Problem 12E

The slope of the graph at $x=3$ is -2

Explanation of Solution

Given:

  $f\left(x\right)=10x-2x^2$ and $\left(3,12\right)$

Calculation:

In order to find the slope of the graph at the point $(c,f(c)),$ use the formula

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

Therefore, using the given function $g(x)$ and plugging in $x=3,$ expand the terms in the difference quotient, and then simplify, divide out, and evaluate the limit by direct substitution to obtain the slope:

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{\left[10(3+h)-2{(3+h)}^2\right]-\left[10(3)-2{(3)}^2\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{\left[30+10h-2\left(h^2+6h+9\right)\right]-30+18}{h}\\ =\underset{h\to 0}{\lim }\frac{-2h^2-2h}{h}\\ =\underset{h\to 0}{\lim }-h-2\\ =-2\left(\text{By direct substitution}\right)\end{array}$

The slope of the graph at $x=3$ is -2

Conclusion:

Therefore, the slope of the graph at $x=3$ is -2