Chapter 11, Problem 49CR

To determine

To evaluate: the formula for the slope of the graph of f at the point (x, f(x)) and then use it to find the slopes at the two specified points.

Answer to Problem 49CR

  $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=3-2x$

Explanation of Solution

Given:

  $f\left(x\right)=3x-x^2$

To find the slope of the graph, using limit definition of the derivative,

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

So,

  $\begin{array}{c}f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{3\left(x+h\right)-{\left(x+h\right)}^2-\left(3x-x^2\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{3x+3h-x^2-2xh-h^2-3x+x^2}{h}\\ =\underset{h\to 0}{\lim }\frac{3h-2xh-h^2}{h}\\ =\underset{h\to 0}{\lim }3-2x-h\\ =3-2x\end{array}$

So,

  $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=3-2x$