Chapter 11.3, Problem 73E

To determine

To find: whether the statement is true or false.

Answer to Problem 73E

The given statement is true.

Explanation of Solution

Given information:

Given statement

The slope of the graph of $f\left(x\right)=x^2$ is different at every point on the graph of $f$ .

Calculation:

Since, the slope of the graph of $f$ at point $\left(x,f\left(x\right)\right)$ is equal to the slope of its tangent line at $\left(x,f\left(x\right)\right)$ is:

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

Use mathematical identity ${\left(a+b\right)}^2=a^2+b^2+2ab$

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

Since the slope contains variable $x$ , So the value of slope is different at different point.

So the above statement is true.