Chapter 11.1, Problem 46E

Solution

To find: given statement is True or False

The given statement is True.

Given information:

The derivative of the function $f$ exists at $x=a$ .

Formula used:

The derivative of the function $f$ at $x=a$ is $f^{\prime }\left(a\right)=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$ .

Calculation:

If $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$ are two points, then the average velocity $\frac{\Delta f}{\Delta t}$ over the timeinterval $\left[a,b\right]$ can be thought of as the slopeof the line connecting the two points, and the slope is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

Note that the instantaneous velocity is the limit of $\frac{\Delta f}{\Delta t}$ and the instantaneous rate of

change of $f$ with respect to $x$ at the point where $x=a$ is called the derivative.

Thus,

  $f^{\prime }\left(a\right)=\underset{t\to 0}{\lim }\frac{f\left(a+t\right)-f\left(a\right)}{\left(a+t\right)-\left(a\right)}$

Since the derivative of $f$ exists at $x=a$ , the limits of slops $\frac{f\left(a+t\right)-f\left(a\right)}{\left(a+t\right)-\left(a\right)}$

for different values of $t$ approaches to 0 is $f^{\prime }\left(a\right)$ , which is the slope of the line passing through the point $\left(a,f\left(a\right)\right)$ . Figure (1), illustrates the existence of the tangent passing through the point $\left(a,f\left(a\right)\right)$ .

  

Figure (1)