Chapter 13.3, Problem 2E

To determine

To evaluate: The value of the average rate of change of function $f$ between the numbers $x$ and $a$ is $\frac{\left(\underset{\_}{}\right)-\left(\underset{\_}{}\right)}{\left(\underset{\_}{}\right)-\left(\underset{\_}{}\right)}$. The limit of average rates of change as x approaches a is the _______rate of change of y with respect to x at $x=a$: this is also the derivative $f^{\prime }\left(\underset{\_}{}\right)$.

Answer to Problem 2E

The value of average rate of change of function $f$ between the numbers $x$ and $a$ is

$\frac{f\left(x\right)-f\left(a\right)}{x-a}$ and when limit of $x$ approaches to $a$ then the rate of change of $y$ with respect to $x$ is known as instantaneous rate and it is also the derivative of the function $f$ which is represented as $f^{\prime }\left(a\right)$.

Explanation of Solution

Section1:

The average rate of change of function $f$ between the numbers $x$ and $a$ is the difference between the two functions is divided by difference between the same two numbers.

Thus, the average rate of change of function $f$ between the numbers $x$ and $a$ is represented as:

$\frac{f\left(x\right)-f\left(a\right)}{x-a}$

Section2:

The limit of the function is the nearest to number $L$ when $x$ is close to $a$.

Similarly, the limit of average rate of change of $y$ with respect to $x$ between the two numbers $x$ and $a$ is represented as:

$\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$

Thus, the limit of average rate is known as instantaneous rate.

Since, the limit of average rate of change of function between two numbers is also the derivative of the function $y=f\left(x\right)$. The derivative of the function between two numbers which is equivalent to slope of the tangent line $m$ is represented as:

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

Thus, the derivative is $f^{\prime }\left(a\right)$.