Question

In Problems 43–52, find the limit as x approaches c of the average rate of change of each function from c to x.

$c=3;f\left(x\right)=x^2$.

Short answer

The answer is 6.

Step-by-step solution

Step. 1 Given Information

$$\begin{gathered} c=3; \\ f\left(x\right)=x^2. \end{gathered}$$

Step. 2 Average rate of change of f.

The average rate of change of f from 3 to x is

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(x\right)-f\left(3\right)}{x-3}=\frac{\left(x^2\right)-\left(3^2\right)}{x-3}=\frac{x^2-9}{x-3}=\frac{(x-3)(x+3)}{(x-3)} \\ =(x+3). \end{gathered}$$

Step. 3 Finding limit

The limit of the average rate of change is

$$\begin{gathered} \underset{x\to 3}{\lim }\frac{∆y}{∆x}=\underset{x\to 3}{\lim }(x+3)=3+3 \\ =6. \end{gathered}$$