Question

Find the average rate of change of f(x) = √x+2/x^2 -3 from x1 = 2 to x2 = 7.

Answer 1

The formula for the average rate of change between two points on a function is:

$r=\frac{f(x_2)-f(x_1)_{}}{x_2-x_1}$

Since we want between:

$\begin{gathered} x_1=2 \\ x_2=7 \end{gathered}$

We need to first calculate the values of the funciton in these points:

$\begin{gathered} f(x_1)=f(2)=\frac{\sqrt[]{2+2}}{2^2-3}=\frac{\sqrt[]{4}}{4-3}=\frac{2}{1}=2 \\ f(x_2)=f(7)=\frac{\sqrt[]{7+2}}{7^2-3}=\frac{\sqrt[]{9}}{49-3}=\frac{3}{46} \end{gathered}$

Now, we input these into the formula:

$r=\frac{f(x_2)-f(x_1)_{}}{x_2-x_1}=\frac{\frac{3}{46}-2}{7-2}=\frac{\frac{3-2\cdot46}{46}}{5}=\frac{\frac{3-92}{46}}{5}=\frac{\frac{-89}{46}}{5}=-\frac{89}{46}\cdot\frac{1}{5}=-\frac{89}{230}$

Thus, the average rate of change is -89/230.