Question

Find the average rate of change for f(x) = x^2+ 3 from x = 3 to x = 4.

Answer 1

The given function is:

`f(x)\text{ = }x^2\text{ + 3}`
`x_1=3,x_2=\text{ 4}`
`\begin{gathered} f(x_1)=f(3)=3^2+3\text{ = 9 + 3 } \\ f(x_1)\text{ = f(3) = 1}2 \end{gathered}`
`\begin{gathered} f(x_2)=f(4)=4^2+3\text{ = 16 + 3 } \\ f(x_2)\text{ = f(4) = 19} \end{gathered}`

The average rate of change is given by the formula for a slope:

`\begin{gathered} (df(x))/(dx)=\text{ }(f(x_2)-f(x_1))/(x_2-x_1) \\ (df(x))/(dx)=\text{ }(f(4)-f(3))/(4-3) \\ (df(x))/(dx)=\text{ }(19-12)/(4-3) \\ (df(x))/(dx)=\text{ }(7)/(1) \\ (df(x))/(dx)=\text{ 7} \end{gathered}`

The average rate of change = 7 units