Question

Find the average rate of change for f(t) =6 t^2 + 10 over the interval [3,3+h]

Answer 1

The formula for the average rate of change from point a to point b is:

$R=\frac{f(b)-f(a)}{b-a}$

In our case:

$\begin{gathered} a=3 \\ b=3+h \\ f(t)=6t^2+10 \end{gathered}$

Let's evaluate the function for a and b:

$\begin{gathered} f(a)=f(3)=6\cdot(3)^2+10=6\cdot9+10=54+10 \\ f(a)=64 \end{gathered}$$\begin{gathered} f(b)=f(3+h)=6\cdot(3+h)^2+10=6\cdot(9+6h+h^2)+10 \\ f(b)=54+36h+6h^2+10 \\ f(b)=6h^2+36h+64 \end{gathered}$

Now, putting into the rate equation:

$\begin{gathered} R=\frac{f(b)-f(a)}{b-a} \\ R=\frac{6h^2+36h+64-64}{3+h-3} \\ R=\frac{6h^2+36h}{h} \\ R=\frac{h(6h+36)}{h} \\ R=6h+36 \end{gathered}$