Question

Find the average rate of change of f(x) = 4x^2 – 3 on theinterval [1, a]

Answer 1

Average rate of change is given by the equation

$\frac{f(x2)-f(x1)}{x2-x1}$

x1=1

x2=a

Find f(1) and f(a)

$\begin{gathered} f(1)=4\cdot(1^2)-3=4-3=1 \\ f(a)=4\cdot a^2-3=4a^2-3 \end{gathered}$

Find the average rate of change

$\frac{(4a^2-3)-1}{a-1}=\frac{4a^2-4}{a-1}$

Factor

$\frac{4a^2-4}{a-1}=\frac{4\cdot(a-1)\cdot(a+1)}{a-1}=4\cdot(a+1)=4a+4$

The average rate of change is 4a + 4 on the interval [1,a]