Question

Given that the average rate of change fory=f(x)over the interval[0,3]is−1, the average rate of change over the interval[2,3]is 5, and the average rate of change over the interval[2,6]is 3, determine the average rate of change over the interval[0,6].

Answer 1

With the given information we can deduce:

$\dfrac{f(3)-f(0)}{3} = -1 \implies f(3)-f(0)=-3$

$\dfrac{f(3)-f(2)}{1} = 5 \implies f(3)-f(2)=5$

$\dfrac{f(6)-f(2)}{4} = 3 \implies f(6)-f(2)=12$

The average rate of change over [0,6] would be

$\dfrac{f(6)-f(0)}{6}$

We can add and subtract the same quantities in order to rewrite this quantity in terms of known quantities:

$\dfrac{f(6)-f(0)}{6} = \dfrac{(f(6)-f(2))-(f(3)-f(2))+(f(3)-f(0))}{6} = \dfrac{12-5-3}{6}=\dfrac{4}{6}=\dfrac{2}{3}$