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:

`(f(3)-f(0))/(3) = -1 \implies f(3)-f(0)=-3`

`(f(3)-f(2))/(1) = 5 \implies f(3)-f(2)=5`

`(f(6)-f(2))/(4) = 3 \implies f(6)-f(2)=12`

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

`(f(6)-f(0))/(6)`

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

`(f(6)-f(0))/(6) = ((f(6)-f(2))-(f(3)-f(2))+(f(3)-f(0)))/(6) = (12-5-3)/(6)=(4)/(6)=(2)/(3)`