Question

Calculate the average rate of change over the interval 2

Answer 1

Your picture unfortunately cuts off the upper limit of the interval, so let's set it to
`b>2`. Then the average rate of change of a function
`y=f(x)` over the interval
`[2,b]` is

`(f(b)-f(2))/(b-2)`

We have

`y=3x+5\implies((3b+5)-(3\cdot2+5))/(b-2)=(3b-6)/(b-2)=(3(b-2))/(b-2)=3`

(where we can cancel the factors of
`b-2` because we assume
`b>2`)

`y=3x^2+1\implies((3b^2+1)-(3\cdot2^2+1))/(b-2)=(3b^2-12)/(b-2)=(3(b-2)(b+2))/(b-2)=3(b+2)`

`y=3^x\implies(3^b-3^2)/(b-2)=(3^b-9)/(b-2)`