Question

What is the rate of change for f(x)=-2cos4x-3 on the interval from x=0 to x=pi over 4

Answer 1

`(16)/(\pi)`

Explanation:

The average rate of a function
`f` on interval
`x=a` to
`x=b` is:

`(f(b)-f(a))/(b-a)`.

`a=0` and
`b=(\pi)/(4)` here for this problem where
`f(x)=-2\cos(4x)-3`.

So we will need to evaluate
`f` for
`x=0 \text{ and } b=(\pi)/(4)`.

To find
`f(0)` we will replace
`x` in
`f(x)=-2\cos(4x)-3` with
`0`:

`f(0)=-2\cos(4(0))-3`

`f(0)=-2\cos(0)-3`

`f(0)=-2(1)-3`

`f(0)=-2-3`

`f(0)=-5`

To find
`f((\pi)/(4))` we will replace
`x` in
`f(x)=-2\cos(4x)-3` with
`(\pi)/(4)`:

`f((\pi)/(4))=-2\cos(4((\pi)/(4)))-3`

`f((\pi)/(4))=-2\cos(\pi)-3`

`f((\pi)/(4))=-2(-1)-3`

`f((\pi)/(4))=2-3`

`f((\pi)/(4))=-1`

So now we need to compute the change in
`f` over the change in
`x`:

`(-1-(-5))/((\pi)/(4)-0)`

`(-1+5)/((\pi)/(4))`

Dividing by
`(\pi)/(4)` is the same as multiplying by
`(4)/(\pi)`:

`(-1+5)\cdot (4)/(\pi)`

`4 \cdot (4)/(\pi)`

`(16)/(\pi)`