Question

Find the average rate of change of each function on the interval specified. Put your answer in simplified factored form. You may want to do your work on paper and submit an image of your written work rather than try to type it all out. To earn full credit please show all steps and calculations. a(t)= \frac{1}{t+4} on the interval [9,9+h]

Answer 1

`a^(\prime)_{\text{avg}}(t)=(-1)/(13(h+13))`

Step-by-step explanation:

The average rate of change of a function a(t) is defined as

`a^(\prime)_{\text{avg}}(t)=(a(t+h)-a(t))/(h)`

Now in our case, we have

`a(t)=(1)/(t+4)`

therefore,

`a(9+h)=(1)/(9+h+4)=(1)/(h+13)`

and

`a(9)=(1)/(9+4)=(1)/(13)`

Hence,

`a^(\prime)_{\text{avg}}(t)=((1)/(h+13)-(1)/(13))/(h)`

which we rewrite to get

`a^(\prime)_{\text{avg}}(t)=(1)/(h)((1)/(h+13)-(1)/(13))`
`\Rightarrow(1)/(h)((13)/(13(13+h))-(h+13)/(13(h+13)))`
`=(1)/(h)((13-(h+13))/(13(h+13)))`
`=(1)/(h)((-h)/(13(h+13)))`
`=(-1)/(13(h+13))`

Hence,

`\boxed{a^(\prime)_{\text{avg}}(t)=(-1)/(13(h+13))}`

which is our answer!