Question

Given the function h(x) = -x2 - 6x +11,determine the average rate of change of thefunction over the interval -4 < x < -1.

Answer 1

The average rate of change of a function f(x) over the interval

`a\le x\le b`

is given by the expression

`\text{ Average rate of change }=(f(b)-f(a))/(b-a)`

So, in this case, you have

`\begin{gathered} a=-4 \\ b=-1 \\ h(x)=-x^2-6x+11 \\ h\mleft(a\mright)=h\mleft(-4\mright)=-(-4)^2-6(-4)+11=-16+24+11=19 \\ \text{ Then} \\ h(-4)=19 \\ h(b)=h(-1)=-(-1)^2-6(-1)+11=-1+6+11=16 \\ \text{ Then} \\ h(-1)=16 \end{gathered}`

Finally, you have

`\begin{gathered} \text{ Average rate of change }=(h(b)-h(a))/(b-a) \\ \text{ Average rate of change }=(h(-1)-h(-4))/(-1-(-4)) \\ \text{ Average rate of change }=(16-19)/(-1+4) \\ \text{ Average rate of change }=(-3)/(3) \\ \text{ Average rate of change }=-1 \end{gathered}`

Therefore, the average rate of change of the function over the given interval is -1.