Question

Exterminator the average rate of change of f(x)3x+2/x+1 as x changes from x=0 to x=2

Answer 1

We can see from the question that we have the following function:

`f(x)=(3x+2)/(x+1)`

And we need to find the rate of change from x = 0 to x = 2.

1. To find the average rate of change, we need to remember the formula to find it:

`\text{ Average rate of change}=\frac{\text{ change in y}}{\text{ change in x}}=(y_2-y_1)/(x_2-x_1)`

And we also have the average rate of change for a function, f(x) between x = a and x = b is given by:

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

2. Then we have that the average rate of change between x = 0 and x = 2 is as follows:

`\begin{gathered} x=0,x=2 \\ \\ \text{ Average rate of change}=(f(2)-f(0))/(2-0) \\ \end{gathered}`

3. However, we need to find the values for the function when f(2) and f(0). Then we have:

`\begin{gathered} f(x)=(3x+2)/(x+1) \\ \\ x=2\Rightarrow f(2)=(3(2)+2)/(2+1)=(6+2)/(3)=(8)/(3) \\ \\ \therefore f(2)=(8)/(3) \end{gathered}`

And we also have:

`\begin{gathered} x=0 \\ \\ f(0)=(3x+2)/(x+1)=(3(0)+2)/(0+1)=(0+2)/(1)=(2)/(1)=2 \\ \\ \therefore f(0)=2 \end{gathered}`

4. Finally, the average rate of change is given by:

`\begin{gathered} A_(rateofchange)=(f(2)-f(0))/(2-0)=((8)/(3)-2)/(2)=((8)/(3)-2)/(2)=((8)/(3)-(6)/(3))/(2)=((2)/(3))/(2)=(2)/(3)*(1)/(2)=(1)/(3) \\ \\ \therefore A_(rateofchange)=(1)/(3) \end{gathered}`

Therefore, in summary, we have that the average rate of change of the function:

`f(x)=(3x+2)/(x+1),\text{ between x = 0 to x =2 is: }(1)/(3)`