Question

Numbers 67,68,and 69 i need a break down of it please im struggling

Answer 1

The given interval is 0 ≤ x ≤ 2. Thus, the two endpoints of the interval are 0 and 2.

Now, the average rate of change of a function over the interval a ≤ x ≤ b is given by the following expression.

$\text{ Average rate of change }=\frac{f(b)-f(a)}{b-a}$

In this case, we have:

$\begin{gathered} a=0 \\ b=2 \end{gathered}$

Let us find f(b) and f(a).

$\begin{gathered} x=0 \\ y=x^2+1 \\ y=0^2+1 \\ y=1 \end{gathered}$$\begin{gathered} x=2 \\ y=x^2+1 \\ y=2^2+1 \\ y=4+1 \\ y=5 \end{gathered}$

Now we can apply the formula to calculate the average rate of change.

$\begin{gathered} a=0\Rightarrow f(a)=1 \\ b=2\Rightarrow f(b)=5 \end{gathered}$$\begin{gathered} \text{ Average rate of change }=\frac{f(b)-f(a)}{b-a} \\ \text{ Average rate of change }=\frac{5-1}{2-0} \\ \text{ Average rate of change }=\frac{4}{2} \\ \text{ Average rate of change }=2 \end{gathered}$

Therefore, the average rate of change of the function over the given interval is 2.