Question

find the average rate of change of the function over the given interval. ( r(\theta) = sqrt{5\theta + 1} ); (0, 7) (\frac{delta r}{delta \theta} = square) (simplify your answer.)

Explanation:

Answer:

To find the average rate of change of the function \( R(\theta) = \sqrt{5\theta + 1} \) over the interval \([0, 7]\), we use the formula for the average rate of change:

\[
\frac{\Delta R}{\Delta \theta} = \frac{R(7) - R(0)}{7 - 0}
\]

First, calculate \( R(7) \):

\[
R(7) = \sqrt{5(7) + 1} = \sqrt{35 + 1} = \sqrt{36} = 6
\]

Next, calculate \( R(0) \):

\[
R(0) = \sqrt{5(0) + 1} = \sqrt{0 + 1} = \sqrt{1} = 1
\]

Now, substitute these values into the average rate of change formula:

\[
\frac{\Delta R}{\Delta \theta} = \frac{6 - 1}{7 - 0} = \frac{5}{7}
\]

So the average rate of change is \(\frac{5}{7}\).