Question

Estimate the average rate of change of r(x) over the interval [2,17]. How does it compare to the estimated rate of change of the function s(x)=^3 square root x-2 over the same interval

Answer 1

The points on the graph are (2,0),(4,2),(8,3.5),(12,4.5) and (17,5.5)

Answer 2

To find the average rate of change of the function \( r(x) \) over the interval \([2, 17]\), we need the values of the function at the endpoints \( r(2) \) and \( r(17) \). The average rate of change is calculated using the formula:

\[
\text{Average Rate of Change} = \frac{r(17) - r(2)}{17 - 2}
\]

However, since you haven't provided the specific function \( r(x) \), I won't be able to compute the exact average rate of change.

Next, for the function \( s(x) = \sqrt[3]{x} - 2 \), we will also calculate the average rate of change over the interval \([2, 17]\).

1. Calculate \( s(2) \):
\[
s(2) = \sqrt[3]{2} - 2 \approx 0.7937 - 2 \approx -1.2063
\]

2. Calculate \( s(17) \):
\[
s(17) = \sqrt[3]{17} - 2 \approx 2.5714 - 2 \approx 0.5714
\]

Now, we can calculate the average rate of change for \( s(x) \) over the interval \([2, 17]\):

\[
\text{Average Rate of Change} = \frac{s(17) - s(2)}{17 - 2} = \frac{0.5714 - (-1.2063)}{15} = \frac{0.5714 + 1.2063}{15} \approx \frac{1.7777}{15} \approx 0.1185
\]

Now, comparing the average rate of change of \( r(x) \) over the interval \([2, 17]\) with the average rate of change of \( s(x) \):

- Find \( r(2) \) and \( r(17) \) to compute \( \frac{r(17) - r(2)}{15} \).
- Compare this value to approximately \( 0.1185 \).

Without \( r(x) \) being defined, I cannot provide a final comparison. If you can provide the specific function \( r(x) \), I'd be glad to help you compute that too!

Answer 3

Thank you for providing the points on the graph of \( r(x) \). We can calculate the average rate of change of \( r(x) \) over the interval \([2, 17]\) using the provided coordinates.

### Step 1: Calculate the Average Rate of Change of \( r(x) \)

Given the points:
- \( r(2) = 0 \)
- \( r(17) = 5.5 \)

The average rate of change of \( r(x) \) over the interval \([2, 17]\) is:

\[
\text{Average Rate of Change} = \frac{r(17) - r(2)}{17 - 2} = \frac{5.5 - 0}{17 - 2} = \frac{5.5}{15} \approx 0.3667
\]

### Step 2: Compare with the Average Rate of Change of \( s(x) \)

From our previous calculation, the average rate of change of \( s(x) = \sqrt[3]{x} - 2 \) over the same interval \([2, 17]\) was approximately \( 0.1185 \).

### Comparison

- Average Rate of Change of \( r(x) \): **\( \approx 0.3667 \)**
- Average Rate of Change of \( s(x) \): **\( \approx 0.1185 \)**

The average rate of change of \( r(x) \) is significantly greater than that of \( s(x) \) over the interval \([2, 17]\). This indicates that \( r(x) \) is increasing at a faster rate than \( s(x) \) over this interval.