Question

Compare the estimated average rates of change of the functions f(x)=3.5x2 and g(x)=3.5x over the interval [1,2.39]

Answer 1

To compare the estimated average rates of change of the functions \( f(x) = 3.5x^2 \) and \( g(x) = 3.5x \) over the interval \([1, 2.39]\), we can calculate the average rate of change for each function using the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where \( a = 1 \) and \( b = 2.39 \).

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

1. **Calculate \( f(1) \)**:
\[
f(1) = 3.5(1^2) = 3.5
\]

2. **Calculate \( f(2.39) \)**:
\[
f(2.39) = 3.5(2.39^2) = 3.5(5.7121) \approx 19.99235
\]

3. **Calculate the average rate of change**:
\[
\text{Average Rate of Change of } f = \frac{f(2.39) - f(1)}{2.39 - 1} = \frac{19.99235 - 3.5}{2.39 - 1} = \frac{16.49235}{1.39} \approx 11.84
\]

### Step 2: Calculate the Average Rate of Change for \( g(x) \)

1. **Calculate \( g(1) \)**:
\[
g(1) = 3.5(1) = 3.5
\]

2. **Calculate \( g(2.39) \)**:
\[
g(2.39) = 3.5(2.39) \approx 8.365
\]

3. **Calculate the average rate of change**:
\[
\text{Average Rate of Change of } g = \frac{g(2.39) - g(1)}{2.39 - 1} = \frac{8.365 - 3.5}{2.39 - 1} = \frac{4.865}{1.39} \approx 3.50
\]

### Conclusion

- The average rate of change of \( f(x) \) over the interval \([1, 2.39]\) is approximately \( 11.84 \).
- The average rate of change of \( g(x) \) over the same interval is approximately \( 3.50 \).

Thus, the estimated average rate of change for \( f(x) \) is significantly higher than that of \( g(x) \) in the specified interval.