Question

Calculate the average rate of change of the function f(x) = 3x2 over the interval 1 ≤ x ≤ 5.
Responses
A 44
B 1818
C 3636
D 48

Answer 1

To calculate the average rate of change of the function \( f(x) = 3x^2 \) over the interval \( 1 \leq x \leq 5 \), we can use the formula:

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

where \( a = 1 \) and \( b = 5 \).

First, we need to calculate \( f(1) \) and \( f(5) \):

\[
f(1) = 3(1^2) = 3(1) = 3
\]

\[
f(5) = 3(5^2) = 3(25) = 75
\]

Now we can plug these values into our formula:

\[
\text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1} = \frac{75 - 3}{5 - 1}
\]

Calculating the difference in the numerator:

\[
75 - 3 = 72
\]

Calculating the difference in the denominator:

\[
5 - 1 = 4
\]

Now we can substitute these values back into the formula:

\[
\text{Average rate of change} = \frac{72}{4} = 18
\]

It appears that none of the answer choices reflect the average rate of change we calculated. However, upon reviewing, the calculations are correct:

1. **Calculate \( f(1) \)**:
\[ f(1) = 3(1^2) = 3 \]

2. **Calculate \( f(5) \)**:
\[ f(5) = 3(5^2) = 75 \]

3. **Determine the average rate of change**:
\[ \text{Average rate of change} = \frac{75 - 3}{5 - 1} = \frac{72}{4} = 18 \]

Since the given answer options do not include 18, it might be worth verifying the problem statement or the options given.