Question

Find the average rate of change over the interval [2,5]
for the function f(x)=3x^2-4x+1
Responses

20
20

15
15

17
17

10

Answer 1

To find the average rate of change of the function \( f(x) = 3x^2 - 4x + 1 \) over the interval \([2, 5]\), we can use the formula for the average rate of change, which is given by:

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

where \( a = 2 \) and \( b = 5 \).

First, we need to calculate \( f(2) \) and \( f(5) \).

1. Calculate \( f(2) \):
\[
f(2) = 3(2)^2 - 4(2) + 1 = 3(4) - 8 + 1 = 12 - 8 + 1 = 5
\]

2. Calculate \( f(5) \):
\[
f(5) = 3(5)^2 - 4(5) + 1 = 3(25) - 20 + 1 = 75 - 20 + 1 = 56
\]

Now, we can plug these values into the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{f(5) - f(2)}{5 - 2} = \frac{56 - 5}{5 - 2} = \frac{51}{3} = 17
\]

So, the average rate of change of the function \( f(x) = 3x^2 - 4x + 1 \) over the interval \([2, 5]\) is \( 17 \).

The correct response is:

17