Question

average rate of change of f(x)=2x^2+3x-1 on the interval 1 is less than or equal to x and x is less than or equal to 2

Answer 1

To find the average rate of change of the function \( f(x) = 2x^2 + 3x - 1 \) on the interval \( [1, 2] \), we use the formula for the average rate of change of a function over the interval \([a, b]\):

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

In this case, \( a = 1 \) and \( b = 2 \).

First, we need to calculate \( f(1) \) and \( f(2) \):

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

2. Calculate \( f(2) \):
\[
f(2) = 2(2^2) + 3(2) - 1 = 2(4) + 6 - 1 = 8 + 6 - 1 = 13
\]

Now we can find the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(2) - f(1)}{2 - 1} = \frac{13 - 4}{2 - 1} = \frac{9}{1} = 9
\]

Thus, the average rate of change of \( f(x) \) on the interval \( [1, 2] \) is \( \boxed{9} \).