Question
what is the average rate of change of the function f(x)=-3x^2+7x+15 over the interval -2 is greater than x and x is greater than 2?
Answers
GPT-4o mini
To find the average rate of change of the function \( f(x) = -3x^2 + 7x + 15 \) over the interval \( -2 < x < 2 \), we will calculate the values of \( f(x) \) at the endpoints of the interval and use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = -2 \) and \( b = 2 \).
1. Calculate \( f(-2) \):
\[
f(-2) = -3(-2)^2 + 7(-2) + 15
\]
\[
f(-2) = -3(4) - 14 + 15
\]
\[
f(-2) = -12 - 14 + 15
\]
\[
f(-2) = -12 - 14 + 15 = -11
\]
2. Calculate \( f(2) \):
\[
f(2) = -3(2)^2 + 7(2) + 15
\]
\[
f(2) = -3(4) + 14 + 15
\]
\[
f(2) = -12 + 14 + 15
\]
\[
f(2) = -12 + 29 = 17
\]
3. Now we can use these values to find the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{17 - (-11)}{2 + 2}
\]
\[
= \frac{17 + 11}{4} = \frac{28}{4} = 7
\]
Thus, the average rate of change of the function \( f(x) \) over the interval \( -2 < x < 2 \) is \( \boxed{7} \).
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = -2 \) and \( b = 2 \).
1. Calculate \( f(-2) \):
\[
f(-2) = -3(-2)^2 + 7(-2) + 15
\]
\[
f(-2) = -3(4) - 14 + 15
\]
\[
f(-2) = -12 - 14 + 15
\]
\[
f(-2) = -12 - 14 + 15 = -11
\]
2. Calculate \( f(2) \):
\[
f(2) = -3(2)^2 + 7(2) + 15
\]
\[
f(2) = -3(4) + 14 + 15
\]
\[
f(2) = -12 + 14 + 15
\]
\[
f(2) = -12 + 29 = 17
\]
3. Now we can use these values to find the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{17 - (-11)}{2 + 2}
\]
\[
= \frac{17 + 11}{4} = \frac{28}{4} = 7
\]
Thus, the average rate of change of the function \( f(x) \) over the interval \( -2 < x < 2 \) is \( \boxed{7} \).