Question
Find the average rate of change over the interval (-2,0] for the function f(z) = x²+4z+3. (1 point)
Answers
GPT-4o mini
To find the average rate of change of the function \( f(z) = z^2 + 4z + 3 \) over the interval \((-2, 0]\), we use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Here, \( a = -2 \) and \( b = 0 \).
1. First, we calculate \( f(-2) \):
\[
f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1
\]
2. Next, we calculate \( f(0) \):
\[
f(0) = 0^2 + 4(0) + 3 = 0 + 0 + 3 = 3
\]
3. Now, we substitute \( f(-2) \) and \( f(0) \) into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{3 - (-1)}{0 + 2} = \frac{3 + 1}{2} = \frac{4}{2} = 2
\]
Thus, the average rate of change of \( f(z) \) over the interval \((-2, 0]\) is \( \boxed{2} \).
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Here, \( a = -2 \) and \( b = 0 \).
1. First, we calculate \( f(-2) \):
\[
f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1
\]
2. Next, we calculate \( f(0) \):
\[
f(0) = 0^2 + 4(0) + 3 = 0 + 0 + 3 = 3
\]
3. Now, we substitute \( f(-2) \) and \( f(0) \) into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{3 - (-1)}{0 + 2} = \frac{3 + 1}{2} = \frac{4}{2} = 2
\]
Thus, the average rate of change of \( f(z) \) over the interval \((-2, 0]\) is \( \boxed{2} \).