Question
Use the formula f(b)-f(a)/b-a to calculate the average rate of change over the interval [-2, -1] given the function table for y = 3^x+2 Express your answer as a integer
The average rate of change is _
The average rate of change is _
Answers
Answered by
GPT-4o mini
To calculate the average rate of change of the function \( f(x) = 3^x + 2 \) over the interval \([-2, -1]\), we'll use the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = -2 \) and \( b = -1 \).
First, we need to find \( f(-2) \) and \( f(-1) \):
1. Calculate \( f(-2) \):
\[
f(-2) = 3^{-2} + 2 = \frac{1}{3^2} + 2 = \frac{1}{9} + 2 = \frac{1}{9} + \frac{18}{9} = \frac{19}{9}
\]
2. Calculate \( f(-1) \):
\[
f(-1) = 3^{-1} + 2 = \frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3}
\]
Now, substituting these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{\frac{7}{3} - \frac{19}{9}}{-1 + 2}
\]
The denominator simplifies to 1:
\[
\text{Average Rate of Change} = \frac{\frac{7}{3} - \frac{19}{9}}{1}
\]
To combine the fractions in the numerator \(\frac{7}{3}\) and \(\frac{19}{9}\), we need a common denominator, which is 9:
\[
\frac{7}{3} = \frac{7 \cdot 3}{3 \cdot 3} = \frac{21}{9}
\]
Now substituting this into the numerator:
\[
\text{Average Rate of Change} = \frac{\frac{21}{9} - \frac{19}{9}}{1} = \frac{21 - 19}{9} = \frac{2}{9}
\]
Since you requested the answer as an integer, the average rate of change over the interval \([-2, -1]\) is:
\(\text{Average Rate of Change} = \frac{2}{9}\), which is not an integer.
If rounded or expressed in integer form directly related to the fractions, we can recognize that the average rate of change in simplified form remains \(\frac{2}{9}\). So, if strictly needing an integer answer, it can also be stated as 0 because the value is less than 1.
To summarize, the average rate of change is:
\[
\text{Average Rate of Change} = 0 \text{ (if looking for an integer representation)}
\]
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = -2 \) and \( b = -1 \).
First, we need to find \( f(-2) \) and \( f(-1) \):
1. Calculate \( f(-2) \):
\[
f(-2) = 3^{-2} + 2 = \frac{1}{3^2} + 2 = \frac{1}{9} + 2 = \frac{1}{9} + \frac{18}{9} = \frac{19}{9}
\]
2. Calculate \( f(-1) \):
\[
f(-1) = 3^{-1} + 2 = \frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3}
\]
Now, substituting these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{\frac{7}{3} - \frac{19}{9}}{-1 + 2}
\]
The denominator simplifies to 1:
\[
\text{Average Rate of Change} = \frac{\frac{7}{3} - \frac{19}{9}}{1}
\]
To combine the fractions in the numerator \(\frac{7}{3}\) and \(\frac{19}{9}\), we need a common denominator, which is 9:
\[
\frac{7}{3} = \frac{7 \cdot 3}{3 \cdot 3} = \frac{21}{9}
\]
Now substituting this into the numerator:
\[
\text{Average Rate of Change} = \frac{\frac{21}{9} - \frac{19}{9}}{1} = \frac{21 - 19}{9} = \frac{2}{9}
\]
Since you requested the answer as an integer, the average rate of change over the interval \([-2, -1]\) is:
\(\text{Average Rate of Change} = \frac{2}{9}\), which is not an integer.
If rounded or expressed in integer form directly related to the fractions, we can recognize that the average rate of change in simplified form remains \(\frac{2}{9}\). So, if strictly needing an integer answer, it can also be stated as 0 because the value is less than 1.
To summarize, the average rate of change is:
\[
\text{Average Rate of Change} = 0 \text{ (if looking for an integer representation)}
\]