Question
How do the average rates of change for the pair of functions compare over the given interval?
f(x)equals=0.70.7xsquared2
g(x)equals=2.82.8xsquared2
55less than or equals≤xless than or equals≤99
Question content area bottom
Part 1
The average rate of change of f(x) over 55less than or equals≤xless than or equals≤99 is enter your response here. The average rate of change of g(x) over 55less than or equals≤xless than or equals≤99 is enter your response here. The average rate of change of g(x) is enter your response here times that of f(x).
(Simplify your answers. Type integers or decimals.)
f(x)equals=0.70.7xsquared2
g(x)equals=2.82.8xsquared2
55less than or equals≤xless than or equals≤99
Question content area bottom
Part 1
The average rate of change of f(x) over 55less than or equals≤xless than or equals≤99 is enter your response here. The average rate of change of g(x) over 55less than or equals≤xless than or equals≤99 is enter your response here. The average rate of change of g(x) is enter your response here times that of f(x).
(Simplify your answers. Type integers or decimals.)
Answers
GPT-4o mini
To calculate the average rate of change of the functions \( f(x) = 0.7x^2 \) and \( g(x) = 2.8x^2 \) over the interval \( [5, 9] \), we can use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( [a, b] \) is the interval over which we are measuring the change.
### Step 1: Calculate \( f(5) \) and \( f(9) \)
1. For \( f(x) = 0.7x^2 \):
- \( f(5) = 0.7(5^2) = 0.7(25) = 17.5 \)
- \( f(9) = 0.7(9^2) = 0.7(81) = 56.7 \)
### Step 2: Calculate the average rate of change for \( f(x) \)
\[
\text{Average Rate of Change for } f(x) = \frac{f(9) - f(5)}{9 - 5} = \frac{56.7 - 17.5}{4} = \frac{39.2}{4} = 9.8
\]
### Step 3: Calculate \( g(5) \) and \( g(9) \)
2. For \( g(x) = 2.8x^2 \):
- \( g(5) = 2.8(5^2) = 2.8(25) = 70 \)
- \( g(9) = 2.8(9^2) = 2.8(81) = 226.8 \)
### Step 4: Calculate the average rate of change for \( g(x) \)
\[
\text{Average Rate of Change for } g(x) = \frac{g(9) - g(5)}{9 - 5} = \frac{226.8 - 70}{4} = \frac{156.8}{4} = 39.2
\]
### Step 5: Compare the average rates of change
Now to find how many times the average rate of change of \( g(x) \) is compared to that of \( f(x) \):
\[
\frac{\text{Average Rate of Change of } g(x)}{\text{Average Rate of Change of } f(x)} = \frac{39.2}{9.8} = 4
\]
### Summary of Results
1. **Average rate of change of \( f(x) \)**: \( 9.8 \)
2. **Average rate of change of \( g(x) \)**: \( 39.2 \)
3. **Average rate of change of \( g(x) \) is \( 4 \) times that of \( f(x) \)**.
To summarize, the final answers are:
- Average rate of change of \( f(x) \): **9.8**
- Average rate of change of \( g(x) \): **39.2**
- Average rate of change of \( g(x) \) is **4** times that of \( f(x)**.
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( [a, b] \) is the interval over which we are measuring the change.
### Step 1: Calculate \( f(5) \) and \( f(9) \)
1. For \( f(x) = 0.7x^2 \):
- \( f(5) = 0.7(5^2) = 0.7(25) = 17.5 \)
- \( f(9) = 0.7(9^2) = 0.7(81) = 56.7 \)
### Step 2: Calculate the average rate of change for \( f(x) \)
\[
\text{Average Rate of Change for } f(x) = \frac{f(9) - f(5)}{9 - 5} = \frac{56.7 - 17.5}{4} = \frac{39.2}{4} = 9.8
\]
### Step 3: Calculate \( g(5) \) and \( g(9) \)
2. For \( g(x) = 2.8x^2 \):
- \( g(5) = 2.8(5^2) = 2.8(25) = 70 \)
- \( g(9) = 2.8(9^2) = 2.8(81) = 226.8 \)
### Step 4: Calculate the average rate of change for \( g(x) \)
\[
\text{Average Rate of Change for } g(x) = \frac{g(9) - g(5)}{9 - 5} = \frac{226.8 - 70}{4} = \frac{156.8}{4} = 39.2
\]
### Step 5: Compare the average rates of change
Now to find how many times the average rate of change of \( g(x) \) is compared to that of \( f(x) \):
\[
\frac{\text{Average Rate of Change of } g(x)}{\text{Average Rate of Change of } f(x)} = \frac{39.2}{9.8} = 4
\]
### Summary of Results
1. **Average rate of change of \( f(x) \)**: \( 9.8 \)
2. **Average rate of change of \( g(x) \)**: \( 39.2 \)
3. **Average rate of change of \( g(x) \) is \( 4 \) times that of \( f(x) \)**.
To summarize, the final answers are:
- Average rate of change of \( f(x) \): **9.8**
- Average rate of change of \( g(x) \): **39.2**
- Average rate of change of \( g(x) \) is **4** times that of \( f(x)**.