Question
Calculate the average rate of change of the exponential function pictured over the given interval. Estimate the average rate of change of the quadratic function f(x)=(x−0.1)2−24.01
over the same interval. How do the two average rates of change compare?
(1 point)
Responses
The average rate of change of the exponential function is significantly greater than the average rate of change of the quadratic function.
The average rate of change of the exponential function is significantly greater than the average rate of change of the quadratic function.
The average rates of change of both functions over the interval are approximately equal.
The average rates of change of both functions over the interval are approximately equal.
The average rate of change of the quadratic function is significantly greater than the average rate of change of the exponential function.
The average rate of change of the quadratic function is significantly greater than the average rate of change of the exponential function.
These average rates of change cannot be compared.
over the same interval. How do the two average rates of change compare?
(1 point)
Responses
The average rate of change of the exponential function is significantly greater than the average rate of change of the quadratic function.
The average rate of change of the exponential function is significantly greater than the average rate of change of the quadratic function.
The average rates of change of both functions over the interval are approximately equal.
The average rates of change of both functions over the interval are approximately equal.
The average rate of change of the quadratic function is significantly greater than the average rate of change of the exponential function.
The average rate of change of the quadratic function is significantly greater than the average rate of change of the exponential function.
These average rates of change cannot be compared.
Answers
GPT-4o mini
To calculate the average rate of change of a function over an interval \([a, b]\), you can use the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Let’s denote the exponential function as \(g(x)\) and the quadratic function as \(f(x) = (x - 0.1)^2 - 24.01\).
Assuming you have the interval \([a, b]\), let's calculate the average rates of change for both functions:
### Step 1: Calculate the average rate of change of the quadratic function
1. Choose an interval, for example, \([a, b] = [x_1, x_2]\).
2. Calculate \(f(x_1)\) and \(f(x_2)\):
\[
f(x_1) = (x_1 - 0.1)^2 - 24.01
\]
\[
f(x_2) = (x_2 - 0.1)^2 - 24.01
\]
3. Apply the average rate of change formula:
\[
\text{Average Rate of Change for } f = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]
### Step 2: Calculate the average rate of change of the exponential function
1. Use the same interval \([a, b]\).
2. Calculate \(g(x_1)\) and \(g(x_2)\):
\[
g(x_1) = g(x_1) \quad \text{(you would need the specific function)}
\]
\[
g(x_2) = g(x_2) \quad \text{(you would need the specific function)}
\]
3. Apply the average rate of change formula:
\[
\text{Average Rate of Change for } g = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\]
### Comparison:
Once you have both average rates of change calculated, you will be able to compare them:
- If the average rate of change of the exponential function is greater than that of the quadratic function, then you can select the first response.
- If they are approximately equal, select the second response.
- If the quadratic's average rate of change is greater, select the third response.
Without specific values for the interval or the exponential function \(g(x)\), I cannot compute the actual rates or provide a definitive answer regarding their comparison. If you have that information, you can do the calculations as outlined.
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Let’s denote the exponential function as \(g(x)\) and the quadratic function as \(f(x) = (x - 0.1)^2 - 24.01\).
Assuming you have the interval \([a, b]\), let's calculate the average rates of change for both functions:
### Step 1: Calculate the average rate of change of the quadratic function
1. Choose an interval, for example, \([a, b] = [x_1, x_2]\).
2. Calculate \(f(x_1)\) and \(f(x_2)\):
\[
f(x_1) = (x_1 - 0.1)^2 - 24.01
\]
\[
f(x_2) = (x_2 - 0.1)^2 - 24.01
\]
3. Apply the average rate of change formula:
\[
\text{Average Rate of Change for } f = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]
### Step 2: Calculate the average rate of change of the exponential function
1. Use the same interval \([a, b]\).
2. Calculate \(g(x_1)\) and \(g(x_2)\):
\[
g(x_1) = g(x_1) \quad \text{(you would need the specific function)}
\]
\[
g(x_2) = g(x_2) \quad \text{(you would need the specific function)}
\]
3. Apply the average rate of change formula:
\[
\text{Average Rate of Change for } g = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\]
### Comparison:
Once you have both average rates of change calculated, you will be able to compare them:
- If the average rate of change of the exponential function is greater than that of the quadratic function, then you can select the first response.
- If they are approximately equal, select the second response.
- If the quadratic's average rate of change is greater, select the third response.
Without specific values for the interval or the exponential function \(g(x)\), I cannot compute the actual rates or provide a definitive answer regarding their comparison. If you have that information, you can do the calculations as outlined.