To compare the average rates of change of the quadratic and exponential functions over the interval [2, 6], we first need to define the functions and then calculate their respective average rates of change.
Let’s define the quadratic function as \( f(x) \) and for the sake of example, let’s assume it is:
\[ f(x) = ax^2 + bx + c \]
Now we need to calculate the average rate of change over the interval [2, 6]. The formula for the average rate of change of a function \( f \) over an interval \([a, b]\) is given by:
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
Assuming we have specific values for \( a, b, \) and \( c \), we can compute:
- Calculate \( f(6) \) and \( f(2) \).
- Plug these values into the formula above to find the average rate of change of the quadratic function.
Next, for the exponential function, given as:
\[ y = 34(2^x) \]
We again use the same formula for Average Rate of Change:
- Calculate \( 34(2^6) \) and \( 34(2^2) \).
- Use these results in the average rate of change formula.
Once we have both average rates of change, we can compare them to find which one is greater and by how much.
Here’s a step-by-step calculation example:
Step 1: Calculate the average rate of change for the quadratic function
Let’s say \( f(x) = x^2 \) (you would replace this with your specific quadratic function).
- \( f(2) = 2^2 = 4 \)
- \( f(6) = 6^2 = 36 \)
Now calculate the average rate of change:
\[ \text{Average rate of change for quadratic} = \frac{36 - 4}{6 - 2} = \frac{32}{4} = 8 \]
Step 2: Calculate the average rate of change for the exponential function
Now for \( y = 34(2^x) \):
- \( y(2) = 34(2^2) = 34(4) = 136 \)
- \( y(6) = 34(2^6) = 34(64) = 2176 \)
Now calculate the average rate of change:
\[ \text{Average rate of change for exponential} = \frac{2176 - 136}{6 - 2} = \frac{2040}{4} = 510 \]
Step 3: Compare the rates of change
Now to compare the two averages:
- Average rate of change for quadratic = 8
- Average rate of change for exponential = 510
To find the difference:
\[ 510 - 8 = 502 \]
Conclusion
The estimated average rate of change of the exponential function is 502 greater than the estimated average rate of change of the quadratic function over the interval [2, 6].
Your Responses
Based on your conclusions and calculations, select the appropriate response based on the calculated differences. None of the responses provided seem to match this calculation, which suggests that the values provided for the quadratic or a different function were incorrect. Please ensure you use the correct quadratic function while performing the calculations for accurate comparison!