To answer these questions, we will calculate the estimated average rates of change for the functions given in the intervals specified.
Question 1
A) For the exponential function \( y = 4^x \) and the quadratic function \( y = 4x^2 \) over the interval \([0,1]\):
-
Exponential function \( y = 4^x \):
- At \( x = 0 \): \( y(0) = 4^0 = 1 \)
- At \( x = 1 \): \( y(1) = 4^1 = 4 \)
- Average rate of change = \( \frac{y(1) - y(0)}{1 - 0} = \frac{4 - 1}{1 - 0} = 3 \)
-
Quadratic function \( y = 4x^2 \):
- At \( x = 0 \): \( y(0) = 4(0^2) = 0 \)
- At \( x = 1 \): \( y(1) = 4(1^2) = 4 \)
- Average rate of change = \( \frac{y(1) - y(0)}{1 - 0} = \frac{4 - 0}{1 - 0} = 4 \)
Conclusion: The quadratic function \( y = 4x^2 \) has a greater estimated average rate of change.
Question 2
A) Compare estimated average rates of change for the quadratic function and the exponential function \( y = 6^x \) over the interval \([0,5]\).
Since we do not have the specific quadratic function or image to derive its values, a general assumption:
- Typically, the exponential function grows more rapidly than quadratic functions for larger intervals.
- Therefore, for \( x = 0 \) to \( x = 5 \):
Conclusion: The average rate of change of the quadratic function is likely less than the estimated average rate of change of the exponential function \( y = 6^x \), thus we denote it as \( < \).
Question 3
A) Compare \( y = 9(13)^x \) (exponential) and \( y = 9x^2 + 3x \) (quadratic) over the interval \([1,3]\):
-
Exponential function \( y = 9(13)^x \):
- At \( x = 1 \): \( y(1) = 9(13)^1 = 117 \)
- At \( x = 3 \): \( y(3) = 9(13)^3 = 9 \cdot 2197 = 19773 \)
- Average rate = \( \frac{19773 - 117}{3 - 1} \) is a positive value.
-
Quadratic function \( y = 9x^2 + 3x \):
- At \( x = 1 \): \( y(1) = 9(1)^2 + 3(1) = 12 \)
- At \( x = 3 \): \( y(3) = 9(3)^2 + 3(3) = 90 \)
- Average rate = \( \frac{90 - 12}{3 - 1} = \frac{78}{2} = 39 \) (positive)
Conclusion: Since both functions are increasing in that interval but change rate varies. However, the exponential function is nearly always increasing more steeply. Therefore, neither function has a negative average rate of change.
Question 4
A) Compare the estimated average rate of change for the quadratic function over the interval \([2,6]\) with the exponential function given by \( y = \frac{34(2^x)}{x} \):
We would need specific values (or an image) to calculate the slopes properly, however:
Assuming standard rates based on the behavior of quadratic vs exponential functions:
The average rate of change of the quadratic function is typically less than that of an exponential function as they grow faster.
Conclusion: Without exact calculations, I would lean towards the exponential function having a greater average rate of change in that interval based on general characteristics.
Suppose the rates were computed and found:
- If confirmed, "The estimated average rate of change of the exponential function is greater than the estimated average rate of change of the quadratic function over the interval [2,6]."
In summary:
- Quadratic \( y = 4x^2 \) has a greater average rate of change.
- Exponential \( y = 6^x \) > Quadratic.
- Neither function is negative on that interval.
- Generally, the exponential function grows faster based on classical properties.