Compare the magnitude of the estimated average rates of change of the exponential function pictured above and the quadratic function f(x)=x^2−20 over the interval [−9,2] and identify which function has a greater rate of change than the other. (1 point) Responses The magnitude of the estimated average rate of change of the exponential function over the interval [−9,2] is less than the magnitude of the estimated average rate of change of f(x) over the interval [−9,2] . The magnitude of the estimated average rate of change of the exponential function over the interval left bracket negative 9 comma 2 right bracket is less than the magnitude of the estimated average rate of change of f of x over the interval left bracket negative 9 comma 2 right bracket. The estimated average rate of change of both the exponential function and f(x) over the interval [−9,2] is approximately 1.5 . The estimated average rate of change of both the exponential function and f of x over the interval left bracket negative 9 comma 2 right bracket is approximately 1 point 5. The estimated average rate of change of the exponential function over the interval [−9,2] is negative, while the estimated average rate of change of f(x) over the interval [−9,2] is positive. The estimated average rate of change of the exponential function over the interval left bracket negative 9 comma 2 right bracket is negative, while the estimated average rate of change of f of x over the interval left bracket negative 9 comma 2 right bracket is positive. The magnitude of the estimated average rate of change of the exponential function over the interval [−9,2] is greater than the magnitude of the estimated average rate of change of f(x) over the interval [−9,2] .

Question

GPT-4o mini · Answer

To compare the estimated average rates of change of the exponential function and the quadratic function \( f(x) = x^2 - 20 \) over the interval \([-9, 2]\), we can use the formula for the average rate of change:

\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]

where \( f(a) \) and \( f(b) \) are the function values at the endpoints of the interval, and \( [a, b] \) is the interval. Let's denote the exponential function as \( g(x) \).

Step 1: Calculate the average rate of change for \( f(x) \)

\( f(-9) = (-9)^2 - 20 = 81 - 20 = 61 \)
\( f(2) = (2)^2 - 20 = 4 - 20 = -16 \)

Now compute the average rate of change:

\[ \text{Average Rate of Change for } f = \frac{f(2) - f(-9)}{2 - (-9)} = \frac{-16 - 61}{2 + 9} = \frac{-77}{11} \approx -7 \]

Step 2: Estimate the average rate of change for the exponential function

Let's assume the exponential function is of the form \( g(x) = a \cdot b^x \). While the exact function isn't given, we can compute its average rate of change similarly.

For \( g(-9) \) and \( g(2) \), we'd need specific values or an exact function, but let's hypothetically choose common values for an exponential. For example:
- \( g(-9) = c_1 \)
- \( g(2) = c_2 \)

We would estimate without specific function values.

Now let’s assume for comparison purposes that \( |g(-9)| \) and \( |g(2)| \) provide numerical values that are not extremely low.

Final Comparison of Magnitudes:

Because the average rate of change for \( f \) was approximately \(-7\), its magnitude is \( 7 \).
The exponential function may have a lesser or greater change depending on its form. If we assume it shows positive growth and that its value notably increases, potentially we could see something larger.
If the exponential even shows a decrease in overall numeric value from a large negative to larger positive, you might get a \(-c\) suggesting it’s less pronounced over such a wide scope.

Conclusion

Without actual values for the exponential function, if it were to yield drastically higher growth from the computed range we could state the exponential function has a greater rate of change in positive terms. Conversely, if clear negative, the quadratic would dominate given broad behavior over intervals up to here.

Thus, the response verbally to which function changes more significantly with range and sign should be asserted as:

The estimated average rate of change of the exponential function over the interval [-9,2] is greater than the magnitude of the estimated average rate of change of \( f(x) \) over the interval [-9,2].