Compare the magnitude of the estimated average rates of change of the exponential function pictured above and the quadratic function f(x)=x2−20

Question

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 greater 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 greater 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 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 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 estimated average rate of change of both the exponential function and f(x)
over the interval [−9,2]
is approximately 1.5
.

Answer 1

To determine the estimated average rates of change for 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 \( a = -9 \) and \( b = 2 \).

Calculating the Average Rate of Change for \( f(x) = x^2 - 20 \)

1. Calculate \( f(-9) \): \[ f(-9) = (-9)^2 - 20 = 81 - 20 = 61 \]

2. Calculate \( f(2) \): \[ f(2) = 2^2 - 20 = 4 - 20 = -16 \]

3. Calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{f(2) - f(-9)}{2 - (-9)} = \frac{-16 - 61}{2 + 9} = \frac{-77}{11} = -7 \]

Estimating the Average Rate of Change for the Exponential Function

To provide a meaningful comparison, we would need an exponential function. A common example would be \( g(x) = e^x \) or similar. Assuming we are comparing the same interval and we use \( g(x) = a^x \) for some base \( a \) (for instance, \( a = e \)), we could say:

1. Calculate \( g(-9) \) and \( g(2) \) using, say, \( g(x) = e^x \):

   \[ g(-9) \approx e^{-9} \quad \text{(which is a very small positive number)} \] \[ g(2) \approx e^2 \approx 7.39 \]

2. Calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{g(2) - g(-9)}{2 - (-9)} = \frac{7.39 - e^{-9}}{11} \] \( e^{-9} \) being negligible means \( \text{average rate} \approx \frac{7.39}{11} \approx 0.67 \)

Conclusion

• The average rate of change of \( f(x) = x^2 - 20 \) over the interval \([-9, 2]\) is \(-7\).
• The average rate of change of a common exponential function like \( g(x) = e^x \) tends to be a small positive value.

As a result, in terms of the provided options, the best conclusion is:

• “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.”

In summary, we can infer:

The estimated average rate of change of the exponential function is less than the rate of change of the quadratic function when evaluated over that interval.