Question

Find 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]. Show the work for both functions.

Answer 1

To find 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 will follow these steps:

1. **Calculate the interval endpoints for both functions.**
2. **Use the formula for average rate of change.**

The average rate of change of a function \( f \) over the interval \([a, b]\) is given by the formula:

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

### For the Quadratic Function \( f(x) = x^2 - 20 \)

1. **Evaluate \( f(-9) \) and \( f(2) \)**:

\[
f(-9) = (-9)^2 - 20 = 81 - 20 = 61
\]

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

2. **Substitute into the average rate of change formula**:

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

### For the Exponential Function

Since the exponential function is not explicitly given, I'll assume a general form \( g(x) = a \cdot b^x \). You need the specific function to evaluate it properly. Let’s say we are working with an example function, such as \( g(x) = e^x \) (or another common base).

1. **Evaluate \( g(-9) \) and \( g(2) \)** (if it's \( g(x) = e^x \)):

\[
g(-9) = e^{-9} \quad \text{(Use a calculator to get a decimal approximation)}
\]
\[
g(-9) \approx 0.0001234
\]

\[
g(2) = e^2 \approx 7.3891
\]

2. **Substitute into the average rate of change formula**:

To find the average rate of change:

\[
\text{Average Rate of Change} = \frac{g(2) - g(-9)}{2 - (-9)} = \frac{7.3891 - 0.0001234}{2 + 9} = \frac{7.3891 - 0.0001234}{11} \approx \frac{7.389}{11} \approx 0.672
\]

### Summary of Results

- **Average rate of change for \( f(x) = x^2 - 20 \) over \([-9, 2]\)**: \(-7\)
- **Average rate of change for the exponential function \( g(x) = e^x \) over \([-9, 2]\)** (if \( g(x) = e^x \)): \(\approx 0.672\)

Please replace \( g(x) \) with the specific exponential function you have to get correct values.