Question

walk me through the steps of finding average rate of change over intervals using y^2-y^1/x^2-x^1

Answer 1

To find the average rate of change of a function over an interval using the formula \(\frac{y_2 - y_1}{x_2 - x_1}\), we follow these steps:

1. **Identify the interval:**
Determine the interval \([x_1, x_2]\) over which you want to find the average rate of change. This gives you the values of \(x_1\) and \(x_2\).

2. **Determine the corresponding \(y\) values:**
For the function \(f(x)\) that you are working with, calculate \(y_1\) and \(y_2\):
- \(y_1 = f(x_1)\)
- \(y_2 = f(x_2)\)

3. **Calculate \(y_2 - y_1\):**
Find the difference between the two \(y\) values:
\[
y_2 - y_1
\]

4. **Calculate \(x_2 - x_1\):**
Find the difference between the two \(x\) values:
\[
x_2 - x_1
\]

5. **Substitute into the formula:**
Use the differences you calculated in steps 3 and 4 to find the average rate of change:
\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

6. **Interpret the result:**
The value you calculated represents the average rate of change of the function \(f(x)\) over the interval from \(x_1\) to \(x_2\). It tells you how much \(f(x)\) changes, on average, per unit change in \(x\) over that interval.

### Example:
Let's work through an example.

**Function:** \(f(x) = x^2\)
**Interval:** \([1, 3]\)

1. Identify the interval: \(x_1 = 1\), \(x_2 = 3\).

2. Calculate the corresponding \(y\) values:
- \(y_1 = f(1) = 1^2 = 1\)
- \(y_2 = f(3) = 3^2 = 9\)

3. Calculate \(y_2 - y_1\):
\[
y_2 - y_1 = 9 - 1 = 8
\]

4. Calculate \(x_2 - x_1\):
\[
x_2 - x_1 = 3 - 1 = 2
\]

5. Substitute into the formula:
\[
\text{Average Rate of Change} = \frac{8}{2} = 4
\]

6. Interpret the result:
The average rate of change of the function \(f(x) = x^2\) over the interval \([1, 3]\) is \(4\). This means that on average, as \(x\) increases from \(1\) to \(3\), the function \(f(x)\) increases by \(4\) units for every 1 unit increase in \(x\).