Rates of Change Notes

Average Rate of Change

The average rate of change is defined as the change in output values divided by the change in input values.
- It can be represented as: \frac{\text{change in output values}}{\text{change in input values}} = \frac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1} = \text{SLOPE}
Example 1:
- Find the average rate of change between the points (1, 2) and (3, 4).
  - x1 = 1, y1 = 2, x2 = 3, y2 = 4
  - Average rate of change = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1
When calculating average rate of change from a table of values, ensure the output values are in the numerator.
The word "per" indicates a rate of change.
- Examples: Miles per gallon, students per classroom, online gamers per server.
- It indicates which variable is dependent (listed first) and which is independent.

The rate of change of a function at a point describes how the output values change as the input values change at that specific point.
Approximating the rate of change at a point:
- Use the average rates of change over small intervals containing that point.
Example 3:
- Estimate the rate of change at x = 1 for the function f(x) = |x^3 - 3x|.
Example 4:
- Approximate the rate of change of g at x = -2 if g(x) = e^x.

Positive Rate of Change:
- As one quantity increases (or decreases), the other quantity increases (or decreases) accordingly.
Negative Rate of Change:
- As one quantity increases, the other quantity decreases.
Example 5:
- As the years increase, a high school student body increases. (Positive rate of change)
Example 6:
- As Mr. Bean’s weight decreases, his running distances increase. (Negative rate of change)

Find the average rate of change of the function on the given interval.
- 1. h(t) = 3t + t^2 over the interval 2 \le t \le 5.
- 2. b(w) = \frac{w}{2} over the interval [-1, 2].
- 3. f(x) = \ln(3x) over the interval 1 \le x \le 4.

Use the information in each table to find the average rate of change on the given interval.
Problem 4:
- Given a table with t (minutes) and d(t) (meters):
- a. 3 \le t \le 55
- b. 10 \le t \le 43
- c. 21 \le t \le 55
Problem 5:
- Given a table with t (months) and d(t) (hair follicles):
- a. 6 \le t \le 48
- b. 12 \le t \le 72
- c. 6 \le t \le 72

Estimate the rate of change of each function at the given point.
- 6. f(x) = \frac{1}{x-1} at x = 4
- 7. f(x) = 2x^2 + 1 at x = -2
- 8. f(x) = 7\sqrt{x} at x = 2
- 9. f(x) = \ln(2x) at x = 3

State whether the situation represents a positive or negative rate of change.
- 1. A candy company uses pints of chocolate to make candy. The more chocolate they use, the more boxes of candy are produced. (Positive)
- 1. The amount of money is Josh’s savings account decreases for each semester he attends college. (Negative)
- 1. As the number of cats Mr. Sullivan owns increases, the number of mice in his barn decreases. (Negative)
- 1. As the amount of water Mr. Brust drinks decreases, the fewer trips to the restroom he needs to make. (Positive)

A continuous function f is defined on the closed interval [5, 6].
Determine how many values of b, where 5 \le b \le 6, make the average rate of change of f on the interval [b, 5] equal to 0.
Reason: The average rate of change being zero implies a horizontal line segment on the graph of f within the given interval.