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: $\frac{\text{change in output values}}{\text{change in input values}} = \frac{\Delta y}{\Delta x} = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1} = \text{SLOPE}$

• Example 1:

  • Find the average rate of change between the points $(1, 2)$ and $(3, 4)$.

    • $x<em>1 = 1, y</em>1 = 2, x<em>2 = 3, y</em>2 = 4$

    • 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.

Rate of Change at a Point

• 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 and Negative Rates of Change

• 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)

Practice Problems: Average 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 $1 \le x \le 4$.

Practice Problems: Using Tables

• 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. $6 \le t \le 72$

Practice Problems: Estimating Rate of Change At a Point

• 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 $x = 3$

Practice Problems: Positive or Negative Rate of Change

• State whether the situation represents a positive or negative rate of change.

  • 10. A candy company uses pints of chocolate to make candy. The more chocolate they use, the more boxes of candy are produced. (Positive)
  • 11. The amount of money is Josh’s savings account decreases for each semester he attends college. (Negative)
  • 12. As the number of cats Mr. Sullivan owns increases, the number of mice in his barn decreases. (Negative)
  • 13. As the amount of water Mr. Brust drinks decreases, the fewer trips to the restroom he needs to make. (Positive)

Test Prep

• 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.