AP Precalculus Lecture Notes: Rates of Change

Section 1.2: Rates of Change

Average Rate of Change

• Slope: Previously learned as change in dependent variable (Y) over change in independent variable (X).
• Delta (Δ): Symbol for change; used in the formula ΔY/ΔX.
• Formula:
  • (\frac{Y_2 - Y_1}{X_2 - X_1})
  • Designate points as ((X_1, Y_1)) and ((X_2, Y_2)).
  • Order doesn't affect result since negatives cancel out.

Example 1: Finding Average Rate of Change

• Given points: ((1, 2)) and ((3, 4)).
• Calculate:
  • (\frac{4 - 2}{3 - 1} = \frac{2}{2} = 1).
• Demonstrated consistency of formula with point order.

Table of Values

• Independent Variables: Typically time, must be on the denominator.
• Example: Calculate slope from a table interval (from (13) to (25)).
  • Designate points: ((1, 76)) and ((25, 30)).
  • Calculate slope: (\frac{30 - 76}{25 - 13} = \frac{-46}{12}).
  • Units: Ensure units match, e.g., meters per second.

Rate of Change at a Single Point

• Approximate by using small intervals around the point.
• Geogebra: Graphing tool to find average rate of change using closely spaced points.
• Example with Function:
  • Points: ((1, 2)) and ((2, 3.5)), slope (1.5).
  • Use smaller intervals, e.g., ((1, 2)) and ((1.1, 2.95)), slope (1.95).

Calculator Tips

• Utilize table and function features for precise values.
• Tricks:
  • Use the function notation for direct input.
  • Use calculator settings to adjust table increments.

Estimating Rate of Change

• Method:
  • Use calculator to find values for very close points, e.g., (1) and (1.001).
  • Calculate slope with small increments.

Positive and Negative Rates of Change

• Positive Rate:
  • Both variables increase; graph ascends from left to right.
• Negative Rate:
  • One variable increases while the other decreases; graph descends.
  • Examples:
    • Positive: Increase in student body over years.
    • Negative: Weight decrease with increased running distance.

Conclusion

• Mastery Check: Understanding concepts of average rate of change, using tables, graphs, and calculator tools.
• Moral: It's important to practice kindness.