Lesson 7 Summary on Average Rate of Change

Lesson 7 Summary on Average Rate of Change

Graphical Representation of Temperature Over Time

  • A graph illustrates the temperature recorded throughout one day as a function of time.
  • The temperature is measured in degrees Fahrenheit (°F) and is plotted against different times of the day.

Key Temperature Readings

  • At 9 a.m.: 35° F
  • At 2 p.m.: 45° F
  • There is a noted temperature increase of 10° F over a duration of 5 hours from 9 a.m. to 2 p.m.

Temperature Change Observations

  • The temperature increase did not occur at a constant rate. Specifically:
    • The temperature rose from 9 a.m. to 10 a.m.
    • The temperature stayed steady for an hour before rising again.

Average Rate of Change Calculation (9 a.m. to 2 p.m.)

  • To find the average rate of change of the temperature between these two times, the following steps are taken:
    1. Find the change in temperature (Δy):
      • Temperature at 2 p.m. (45° F) – Temperature at 9 a.m. (35° F) = 10° F
    2. Determine the time interval (Δx):
      • Time from 9 a.m. to 2 p.m. = 5 hours
    3. Calculate the average rate of change using the formula:
      • average rate of change=ΔyΔx=105=2° F per hour\text{average rate of change} = \frac{\Delta y}{\Delta x} = \frac{10}{5} = 2 \text{° F per hour}
  • Conclusion:
    • On average, the temperature increased by 2° F per hour between 9 a.m. and 2 p.m.

Average Rate of Change Calculation (2 p.m. to 8 p.m.)

  • To analyze how quickly the temperature was falling during this time period, similar steps are followed:
    1. Find the change in temperature (Δy):
      • Temperature at 8 p.m. (30° F) – Temperature at 2 p.m. (45° F) = -15° F
    2. Determine the time interval (Δx):
      • Time from 2 p.m. to 8 p.m. = 6 hours
    3. Calculate the average rate of change:
      • average rate of change=ΔyΔx=−156=−2.5° F per hour\text{average rate of change} = \frac{\Delta y}{\Delta x} = \frac{-15}{6} = -2.5 \text{° F per hour}
  • Conclusion:
    • On average, the temperature decreased by 2.5° F per hour between 2 p.m. and 8 p.m.

General Formula for Average Rate of Change

  • The average rate of change of a function f between two input values a and b can be calculated using the following formula:
    • average rate of change=f(b)−f(a)b−a\text{average rate of change} = \frac{f(b) - f(a)}{b - a}
  • Here, if there are two points on the graph of the function:
    • Point 1: (a, f(a))
    • Point 2: (b, f(b))
  • The average rate of change is represented as the slope of the line that connects these two points on the graph.

Glossary

  • Average Rate of Change: The rate at which a quantity changes, calculated as the difference in the quantity divided by the difference in input values over a specified interval.