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Exploring Rates of Change: Average and Instantaneous Calculations in Calculus

This lesson focuses on understanding and calculating average and instantaneous rates of change using calculus principles. Students will work collaboratively to predict height changes in a falling object over time. Through data analysis, scatter plots, and algebraic methods (including regression equations), learners will determine the best estimates for instantaneous rates of change at specific intervals. The lesson emphasizes graphical, numerical, and algebraic approaches, facilitating insights into real-world applications of calculus.

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Exploring Rates of Change: Average and Instantaneous Calculations in Calculus

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  1. Unit B - Differentiation B3.3 - Rates of Change Calculus - Santowski

  2. Lesson Objectives • 1. Calculate an average rate of change • 2. Calculate an instantaneous rate of change using difference quotients and limits • 3. Calculate instantaneous rates of change numerically, graphically, and algebraically • 4. Calculate instantaneous rates of change in real world problems

  3. (A) Exploration • To investigate average and instantaneous rates of change, you will complete the following investigation with your group in your own • Ask questions of me only if the instructions are not clear

  4. Rates of Change - An Investigation • PURPOSE  predict the rate at which the height is changing at 2 seconds • Consider the following data of an object falling from a height of 45 m.

  5. Table of Data

  6. Table of Data

  7. Scatterplot and Prediction • 1. Prepare a scatter-plot of the data. • 2. We are working towards finding a good estimate for the rate of change of the height at t = 2.0 s. • So from your work in science courses like physics, you know that we can estimate the instantaneous rate of change by drawing a tangent line to the function at our point of interest and finding the slope of the tangent line. • So on a copy of your scatter-plot, draw the curve of best fit and draw a tangent line and estimate the instantaneous rate of change of height at t = 2.0 s. • How confident are you about your prediction. • Give reasons for your confidence (or lack of confidence).

  8. Algebraic Estimation – Secant Slopes • To come up with a prediction for the instantaneous rate of change that we can be more confident about, we will develop an algebraic method of determining a tangent slope. • So work through the following exercise questions: • We will start by finding average rates of change, which we will use as a basis for an estimate of the instantaneous rate of change. • Find the average rate of change of height between t = 0 and t = 2 s • Mark both points, draw the secant line and find the average rate of change. • Now find the average rate of change of height between • (i) t = 0.5 and t = 2.0, • (ii) t = 1.0 and t = 2.0, • (iii) t = 1.5 and t = 2.0, • Draw each secant line on your scatter plot

  9. Prediction Algebraic Basis • Now using the work from the previous slide, we can make a prediction or an estimate for the instantaneous rate of change at t = 2.0 s. (i.e at what rate is the height changing 1990) • Explain the rationale behind your prediction. • Can we make a more accurate prediction??

  10. Algebraic Prediction – Regression Equation • Unfortunately, we have discrete data in our example, which limits us from presenting a more accurate estimate for the instantaneous rate of change. • If we could generate an equation for the data, we may interpolate some data points, which we could use to prepare a better series of average rates of change so that we could estimate an instantaneous rate of change. • So now find the best regression equation for the data using technology. • Justify your choice of algebraic model for the height of the object.

  11. Algebraic Prediction – Regression Equation

  12. Algebraic Prediction – Regression Equation

  13. Algebraic Prediction – Regression Equation • Now using our equation, we can generate interpolated values for time closer to t = 2.0 s (t = 1.6, 1.7, 1.8, 1.9 s). Now determine the average rates of change between • (i) t = 1.6 s and t = 2.0 s, • (ii) t = 1.7 s and t = 2.0 s etc... • We now have a better list of average rates of change so that we could estimate an instantaneous rate of change.

  14. Algebraic Prediction – Regression Equation

  15. Algebraic Prediction – Regression Equation

  16. Best Estimate of Rate of Change • Finally, what is the best estimate for the instantaneous rate of change at t = 2.0 s? • Has your rationale in answering this question changed from previously? • How could you use the same process to get an even more accurate estimate of the instantaneous rate of change?

  17. Another Option for Exploration • One other option to explore: • Using our equation, generate other interpolated values for time close to but greater than t = 2.0 s (i.e. t = 2.5, 2.4, 2.3, 2.2, 2.1 s). • Then calculate average rates of change between • (i) t = 2.5 and t = 2.0, • (ii) t = 2.4 and t = 2.0, etc.... which will provide another list of average rates of change. • Provide another estimate for an instantaneous rate of change at t = 2.0 s. • Explain how this process is different than the option we just finished previously? • How is the process the same?

  18. A Third Option for Exploration • Another option to explore is as follows: • (i) What was the average rate of change between t = 1.0 and t = 2.0? • (ii) What was the average rate of change between t = 3.0 and t = 2.0? • (iii) Average these two rates. Compare this answer to your previous estimates. • (iv) What was the average rate of change between t = 1.0 and t = 3.0? Compare this value to our previous estimate. • (v) Now repeat the process from Question (iv) for the following: • (a) t = 1.5 and t = 2.5 • (b) t = 1.75 and t = 2.25 • (c) t = 1.9 and t = 2.1 • (vi) Explain the rational (reason, logic) behind the process in this third option

  19. Summary • From your work in this investigation: • (i) compare and contrast the processes of manually estimating a tangent slope by drawing a tangent line and using an algebraic approach. • (ii) Explain the meaning of the following mathematical statement: • slope of tangent = • Or more generalized, explain

  20. Summary

  21. Summary

  22. Ex 1. The point (2,2) lies on the curve f(x) = 0.25x3. If Q is the point (x,f(x)), find the average rate of change (or the secant slope of the segment PQ) of the function f(x) = 0.25x3 if the x co-ordinate of P is: Then, predict the instantaneous rate of change at x = 2 (i) x = 3 (ii) x = 2.5 (iii) x = 2.1 (iv) x = 2.01 (v) x = 2.001 (vi) x = 1 (vii) x = 1.5 (viii) x = 1.9 (ix) x = 1.99 (x) x = 1.999 (B) Instantaneous Rates of Change - Numeric Calculation

  23. Ex 2. For the function f(x), determine instantaneous rate of change at x = 0 Pick some appropriate x values to allow for a valid numerical investigation of (B) Instantaneous Rates of Change - Numeric Calculation

  24. (C) Instantaneous Rates of Change - Algebraic Calculation • Find the instantaneous rate of change of the function f(x) = -x2 + 3x - 5 at x = -4

  25. Make use of the limit definition of an instantaneous rate of change to determine the instantaneous rate of change of the following functions at the given x values: (C) Instantaneous Rates of Change - Algebraic Calculation

  26. (D) Internet Links - Applets • The process outlined in the previous slides is animated for us in the following internet links: • Secants and tangent • A Secant to Tangent Applet from David Eck • JCM Applet: SecantTangent • Visual Calculus - Tangent Lines from Visual Calculus – Follow the link for the Discussion

  27. (E) Homework • S3.3, p173-178 • (1) average rate of change: Q1,3,6,8 • (2) instantaneous rate of change: Q9,12,14,15 (do algebraically) • (3) numerically: Q17,20 • (4) applications: Q27-29,32,34,39,41 • NOTE: I AM MARKING EVEN #s

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