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Unit B - Differentiation

Unit B - Differentiation. B3.3 - Rates of Change Calculus - Santowski. Lesson Objectives. 1. Calculate an average rate of change 2. Calculate an instantaneous rate of change using difference quotients and limits

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Unit B - Differentiation

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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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