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A1 – Rates of Change

A1 – Rates of Change. IB Math HL&SL - Santowski. (A) Average Rates of Change. Use graphing technology for this investigation (Winplot/Winstat/GDC) PURPOSE  predict the rate at which the world population is changing in 1990 Consider the following data of world population over the years.

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A1 – Rates of Change

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  1. A1 – Rates of Change IB Math HL&SL - Santowski

  2. (A) Average Rates of Change • Use graphing technology for this investigation (Winplot/Winstat/GDC) • PURPOSE  predict the rate at which the world population is changing in 1990 • Consider the following data of world population over the years

  3. (B) Table of World Population Data

  4. (C) Scatterplot and Prediction • 1. Prepare a scatter-plot of the data. Use t = 0 for 1900 • 2. We are working towards finding a good estimate for the rate of change of the population in 1990. • 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 population in 1990. • How confident are you about your prediction. • Give reasons for your confidence (or lack of confidence).

  5. (D) Algebraic Estimation – Secant Slopes • 3. To come up with a prediction for the instantaneous rate of change that we can be confident about, we will develop an algebraic method of determining a tangent slope. • So work through the following exercise questions 3 through 10 • 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 the population between 1950 and 1990 • Mark both points, draw the secant line and find the average rate of change. • Now find the average rate of change for the population between • (i) 1960 and 1990, • (ii) 1970 and 1990, • (iii) 1980 and 1990, • (iv) 2000 and 1990. • Draw each secant line on your scatter plot

  6. (E) Prediction Algebraic Basis • 4. Now using the work from Question 3, we can make a prediction or an estimate for the instantaneous rate of change in 1990. (i.e at what rate is the population changing 1990) • Explain the rationale behind your prediction.

  7. (F) Algebraic Prediction – Regression Equation • 5. 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 (GDC or WINSTAT). • Justify your choice of algebraic model for the population.

  8. (F) Algebraic Prediction – Regression Equation • 6. Now using our equation, we can generate interpolated values for years closer to 1990 (1985, 1986, 1987, 1988, 1989). Now determine the average rates of change between • (i) 1985 and 1990, • (ii) 1986 and 1990 etc... • We now have a better list of average rates of change so that we could estimate an instantaneous rate of change.

  9. (G) Best Estimate of Rate of Change • 7. Finally, what is the best estimate for the instantaneous rate of change in 1990? • Has your rationale in answering this question changed from previously? • How could you use the same process as in Question 6 to get an even more accurate estimate of the instantaneous rate of change?

  10. (H) Another Option for Exploration • 8. One other option to explore: • Using our equation, generate other interpolated values for populations close to but greater than 1990 (1995, 1994, 1993, 1992, 1991). • Then calculate average rates of change between • (i) 1995 and 1990, • (ii) 1994 and 1990, etc.... which will provide another list of average rates of change. • Provide another estimate for an instantaneous rate of change in 1990. • Explain how this process in Question 8 is different than the option we just finished in Question 6? • How is the process the same?

  11. (I) A Third Option for Exploration • 9. Another option to explore is as follows: • (i) What was the average rate of change between 1980 and 1990 (see work in Question 3)? • (ii) What was the average rate of change between 1990 and 2000 (see work in Question 3)? • (iii) Average these two rates. Compare this answer to your estimate from Question 7 and 8. • (iv) What was the average rate of change between 1980 and 2000? Compare this value to our estimate from Question 9(iii) and from Question 7 and from 8. • (v) Now repeat the process from Question 9(iv) for the following: • (a) 1986 and 1994 • (b) 1988 and 1992 • (c) 1989 and 1991 • (vi) Explain the rational (reason, logic) behind the process in this third option

  12. (J) Summary • 10. From your work in Questions 3 through 9: • (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

  13. (K) Homework • Stewart, 1989, Calculus – A First Course, Chap 1.1, p9, Q2,3,7,9,10

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