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 YearPopulation (in millions) 19001650 19101750 19201860 19302070 19402300 19502520 19603020 19703700 19804450 19905300 19965770 19996000 20026200 20046400

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