1. Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 2 Ingredients of Change: Nonlinear Models

2. Copyright © by Houghton Mifflin Company, All rights reserved. Chapter 2 Key Concepts Exponential ModelsExponential ModelsExponential ModelsExponential Models Log ModelsLog ModelsLog ModelsLog Models Logistic ModelsLogistic ModelsLogistic ModelsLogistic Models Quadratic and Cubic ModelsQuadratic and Cubic ModelsQuadratic and Cubic ModelsQuadratic and Cubic Models Shifting DataShifting DataShifting DataShifting Data Choosing a ModelChoosing a ModelChoosing a ModelChoosing a Model

3. Copyright © by Houghton Mifflin Company, All rights reserved. Exponential Models Q(t) = ab t, b > 0, b  1Q(t) = ab t, b > 0, b  1 Constant percentage of changeConstant percentage of change GraphsGraphs Exponential Growth Q(t) = ab t with b > 1 Exponential Decay Q(t) = ab t with b < 1

4. Copyright © by Houghton Mifflin Company, All rights reserved. Exponential Models: Example Year Balance (dollars) 0123401234 100.00 105.00 110.25 115.76 121.55 Constant Percentage Change

5. Copyright © by Houghton Mifflin Company, All rights reserved. Exponential: Exercise 2.1 #23 Year Million of CD Singles 1993 1994 1995 1996 1997 7.8 9.3 21.5 43.2 66.7 Determine the yearly percentage growth of CD singles sales.

6. Copyright © by Houghton Mifflin Company, All rights reserved. Exponential Models: Example Year Balance (dollars) 0123401234 100.00 105.00 110.25 115.76 121.55 Use regression to find an exponential model. B(t) = 100(1.05) t dollars after t years.

7. Copyright © by Houghton Mifflin Company, All rights reserved. Exponential: Exercise 2.1 #23 Year Million of CD Singles 1993 1994 1995 1996 1997 7.8 9.3 21.5 43.2 66.7 Use regression to find an exponential model. C(t) = 6.673(1.791) t million CD singles t years after 1993.

8. Copyright © by Houghton Mifflin Company, All rights reserved. Log Models f(x) = a + b ln x with b > 0 f(x) = a + b ln x with b < 0 f(x) = a + b ln x, x > 0f(x) = a + b ln x, x > 0 Inverse of exponential function y = e -a/b (e 1/b ) xInverse of exponential function y = e -a/b (e 1/b ) x GraphsGraphs

9. Copyright © by Houghton Mifflin Company, All rights reserved. Log Models: Example Time to Maturity (years) Bond Rate (percent) 1234512345 3.60 4.10 4.25 4.40 4.50 Use regression to find a log model and predict the bond rate for 20 years time to maturity. R(t) = 3.64 + 0.551 ln t percent at t years to maturity. R(20) = 3.64 + 0.551 ln 20 = 5.29 percent = 5.29 percent

10. Copyright © by Houghton Mifflin Company, All rights reserved. Log Models: Exercise 2.1 #32 Age beyond 2 weeks (weeks) Weight (grams) 1357913579 11 20 23 26 27 Use regression to find a log model and predict the weight of the mice when they are 4 weeks old. W(t) = 11.298 + 7.372 ln t percent at t + 2 weeks old. R(2) = 11.298 + 7.372 ln 2  16 grams at 4 weeks  16 grams at 4 weeks

11. Copyright © by Houghton Mifflin Company, All rights reserved. Logistic Models Approaches a fixed value as x   or - Approaches a fixed value as x   or -  GraphsGraphs

12. Copyright © by Houghton Mifflin Company, All rights reserved. Logistic Models: Example Time (hours after 8) Total people infected 012345012345 124689124689 Using logistic regression,

13. Copyright © by Houghton Mifflin Company, All rights reserved. Logistic Models: Example The graph of the model

14. Copyright © by Houghton Mifflin Company, All rights reserved. Logistic Models: Exercise 2.3 #24 Year Emissions (millions of tons) 1970 1975 1980 1985 1990 1995 220.9 159.7 74.2 22.9 5.0 3.9 Use regression to find a logistic model. What is the end behavior of the function as time increases? As t increases, E(t) approaches 0 million tons. t years after 1970.

15. Copyright © by Houghton Mifflin Company, All rights reserved. Quadratic/Cubic Models Quadratic: f(x) = ax 2 + bx + cQuadratic: f(x) = ax 2 + bx + c GraphsGraphs a > 0 a < 0

16. Copyright © by Houghton Mifflin Company, All rights reserved. Quadratic/Cubic Models Cubic: g(x) = ax 3 + bx 2 + cx + dCubic: g(x) = ax 3 + bx 2 + cx + d GraphsGraphs a > 0 a < 0

17. Copyright © by Houghton Mifflin Company, All rights reserved. Quadratic/Cubic Models: Example Year Fuel Consumption (gallons per vehicle / year) 1970 1975 1980 1985 1990 1995 830 790 712 685 677 700 F(t) = 0.425t 2 - 16.431t + 840.321 gallons per vehicle per year where t is the number of years since 1970.

18. Copyright © by Houghton Mifflin Company, All rights reserved. Quadratic/Cubic: Exercise Year Natural Gas Cost per 1000 Cubic Feet (dollars) 1980 1982 1985 1990 1995 1996 1997 3.68 5.17 6.12 5.77 6.06 6.34 6.93 Find and graph the cubic model that best fits the data. P(x) = 0.004x 3 + 0.118x 2 + 0.967x + 3.681

19. Copyright © by Houghton Mifflin Company, All rights reserved. Shifting Data Aligning inputs shifts data horizontallyAligning inputs shifts data horizontally Aligning outputs shifts data verticallyAligning outputs shifts data vertically Align data in order to:Align data in order to: –reduce the magnitude of coefficients –introduce vertical and horizontal shifts to compensate for calculator model deficiencies (exponential and logistic)

20. Copyright © by Houghton Mifflin Company, All rights reserved. Shifting Data: Example xf(x) 0246802468 101 104 116 164 356 Exponential regression yields f(x) = 81.6597(1.1604) x.

21. Copyright © by Houghton Mifflin Company, All rights reserved. Shifting Data: Example xf(x) - 100 0246802468 1 4 16 64 256 Exponential regression on aligned data yields f(x) = 2 x. Our final equation is g(x) = 2 x + 100

22. Copyright © by Houghton Mifflin Company, All rights reserved. Shifting Data: Exercise 2.4 #9 Years (since 1990) Permit Renewals 123456123456 146 195 215 328 476 831 Find and graph an exponential model. Then shift the renewals down by 135 and find and plot the new model. p(x) = 93.5575(1.4007) x

23. Copyright © by Houghton Mifflin Company, All rights reserved. Shifting Data: Exercise 2.4 #9 Years (since 1990) Permit Renewals 123456123456 11 60 80 193 341 696 Find and graph an exponential model. Then shift the renewals down by 135 and find and plot the new model. R(x) = 7.9194(2.1524) x

24. Copyright © by Houghton Mifflin Company, All rights reserved. Choosing a Model Look at the curvature of the data scatter plotLook at the curvature of the data scatter plot Look at the fit of the possible equationsLook at the fit of the possible equations Look at the end behavior of the scatter plotLook at the end behavior of the scatter plot Consider that there may be more than one good model for a particular set of dataConsider that there may be more than one good model for a particular set of data

25. Copyright © by Houghton Mifflin Company, All rights reserved. Choosing a Model: Example Years (since 1960) Age at Marriage 0102030 20.320.822.023.9 Exponential Model E(x) = 20.00(1.005) x

26. Copyright © by Houghton Mifflin Company, All rights reserved. Choosing a Model: Example Years (since 1960) Age at Marriage 0102030 20.320.822.023.9 Aligned Exponential Model (Data shifted down by 20) S(x) = 0.321(1.090) x + 20

27. Copyright © by Houghton Mifflin Company, All rights reserved. Choosing a Model: Example Years (since 1960) Age at Marriage 0102030 20.320.822.023.9 Quadratic Q(x) = 0.0035x 2 + 0.015x + 20.3 The quadratic model is the best fit.

28. Copyright © by Houghton Mifflin Company, All rights reserved. Choosing a Model: Exercise 2.6 #2 What functions are candidates to fit the data? quadratic and exponential The following are not good candidates: cubic: no inflection point logistic: no inflection point logarithmic: not increasing at a decreasing rate