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

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

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

Copyright © by Houghton Mifflin Company, All rights reserved. Exponential Models: Example Year Balance (dollars) Constant Percentage Change

Copyright © by Houghton Mifflin Company, All rights reserved. Exponential: Exercise 2.1 #23 Year Million of CD Singles Determine the yearly percentage growth of CD singles sales.

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

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

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

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

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

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

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

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

Copyright © by Houghton Mifflin Company, All rights reserved. Logistic Models: Exercise 2.3 #24 Year Emissions (millions of tons) 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.

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

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

Copyright © by Houghton Mifflin Company, All rights reserved. Quadratic/Cubic Models: Example Year Fuel Consumption (gallons per vehicle / year) F(t) = 0.425t t gallons per vehicle per year where t is the number of years since 1970.

Copyright © by Houghton Mifflin Company, All rights reserved. Quadratic/Cubic: Exercise Year Natural Gas Cost per 1000 Cubic Feet (dollars) Find and graph the cubic model that best fits the data. P(x) = 0.004x x x

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)

Copyright © by Houghton Mifflin Company, All rights reserved. Shifting Data: Example xf(x) Exponential regression yields f(x) = (1.1604) x.

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

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

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

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

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

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

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

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