MODELING WITH EXPONENTIAL GROWTH AND DECAY Sec. 3.1c PRACTICE PROBLEMS Using the given data, and assuming the growth is exponential, when will the population of San Jose surpass 1 million persons? Population of San Jose, CA Year Population 1990 2000 782,248 894,943 Let P(t) be the population of San Jose t years after 1990. General Equation: P t P0 b t Initial Population Growth Factor PRACTICE PROBLEMS Using the given data, and assuming the growth is exponential, when will the population of San Jose surpass 1 million persons? Population of San Jose, CA Year Population 1990 2000 782,248 894,943 Let P(t) be the population of San Jose t years after 1990. To solve for b, use the point (10, 894943): P 10 782, 248b 894,943 894,943 b 10 1.0135 782, 248 10 PRACTICE PROBLEMS Using the given data, and assuming the growth is exponential, when will the population of San Jose surpass 1 million persons? Population of San Jose, CA Year Population 1990 2000 782,248 894,943 Let P(t) be the population of San Jose t years after 1990. P t 782, 248 1.0135t Now, graph this function together with the line y = 1,000,000 t 18.31 The population of San Jose will exceed 1,000,000 in the year 2008 PRACTICE PROBLEMS The number B of bacteria in a petri dish culture after t hours is given by B 100e 0.693t 1. What was the initial number of bacteria present? B 0 100 bacteria 2. How many bacteria are present after 6 hours? B 6 6394 bacteria PRACTICE PROBLEMS Populations of Two Major U.S. Cities 1990 Population 2000 Population City Flagstaff, AZ Phoenix, AZ 44,545 3,455,902 67,880 4,003,365 Assuming exponential growth (and letting t = 0 represent 1990), what will the population of Flagstaff be in the year 2030? PF t 44,545 1.0430 PF 40 240,199 t PRACTICE PROBLEMS Populations of Two Major U.S. Cities 1990 Population 2000 Population City Flagstaff, AZ Phoenix, AZ 44,545 3,455,902 67,880 4,003,365 Assuming exponential growth (and letting t = 0 represent 1990), when will the population of Phoenix exceed 4.5 million? PP t 3, 455,902 1.0148 t t 17.95 In the year 2007 PRACTICE PROBLEMS Populations of Two Major U.S. Cities 1990 Population 2000 Population City Flagstaff, AZ Phoenix, AZ 44,545 3,455,902 67,880 4,003,365 Will the population of Flagstaff ever exceed that of Phoenix? If so, in what year will this occur? Is this a realistic estimate in answer to this question? The two curves intersect at about t = 158.70, so according to these models, the population of Flagstaff will exceed that of Phoenix in the year 2148… LOGISTIC FUNCTIONS AND GROWTH NOW LET’S CONSIDER THIS GRAPH… f x 5 0.8 , Range: , 0 Domain: Continuous Decreasing on x No Local Extrema H.A.: y = 0 V.A.: None , No Symmetry Bounded Above by y = 0 End Behavior: lim f x 0 x lim f x x So far, we’ve looked primarily at exponential growth, which is unrestricted. (meaning?) However, in many “real world” situations, it is more realistic to have an upper limit on growth. (any examples?) In such situations, growth often starts exponentially, but then slows and eventually levels out………………………does this remind you of a function that we’ve previously studied??? DEFINITION: LOGISTIC GROWTH FUNCTIONS Let a, b, c, and k be positive constants, with b < 1. A logistic growth function in x is a function that can be written in the form c f x x 1 a b or c f x kx 1 a e where the constant c is the limit to growth. Note: If b > 1 or k < 0, these formulas yield logistic decay functions (unless otherwise stated, the term logistic functions will refer to logistic growth functions). With a = c = k = 1, we get our basic function!!! 1 f x x 1 e BASIC FUNCTION: THE LOGISTIC FUNCTION 1 f x x 1 e Domain: All reals y=1 Continuous Range: (0, 1) Increasing for all x Symmetric about (0, ½), but neither even nor odd (0, ½) Bounded (above and below) No local extrema H.A.: y = 0, y = 1 V.A.: None End Behavior: lim f x 0 lim f x 1 x x GUIDED PRACTICE Graph the given function. Find the y-intercept, the horizontal asymptotes, and the end behavior. 8 h x x 1 3 0.7 y-intercept: 8 8 2 h 0 0 1 3 1 3 0.7 End Behavior: The limit to growth is 8, so: H.A.: y = 0, y = 8 lim h x 0 lim h x 8 x x GUIDED PRACTICE Graph the given function. Find the y-intercept, the horizontal asymptotes, and the end behavior. 20 g x 3 x 1 2e y-intercept: End Behavior: 20 3 H.A.: y = 0, y = 20 lim g x 0 lim g x 20 x x GUIDED PRACTICE Based on recent census data, a logistic model for the population of Dallas, t years after 1900, is 1,301, 614 P t 0.05055 t 1 21.603e According to this model, when was the population 1 million? Graph this function together with the line y = 1,000,000 …where do they intersect??? t 84.50 The population of Dallas was 1 million by the end of 1984 GUIDED PRACTICE Based on recent census data, the population of New York state can be modeled by 19.875 P t 1 57.993e 0.035005 t where P is the population in millions and t is the number of years since 1800. Based on this model, (a) What was the population of New York in 1850? 19.875 P 50 1.794558 0.035005 50 1 57.993e The population was about 1,794,558 people in 1850 GUIDED PRACTICE Based on recent census data, the population of New York state can be modeled by 19.875 P t 1 57.993e 0.035005 t where P is the population in millions and t is the number of years since 1800. Based on this model, (b) What will New York state’s population be in 2020? 19.875 P 220 19.366967 0.035005 220 1 57.993e The population will be about 19,366,967 in 2020 GUIDED PRACTICE Based on recent census data, the population of New York state can be modeled by 19.875 P t 1 57.993e 0.035005 t where P is the population in millions and t is the number of years since 1800. Based on this model, (c) What is New York’s maximum sustainable population (limit to growth)? lim P t 19.875 = 19,875,000 people t