Modeling with Exponential Growth and Decay Sec. 3.1c Homework: p. 288 41-57 odd

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 Let P(t) be the population of San Jose t years after 1990. Year Population 1990 782,248 General Equation: 2000 894,943 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 Let P(t) be the population of San Jose t years after 1990. Year Population 1990 782,248 To solve for b, use the point (10, 894943): 2000 894,943

The population of San Jose will 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 Let P(t) be the population of San Jose t years after 1990. Year Population 1990 782,248 2000 894,943 Now, graph this function together with the line y = 1,000,000 The population of San Jose will exceed 1,000,000 in the year 2008

Practice Problems bacteria bacteria The number B of bacteria in a petri dish culture after t hours is given by 1. What was the initial number of bacteria present? bacteria 2. How many bacteria are present after 6 hours? bacteria

Practice Problems Populations of Two Major U.S. Cities City 1990 Population 2000 Population Flagstaff, AZ 44,545 67,880 Phoenix, AZ 3,455,902 4,003,365 Assuming exponential growth (and letting t = 0 represent 1990), what will the population of Flagstaff be in the year 2030?

Practice Problems  In the year 2007 Populations of Two Major U.S. Cities City 1990 Population 2000 Population Flagstaff, AZ 44,545 67,880 Phoenix, AZ 3,455,902 4,003,365 Assuming exponential growth (and letting t = 0 represent 1990), when will the population of Phoenix exceed 4.5 million?  In the year 2007

Practice Problems The two curves intersect at about t = 158.70, so Populations of Two Major U.S. Cities City 1990 Population 2000 Population Flagstaff, AZ 44,545 67,880 Phoenix, AZ 3,455,902 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… Domain: No Local Extrema H.A.: y = 0 Range: V.A.: None Continuous End Behavior: Decreasing on No Symmetry Bounded Above by y = 0

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 or 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!!!

Basic Function: The Logistic Function Domain: All reals Range: (0, 1) Continuous Increasing for all x y = 1 Symmetric about (0, ½), but neither even nor odd (0, ½) Bounded (above and below) No local extrema End Behavior: H.A.: y = 0, y = 1 V.A.: None

Guided Practice Graph the given function. Find the y-intercept, the horizontal asymptotes, and the end behavior. y-intercept: The limit to growth is 8, so: H.A.: y = 0, y = 8 End Behavior:

Guided Practice 20 y-intercept: H.A.: y = 0, y = 20 3 End Behavior: Graph the given function. Find the y-intercept, the horizontal asymptotes, and the end behavior. 20 y-intercept: H.A.: y = 0, y = 20 3 End Behavior:

Guided Practice Based on recent census data, a logistic model for the population of Dallas, t years after 1900, is 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??? The population of Dallas was 1 million by the end of 1984

Guided Practice The population was about 1,794,558 people in 1850 Based on recent census data, the population of New York state can be modeled by 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? The population was about 1,794,558 people in 1850

Guided Practice The population will be about 19,366,967 in 2020 Based on recent census data, the population of New York state can be modeled by 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? The population will be about 19,366,967 in 2020

Guided Practice = 19,875,000 people Based on recent census data, the population of New York state can be modeled by 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)? = 19,875,000 people