MODELING WITH EXPONENTIAL GROWTH AND DECAY Sec. 3.1c Homework: p. 288 41-57 odd.

MODELING MODELING WITH WITH EXPONENTIAL EXPONENTIAL GROWTH AND GROWTH AND DECAYDECAY

Sec. 3.1cSec. 3.1c

Homework: p. 288 41-57 oddHomework: p. 288 41-57 odd

PRACTICE PROBLEMSPRACTICE PROBLEMS

0tP t P b

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, CAYear Population

1990 782,248

2000 894,943

Let P(t) be the population ofSan Jose t years after 1990.

General Equation:

Initial Population

Growth Factor


1010 782,248 894,943P b



1990 782,248

2000 894,943


To solve for b,use the point (10, 894943):

10894,943

782,248b 1.0135


782,248 1.0135tP t



1990 782,248

2000 894,943


Now, graph this function together with the line y = 1,000,000

18.31t The population of San Jose willThe population of San Jose willexceed 1,000,000 in the year 2008exceed 1,000,000 in the year 2008


The number B of bacteria in a petri dish culture after t hours isgiven by 0.693100 tB e1. What was the initial number of bacteria present?

2. How many bacteria are present after 6 hours?

0 100B bacteria

6 6394B bacteria


44,545 1.0430tFP t

Populations of Two Major U.S. CitiesCity 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?

40 240,199FP


3,455,902 1.0148tPP t

Populations of Two Major U.S. CitiesCity 1990 Population 2000 Population


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?

17.95t In the year 2007In the year 2007

PRACTICE PROBLEMSPRACTICE PROBLEMSPopulations of Two Major U.S. Cities

City 1990 Population 2000 Population


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 estimatein answer to this question?

The two curves intersect at about The two curves intersect at about t t = 158.70, so= 158.70, soaccording to these models, the population of Flagstaffaccording to these models, the population of Flagstaffwill exceed that of Phoenix in the year 2148…will exceed that of Phoenix in the year 2148…

5 0.8x

f x

Domain: , Range:

Continuous

Decreasing on

No Symmetry

Bounded Above by y = 0

No Local Extrema

H.A.: y = 0

V.A.: None

End Behavior:

lim 0x

f x

,0

,

limx

f x

So far, we’ve looked primarily at exponentialSo far, we’ve looked primarily at exponentialgrowth, which is growth, which is unrestrictedunrestricted. (meaning?). (meaning?)

However, in many “real world” situations, it isHowever, in many “real world” situations, it ismore realistic to have an upper limit on growth.more realistic to have an upper limit on growth.(any examples?)(any examples?)

In such situations, growth often startsIn such situations, growth often startsexponentially, but then slows and eventuallyexponentially, but then slows and eventuallylevels out………………………does this remind youlevels out………………………does this remind youof a function that we’ve previously studied???of a function that we’ve previously studied???

Let a, b, c, and k be positive constants, with b < 1. A logisticgrowth function in x is a function that can be written in the form

or 1 x

cf x

a b

1 kx

cf x

a e

where the constant c is the limit to growth.

Note: If b > 1 or k < 0, these formulas yield logistic decayfunctions (unless otherwise stated, the term logistic functionswill refer to logistic growth functions).

With a = c = k = 1, weget our basic function!!! 1

1 xf x

e

y = 1

1

1 xf x

e

(0, ½)

Domain: All reals Range: (0, 1)

Continuous Increasing for all x

Symmetric about (0, ½), butneither even nor odd

Bounded (above and below)

No local extrema

H.A.: y = 0, y = 1

V.A.: None

End Behavior:

lim 0x

f x

lim 1x

f x

Graph the given function. Find the y-intercept, the horizontalasymptotes, and the end behavior.

y-intercept:

8

1 3 0.7xh x

0

80

1 3 0.7h

8

21 3

The limit to growth is 8, so:

H.A.: y = 0, y = 8

End Behavior: lim 0x

h x

lim 8xh x

Graph the given function. Find the y-intercept, the horizontalasymptotes, and the end behavior.

y-intercept:

3

20

1 2 xg x

e

End Behavior: lim 0x

g x

lim 20xg x

20

3H.A.: y = 0, y = 20

Based on recent census data, a logistic model for the populationof Dallas, t years after 1900, is

0.05055

1,301,614

1 21.603 tP t

e

The population of Dallas was 1 million by the end The population of Dallas was 1 million by the end of 1984of 1984

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??? 84.50t

Based on recent census data, the population of New York statecan be modeled by

The population was about 1,794,558 people in 1850The population was about 1,794,558 people in 1850

0.035005

19.875

1 57.993 tP t

e

where P is the population in millions and t is the number of yearssince 1800. Based on this model,

(a) What was the population of New York in 1850?

0.035005 50

19.87550

1 57.993P

e

1.794558


The population will be about 19,366,967 in 2020The population will be about 19,366,967 in 2020

0.035005

19.875

1 57.993 tP t

e


(b) What will New York state’s population be in 2020?

0.035005 220

19.875220

1 57.993P

e

19.366967


= 19,875,000 people= 19,875,000 people

0.035005

19.875

1 57.993 tP t

e


(c) What is New York’s maximum sustainable population (limit togrowth)?

lim 19.875tP t

MODELING WITH EXPONENTIAL GROWTH AND DECAY Sec. 3.1c Homework: p. 288 41-57 odd.

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