Quick Review
2
3
2
1. Use polynomial division to write the rational
( )function in the form ( ) , where the degree
( )
of is less than the degree of .
a. 13
b. 1
R xQ x
D x
R D
x x
xx
x
1
22
x
x
1
3
x
xx
Quick Review
-0.1
x
x
302. Let ( ) .
1 5a. Find where ( ) is continuous.
b. Find lim ( ).
c. Find lim ( ).
d. Find the y-intercept of the graph of .
e. Find all horizontal asymptotes of the graph of .
xf x
ef x
f x
f x
f
f
,
30
0
5
30 and 0 yy
What you’ll learn about How Populations Grow Partial Fractions The Logistic Differential Equation Logistic Growth Models
Essential QuestionHow do we use logistic growth models inCalculus to help us find growth in the realworld?
Partial Fraction Decomposition with Distinct Linear Denominators
If where P and Q are polynomials with the degree
of P less than the degree of Q, and if Q(x) can be written as a product of distinct linear factors, then f (x) can be written as a sum of rational functions with distinct linear denominators.
,xQ
xPxf
Example Finding a Partial Fraction Decomposition
122
xxx 486221212 2 xxxxCxxBxxA
14A 10B 42Set x 1 So A
14
4862
2
xx
xxxf1. Write the function as a sum of
rationalfunctions with linear denominators.
122
486 2
xxx
xxxf
122
x
C
x
B
x
A
12 xxA 12 xxB 22 xxC
40C 30 A 34 B 362Set x 3 So B 04C 03A 01 B 61Set x 2 So C 31C
122
486 2
xxx
xxxfTherefore
2
1
x 2
3
x 1
2
x
Example Antidifferentiating with Partial Fractions
2ln x
Cxxx 2312 2 ln
122
486 2
dxxxx
xx
2. Find
dxxxx
xx
122
486 2
dxxxx
1
2
2
3
2
1
2ln3 x 1ln2 x C
Example Antidifferentiating with degree higher in Numerator
4
22
3
dxx
x
3. Find
00024 232 xxxxx2
xxx 802 23 x8
4
82
2 x
xx
dxx
x
4
22
3
dx
x
xx
4
82
2
2x dxx
x
4
82
Example Antidifferentiating with degree higher in Numerator
4
22
3
dxx
x
3. Find
2x dxx
x
4
82
22
8
xx
x
22
x
B
x
A
22
xx 2xA 2 xB xxBxA 822
4A 0B 162Set x 4 So A 0A 4 B 162Set x 4 So B
dxxx
x 2
4
2
42
2x 2ln4 x 2ln4 x
Cxxx 2 2 ln42
C
What you’ll learn about How Populations Grow Partial Fractions The Logistic Differential Equation Logistic Growth Models
Essential QuestionHow do we use logistic growth models inCalculus to help us find growth in the realworld?
Logistic Differential Equation
Exponential growth can be modeled by the differential equation
for some k > 0.kP
dt
dP
If we want the growth rate to approach zero as P approaches a maximal carrying capacity M, we can introduce a limiting factor of M – P :
This is the logistic differential equation.
PMkPdt
dP
Population is growing the fastest.Slope is at its steepest.
Maximum Capacity
Example Logistic Differential Equation
4. The growth rate of a population P of bears in a newly established wildlife preserve is modeled by the differential equation
,100008.0 PPdt
dP where t is measured in years.
a. What is the carrying capacity for the bears in
this wildlife preserve?
b. What is the bear population when the
population is growing the fastest?
c. What is the rate of change of the population
when it is growing the fastest?
bears 100
bears 50
008.0dt
dP 50 50100 20 yearper bears
Example Logistic Differential Equation
4. The growth rate of a population P of bears in a newly established wildlife preserve is modeled by the differential equation
,100008.0 PPdt
dP where t is measured in years.
d. Solve the differential equation with the initial
condition P(0) = 25.
dtPP
dP008.0
100
dP
dP
P
B
P
A
100 dt .0080
1100 PBPA1100 A0Set P 01.0 So A1100 B100Set P 01.0 So B
dtdPPP
008.0 100
01.001.0
dtdPPP
8.0 100
11
Pln P 100ln Ct 8.0 P100ln Pln Ct 8.0
P
P100ln Ct 8.0
1100
P
Cte 8.0
P
100 Cte 8.01
Example Logistic Differential Equation
4. The growth rate of a population P of bears in a newly established wildlife preserve is modeled by the differential equation
,100008.0 PPdt
dP where t is measured in years.
d. Solve the differential equation with the initial
condition P(0) = 25.
P
100 Cte 8.01
25
100 Ce1
Ce3
P
100 ct ee 8.01
teP
8.031100
te
P8.031
1
100
teP
8.031
100
The General Logistic FormulaThe solution of the general logistic differential equation
is
PMkPdt
dP
tkMeA
MP
1
where A is a constant determined by an appropriate initial condition. The carrying capacity M and the growth constant k are positive constants.
Example Logistic Differential Equation
5. The table shows the population of Aurora, CO for selected years between 1950 and 2003.
a. Use logistic regression to find a logistic curve to model the data and superimpose it on a scatter plot of population against years after 1950?
b. Based on the regression equation, what will the Aurora population approach in the long run?
c. Based on the regression equation, when will the population of Aurora first exceed 300,000 people?
d. Write a logistic differential equation in the form dP/dt = kP(M – P) that models the growth of the Aurora data in the table.
teP
1026.0577.231
7.316440
people 441,316
te 1026.0577.231
7.316440000,300
0548.1577.231 1026.0 te
0548.577.23 1026.0 te
00232.1026.0 te
1.59t2009in
dt
dP1026.0Mk
7.316440M71024.3 k 71024.3 P P7.316440