Drill. Find the integral using the given substitution. Lesson 6.5: Logistic Growth. Day #1: P. 369: 1-17 (odd) Day #2: P. 369/70: 19-29; odd, 37. Partial Fraction Decomposition with Distinct Linear Denominators.
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Day #1: P. 369: 1-17 (odd)
Day #2: P. 369/70: 19-29; odd, 37
Because the numerator’s degree is larger than the denominator’s degree, we will need to use polynomial division.
When x = -1 Denominators
A(0) + B(-2) = 4; B = - 2
When x = 1
A(2) + B(0) = 4; A= 2
When x = -2
12B = 36
B = 3
When x = 1
-3C = -6
C = 2
When x = 2
When x = -7, B(-7) = -21, B = 3
When x = 0, 7A = 14, A = 2
Remember that exponential
growth can be modeled by
dP/dt = kP, for some k >0
If we want the growth rate
to approach 0 as P approaches
a maximal carrying capacity M,
we can introduce a limiting
factor of M – P.
Logistic Differential Equation
dP/dt = kP(M-P)
Where A is a constant determined by an appropriate initial condition. The carrying capacity M and the growth constant k are positive constants.
When will the population first exceed 300,000? Denominators
Write a logistic equation in the form of dP/dt = kP(M-P) that models the data.
We know that M = 316440.7 and Mk =.1026
Therefore 316440.7k=.1026, making k = 3.24 X 10-7
dP/dt= 3.24 X 10-7 P(316440.7-P)
Remember that M is the carrying capacity, and according to the equation, M = 100 bears
The growth rate is always
maximized when the
population reaches half
the carrying capacity.
In this case, it is 50 bears.
dP/dt = .008P(100-P)
dP/dt = .008(50)(100-50)
dP/dt =20 bears per year
Generate a slope field. The window should be x: [0, 25] and y: [0, 1000]
We first need to separate the variables. the fastest?
Then we will integrate one side by using partial fractions.
Multiply both sides by 1000 to clear decimals. the fastest?
Multiply both sides by -1 to put 1000-P on the top
Remember that ln(x) =y
Can be rewritten as ey = x
Solve for P P(0) = 61