AP Calculus AB/BC 6.5 Logistic Growth, p. 362

Example 1: This would be a lot easier if we could re-write it as two separate terms. These are called non-repeating linear factors.

Example 1: The short-cut for this type of problem is called the Heaviside Method, after English engineer Oliver Heaviside. Multiply by the common denominator. Let x = −1 so we can solve for B. Let x = 3 so we can solve for A.

Example 1: The short-cut for this type of problem is called the Heaviside Method, after English engineer Oliver Heaviside.

Example 2: Note the degree of the numerator is larger than the degree of the denominator. So, use long division to divide the polynomials. 2x 2x3 – 8x 8x

Example 2:

Example 3: Solve the differential equation.

Example 4: If the degree of the numerator is higher than the degree of the denominator, use long division first. (from example one)

Good News! p The AP Exam only requires non-repeating linear factors! The more complicated methods of partial fractions are good to know, and you might see them in college, but they will not be on the AP exam or on my exam. p

We have used the exponential growth equation to represent population growth. The exponential growth equation occurs when the rate of growth is proportional to the amount present. If we use P to represent the population, the differential equation becomes: The constant k is called the relative growth rate.

The population growth model becomes: However, real-life populations do not increase forever. There is some limiting factor such as food, living space or waste disposal. There is a maximum population, or carrying capacity, M. A more realistic model is the logistic growth model where growth rate is proportional to both the amount present (P) and the carrying capacity that remains: (M-P)

The equation then becomes: Logistics Differential Equation

Example 5: Using the logistic equation dP/dt = 0.0002P(1200 – P) where t is measured in years, find the: Carrying capacity of the population Size of the population when it is growing the fastest Rate at which the population is growing when it is growing the fastest. (a) The carrying capacity is 1200. (b) Size of the population when it is growing the fastest is 1200/2 = 600. (c) dP/dt = 0.0002(600)(1200 – 600) = 72 this year.

Logistics Differential Equation We can solve this differential equation to find the logistics growth model. Logistics Growth Model

Logistic Growth Model Example: Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears. Assuming a logistic growth model, when will the bear population reach 50? 75? 100?

Ten grizzly bears were introduced to a national park 10 years ago Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears. Assuming a logistic growth model, when will the bear population reach 50? 75? 100?

At time zero, the population is 10.

After 10 years, the population is 23.

p We can graph this equation and use “trace” to find the solutions. Years Bears We can graph this equation and use “trace” to find the solutions. y=50 at 22 years y=75 at 33 years y=100 at 75 years p