1. 6.5 Part I: Partial Fractions

4. This would be a lot easier if we could re-write it as two separate terms. 1 These are called non- repeating linear factors. You may already know a short-cut for this type of problem. We will get to that in a few minutes.

5. This would be a lot easier if we could re-write it as two separate terms. Multiply by the common denominator. Set like-terms equal to each other. Solve two equations with two unknowns. 1

6. This technique is called Partial Fractions 1

7. The short-cut for this type of problem is called the Heaviside Method, after English engineer Oliver Heaviside. Multiply by the common denominator. 1 Let x = - 1 Let x = 3

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

9. Ex:

11. Good News! 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.

12. Repeated roots: we must use two terms for partial fractions. 2

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

16. irreducible quadratic factor repeated root first degree numerator A challenging example:

19. We can do this problem on the TI-89: expand ((-2x+4)/((x^2+1) * (x-1)^2)) Of course with the TI-89, we could just integrate and wouldn’t need partial fractions!  3F2

20. 6.5 Part II Logistic Growth Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004 Columbian Ground Squirrel Glacier National Park, Montana

21. 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.

22. 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 )

23. The equation then becomes: Logistics Differential Equation We can solve this differential equation to find the logistics growth model.

24. Partial Fractions Logistics Differential Equation

27. Logistics Growth Model

28. Example: Logistic Growth Model 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?

29. 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?

30. At time zero, the population is 10.

31. After 10 years, the population is 23.

32. 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 

33. Gorilla Population A certain wild animal preserve can support no more than 250 lowland gorillas. Twenty-eight gorillas were known to be in the preserve in 1970. Assume that the rate of growth of the population is Where time t is in years. a)Find a formula for the gorilla population in terms of t. b)How long will it take the gorilla population to reach the carrying capacity of the preserve?

37. Gorilla Population A certain wild animal preserve can support no more than 250 lowland gorillas. Twenty-eight gorillas were known to be in the preserve in 1970. Assume that the rate of growth of the population is Where time t is in years. a)Find a formula for the gorilla population in terms of t. b)How long will it take the gorilla population to reach the carrying capacity of the preserve?