1. Guillaume De l'Hôpital 1661 - 1704 Actually, L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. De L’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons. 8.1 L’Hôpital’s Rule

2. Zero divided by zero can not be evaluated, and is an example of indeterminate form. Consider: If we try to evaluate this by direct substitution, we get: In this case, we can evaluate this limit by factoring and canceling: 8.1 L’Hôpital’s Rule

3. If we zoom in far enough, the curves will appear as straight lines. The limit is the ratio of the numerator over the denominator as x approaches 2. 8.1 L’Hôpital’s Rule

4. As becomes: 8.1 L’Hôpital’s Rule

6. L’Hôpital’s Rule: If is indeterminate, then: 8.1 L’Hôpital’s Rule

7. We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative: 8.1 L’Hôpital’s Rule

8. Example: If it’s no longer indeterminate, then STOP! If we try to continue with L’Hôpital’s rule: which is wrong, wrong, wrong! 8.1 L’Hôpital’s Rule

9. On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate: not 8.1 L’Hôpital’s Rule

10. L’Hôpital’s rule can be used to evaluate other indeterminate forms besides. The following are also considered indeterminate: The first one,, can be evaluated just like. The others must be changed to fractions first. 8.1 L’Hôpital’s Rule

11. This approaches 8.1 L’Hôpital’s Rule

12. Indeterminate Forms: L’Hôpital applied 8.1 L’Hôpital’s Rule

13. The function grows very fast. If x is 3 inches, y is about 20 inches: We have gone less than half- way across the board horizontally, and already the y- value would reach the Andromeda Galaxy! At 64 inches, the y-value would be at the edge of the known universe! (10.5 billion light-years) 8.2 Relative Rates of Growth

14. The function y = ln x grows very slowly. We would have to move 2.6 miles to the right before the line moves a foot above the x-axis! By the time we reach the edge of the universe again (10.5 billion light-years) the chalk line will only have reached 64 inches! The function y = ln x increases everywhere, even though it increases extremely slowly. 8.2 Relative Rates of Growth

15. Definitions: Faster, Slower, Same-rate Growth as Let f ( x ) and g ( x ) be positive for x sufficiently large. 1. f grows faster than g (and g grows slower than f ) as if or 2. f and g grow at the same rate as if 8.2 Relative Rates of Growth

16. WARNING Please temporarily suspend your common sense. 8.2 Relative Rates of Growth

17. According to this definition, y = 2x does not grow faster than Since this is a finite non-zero limit, the functions grow at the same rate! The book says that “ f grows faster than g ” means that for large x values, g is negligible compared to f. 8.2 Relative Rates of Growth

18. Which grows faster, or ? This is indeterminate, so we apply L’Hôpital’s rule. Still indeterminate. grows faster than. We can confirm this graphically: 8.2 Relative Rates of Growth

19. “Growing at the same rate” is transitive. In other words, if two functions grow at the same rate as a third function, then the first two functions grow at the same rate. 8.2 Relative Rates of Growth

20. Show that and grow at the same rate as. 8.2 Relative Rates of Growth

21. f and g grow at the same rate. 8.2 Relative Rates of Growth

22. Definition f of Smaller Order than g Let f and g be positive for x sufficiently large. Then f is of smaller order than g as if We write and say “ f is little-oh of g.” Saying is another way to say that f grows slower than g. 8.2 Relative Rates of Growth

23. Saying is another way to say that f grows no faster than g. Definition f of at Most the Order of g Let f and g be positive for x sufficiently large. Then f is of at most the order of g as if there is a positive integer M for which We write and say “ f is big-oh of g.” for x sufficiently large 8.2 Relative Rates of Growth

24. Until now we have been finding integrals of continuous functions over closed intervals. Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals. 8.3 Improper Integrals

25. The function is undefined at x = 1. Since x = 1 is an asymptote, the function has no maximum. Can we find the area under an infinitely high curve? We could define this integral as: (left hand limit) We must approach the limit from inside the interval. 8.3 Improper Integrals

27. This integral converges because it approaches a solution. 8.3 Improper Integrals

28. This integral diverges. 8.3 Improper Integrals

29. The function approaches when. 8.3 Improper Integrals

31. What happens here? If then gets bigger and bigger as, therefore the integral diverges. If then b has a negative exponent and, therefore the integral converges. (P is a constant.) 8.3 Improper Integrals

32. Converges 8.3 Improper Integrals

33. Does converge? Compare: to for positive values of x. For 8.3 Improper Integrals

34. For Since is always below, we say that it is “bounded above” by Since converges to a finite number, must also converge! 8.3 Improper Integrals

35. Direct Comparison Test: Let f and g be continuous on with for all, then: 2 diverges if diverges. 1 converges if converges. 8.3 Improper Integrals

36. The maximum value of so: on Since converges, converges. 8.3 Improper Integrals

37. for positive values of x, so: Since diverges, diverges. on 8.3 Improper Integrals

38. If functions grow at the same rate, then either they both converge or both diverge. Does converge? As the “1” in the denominator becomes insignificant, so we compare to. Since converges, converges. 8.3 Improper Integrals

41. 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. 8.4 Partial Fractions

42. Solve two equations with two unknowns. This technique is called Partial Fractions 8.4 Partial Fractions

43. 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. 8.4 Partial Fractions

44. Repeated roots: we must use two terms for partial fractions. 8.4 Partial Fractions

45. If the degree of the numerator is higher than the degree of the denominator, use long division first. (from example one) 8.4 Partial Fractions

46. irreducible quadratic factor repeated root 8.4 Partial Fractions

47. These are in the same form. 8.4 Partial Fractions

48. This is a constant. 8.4 Partial Fractions

49. If the integral contains, we use the triangle at right. If we need, we move a to the hypotenuse. If we need, we move x to the hypotenuse. 8.4 Partial Fractions

52. We can get into the necessary form by completing the square. 8.4 Partial Fractions

54. Complete the square: 8.4 Partial Fractions

56. Here are a couple of shortcuts that are result from Trigonometric Substitution: These are on your list of formulas. They are not really new. 8.4 Partial Fractions