1. Calculus AB Quick Facts Part Two

2. First let’s talk about what the integral means! Can you list some interpretations of the definite integral?

3. Here’s a few facts: 1. If f(x) > 0, then returns the numerical value of the area between f(x) and the x-axis (area “under” the curve) 2. = F(b) – F(a) where F(x) is any anti-derivative of f(x). (Fundamental Theorem of Calculus) 3. Basically gives the total cumulative change in f(x) over the interval [a,b]

4. What is a Riemann Sum? Hint: Here’s a picture!

5. A Riemann sum is the area of n rectangles used to approximate the definite integral. = area of n rectangles As n approaches infinity… and So the definite integral sums infinitely many infinitely thin rectangles!

6. The indefinite integral = ?

7. Well…hard to write; easy to say The indefinite integral equals the general antiderivative… = F(x) + C Where F’(x) = f(x)

8. Now let’s see if you’ve memorized specific anti-derivatives that you will need to know quickly during the AP exam….

9. sike! I just made that one up to scare you…now the rest will seem easy!

10. = ?

11. ax + C I hope you got that one!

12. = ?

13. + C Ready?

14. = ??

15. - cos x + C Don’t forget we are going backwards! So if the derivative was positive, the anti-derivative is negative.

16. =?

17. sin x + C Got the negative/positive situation straight?? Good!

18. = ???

19. OK that’s a hard one! ln|tanx+sec x|+C If you got it right, you deserve a little treat!

20. = ?

21. tan x + C That should have been easy!

22. = ??

23. If you forget this one think: “tan x = sin x / cos x” (then let u = cos x, du = - sin x dx, etc.) -ln(cos x) + C or ln(sec x) + C

24. =??

25. ln |x| +C You need the absolute value in case x<0

26. where n > 1 Hint:

27. 1/x n = x -n sooooooo……. the answer is: + C You didn’t say ln(x n ) did ya??

28. = ?

29. e x + c Easiest anti-derivative in the universe, eh?

30. = ?

31. sec x + C Another easy peasy as a daisy anti-derivative!

32. = ?

33. Not toooo difficult? -cot x + C Safe landing?

34. = ??

35. -csc x + C How are you holding up? Bored out of your gourd? Suck it up! You’ll thank me when you test out of college calculus!

36. = ???

37. + C Grin and bear it!!

38. ?

39. tan -1 x + C Keep it going!!

40. ?

41. sin -1 x + C

42. ?

43. sec -1 x + C It’s all down hill now!!!!

44. I said you are done! Stop clicking.

45. How do you compute the average value of ?

46. ______________________ b - a dx Note: This is also known as the Mean (average) Value Theorem for Integrals

47. If = ky What does y = ?

48. Pre-Calculus trivia: doubling time is =

49. What’s general formula for a Riemann Sum?

50. or…more specifically Calculus trivia: as n (number of rectangles) goes to the summation sign becomes the integral sign and x becomes dx

51. What’s the Trapezoidal Rule?

52. The Trapezoidal Rule is the formula for estimating a definite integral with trapezoids. It is more accurate than a Riemann Sum which uses rectangles. Notice that all the y-values except the first and last are doubled. Do we need to take a short break?

53. Back already?

54. What is L’Hopital’s Rule?

55. Given that as x both f and g or both f and g then the limit of = the limit of as x L’Hopital’s Rule: ^

56. What is the Fundamental Theorem of Calculus???

57. where F ‘(x) = f(x) Do you know the other form? The one that is less commonly “used”? The FUN damental Theorem of Calculus:

59. What about if the question looks like this?

60. Did you remember Chain Rule?

61. What is the general integral for computing volume by slicing? (Assume we are revolving f(x) about the x-axis)

62. What if we revolve f(x) around y=a ?

64. What if we revolve the area between 2 functions: f(x) and g(x) around the x-axis?

65. Be sure to square the radii separately!!! (and put the larger function first)

66. 1. How do you compute displacement? (distance between starting & ending points) 2. How do you compute total distance traveled?

67. displacement: total distance:

68. Yea!!! That’s all folks!