1. 4.9 Antiderivatives Wed Feb 4 Do Now Find the derivative of each function 1) 2)

2. Antiderivatives Antiderivative - the original function in a derivative problem (backwards) F(x) is called an antiderivative of f(x) if F’(x) = f(x) Antiderivatives are also known as integrals

3. Integrals + C When differentiating, constants go away When integrating, we must take into consideration the constant that went away

4. Indefinite Integral Let F(x) be any antiderivative of f. The indefinite integral of f(x) (with respect to x) is defined by where C is an arbitrary constant

5. Examples Examples 1.2 and 1.3

6. The Power Rule For any rational power 1) Exponent goes up by 1 2) Divide by new exponent

7. Examples Examples 1.4, 1.5, and 1.6

8. The integral of a Sum You can break up an integrals into the sum of its parts and bring out any constants

9. EX

10. Closure Hand in: Integrate the following function HW: p. 280 #1-2 11-23 odds

11. 4-9 Integrals of Trig, e, lnx Thurs Feb 5 Do Now Integrate the following: 1) 2)

12. HW Review: p.280 #1, 2, 11- 23 odds 1) 23) 2) 11) 13) 15) 17) 19) 21)

13. Trigonometric Integrals These are the trig integrals we will work with:

14. Examples Ex 1.7

15. Exponential and Natural Log Integrals You need to know these 3:

16. Example Ex 1.8

17. You try Integrate the following: 1) 2) 3)

18. Closure Hand in: Integrate the following HW: p. 280 #3-9 odds 26-29 all 36

19. 4-9 Integrals of the form f(ax) Fri Feb 6 Do Now Evaluate the following integrals

20. HW Review p.280 3-9 26-29 36 3) 5) 2sinx + 9cosx + C 36) 4lnx – e^x + C 7) 9) a-iib-iiic-id-iv 26) 27) 12sec x + C 28) 29) –csc t + C

21. Integrals of the form f(ax) We have now seen the basic integrals and rules we’ve been working with What if there’s more than just an x inside the function? Like sin 2x?

22. Integrals of Functions of the Form f(ax) If, then for any constant, Step 1: Integrate using any rule Step 2: Divide by a

23. Examples Ex 1.9

24. You Try Evaluate the integrals

25. Closure Hand in: Integrate the following HW: p.281 #31-39 odds, 30 38 Quiz Next Thurs

26. 4-9 Finding original functions through integrating Mon Feb 9 Do Now Integrate 1) 2)

27. HW Review p.281 #30-39 30) 31) 33) 35) 37) 38) 39)

28. Revisiting the + C Recall that every time we integrate a function, we need to include + C Why?

29. Solving for C We can solve for C if we are given an initial value. Step 1: Integrate with a + C Step 2: Substitute the initial x,y values Step 3: Solve for C Step 4: Substitute for C in answer

30. Examples

31. You try Find the original function

32. Closure Hand in: Find the original function of HW: p.281-282 #47-61 odds 4.9 Quiz Thurs Feb 12

33. 4-9 Working from the 2 nd derivative Tues Feb 10 Do Now Integrate and find C 1) 2)

34. HW Review p.281-2 #47-61 47) 49) 51) 53) 55) 57) 59) 61)

35. Finding f(x) from f’’(x) When given a 2 nd derivative, use both initial values to find C each time you integrate EX: f’’(x) = x^3 – 2x, f’(1) = 0, f(0) = 0

36. Acceleration, Velocity, and Position Recall: How are acceleration, velocity and position related to each other?

37. Integrals and Acceleration We integrate the acceleration function once to get the velocity function –Twice to get the position function. Initial values are necessary in these types of problems

38. Example 1 If a space shuttle’s downward acceleration is given by y’’(t) = -32 ft/s^2, find the position function y(t). Assume that the shuttle’s initial velocity is y’(0) = -100 ft/s, and that its initial position is y(0) = 100,000 ft.

39. Ex 2 A car traveling with velocity 24m/s begins to slow down at time t = 0 with a constant deceleration of a = -6 m/s^2. When t = 0, the car has not moved. Find the velocity and position at time t.

40. Closure Hand in: Determine the position function if the acceleration function is a(t) = 12, the initial velocity is v(0) = 2, and the initial position is s(0) = 3 HW: p.282 #63-69 odds 4.9 Quiz Thurs Feb 12

41. 4.9 Review Wed Feb 11 Do Now If a ball is thrown up into the air and begins to fall, it has an acceleration function of a(t) = -32 ft/s^2. Find the position function if the initial velocity is v(0) = 0, and its initial position is s(0) = 20 ft

42. HW Review p.282 #63-69 63) 65) 67) 69)

43. Integral Quiz Review What to know: –Power Rule –Trig Rules (sinx, cosx, sec^2 x) –The two exponential rules –Ln x –Sums and differences of integrals –Integral of f(ax) –Solving for C 2 nd deriv / Acceleration may be included in this section

44. Review Worksheet p.332 #1-24 27 29-32 #55-60 65-68 +C

45. Closure Journal Entry: What is integration? How are integrals and derivatives related? HW: Finish worksheet Quiz Thurs Feb 12

46. 4.9 Review Tues Feb 11 Do Now Given f ’’(x) = -32, f ‘ (0) = 2, and f(1) = 5, find f(x)

47. HW Review: p.332 #5,7,10,11,12 5) 7) 10) 11) 12)

48. HW Review p.332 #15 16 19 21 23 15) -2cosx + sinx + C 16) 3sinx + cosx + C 19) 5tanx + C 21) 23) 3sinx - ln|x| + C

49. HW Review p.332 #27 29 31 34 39 27) 29) 31) 34) tan3x + C 39)

50. HW Review p.333 #55-60 55) 56) 57) 58) 59) 60)

51. HW Review p.333 #65-68 65) 66) 67) 68)