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AB & BC Material

Below are answers for practice tests handed out in class. These were updated 4/18. There were some mistakes when they were published originally on 4/17. Answers for the practice tests in the preceding post are in those files already.

Practice Test 1
1. C 11. B 21. E 31. B 41. A
2. C 12. A 22. E 32. B 42. C
3. D 13. E 23. E 33. C 43. A
4. B 14. A 24. C 34. B 44. C
5. A 15. E 25. C 35. A 45. E
6. B 16. E 26. D 36. D
7. E 17. C 27. E 37. E
8. D 18. C 28. E 38. B
9. C 19. B 29. B 39. E
10. D 20. E 30. C 40. B

Practice Test 2
1. C 11. B 21. C 31. A 41. D
2. E 12. B 22. C 32. B 42. C
3. B 13. C 23. D 33. D 43. D
4. C 14. D 24. A 34. B 44. D
5. D 15. A 25. B 35. C 45. C
6. D 16. E 26. D 36. E
7. C 17. D 27. C 37. D
8. A 18. E 28. B 38. B
9. E 19. A 29. C 39. D
10. B 20. B 30. A 40. C

Practice Test 3
1. B 11. A 21. A 31. A 41. E
2. C 12. D 22. B 32. B 42. E
3. A 13. E 23. C 33. D 43. C
4. D 14. C 24. C 34. B 44. D
5. B 15. B 25. E 35. C 45. C
6. D 16. B 26. A 36. C
7. D 17. D 27. D 37. D
8. D 18. B 28. D 38. A
9. B 19. C 29. E 39. E
10. C 20. C 30. C 40. C

Practice AB Tests

Practice AB Test 01
Practice AB Test 02

Area Between Two Curves

Larson 7.0 Ch 6 Section 1 1-29 odds

Slope fields, law of exponential change and separation of variables

Work on slope field problems, and law of exponential change problems. You also find it helpful to work on some separation of variables problems.

Integration By Substitution

Please see the videos below for a review of today's lesson and then some. This is a critical topic and one that just needs some practice. Please complete all of the hw assigned by Monday. As always, worked out solutions can be found in the online textbook.

We will have a test on this material the first time we meet next week. There are only two weeks left before break after this week and we have a few important topics to learn prior to that. Thanks for your hard work and understanding! We are almost there.

Videos for Integration by Substitution:
1. Introduction to Integration by Substitution
2. Basic Examples of Integration by Substitution
3. More Complex Examples of Integration by Substitution
4. Change of Variables with Integration by Substitution
5. Examples of Integration by Substitution for Definite Integrals
6. More Complex Example of Integration by Substitution for Definite Integrals

Larson 7.0 Ch 4 Section 5 #7-33 odds


Larson 7.0 Ch 4 Section 5 #41-53 odds


Larson 7.0 Ch 4 Section 5 #57-75 odds. When it says as in example 5, it means using the change of variable strategy discussed in video 4 above.

the Fundamental Theorem of Calculus and the Mean Value Theorem for Integrals

In the previous post, we have already worked with the FTC Part I or the First FTC. I include it again in this section with some additional problems to practice from Larson 7 Ch 4 Section 4.





Related to this is the Mean Value Theorem for Integrals and the Average Value of a Function. You can see that the two are just slight variations of each other. In both f(c) refers to the average value of f(x) on the closed interval [a,b].









Then we have the FTC Part 2 or the Second FTC. Once you get the idea it is fairly straight forward. When the upper limit of integration is a function of x and not just x, remember to apply the chain rule and multiply by the derivative of the upper limit.





And finally a few problems to put it all together:

Definite Integrals

There are two special properties of the definite integral that are explained in the following graphic. Examples are also provided.



Below are exercises related to solving definite integrals algebraically. This formula is known as the Fundamental Theorem of Calculus Part I. Guidelines for evaluating definite integrals using the FTC Part I are below:





Below are exercises that utilize the three different RAM methods.



Below are exercises that utilize what is called the trapezoidal approximation method or the trapezoidal rule. The definition of the trapezoidal rule is shown below:



And then an example of the trapezoidal rule in action:



And an assignment: