AP CALCULUS BC, Independent Study

This is a weekly schedule to complete the AP Calculus BC curriculum, for a student who has already successfully completed AP Calculus AB.

TEXT: Calculus: Graphical, Numerical, Algebraic, by Finney, Thomas, Demana, Waits; Addison-Wesley.

The course is pass/fail; there will be several major tests, as indicated in the weekly schedule below.

Students will learn the material independently, setting up appointments and asking Dr. Fisher questions on an "as-needed" basis.

You should do as many odd-numbered homework problems in each section as time permits, checking answers in the back of your text.

1. WEEK #1, August 28–29
Study these Sample Prerequisite problems and solutions, to review foundational material.
QUIZ NEXT WEEK

2. WEEK #2, September 2–5
Study the "AP CALCULUS AB in-a-nutshell" sheets (supplied by Dr. Fisher) to review all the important ideas in AP Calculus AB.
QUIZ NEXT WEEK

3. WEEK #3, September 8–12
6.4 Lengths of Curves in the Plane
7.5 Indeterminate Forms and l'Hopital's Rule

4. WEEK #4, September 15–19
8.2 Integration by Parts
8.5 Rational Functions and Partial Fractions

5. WEEK #5, September 22–26
8.6 Improper Integrals

6. WEEK #6, September 29–30, October 1–3
8.7 Differential Equations (includes the logistic growth model, Euler's method)

7. WEEK #7, October 6–10
9.1 Limits of Sequences of Numbers
9.2 Infinite Series

8. WEEK #8, October 14–17
9.3 Series without Negative Terms: Comparison and Integral Tests

9. WEEK #9, October 20–24
TEST OVER FIRST QUARTER MATERIAL (from odd-numbered homework problems)

10. WEEK #10, October 27–31 (end of first quarter)
9.4 Series with Nonnegative Terms: Ratio and Root Tests
9.5 Alternating Series and Absolute Convergence

11. WEEK #11, November 3–7
9.6 Power Series
9.7 Taylor Series and Maclaurin Series

12. WEEK #12, November 17–21
9.8 Further Calculations with Taylor Series

13. WEEK #13, November 24–25
10.3 Parametric Equations for Plane Curves
10.4 The Calculus of Parametric Equations

14. WEEK #14, December 1–4
10.5 Polar Coordinates
10.6 Graphing in Polar Coordinates

15. WEEK #15, December 8–12
10.7 Polar Equations of Conic Sections
10.8 Integration in Polar Coordinates

16. WEEK #16, December 15–19
11.1 Vectors in the Plane
11.2 Cartesian (Rectangular) Coordinates and Vectors in Space

17. WEEK #17, January 5–9, 2009
TEST OVER SECOND QUARTER MATERIAL (from odd-numbered homework problems)

18. WEEK #18, January 20–23 (end of second quarter)
11.3 Dot Products
11.4 Cross Products

19. WEEK #19, January 26–30
11.5 Lines and Planes in Space
12.1 Vector-valued Functions and Curves in Space; Derivatives and Integrals

20. WEEK #20, February 2–6
12.2 Modeling Projectile Motion
12.3 Directed Distance and the Unit Tangent Vector T

21. WEEK #21, February 9–13
Make up your own "in-a-nutshell" sheet for all the important BC concepts

22. WEEK #22, February 23–27
TEST OVER ALL BC MATERIAL (from odd-numbered homework problems)

23. WEEK #23, March 2–6
TEST OVER both AB and BC "in-a-nutshell" sheets

24. WEEK #24, March 9–13
sample tests (AB and BC)

25. WEEK #25, March 16–20
sample tests (AB and BC)

26. WEEK #26, March 23–27
sample tests (AB and BC)

27. WEEK #27, March 30–31, April 1–3 (end of third quarter)
sample tests (AB and BC)

28. WEEK #28, April 6–9
sample tests (AB and BC) (one graded)

29. WEEK #29, April 13–17
sample tests (AB and BC) (one graded)

30. WEEK #30, April 27–30, May 1
sample tests (AB and BC) (one graded)

31. WEEK #31, May 4–8 (week of AP CALCULUS BC exam)

The BC test covers all the topics on the AB test.
Here are the additional topics on the AP Calculus BC Exam:
• Parametric, polar, and vector functions: The analysis of planar curves includes those given in parametric form, polar form, and vector form.
• Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration
• Numerical solution of differential equations using Euler's method
• L'Hospital's Rule, including its use in determining limits and convergence of improper integrals and series
• Derivatives of parametric, polar, and vector functions
• Applications of integrals: Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, the length of a curve (including a curve given in parametric form), and accumulated change from a rate of change.
• Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only)
• Improper integrals (as limits of definite integrals)
• Solving logistic differential equations and using them in modeling
• Concept of series: A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence and divergence.
• Series of constants: Motivating examples, including decimal expansion
• Geometric series with applications
• The harmonic series
• Alternating series with error bound
• Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series
• The ratio test for convergence and divergence
• Comparing series to test for convergence or divergence
• Taylor series: Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve)
• Maclaurin series and the general Taylor series centered at x = a
• Maclaurin series for the functions e^x, sin x, cos x, and 1/(1-x)
• Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series
• Functions defined by power series
• Radius and interval of convergence of power series
• Lagrange error bound for Taylor polynomials