HOW TO: Self-Studying Calculus BC while taking Calc AB

Equilibrium
Mon, 20 Jun 2011 00:48:13 +0000

A year ago, I self-studied Calc BC while doing Calc AB. I felt it was a highly worthwhile experience - my AB class was going way too slow and calc BC made things interesting. It wasn’t extremely difficult, but it was still pretty challenging. However, the experience was highly worth it. Not spending hundreds of hours doing Calc II in college + saving thousands of dollars = very well worthwhile experience.

AP Calc AB vs. Calc BC
In college, there are two calculus courses: Calculus I and Calculus II. Calc AB tests on Calculus I ONLY, and Calc BC tests BOTH Calculus I and Calculus II – Approx. 60% of the BC exam is the same as the AP Calculus AB exam, while the other 40% is on Calc II material.

Should I do Calculus BC?
It is a very worthwhile endeavor… anyone talented at math should definitely consider doing it. Calculus BC doesn’t have a TON of new material, and you could also save thousands of dollars and skip an entire semester of calculus in college. Calc BC isn’t too tough – think of it as like taking an additional honors-level math course during your senior year. However, generally, you should be very competent at math to independently study BC on your own. By competent, I mean having mastered the math PRIOR to calculus, and having strong analytical and problem-solving skills. I do NOT mean being innately good or possessing natural math talent. You also need to be resourceful and independent… can you trust yourself to stay caught up without a teacher forcing deadlines on you? If you’re still unsure, wait and try out Calc AB - if it’s tough, don’t worry about it. If you’re coasting through Calc AB, then consider doing BC!

Should I Skip Calculus II in College?
Unless you are completely clueless, do it. Calculus 2 is mostly an amalgam of stuff you will never see again. As long as you can integrate pretty well, and understand how to do calculus with parametric/polar functions, you will be fine for Calculus III (i.e. vector calculus)

Should I get a textbook?
Just use your school textbook, and your review book (see below for details on review books) and you honestly should be fine.

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Here are the topics in Calculus BC that are NOT in Calc AB: I also attached some commentary to each topic.

Unit 1 – Pre-Requisites for Calculus
Completing the Square
Complete Mastery of Trigonometric Identities!!
Partial Fractions Separation Technique (VERY IMPORTANT)
Vectors in a Plane (Basically, understand know what a vector is – you DO NOT NEED TO UNDERSTAND THE DOT/CROSS PRODUCT OR VECTOR FUNCTIONS.)
Parametric and Polar Equations (VERY IMPORTANT TOPIC)

Unit 2 – Derivatives
Using Parametric Equations to model the position, velocity, and acceleration of a moving object (i.e. projectile motion) - (VERY SIMPLE AND VERY IMPORTANT)
Finding Tangent Lines to Polar and Parametric Functions (SOMEWHAT CHALLENGING, BUT VERY IMPORTANT)
L^Hopital’s Rule (VERY EASY AND EXTREMELY IMPORTANT)
Finding the Derivative / Rates of Change of Polar and Parametric Functions (IMPORTANT)
Finding the Second Derivative of Parametric Functions (NOT VERY IMPORTANT)
Knowing the Derivatives of ALL the functions (like arcsec(x), arccsc(x)) (This may or may not be on the AP. Still, it’s very easy. You just gotta memorize them)

Unit 3 – Integrals
Improper Integrals (Not too important… maybe 1-2 questions on it at most… somewhat difficult topic)
Integrating by Partial Fractions (VERY VERY VERY IMPORTANT)
Integration by Parts (EXTREMELY Important!!! You’d better know it!)
Knowing Inverse Trig / Normal Trig Integrals (memorization here… just do it)
Trig Integrals (i.e. integrate sin^3(x)) - (Probably not on AP… skip if short on time… also very difficult)
Arc Length of Parametric Functions (straightforward topic)
Arc Length of Regular Functions (WILL be on AP. This is a must-know topic. Somewhat difficult)
Areas of Polar Function Graphs (This may be on the AP, but it is very challenging. Try your best to know it)

Unit 4 – Differential Equations
Euler’s Method – (I can promise you this will be on the AP. This is also a simple topic, so learn it! Easy way to earn points here!)
Logistic Differential Equations (easy way to earn pts here – fairly straightforward topic)

Unit 5 – Sequences / Series
See the syllabus for full description. However, the most important concepts are: The ratio test, power series, taylor/maclaurin series, and lagrange error bounds. You MUST be sure to know how to test power series for convergence / divergence using the ratio test, and also be sure to know how to test convergence of power series at END POINTS!! You also need to be able to differentiate and integrate power series.

See collegeboard AP Calculus BC Course Description for the entire listing of topics
The link: Supporting Students from Day One to Exam Day – AP Central | College Board;

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Tips / Advice for Preparing - A list of advice in no particular order:

1. Find other students who are also studying Calc BC on their own to work with you – it frequently helps to have buddies who will keep you on track.

2. START VERY VERY VERY EARLY!!! (i.e. September/October) - this is a test that you absolutely cannot cram for!

3. Get a review book. The two best books are by Princeton Review and Peterson’s. The Peterson book is a little advanced, but is pretty clear and also helps you prepare for college calculus. The Princeton Review is extremely clear but a little too basic at times. When you get your review book, get it off amazon.com to save money, NOT Barnes and Noble. Also, get an old version (2004-2005 is fine, but 1999 is NOT). The new versions are basically the same as the old versions. I’ve also heard Peterson’s is very expensive, and if that’s the case Princeton Review is absolutely fine!

4. Here are a few useful websites that I really liked:
[PatrickJMT[/url]
[url=Math Study Guides - SparkNotes]Math Study Guides - SparkNotes](http://patrickjmt.com/)
[Pauls Online Math Notes](http://tutorial.math.lamar.edu/)
[Calculus-Help.com: Survive calculus class! - Calculus-Help.com](http://www.calculus-help.com/)
[Mrs. Roberts’ Calculus Page](http://mrsroberts.com/MrsRoberts/Calculus/calculus.htm)
[Khan Academy](http://www.khanacademy.org/)
[AP Central - AP Calculus BC Course Home Page](AP Calculus BC Course – AP Central | College Board)
[Study Hacks Blog Archive How to Ace Calculus: The Art of Doing Well in Technical Courses](How to Ace Calculus: The Art of Doing Well in Technical Courses - Cal Newport)
[AP Calculus BC Notes](http://www.saragriffin.com/apcalculusbcnotes.html)
[AP Calculus BC Notes and Handouts](http://www.bellaire.org/mazzoni/apcalc/notes/)
[Karl’s Calculus Tutor: Table of Contents](http://www.karlscalculus.org/calculus.html)

Practice MC Test Questions: http://asmsa.org/math/marizza/Calculus/APTEST/ap04_calcmc_collection_final_4_12_05.pdf

5) Plan your study schedule…I highly recommend creating a calendar, and sticking to it. For each new concept, schedule a time to learn it, and also time to do a lot of practice problems that contain the concept.
6) If you’d like a textbook, I suggest simply using your school’s textbook. However, the review book honestly should be fine for a 5 unless you really really want to master the material in Calc BC.
7) Studying for Calc BC while doing Calc AB can be very difficult as you frequently need to understand Calc AB before you can move on to BC topics. However, it is doable. After you cover the pre-requisite topic in Calc AB, IMMEDIATELY start working on the BC topics that are related to the AB topics. So, for example, once you have finished the unit on derivatives, IMMEDIATELY start working on differentiating polar and parametric functions.
8) Start to learn series ASAP (i.e. in September / October) – As long as you have a basic understanding of the derivative, you can begin to work on power series, which is VERY important for the exam. So, start early. You will not regret it.
9) Be sure not to just glance over the concepts / formulas – be sure to do lots and lots of problems involving the concepts. For the AP exam, being able to solve problems on your own should be your “gold standard” of self-assessment.
10) Try a variety of approaches - If you’re struggling to understand one guide’s particular explanation, then use another resource to learn the topic. Or maybe try reading the textbook, or maybe try watching video lectures on the topic. Or maybe try to seek out your calculus teacher for help.

11) Here is a brief timeline of the preparation process:
a) August-October: Make sure you understand all the course pre-requisites, especially parametric/polar equations. Start working on learning to apply derivatives to calculus.
b) October-December: Start working on power series NOW. Try to get a basic understanding of them. Also, make sure you completely master the Calc BC topics on derivatives!!
c) January- March: I really hope you’ve covered integrals by now. If you haven’t, then make sure you completely understand the BC topics in derivatives + you have some basic understanding of power series. If you have some basic knowledge of integrals, start working on all the integration topics mentioned above. If you’re pressed for time, skip the less-tested topics (i.e integrating trigonometric power functions, such as sin^3(x).) You should also FINISH COMPLETELY THE UNIT ON POWER SERIES during this time period!!
d) March-May: Make sure you understand the rest of the BC topics, and do a quick review. Then start doing lots and lots of old AP questions – I recommend working at least 5 years of old AP FRQs and at least two MC practice tests.

12) Don’t forget to work hard in Calc AB. After all, the material from Calculus AB is 60-65% of the AP Calculus BC Exam!!
13) Whenever you are confused on something, or see a problem that you’re struggling with, make note of that so you don’t forget. Basically, keep track of what areas you struggle in, so you can actually focus on those areas.
14) In total, you should expect to put 60-80 hours into this independent study.

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Post-AP Situation:
For those of you who really want to skip Calc II in college + are considering an engineering/science/math major, I recommend learning these topics after doing Calc BC:
Work, and how to compute it using calculus
Conics (i.e. hyperbolas, ellipses)
Moments of Inertia
Center of Mass of Objects
Hyperbolic Trigonometry
Trig Substitutions
Simpson’s Rule
The Binomial Series
Vector dot product, Vector cross product

Hope it helps - best of luck! :)

If you do have any questions, please make a post AND MESSAGE me.