Introduction:
So you want an “A” on your next Calculus test. All right, I can help you with that.
Note: Please read our article “How to learn calculus in 2022 (6 easy steps)” to understand the context in which I have written this blog post.
Many of you have written comments and messages indicating that you are taking my advice on studying and being prepared for tests. I appreciate every one of you who has done so, even though I’m not able to respond personally to all the emails. Keep it up!
Some comments:
I got a few comments from some people who were doing poorly on their tests, but they weren’t sure if they were studying the right way or practicing enough problems. One person sent me an email that said, “Crap, am I just stupid?” My answer was that if this student wasn’t getting anything out of his professors’ review sessions, then probably not. If he doesn’t understand something as simple as the chain rule, as explained by his professor in a 10-minute pre-test review session after going over it again and again in class – then no, he’s probably not stupid. If this student wants to see how smart he is, then the best thing to do would be to take another class – maybe a math course at a community college – and see how well they do there.
Some feedback:
I got feedback from some of my readers who scored 100% on their test using this study guide and advice: Some people said the problem sets were too easy, and others repeatedly questioned why we went over certain topics. The best advice I have is to try the problem sets first. If you get a problem correct but feel a little uncomfortable doing it – then maybe go over that topic a few more times before going on. Remember – if you plan to use this as your primary test preparation, you should be spending at least an hour or two a day on this stuff.
The most common complaint from people was some variation of “But I don’t know how to solve some of these problems!” To those people, I had this advice: Go back and re-study those chapters! The book has been out since 1965 for a reason – Calculus isn’t particularly hard if students can dedicate some time and energy into building up their skills with practice problems.
I will reiterate that this study guide is built for someone who either 1) wants to get an A on the calculus test or 2) someone who already has a firm grasp of Calculus and just wants to review everything again for their test. If you are neither of these two things, I don’t know how well this will work out for you! But if you are one of those people, read on-
Calculus AB/BC Review Guide: From Chapter One To Chapter Twelve (The Most Commonly Mentioned Topics)
Chapter One: Limits & Continuity – Page 40-49 & 51-62.
Do all the practice problems in Section 1-8 to better prepare for the calculus test. You should have a firm grasp of what continuity is and how to find limits, including determining which functions are continuous and which aren’t.
Chapter Two: Differentiation – Page 63-73 & 75-91.
In this chapter, you might need a little more practice because many students didn’t know about finding derivatives from graphs or table values. The Chain Rule was another topic that seemed to give some people trouble, so did all the problems in Section 2-7. Be sure you understand how to take higher-order derivatives, as well as the Second Order Derivative Test. Don’t forget Rules 1 & 2 on differentiating trig functions! A calculus tutor can help you with these topics.
Chapter Three: Applications of Differentiation – Page 92-105.
In this chapter, you should pay particular attention to the examples that don’t use graphs or table values when determining a derivative. Also, the hardest topic in this entire book is rates, and it makes students cry. So do all of Section 3-5 if you want a good grade on your calculus test!
Chapter Four: Integrals – Page 106-121 & 127-150.
You might need a little more practice for this section since most people were confused about finding antiderivatives from graphs. Here are some other common pitfalls from Section 4-3: Evaluating improper integrals, recognising geometric interpretations for antiderivatives, and many people tried to figure out the area under y = x 2 . Chapter 4-5 is the chapter people most often skip over because they think it’s not very important. Of course, derivatives are integrals, and understanding integration by parts, substitution, and integrating trig functions (3 of the most common integrals) will help make sure you get an “A” on this calculus test!
Chapter Five: Applications of Integral Calculus – Page 152-159 & 161-172.
Without a doubt, the most commonly skipped topics were related rates in Chapters 5-6, so I recommend you go back to that section if needed. You might also need to review basic integration techniques like using u-substitution or integrating with partial fractions. As for some tips on taking this test, check out my blog post, which goes into some additional detail on how to study for this test.
Chapter Six: Infinite Sequences and Series – Page 173-186 & 188-200.
When it comes to series, I recommend you go over Examples 1, 2, 5, 6, 7a+b, 8a+b, 11 & 12. These are the kinds of examples that might be asked on the calculus test because students rarely appreciate how much Calculus is involved in analyzing infinite series! Some students lacked confidence when working with special series like p-series, telescoping series, divergent, and two different alternating series (one of which has a cool geometric interpretation). Don’t forget that you can use the series to find the value of a function, even if it is divergent!
Chapter Seven: Vectors – Page 201-208.
Most people tend to forget about three-dimensional vectors or finding the magnitude of a vector. In addition, they also had some problems with the basics of vector addition, subtraction, multiplication, and dot/cross products.
Chapter Eight: Partial Derivatives – Page 209-227.
Most people have trouble understanding what a partial derivative represents, so be sure to review it earlier in the book if necessary! Also, watch out for some common mistakes like not telling if a function is differentiable by examining the first and second derivatives or not understanding that to take the directional derivative of a vector, you need to know which direction the vector is pointing!
Chapter Nine: Multivariable Differentiation – Page 228-259.
This section was just like taking Partial Derivatives again, so many students had issues with the chain rule, differentiating functions of several variables, and taking a partial derivative when there is a vector in the denominator! Many forgot assume theorem from Section 9-5 was, so try to go over that section if it seems familiar. This will help you in your test in calculus immensely.
Chapter Ten: Multivariable Integration – Page 260-281.
In this section, many people’s only issues were determining the volume generated by rotating one function around another or integrating parts. Another common problem was understanding how changing parameters affects double definite integrals, triple integrals, and iterated integrals. There are three things I recommend you know very well on this test; First, make sure you understand all the properties of multiple integrations (i.e. integrating x dy = x dx + y dy) so you don’t are not flustered by the problems on page 280-281. Also, be able to solve multiple integrals using u-substitution and finally, know how to handle iterated integrals.
Chapter Eleven: Vector Calculus – Page 282-297.
The Chain Rule was the most important thing tested in this section (many people also didn’t realize that dy/dx is a vector). Those who had trouble with this were stumped when it came to understanding how to take directional derivatives of functions involving vectors like v = x i + y j. Some students also needed help remembering what Stokes’ theorem does (it tells you about line integrals), and there were some issues related to the surface area.
Chapter Twelve: Second Order Differential Equations – Page 298-310.
The problems here were all about separable equations, finding particular solutions to homogeneous second order differential equations, and then using these results to find the general solution when it is possible. People hated this chapter because it was very abstract with little applied geometry or physical reasoning, so make sure you review it if necessary!
Chapter Thirteen: Applications of Partial Derivatives – Page 311-322.
Most students had trouble interpreting double integrals as volumes (or incorrectly thinking this type of integral was only used for computing arc length) and triple integrals as a way to find the center of mass. Also, make sure you understand what is the meaning of a level surface and how to find it; people confused this by finding the plane’s equation that intersects a given surface.
Chapter Fourteen: Vector Calculus – Page 323-325.
The problems in this section will focus on understanding vector calculus from a geometric perspective rather than relying only on the differentials. Make sure you can identify where lines, planes, and spheres are tangent from within an example problem! Also, make sure you know all the basics about line integrals (what is meaning of conservative fields) and get comfortable using Green’s theorem! We know that this comes under calculus 3 and is immensely complex for many students. You can get help here.
The final exam should be much more difficult than any practice test your instructor gives you, so if you haven’t taken one yet, I would suggest using some of these resources. Please make sure you give yourself enough time to study and understand all the concepts before taking a practice test because it is much more helpful if your homework can come from a source other than the professor! Also, don’t forget that the final exam will not be this easy this year, some subjects or topics will overlap, but some may not!
If you have any ideas for things that I should to this list, please let me know in the comments below! Thanks for reading, and good luck with your calculus test 🙂
