[High School - Calculus] Preparing for the AP Exam

[Relambrien]

So, I'm taking the AP Calc exam (Calc AB or whatever the first one is) on the seventh. I need to know what sort of things I'm going to have to be good at, what tips or tricks I could use, etc. If anyone could provide any help on this, it would be great.

The reason I'm asking here instead of in class is because my math teacher...is a terrible teacher. I know because he's exactly like me: he can understand things well, but can't explain them at all. Still, I understand the material better than anyone else in the class, and I'm the only one that really has a shot at passing. Considering no one in my school's five-year history has passed the AP Calc exam before, I'm taking this very seriously.

Now, I know the basic tenets of the exam. Limits, differentiation, integration. I'll need to know common derivatives and derivative properties, like these:

Sum and difference rule for differentiation/integration
Product/division rule for differentiation
Power rule for differentiation/integration
Chain rule for differentiation

Derivatives of the six trigonometric functions and their inverses
Derivative of the natural log function
Derivatives of functions involving e

I assume I'll also need to know l'Hopital's Rule, for 0/0, inf^0, 1^inf, inf/inf, etc.

Are there any other topics I missed?

Now, the integration is where I get a little bit confused. How much of the exam involves u-substitution, or the Fundamental Theorem? What antiderivatives do I need to know? What sorts of integration will be allowed with a calculator?

Speaking of calculators, I have a TI-89 Titanium. I know -how- to use it, and I can use it well, but what I don't know is what sort of strategies I can use with it. The solve, differentiate, and integrate commands are awesome, but is there anything else I could apply in obscure ways to help bring about the answer?

And then, just general exam tips. How many questions, how many sections, etc. How many multiple-choice, how many grid-in? Does it work like the SAT in which an incorrect answer takes points away? Which sections will allow the use of a calculator?

If anyone could help me out, I'd really appreciate it. I need to do everything I can to prepare for this exam. Like I said, I'm the first person in my school's history to be likely to pass, and beyond that, I need a 5 for the colleges I want to attend to give me credit for the course. I just want to do the best I can.

[MrRubix]

IMO the calculus on that exam isn't too bad at all. If you have a firm grasp of how to manipulate derivatives and integrals and trig functions, you're set. You can also try getting a prep book with practice problems. A lot of the stuff you're talking about is very formulaic and easy to follow. IMO, the hardest part of calculus is realizing when to apply which rules. Sometimes a complex integral needs to be solved in a creative way, but there aren't too many questions of that type on the AP exam.

I think it would be useful to have a list of trig functions and radian conversions, as well as basic examples and concepts (chain rule, product rule, quotient rule, integration by parts, u-substitution, trig substitution, indeterminate forms (or whatever they're called), how and when to use l'Hospital's Rule, basic trig identities, integrals and derivatives of common functions, how to manipulate e, how to manipulate logs (and all the arithmetic logarithmic functions such as how to add/subtract/multiple/divide logs), blah blah blah.

So uh yeah one of my majors is math so if you need any help understanding any of this, just paste some questions here and I'll give you some advice. AP Calc is an easy 5 if you have a firm grasp on the fundamentals.

IMO one of the best things you can do is take a few practice tests. Either get a prep book from the library and hammer out some practice questions, or look around online for some free samples. I have perfect scores on practically every standardized/AP/IB test I've ever taken with the exception of SAT Verbal (790) and a lot of this was due to the fact that I spent a lot of time looking at practice material. I took the SAT twice and was able to bring a no-prep 1530 or so to a prepped 1590, which isn't bad at all. Looking at practice material beforehand can make the difference between a 4 and a 5 on an AP exam. You should be spending most of your time figuring out how to solve problems, not how the test is set up. Therefore, it's good to "learn the battlefield" before marching in.

[leo172519]

When i took it there was a lot to do with general trig properties with derivatives. But it may have changed :-/ gl

[ieatyourlvllol]

Yeah, and other than knowing the aforementioned concepts and methods, you need to be able to manipulate variables of differentiation and integration in order to solve problems that aren't explicitly covered by the standard framework.

[Relambrien]

Originally Posted by MrRubix

IMO the calculus on that exam isn't too bad at all. If you have a firm grasp of how to manipulate derivatives and integrals and trig functions, you're set. You can also try getting a prep book with practice problems. A lot of the stuff you're talking about is very formulaic and easy to follow. IMO, the hardest part of calculus is realizing when to apply which rules. Sometimes a complex integral needs to be solved in a creative way, but there aren't too many questions of that type on the AP exam.

Alright, I'll make sure I do that. And yeah, I've noticed that the creativity necessary in calculus is by far the hardest part.

Originally Posted by MrRubix

I think it would be useful to have a list of trig functions and radian conversions, as well as basic examples and concepts (chain rule, product rule, quotient rule, integration by parts, u-substitution, trig substitution, indeterminate forms (or whatever they're called), how and when to use l'Hospital's Rule, basic trig identities, integrals and derivatives of common functions, how to manipulate e, how to manipulate logs (and all the arithmetic logarithmic functions such as how to add/subtract/multiple/divide logs), blah blah blah.

Yep, my teacher gave us sheets with all the common derivative laws and trig derivatives on it. I also have a copy of the unit circle to study. But integration by parts is something we never learned, so that's troubling.

Originally Posted by MrRubix

IMO one of the best things you can do is take a few practice tests. Either get a prep book from the library and hammer out some practice questions, or look around online for some free samples. I have perfect scores on practically every standardized/AP/IB test I've ever taken with the exception of SAT Verbal (790) and a lot of this was due to the fact that I spent a lot of time looking at practice material. I took the SAT twice and was able to bring a no-prep 1530 or so to a prepped 1590, which isn't bad at all. Looking at practice material beforehand can make the difference between a 4 and a 5 on an AP exam. You should be spending most of your time figuring out how to solve problems, not how the test is set up. Therefore, it's good to "learn the battlefield" before marching in.

Now my no-prep 1470 sounds inadequate

But yeah, I'll take a look at doing a few practice tests.

Thanks a bunch so far guys, this is really useful.

[MrRubix]

Integration by parts is actually really easy. All you really need to know is the whole, like

integral of udv = uv - integral of vdu (just to clarify, udv means u times the derivative of v)

Basically you use it when you're being asked to find the integral of (usually) the product of two functions that are (usually) of different forms such that u-sub and other types of integration don't seem appropriate. For example, if I asked you to find the integral of e^x*x^2, you might wanna use int. by parts.

In such cases, you usually set up the following four things: u, du, v, and dv. You generally want to set u to something with a simple derivative and you want to set dv to something that has an easy integral. So in this case, you might want to make u=x^2 and dv=e^x. Therefore du is 2x and v is e^x. We wouldn't really want to set u=e^x and dv=x^2 because du is still e^x, whereas v would become x^3/3, which isn't really any simpler than before. So choose your u and dv wisely.

integral of udv = uv - integral of vdu
integral of x^2*e^x dx = x^2*e^x - integral of e^x * 2x dx

The tricky part about int by parts questions is that sometimes the integral on the righthand side is, in itself, ANOTHER int by parts question. So looking at that integral separately, let's treat it as a new problem.

integral of e^x * 2x -> set u=2x and dv=e^x, so du=2 and v=e^x
integral of e^x * 2x dx = 2x*e^x - integral of e^x * 2 dx

We can always bring constants out of integrals:
integral of e^x * 2x dx = 2x*e^x - 2* integral of e^x dx, and we already know the integral of e^x, so:
integral of e^x * 2x dx = 2x*e^x - 2* e^x

So now we plug that back into our first equation:
integral of udv = integral of x^2*e^x dx = x^2*e^x - integral of e^x * 2x dx
integral of udv = integral of x^2*e^x dx = x^2*e^x - (2x*e^x - 2* e^x)
= e^x * (x^2-2x+2)

Anyways that's a basic example of int by parts. I typed that out of my ass so uh let me know if something doesn't make sense.

[tsugomaru]

People get 5s on the AP Calc AB exam all the time at my school. BC is the one people have trouble with.

~Tsugomaru

[FRO]

Hey Relambrien, I'm in the same situation as you. Fortunately for me though, I have a calculus and physics teacher who can get almost every single one of his students to pass the AP exams in calculus and physics, and a couple of years has had 100% of his students pass.

First of all, don't worry about the integration by parts that a couple of people above have mentioned. My teacher already let the class know that the AP exam makers took integration by parts off.

I've taken quite a few practice exams (both MC and free-response), and I don't think you'll have any problem with the MC part. I don't know if you've taken any practice MC's yet, but they're really easy and you only need to get about 75% right to get a 5, not a 3, on the test (albeit you have to do well on the free response section). It seems to me that the biggest topics that you need to know on the free response section are 1) be able to interpret position/velocity/acceleration graphs in every possible manner. Know when to use area under the curve, when to use slope, and when to use y-value 2) know how to find area between two curves, the volume of a shape rotated around a horizontal or vertical line, etc 3) derivatives and integrals, obviously 4) differentiable equations.

I'm sure there are a couple of others I forgot to mention, but these are the big ones that you must know to do well on the test.

Very surprising nobody's passed at your school before. Hopefully you're the first though! Good luck

[igotrhythm]

Differential equations? Dude, I was the only junior in my AP Calc BC class (nothing compared to the Indian guy a year below me who was taking multiple-variable calculus at a community college HIS junior year, though) and even though I waited until my senior year to take the actual test, I still got a 3, and there was absolutely nothing about differential equations. Unless you've had to study them, I wouldn't worry about them too much. From what I hear, though, those things can be devilish.

[flipsta_lombax]

I'm taking the test along with you. Applications of derivatives and antiderivatives concern me alot more though since they are the ones that get you thinking.

First, to answer your question about the u-substitution, my teacher always tell us to talk to ourselves: Is one of the factors the derivative of the other?

For example, I am taking is from the practice Calc AB test we did for homework:

Intergral of (sin3x)(cos3x)dx

u= sin3x (take the derivative)
du= 3 (cos3x) dx (move the constant to the other side)
1/3 du = (cos3x) dx

Plug this back in for the replacement of the original integral (hence substitution):

1/3 Integral of u du.
1/3 u^2/2 + C. (Replace what u was and simplify)
1/6 (sin3x)^2 + C (The ' + C ' is important if there is no limits of integration)


When there ARE limits of integration however, you don't need the ' + C ' and you need to use the Fundamental Theorem of Calculus. Let's say the previous integral had limits from pi/3 to 2pi/3:

1/6 (sin3(2pi/3))^2 - 1/6 (sin3(1pi/3))^2 = 0

And there we go.

At school, we have the best Calc teacher (she has only been teaching for 60+ years of math).

What we've been focusing on for the AP test is all those applications like particle motion, volume of solids in revolution and area under the curve, and for derivatives like optimization and rates of change.

My tip:
=> Find a group to study with and do practice problems. Try to find something because I promise you practicing these things make it so much easier than it has to be.

=> Don't practice this the day before the test... Never works.



I do wish you luck on getting that 5. Hope I helped.

[sumzup]

Yay Calculus!!! (sorry, I'm on a high after having taken my Calc 3 [multi-variable] final; I think I aced it)

I'm kinda in the same boat here...I'm doing the BC test come May. AB really isn't that hard; a bit of work and you've got a pretty easy 5.

Check out the College Board's list of topics and sample questions.

One thing that's important to know is the volume of solids formed by rotating graphs around axes. Do you know how to do that?

I'll be back with more tips/info after I go and do my Astronomy final.

[igotrhythm]

*remembers filling 3 pages of paper with the proof that Gabriel's Horn has a finite volume but an infinite surface area...that was a pain in the ass that ended up making it necessary to look at tables of integrals*

[Ramengan]

Originally Posted by MrRubix

I think it would be useful to have a list of trig functions and radian conversions, as well as basic examples and concepts (chain rule, product rule, quotient rule, integration by parts, u-substitution, trig substitution, indeterminate forms (or whatever they're called), how and when to use l'Hospital's Rule, basic trig identities, integrals and derivatives of common functions, how to manipulate e, how to manipulate logs (and all the arithmetic logarithmic functions such as how to add/subtract/multiple/divide logs), blah blah blah.

I agree with just about everything on this list, as almost all of it was necessary when I took the exam last year. I say almost because I don't remember integration by parts on it at all, or even learning it in class. Are you sure it was a calc AB topic? U-subbing and log rules are definitely important, as of course are all the basic derivative rules and popular functions.
I'd add in implicit differentiation (if that's already been said, I apologize for not seeing it), and I remember doing some basic work with slope fields, but I can't remember where that would fall on a scale of importance.

@Relambrien: Get on AIM this weekend and we can work through some more practice problems. The fundamental theorem and rotational solids are kinda big, so we'll focus on those. Also the infamous "pool" problem, and other related rates stuff.

[ledwix]

I also have a TI-89 Titanium, Relambrien. And I'm taking the AB test next Wednesday.

That IS, in fact, a permissible calculator, right? I don't want to have to borrow someone's 84. I know I should have confirmed a long time ago and am stupid for not doing so, but I just assumed based on being able to use it on some other AP tests that I could use it on this one.

I'll probably end up re-reading some of these posts for reference before then, so thanks to the people who contributed to the thread.

I would also recommend not worrying about passing the test. (As Rubix said, it's an easy 5 with some decent studying) My guess is you'll get a comfortable 5 considering that you'll be studying hard.

[flipsta_lombax]

Product summary of the TI-89 Titanium:

"The TI-89 Titanium's advanced functionality and 3-D graphing make problem-solving for AP*, advanced mathematics, and engineering courses infinitely easier. It's the most powerful TI graphing calculator allowed for use on the AP* Calculus, AP Statistics, AP Physics, AP Chemistry, PSAT/NMSQT**, SAT® I , SAT II, and Math IC & IIC exams. "

I believe it is then. :]

[MrRubix]

Trust me you do *not* need a hardcore calculator to do the test. I used a TI-82 (yes, an 82) only because I liked being able to modify my equations easily with the arrow buttons. Screw the fancy functions as you don't want to find yourself relying on them. A calculator's meant to augment, not replace

[Relambrien]

Okay. So the AP exam was a lot easier than I thought it was going to be. Still challenging, but I didn't need to know half of what I thought I did.

In fact, the only time I used trigonometric derivatives was on one question involving the arctangent. I didn't need any knowledge of the unit circle at all. There was a -lot- more stuff involving the Euler number and natural logs than trig calculus.

I'm pretty confident I got easily enough on the multiple choice section to score a 5, but I missed two parts of a question on each part of the free-response section. Meaning, of six questions with four parts each, I missed at least four. And since the parts aren't weighted equally, I'm hoping the ones I -did- miss weren't very heavy.

Also, there was only one question involving differential equations, which I really didn't get. I understand slope fields, but I couldn't remember how to solve differential equations analytically, so I know I missed that.

I'll be looking forward to the third week in July, when I get my scores back. If I got a 5, then awesome, I'll take BC next year. If not, I'll just take AB again. I need the 5 for the colleges I want to attend to give me credit for the class.

[flipsta_lombax]

So my teacher was wrong about integration by parts being on this test. Everything was relatively easy. The student response without the calculator was the trickiest on my part, but everything else was easier to solve.

[sumzup]

BC was pretty good; I did well enough to get a 5, I think, but I could see where getting a 4 would be reasonable. I didn't have nearly enough time to study...I studied on the plane back from Washington, D.C. to Washington State (Sci Bowl) so it was pretty grueling.

I basically needed some more time, but as it is I think I missed 3 free response subparts out of around 24. Multiple choice was easy enough, except for lame Calc AB stuff on 2 questions I didn't remember.

ALSO; to Relambrien: Don't retake the AB test if you don't get a 5; just take the BC next year, because you'll get an AB subscore that is equivalent to having taken AB.

[Relambrien]

Originally Posted by sumzup

ALSO; to Relambrien: Don't retake the AB test if you don't get a 5; just take the BC next year, because you'll get an AB subscore that is equivalent to having taken AB.

Hmm, I did not know that. Thanks for telling me