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# The Ultimate List of AP Calculus Tips

Learn the language and vocabulary, because half of the battle is knowing what a question is asking and/or telling you! Give yourself the benefit of analyzing the information presented in a question, because then you can execute on the knowledge and skills that you have practiced!

In order for you to score a 4 or 5 on the AP Calculus exam (AB or BC), it is important for you to follow the tips outlined below.  So far in 2015, only 38.4% of students who took the AP Calculus AB exam received a grade of 4 or 5.  However, 61.5% of students that took the AP Calculus BC exam in 2015 received a 4 or 5 score.  The AP Calculus BC exam covers all of the topics in the AP Calculus AB exam plus some additional ones as well.  Take the time to review the following tips and you’ll be well on your way to earning the highest possible score on your AP Calculus exam.  Relax, read and absorb the tips as you go!  Good luck!

We recommend you supplement your AP Calculus preparation with Albert’s online practice questions. If you prefer going old school, you can read about the best review books for AP Calculus here.

## How to Study for AP Calculus Exam Tips

1. Know the exam’s content: The AP Calculus exam will cover a number of specific concepts besides just differentiation (Derivatives) and integration (Integrals).  See the following for a comprehensive list of topics and concepts covered in the exam:

List of Topics and Concepts on the AP Calculus Exam

#### – Integrals

Interpretations

Properties

Applications

Techniques

Numerical approximations

Definite

Indefinite

Areas under curve

Euler’s Method

Integration by parts

#### – Derivatives

Concept

At a point

As a function

Applications

Higher Order derivatives

Techniques

Max/Min problems

#### – Asymptotes

As Calculus is a systematic assembly of concepts, take the time to familiarize yourself with the terminology, and, more importantly, understand the actual concepts.  Calculus is a highly conceptualized subject matter and it is extremely important that you have a firm grasp on all of the major concepts.  Once you have mastered the individual concepts you are ready for the next step (Tip 2).

2. Practice makes perfect: In order to score a 4 or 5 on your AP Calculus exam you will need to practice lots of problems.  The more problems you do, the more adept you will be at deciphering the way in which each type of problem is presented.  There are numerous textbooks on calculus that are suggested study guides for students.  Get one or two of these textbooks and start doing the problems in each chapter.  You may also use online study guides for the same purpose.  One such valuable resource is available at the Albert.io website.  Do yourself a big favor and search for additional online resources as time permits.  There are plenty of them!  If you have extra time, we highly recommend visiting College Board’s AP Calculus course homepage.

3. Assess the exam and gauge your time appropriately: There are two multiple-choice and two Free-Response Question (FRQ) sections of the exam.  Both the multiple-choice and FRQ sections are broken up into four separate time segments.  You will have two segments where you will be allowed to use a calculator and two where you will not be allowed to use a calculator.  For each FRQ (there are a total of six) your time allotment should be 15 minutes.  Try your best to pace your time on these questions.  Since there are a total of 45 multiple-choice questions and a total time of 105 minutes, you should pace yourself at two minutes per multiple-choice question.  By skimming the exam before you start, you can quickly tackle the problems you feel the most comfortable with.  Then, you can tackle the more difficult questions.

4. Pay attention to detail: Especially when working with the AP Calculus FRQs, it is important that you are detail oriented in solving the more complex problem of an FRQ.  In other words, make sure that you detail every step in the process.  For example, if they have two or three equations that require differentiation or integration at some point, then show this work in detail.  You should draw a box around any important interim equations that you will need for figuring out the final answer.  The picture below is an excellent example illustrating the use of this tip:

Also, make sure that your final answer to a FRQ is clearly identified as such.  If you are having problems with a particular multiple-choice question use the process of elimination.  Try your best to narrow down your answer to one or two of the multiple-choice selections.  For example, if you know that the answer needs to be in the derivative form then you can eliminate any and all options containing the integral form.

5. Review and double-check your final answers: We cannot emphasize this tip enough.  As time permits, always perform a quick review of an answer.  OK, if you are so totally confident of a particular answer then don’t waste the time on re-checking its correctness.  Move onto the next question and so on.  For problems that you are not able to answer during your first skim, place a check mark or circle them so that it will be easy for you to find them later on during the exam.  This will certainly be time well spent.

6. Review important trigonometric derivatives: Make sure that you know the common trigonometric derivatives and inverse trig derivatives. You will most definitely need them for many of the problems on the AP Calculus exam.  Here is a list of the trig derivatives you should know by memory:

#### Inverse Trig Derivatives

$\frac{d}{dx}\sin{x}=\cos{x}$ $\frac{d}{dx}\arcsin{x}=\dfrac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\cos{x}=-\sin{x}$ $\frac{d}{dx}\arccos{x}=\dfrac{-1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\tan{x}=\sec ^2{x}$ $\frac{d}{dx}\arctan{x}=\dfrac{1}{x^2+1}$
$\frac{d}{dx}\cot{x}=-\csc^2{x}$ $\frac{d}{dx}\text{ arccot }{x}=\dfrac{-1}{x^2+1}$
$\frac{d}{dx}\sec{x}=\sec {x}\cdot \tan{x}$ $\frac{d}{dx}\text{ arcsec }{x}=\dfrac{1}{\left|x\right|\sqrt{x^2-1}}$
$\frac{d}{dx}\csc{x}=-\csc{x}\cdot \cot{x}$ $\frac{d}{dx}\text{ arccsc }{x}=\dfrac{-1}{\left|x\right|\sqrt{x^2-1}}$

7. Show all of your work: Make sure that when performing solutions to either the multiple-choice or FRQ problems that you show all your work.  This is especially important for the FRQs as partial credit will apply.  Multiple-choice questions do not have partial credit.  They are correct or wrong, so there is no in-between.

8. Write down the equations: If you are using your calculator to solve an equation, be sure to write down the equation first.  An answer without an equation may not get full credit even if it is correct.  Also, if you use your calculator to find the value of a definite integral or derivative, write down the integral or derivative equation first.

9. Use your calculator only for the following: Graphing functions, computing numerical values for derivatives and definite integrals, and for solving complex equations.  In particular, do not use your calculator to determine max/min points, concavity, inflection points, increasing/decreasing, domain, and range. (You can explore all these with your calculator, but your solution must stand alone in order to receive credit).

10. Know both the product and quotient rules for derivatives: These are some of the most frequently used rules in all of Calculus.  You must know them well for the AP exam.  We have always used the following to remember the product rule: The derivative of the product of two functions of the same variable is equal to the first function times the derivative of the second function plus the second function times the derivative of the first.  Symbolically the product rule is:

$\dfrac{d}{dx}\left(g\left(x\right) h\left(x\right)\right)=g\left(x\right)\dfrac{d}{dx} h\left(x\right)+h\left(x\right)\dfrac{d}{dx}g\left(x\right)$

The quotient rule is more complex and can be memorized as follows:

If:

$f\left(x\right)=\dfrac{g\left(x\right)}{h\left(x\right)}$

…then:

$f'(x)=\dfrac{g'(x)h(x)-g\left(x\right)h'(x)}{{\left[h(x)\right]}^{2}}$

In words you can memorize the quotient rule as follows:  The derivative of a quotient of two functions of the same variable is equal to the derivative of the top function times the bottom function MINUS the top function times the derivative of the bottom function ALL over the bottom function squared.

11. Indefinite Integrals: If you are ever asked to integrate a specific function always remember to add + C as part of the answer after the equal sign.  Leaving it out will cost you points.  For example, what is the integral of:

$Y=f\left(x\right)={x}^{ 3}+{x}^{2}$ ?

$\int{Ydx}=\dfrac{{x}^{4}}{4}+\dfrac{{x}^{ 3}}{3}+C$

## AP Calculus Multiple-Choice Review Tips

In this section we review some specific tips regarding the multiple-choice portions of the AP Calculus exam.  Although some of these may also be extended to the FRQ portion, these tips are more tailored to the multiple-choice questions on the test.

1. Understand what the question is asking: Make sure that you read the question carefully and understand what they are asking for.  Usually this is easier with mathematics type exams since math is more symbolic in nature.  So it should be quite easy and apparent when you are trying to understand an AP Calculus question.

For example: What is the value of $f^{'}\left(x\right)$ for the function $f\left( x \right) =3{x}^{3}+{x}^{2}+4$ at the point $x =4$?

In this example they are asking for the numerical value of the first derivative of the function at a particular point.  So you will first need to find the first derivative of the function $f\left(x\right)=3{x}^{3}+{x}^{2}+4$ which is $f^{'}\left(x\right)=9{x}^{ 2}+2x$ and then plug in the value $4$ for $x$ into the first derivative equation.  Therefore, the answer is $f^{'}\left(x\right)=152$ at $x = 4$.

2. The Process of Elimination Technique (POE): If you really know your stuff you will normally not need to use this technique.  However, there are almost always a few problems that you may be having trouble with on the exam and this technique would definitely come in very handy.  If there are answers that are obviously wrong it would be a good idea to place a mark on those answers telling you that they are incorrect.  Since there are five possible answers to each multiple-choice question, eliminating one answer increases your odds by 20%.  Not too shabby!  Eliminating two answers increases your odds of a correct response by 40% and so on.  Again, use this technique only if necessary.  Otherwise, solve the problem and choose the correct answer.

4. Use an AP Calculus study guide or textbook to prepare: Using a study guide or a good textbook can be a tremendous help in getting a high score on your AP Calculus exam.  The reason for this is that many of the types of problems that occur on the AP Calculus exam come from these study guides.  You can find many of them on the internet by doing a simple internet search. The most important thing to remember is that the more problems you do at home or with a study partner, the better your chances will be for getting a high score.  Using tip will also give you exposure to the types of problems that will appear on the AP Calculus exam.  This leads us to the next tip.

5. Know the different types of calculus problems: There are many different types of  Calculus problems.  One of the most common is the min/max problem.  These problems require you to find the derivative of a specified function and set it equal to zero. You would then solve the resulting equation (the derivative) to find its roots and apply these roots to the original function to determine the min or max.  Another common type of problem is that of finding limits of functions.  Starting in 2016, the Calculus AP exam will contain problems relating to L’Hôpital’s rule.  This is a great way of determining the limit of a function divided by another function.  You can visit this page to understand how to use L’Hôpital’s rule when finding the limit of a quotient of two functions.  The other types of problems found on the AP exam are the following: continuity of functions, asymptotic functions, antidifferentiation and the Fundamental Theorem of Calculus.

6. Understand and master the Chain Rule: It is very important that you know how to use the Chain Rule.  This rule is used for the computation of derivatives of the composition of two or more functions.  See this page for more information on how to use the Chain Rule.  Knowing this rule will allow you to easily calculate the derivative of a multi-functioned problem containing at least two or more functions and their respective variables.

7. Identify your weaknesses: If there are certain types of AP Calculus problems you generally have issues with, practice doing more of those before taking the AP Calculus exam.  Focus on the underlying concept of these (or any other) problems.  Mastering and understanding the underlying and fundamental concepts of AP Calculus will greatly help you improve your score.

8. Review AP Calculus practice multiple choice questions online: One of the best ways to prepare for the multiple choice portion of the AP Calculus exam is to use online practice sites.  These sites either have actual past AP Calculus multiple choice questions or they are modeled to be strikingly similar to past exams. Here is one such site with the 2006 AP Calculus practice exams along with the key to the answers.  Going through each and every problem on this one website is like the equivalent of taking the AP Calculus test a total of five times!  You get a real flavor of what to expect on the actual AP Calculus exam if you go through these problems one by one.  In this manner, you will also be keen as to what types of problems you can expect to find when you take your own exam.

9. Study the multiple-choice problems in the 2016 College Board AP Calculus course description: The course description manual published by College Board AP covers both AP Calculus AB and AP Calculus BC exams.  It contains just about all of what you’ll need in order to get the best possible idea of taking the actual exam, the philosophy and principles behind it and a number of multiple choice and FRQ examples.  We strongly recommend that you read and understand the entire document and then do the sample questions.  Better still, we also recommend that you time yourself.  Part A of the multiple-choice section contains 30 questions (60 minutes) and Part B contains 15 questions in (45 minutes).

## AP Calculus Free Response Question Tips

The following tips are specifically designed to help you master the FRQ section of the exam.  An FRQ is like a series of multiple-choice questions with the caveat that the FRQ is fashioned in the form of a chain of reasoning exercise.  In essence, the FRQ is digging deeper in attempting to assess your more global understanding of Calculus.

1. Practice released FRQs: Visit AP Central here for examples of the AP Calculus Free Response Questions from 1999-2016. These will give you the best idea of what to expect on your exam. If you have trouble or need more advice, review these walkthroughs and tips from Stacey Roshan.

2.Underline key equations: FRQs for the AP Calculus exam are essentially word problems that come in multiple parts.  They are usually between 3-5 parts per FRQ.    The first thing you should do is underline any key equations that are given.  FRQs are designed to test your ability for an “extended chain of reasoning”.

3. Show all your work: Since partial credit is given for FRQ’s it is especially important to show all of your work.  For example, you may be given a function $f\left(x\right)$ and will need its derivative $f^{'}\left(x\right)$.  Make sure that you actually write down the first derivative $f^{'}\left(x\right)$ and underline or box it in since it will be an important equation you will need for the other parts of the problem. Here is an actual example from the College Board website.  Your work should be clearly written in the space provided and your answers also should be provided in the proper space.  Sometimes you will be asked to justify your answer.  In this case briefly describe how you arrived at the answer.  Indicate what concepts or equations you used to get to the correct answer.  We recommend placing the words “Final Answer” or the letters “FA” right next to the boxed in final answers.  This will make it very clear to the graders of your exam that this is your final answer.

4. Budget your time: You will have a total of six FRQs.  Part A will contain two FRQs and is 30 minutes long (calculator permitted).  Part B will contain four FRQs (calculator not permitted) and is one hour long.  Important note!  If you still have time when you finish Part B, you are allowed to go back and finish Part A if you need to.  Try to spend no more than 10 minutes per FRQ.  This will allow you some time to re-visit a particular question.

5. Read the data and question parts slowly and carefully: The AP Calculus Free Response Question has essentially two parts.  The first part is he data or information needed to solve the individual problems.  The second part is the individual questions.  You will need to understand both parts in order to correctly answer and provide solutions to all of the questions.

6. Be specific and brief in your justification answers: If you are asked to justify your answer, don’t write a book about it!  Be brief and to the point.  Make sure you include all pertinent aspects of how you arrived at the solution.  For example, if you are asked to provide justification on how you determined that a certain polynomial function has a max or a min at a certain point you must show the individual steps you took in order to arrive at the solution.  To do this, make a table showing the original function, the function’s first derivative and its roots and all of the critical points.  Then finally show how you determined that the point was a min or max.  See the example below:

Find the absolute maxima and minima of the function:

$f\left(x\right) ={x}^{3}-6{x}^{2}+9x+2,\quad 0\le x\le 4$ .

Take the first derivative:

$f^{'}\left(x\right) =3{x}^{2}-12x+9$ .

Find the roots which are:

$x = 1$ and $x = 3$.

These are the only critical points of $f$. We consider the following table of the endpoints and the critical points of $f$:

#### $f(x)$

$0$ $2$
$4$ $6$
$1$ $6$
$3$ $2$

the absolute maximum occurs at both $x = 1$ and $x = 4$ and is $6$ and the absolute minimum occurs at both $x = 0$ and $x = 3$ and is $2$.

7. Pay attention to details: Don’t forget to give units if required.  For example, if they ask you how many cubic feet of water are flowing through a pipe at a certain time, make sure that your final answer includes both the number and the units.  In this case the units would be ft3.  Another example would be to include the constant C whenever you integrate a given function.  Leaving the C out will cost you points.  Also, and this is especially important, make sure that your standard Calculus notations are correct and complete.  For example, $\frac{{d }^{2}}{{dx}^{2}}f\left(x\right) =\frac{d}{dx} f^{ '}\left(x\right) =f^{''}\left(x\right)$

8. Rounding of numerical answers: Make sure that any numerical answers are given to the nearest thousandth (3 places after the decimal point).  Also, store any interim values in your calculator and use those numbers to calculate the final answer.  You will lose points if your answers are not properly rounded.

9. Be neat when working on FRQs: Try to be as neat as possible when showing your work or answers on Free Response Questions.  This is good for two important reasons.  Firstly, if the graders cannot read your work they will not give you partial credit.  Secondly, it is good for you because if you have made a mistake on the early part of a question, say section (a), then your answers for sections (b), (c) and (d) will be wrong as well since the FRQs are designed to challenge your chain of reasoning.

10. Make up your own problems and then solve them: This may sound silly but it will actually test your true aptitude in Calculus.  Start with making up your own polynomial equation and then find its first derivative.  Then, in order to prove to yourself that you understand the Fundamental Theorem of Calculus, take the integral of the derivative you just found.  You should, of course, come up with the original polynomial.  Take this a step further and make the original function more complex.  Take a polynomial and multiply or divide it by a trigonometric function.  Write it down and try taking its derivative.  Once you have gotten an answer, plug in some numerical values and then check it with your calculator.  Just think of how many possibilities you could come up with.  Stick with reasonable expressions and don’t overdo it.  Write all of your work down on a piece of paper and keep it as your own personal study guide.

11. Never forget +C. When integrating a function, remember that you need to account for an arbitrary constant C. It may seem frivolous after you’ve done more complex calculations, but it’s not. Check through your answers at the end of the section to make sure you have included a constant when necessary. I often forgot this during the school year, so I made sure to commit it to memory while studying. When the test arrived, I was incredibly glad I had. C is a crucial part of the equation and you will be marked down if you forget it!

[bctt tweet=”When integrating a function, remember that you need to account for an arbitrary constant C.”]

12. Don’t doodle! Many students feel discouraged and think it funny to draw instead of answering the question, try not to do this. Doodling is the equivalent of leaving a multiple choice question blank — that is, you will get zero points no matter what. Even if you’re stumped, try rewriting what you know and brainstorming some ways you might be able to solve it. You might get a couple of points anyway. Better yet, you might have an insight that helps you work through the problem.

13. Memorize the key derivatives and integrals of common trigonometric and other functions: This tip is very important because it will save you time by not having to explicitly derive already known expressions.  They will almost certainly appear on the AP Calculus exam so it is best that you know them by heart much like you learned multiplication tables in grade school using flash cards.  You can also memorize these key derivatives and integrals by simply writing them down in your personal notes.  Whenever you study for the AP Calculus exam, take the time to memorize, but more importantly, to understand the following:

## Tips by AP Calculus Teachers

The following are a collection of tips specifically designed for the student to optimize his/her performance on the AP Calculus exam.  Most all of the tips are from the writer of this article.  I first learned Calculus at the age of 16 and was self-taught in the beginning.  I also searched the internet extensively as it related to the AP examination process in order to get more authoritative sources such as College Board AP Central.

1. Relax and Enjoy! I know it sounds somewhat counterintuitive to enjoy taking a test but it’s actually very possible to do.  Remember, you are taking this exam because you are smart and because you may even enjoy Calculus.  Think of taking the exam as an opportunity and not a chore or commitment.  You should put yourself in the frame of mind that it is an honor for you to be taking the exam in the first place and you now have that great opportunity to demonstrate your advanced skills.  Stay positive in your thinking and simply focus on each individual question as if it were like any other test you have taken in school.

2. Prepare yourself both mentally and physically: A few days before you actually take the exam, try not to think of it constantly.  Doing so will only increase your anxiety level.  Make sure that you relax your mind and get a good night’s sleep the night before the exam.  Remember that taking the exam is an opportunity and not a necessity.  While taking the exam keep this theme in mind and focus only on getting the correct answers.  That should be your only goal.

3. Do as many problems as possible beforehand: Six to eight weeks before the exam start your preparation by doing a few Calculus problems each day.  Make a habit of it.  It will only take an hour a day and you will be ready and ultimately rewarded with a high score on the AP exam.  You must do your due diligence and get in the groove of doing the problems each day during this time period.  Don’t worry, be patient, practice daily and you’ll be well on your way to success.  The score of 4 or 5 on the Calculus AP exam will happen by itself if you take this advice and persevere.

4. Be extremely confident in yourself: In order to get a good grade on any important standardized exam, you need to be extremely confident in yourself.  Think positively by saying to yourself, “I know I can get a 4 or 5 on this AP Calculus test because I have prepared so well and for so long to get this far”.  Also, say to yourself, “I’ve done so well on all the other exams I’ve taken so why not just treat this exam like those?”  Speak with your friends and tell them how much you are looking forward to taking the exam.  Their responses will not only reinforce your confidence levels but also will make you feel better about taking the exam with the expectation of achieving a 4 or 5.

5. Focus on learning and understanding fundamental concepts: This is the most important tip I give to all my students.  Calculus is all about solid, valid and proven concepts.  Know that the derivative of a function at a certain point is the slope of the line that is tangent to that curve at that point.  Know that integration is the inverse of differentiation and vice versa.  Also know that the net area bounded by the curve between any two points on the curve is equal to the definite integral between those two points.  See Wikipedia.

6. Review past math concepts: It is also a great idea to make sure that you are still fluent with the other math courses and concepts you have learned in the past.  Surely you will want to be familiar with advanced algebra and graphing polynomials.  You will also want to make sure that your trigonometry is up to par.  Many Calculus problems will use trig functions so it would be best for you to review trigonometric identities and common trigonometric formulas.  Thanks for the tip from the folks at the College Board.

7. Study with friends or other students: If any of your fellow friends or students will also be taking the AP Calculus exam, it would be a great idea for you to get together with them every so often and do some problems together.  Remember the old saying that “Two heads are better than one”.  Studying with other people with the same goals is great because it adds an element of personality to your academic understanding of a particular subject matter.  Most serious students do this during High School or College and it also helps take the stress out of the anticipation of taking such a high-level exam such as AP Calculus.  Thanks for the tip from the folks at the College Board.

8. Use outside resources: You should always use outside sources for preparing for the AP Calculus exam. Relying on only one textbook would give you a narrower understanding of the types of problems that might occur on the AP test.  Remember that the AP exam is prepared by a committee that likely uses multiple resources to create the problems.  You should also diversify your portfolio, so to speak, and use the internet or other good textbooks on Calculus.  There are scores of websites devoted to sample Calculus problems.  I would visit as many as possible without over doing it of course. Thanks for the tip from the folks at the College Board.

9. Ask for help: If you are having difficulty with certain types of problems then it would be a good idea to seek help.  You might want to schedule some time with your teacher at school or perhaps look into hiring a good Calculus tutor.  Thanks for the tip from the folks at the College Board.

10. Help other students: Besides just getting together with other students or friends for a Calculus studying session, it would be good for you to help other students that may be having problems with understanding Calculus.  If you are very comfortable with having an above average understanding and knowledge of Calculus, it would be great for you to help someone else out.  This would be a win-win situation for both of you since teaching a subject matter always reinforces the concepts with yourself.  You will gain additional confidence and understanding of the basic principles of Calculus by teaching someone else.  Thanks for the tip from the folks at the College Board.

11. Learn from students that have taken the test before: You may already know some people that have taken the AP Calculus exam before.  If so, speak with them about their experience in taking the exam.  They may have some tips for you as well.  Ask them questions like, “What types of problems were on the test?” or “Did you have enough time to finish all of the problems?”.  You may also ask them if they thought the test was easy or difficult.  Try to get as much information from them as you can.  It certainly can’t hurt. Thanks for the tip from the folks at the College Board.

12. Don’t procrastinate: Some students are better than others at not procrastinating.  Make sure that you are serious about preparing for the exam way in advance and start doing problems on a daily or otherwise frequent basis.  It is important that you stick to a plan and execute it properly.  Don’t say to yourself, “Oh, I already know this stuff so I will put it off until the week before the exam”.  If you do that, it will definitely ensure that you will not be fully prepared for the exam.  Always remember that practice makes perfect and just make sure that you do a lot of practicing.  Again, like I said before, the more you practice, the better are your chances of getting a 4 or 5 on the AP Calculus exam.  Good luck!

13. Answer all questions: When you are finished with the exam, make sure that all questions have an answer.  If you leave a question blank you will not get any points for that question.  This could make the difference between a 4 or a 5 score if you leave too many questions unanswered.  Thanks for the tip from the folks at the College Board.

14. Wear a watch: Although there may be a clock on the wall in the room, there is no guarantee.  Even if they do have a clock in the room it would be a lot better if you were to simply look down at your wrist rather than looking up to the  wall clock all the time.  Thanks for the tip from the folks at the College Board.

15. If you feel overwhelmed or your hand cramps, take a short break. Taking thirty seconds to shake out your wrist after all that writing and look up from your paper will not be harmful. It’s a timed test, but it’s not a time trial. Taking short breaks when you feel confused allow you to clear your head and relax.

16. Remember, you don’t need to score 100% to pass. The AP Calculus test is supposed to be difficult. While you might have done very well on unit tests in your Calculus class, most people do not score nearly as well on the AP test. This is how it is designed. To receive a 5 in recent years, examinees have only needed to answer 63% of material correctly. This is not a test to ensure you know everything, but to measure what you do.

17. When it’s done, relax and celebrate! Once you have turned in your exam, there is nothing more you can do for a couple of months. Do not worry too much about your results until they come out. You have worked hard all year to earn college credit, and now your studying and practice has paid off!

18. Work AP problems ALL YEAR LONG! Don’t wait until the end of the year for the review. We review by chapter, the day before or after each test. Thanks for the tip from Ms. Gisella C. from Boca Raton Community High.

19. When things get difficult, focus on what you DO know as a starting point–not what you don’t. Pull together the ideas of what you DO know and good things will happen. Thanks for the tip from Aaron P. from Crystal Lake Central High.

20. Be sure to label all work correctly. For instance, it you taking the derivative be sure to label f prime (x) . Setting up an integral use the integral symbol etc. As a reader it is amazing how many students work a problem with no labeling or worse mix f and f prime. Thanks for the tip from Mr. Waddell.

21. Review all formulas the night before the exam and then go to sleep. No need to try to cram…you just can’t for this test. It’s what I always tell them. Well, that and to learn every little detail about every little Calculus theorem there is. Thanks for the tip from Pamela L. from Ralph L. Fike High.

22. In doing derivatives, I always stress the difference between $y = x^n$, $y = a^x$, $y = a^n$, and $y = x^x$All of them look similar, but different rules apply: Power Rule, Exponential Rule, Constant Rule, and Logarithmic Differentiation. I usually do this by presenting 4 specific problems on the board, such as the ones with n = e, a = pi. I ask them which rules apply. They always try to use the power rule on all 4. However, with practice at the board, they get the difference. Thanks for the tip from Ron T.

23. If a student boxed in an answer on the FRQ’s, that was the only answer we could consider. Too many students had the correct answer elsewhere and I could read that answer, unless they had a boxed in answer. Thanks for the tip from Laura S. from Chatham High.

24. Learn the language and vocabulary, because half of the battle is knowing what a question is asking and/or telling you! Give yourself the benefit of analyzing the information presented in a question, because then you can execute on the knowledge and skills that you have practiced! Thanks for the tip from Brittany A. at YesPrep Public Schools.

25. The best thing I can think is for students to practice “AP level” problems as much as possible. Students should find past AP tests, the AP course description or review books and do as many problems as they can. Thanks for the tip from Richard S.

26. There may be stuff on the test that you do not know. That is okay. Just don’t panic and do well on the stuff you do know how to do. Thanks for the tip from Ned E. from Lebanon High.

27. Relax and don’t be afraid of making mistakes…you are sharper than you think. Thanks for the tip from Ms. L.

28. Have the unit circle memorized so that you can fill in a blank one in 5 minutes or less. It will save you precious time on the AP exam. Thanks for the tip from Liesa K. at Art College Prep, Jill P., and Michelle J.

29. One tip I give them to be successful on the AP exam is to mentally get after it. They are a little confused at what I mean by that at first so I relate it to sports. If you are in a wrestling match, you ‘get after it’ physically. On the exam you get after it mentally. There are no times where you lose focus. Your mind is working it’s tail off. If you get tired or start to lose focus, you have to pep yourself up and get after it even harder. When the test is over, you will feel drained but proud of your efforts. Thanks for the tip from Brad S.

30. When you have worked the problem, do not write the answer until you go back and read the question. This applies to both parts of the test. Thanks for the tip from Mr. B.

31. The best tip I could give for the multiple choice sections is called “Triage.” On the first pass through the multiple choice problems, do only those you immediately and absolutely know how to do. Circle the ones that you cannot do without some effort. Make a second pass through, this time skipping those you feel you have no idea how to do. Go through the problems one last time and attempt those problems you believe to be most difficult. This makes the best use of your time and keeps you from expending mental energy early in the test on problems that are difficult. Thanks for the tip from Mr. G.

32. Do not simplify a numerical answer ! If you get 1(1) + 2(3) + 5cos 45…….stop!!! You are finished. I took away over 100 points when I graded the exam because students told me 1(1) = 2. Thanks for the tip from KC H. at Freedom Area High.

33. Don’t let AP readers guess at what you mean. Write the responses to your Free Response questions completely, leaving nothing to the readers imagination. Thanks for the tip from Ryan H. at Southern Lehigh High.

34. Don’t simplify your answers for the FRQs. Thanks for the tip from David O. at St. Johns Country Day School.

35. Questions have to do with one of the following areas: limits, derivatives, or integrals (definite or indefinite). If you get stuck on a problem, just ask yourself which of these the question is asking you to find. Thanks for the tip from Dan M. at Broad Run High.

36. AP stands for Always Practicing. You can not cram prior to the exam to be successful on any AP test. You must put the time and work in all year long. This includes weekends, holidays, and even snow days. If you have consistently done this, your scores will reflect your efforts. Thanks for the tip from Dan M. at Broad Run High.

37. The value of a limit of a function as x approaches “a” is NEVER the value of the y coordinate of an isolated “dot”. Thanks for the tip from James M.

38. EVERY point on the free response is its own little test. Write down all that you know and don’t get freaked out about stuff in an earlier part of a question you may NOT have known. STAY IN THE GAME! Thanks for the tip from Bo G.

39. THINK GRAPHICALLY. A picture IS worth a thousand words, and your ability to picture what is happening in the context of the problem will help you understand if the derivative should be negative or positive, if the value should be big or small, if the second derivative (acceleration in many cases) is positive or negative, if your integral value should be positive or negative, etc. To reason with a visual supports your algebra in many ways. Thanks for the tip from Chris L.

40. Sit on your ego! Know that you will be challenged and that it is not a reflexion on you! Thanks for the tip from Damien J.

41. The more you practice a variety of problems from different resources the better you begin to recognize and understand types of problems. Thanks for the tip from Kristy H.

42. Bring extra batteries for your calculator! Thanks for the tip from David P. at Torrey Pines High.

43. You must know the why and how, not just the what. Thanks for the tip from John S.

44. Don’t forget your calculator on exam day and be sure to put fresh batteries in it. Thanks for the tip from Dawn D. at FCHS.

45. Make sure your calculator is in radians (check the settings in general). Thanks for the tip from Dawn D. at FCHS.

46. Use your calculator when there are shortcuts to solving a problem rather than performing the calculus by hand. Thanks for the tip from Dawn D. at FCHS.

47. Answer only the question asked, don’t waste time doing something that isn’t needed or required to a problem. Thanks for the tip from Dawn D. at FCHS.

48. If a graph is given, be sure to identify which function it represents….is it the function or a derivative? Thanks for the tip from Dawn D. at FCHS.

49. Don’t oversimplify. The test doesn’t require it, and oversimplifying can use valuable time! Thanks for the tip from Misty P. at Virtual Virginia.

50. I would tell students to know their major theorems, both hypothesis and conclusion, cold. To me these are IVT, EVT, Rolle’s, MVT (derivatives), FTC I (including its use for Net Change on a rate), FTC II, and MVT (integrals). After that, practice justifying numerical and graphical problems on previous FRQs using the information given (Ex: Given a graph of f(x), do not simply say f'(x) is positive, say f'(x) is positive as the given graph, f(x), is increasing, thereby connecting your justification to the information given). Finish by checking your work with the sample 9/9 FRQ student response provided by the College Board. Thanks for the tip from Shaun B. at James Clemens High.

51. Use 3 decimal places. Thanks for the tip from Keith L. at SHHS.

52. Make sure students know that a critical number must be in the domain of the function. So in f(x) = (x + 5)/(x – 3), 3 cannot be a critical number. Thanks for the tip from Jane W.

53. On application questions with lots of information, highlight the units on the given functions (equation, graph, or table) in the problem and ask yourself what needs to be done (integrate or differentiate) in order to arrive at the correct units in your answer. Thanks for the tip from Joseph S.

54. Learn to do mental math. You save time on the Multiple Choice, no calculator part. Thanks for the tip from Leena G.

55. My tip is to never leave a free response question blank and if you are unsure what to do, set the given equation equal to zero and solve it and set the derivative of the given equation equal to zero and solve it. Thanks for the tip from Craig G.

56. Do Saturday work sessions by topic and give mock exams. Thanks for the tip from Valerie P.

57. On free response questions especially, practice understanding in words what you are doing mathematically. Like knowing by the words if they want a average rate of change or the average of the integral. Knowing how those are asked and what the answers mean, make it easier not only to answer the question but to justify the answer as well. Thanks for the tip from Natalie B. from Memorial Early College High.

58. Time can be your enemy. When taking practice exams, set your timer to give yourself 5 fewer minutes than you would when taking the actual exam. You’ll get used to working at a quicker pace and you’ll have extra time to check your work before time is called. Thanks for the tip from Mary B.

59. Realize this is not a UIL competition test. They are looking for a reason to give you credit for Calculus – not trying to make a test no one can make a 100 on. Practice being thorough in your answers so they can see you understand the concept. Thanks for the tip from Michael B.

60. Starting In January do one free response question a night. This attachment has the problems by topic, year, and number. Thanks for the tip from Eric H.

61. Continue to ask yourself WHY? as you progress through the topics. If you are doing that, and able to answer that, then you will continue to understand the concepts deeply, and the unique problems from the AP will be easily tackled. Thanks for the tip from Eric S.

62. Answer every part of a FRQ. Label each part a), b), c), etc. If there are multiple answers, roots, solutions, points to a single question, make sure to sum up your answers in one Box. Thanks for the tip from Mary S. from Grafton High.

63. The best tip I have for my students is to keep up with the current homework, homework is where most of your learning takes place. If you have questions make sure to ask them. Work with a buddy or buddies if you can and discuss your ideas. Catching up is very difficult. Thanks for the tip from Carolanne F.

64. Read the question carefully and answer it completely. – Many FR questions require the student to complete multiple tasks. Many students lose points because they do not answer the questions appropriately. Thanks for the tip from Jessica S. from Cypress Bay High.

65. Know the big concepts, but pay attention to the details. Show all work and do not skip steps. Thanks for the tip from Beth P. from Martin Luther King, Jr. Magnet School.

66. Here’s a tip I use to try to get kids to take a breath before starting a free response question: Basically, Calculus is derivatives and integrals, so ask yourself before doing any work, “Is this a derivative or an integral question?” Then, at least, they have somewhere to start. Thanks for the tip from Steve S. from Kents High.

67. The mean value theorem and its converse must be understood at all levels and one must be able to provide a simplistic physics example to explain it. Thanks for the tip from Jonathan H.

68. Limits must be understood to bind the whole of calculus together and must not be treated as the red headed step child, and this includes graphing them by hand using precalculus and algebra 2 rules and not being reliant on a calculator. Thanks for the tip from Jonathan H.

69. There are 3 phases of calculus: position, velocity, and acceleration. You must know which phase the problem is in currently and which phase you need to transform it to. This will help you decide if you are going to integrate or differentiate. Thanks for the tip from Rachael A.

70. Understanding graphs: keep in mind that a function can decrease while its derivative increases. Although tangent lines are getting less steep, their slopes are becoming less negative. Thanks for the tip from Mr. S.

71. Practice, practice, practice, and more practice.  There are so many non-secure, practice multiple-choice and free-response questions available that there is no reason not to take full advantage of them in preparation for the exam.  Exposure to the question formats, common notation/terminology, etc. is key to applying content knowledge for applicable scenarios and, ultimately, being successful on the actual exam. Thanks for the tip from Dominic B.

72. Even if you have the correct answer, linkage errors and presentation errors can make students lose points in the grading process. Go back and check for these before time is up. Linkage error example: 10 * 5 = 50 / 2 = 25 is a linkage error because 10 * 5 does not equal 50 / 2. You must start a new line rather than link them together; so 10 * 5 = 50; 50 / 2 = 25 Presentation error example: When doing limit problems, not putting the limit as h approaches 0 in front of each expression along the way to the answer results in a presentation error.Thanks for the tip from Doreen V.

Are you a teacher or student? Do you have an awesome tip? Let us know!

## Frequently Tested Concepts for the AP Calculus Exam

Differential Calculus

Properties of limits, Properties of derivatives, Domain and range, One-sided limits, Limits at infinity, Continuity, 3 Types of Discontinuities, Product rule, Quotient rule, Power rule, Chain rule, Even functions, Odd functions, Periodic functions, Trig derivatives and inverse trig derivatives, Implicit differentiation, Higher order derivatives, Mean Value Theorem, L’Hopital’s Rule, Tangent lines, Extreme Value Theorem, Newton’s method of approximation.

Integral Calculus

Indefinite integrals, U-Substitution Integration by parts, Exponential growth and decay, Definite integrals, Riemann sums, Trapezoidal method, Fundamental Theorem of Calculus #1, Fundamental Theorem of Calculus #2, Average Value, Disk method, Washer method, Shell method.

This Ultimate List of AP Calculus Tips was written for the express purpose of giving the AP Calculus student a strong competitive advantage when taking the exam.  The academic world is especially competitive these days and it is critical that even the best of students read and understand these tips. Make sure to bookmark this page so that you can refer to it throughout the school year as you navigate your way through AP Calculus AB/BC. Best of luck!

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