Introduction: The Ultimate List of AP® Calculus Tips
Are you preparing to take the AP® Calculus exam? The AP® Calculus AB and BC exams are some of the most difficult AP® exams out there. But, fear not! Albert is here to help. We outline essential studying tips and tricks to help you prepare to do your best on exam day.
Whether you’re studying for the AP® Calculus AB or BC exam, the study tips below will help you earn a that 4 or a 5. If you’re not sure what the difference is between the two exams, the AP® Calculus BC exam covers all of the same 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!
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What We Review
Overall How To Study for AP® Calculus: 7 Tips for 4s and 5s
1. Practice makes perfect.
In order to score a 4 or 5 on your AP® Calculus exam you will need to practice a lot of different problems. The more problems you do, the more adept you will be at deciphering the way in which each type of problem is presented.
In the weeks leading up to the exam, create a study schedule for yourself where you’re practicing at least 10 problems each day.
Albert is a great resource for this. We have practice problems organized by unit and concept, a variety of free response questions, two AP® Calculus AB practice exams, and two AP® Calculus BC practice exams.
2. Budget 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.
Use any left over time that you have to review and double-check your answers. 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.
3. Show all of your work.
Make sure that when performing solutions to both the multiple-choice and FRQ problems that you show all your work.
This is especially important for the FRQs, as partial credit will apply. If you don’t get every aspect of the problem correct, you will earn points for what you did do correctly.
Even though multiple-choice questions do not have partial credit, showing your work as you solve the problem will help lead you to the correct answer. It also allows you to backtrack and re-check each step if the answer you got wasn’t a multiple choice option.
Remember to write down every equation if you’re using a calculator to solve it. An answer without an equation may not get full credit, even if it is correct.
For the same reason, if you use your calculator to find the value of a definite integral or derivative, write down the integral or derivative equation first.
4. Use your calculator only for the following actions.
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. Even though you can explore all these with your calculator, your solution must stand alone in order to receive credit.
5. 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:
Trig Derivatives | Inverse Trig Derivatives |
\dfrac{d}{dx}sin{x}=cos{x} | \dfrac{d}{dx}arcsin{x}=\dfrac{1}{\sqrt{1-x^2}} |
\dfrac{d}{dx}cos{x}=-sin{x} | \dfrac{d}{dx}arccos{x}=\dfrac{-1}{\sqrt{1-x^2}} |
\dfrac{d}{dx}tan{x}=sec ^2{x} | \dfrac{d}{dx}arctan{x}=\dfrac{1}{x^2+1} |
\dfrac{d}{dx}cot{x}=-csc^2{x} | \dfrac{d}{dx}\text{ arccot}{x}=\dfrac{-1}{x^2+1} |
\dfrac{d}{dx}sec{x}=sec{x}\cdot tan{x} | \dfrac { d }{ dx } { arcsec }{ x }=\dfrac { 1 }{ \left( x \right) \sqrt { x^{ 2 }-1 } } |
\dfrac{d}{dx}csc{x}=-csc{x}\cdot cot{x} | \dfrac{d}{dx}\text{ arccsc }{x}=\dfrac { -1 }{ \left( x \right) \sqrt { x^{ 2 }-1 } } |
6. 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.
7. 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.
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?”
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AP® Calculus Multiple-Choice 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. Use 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, cross them out immediately. Since there are four possible answers to each multiple-choice question, eliminating one answer increases your odds by 25%. Not too shabby!
Eliminating two answers increases your odds of a correct response by 50% and so on. Again, use this technique only if necessary. Otherwise, solve the problem and choose the correct answer.
2. Learn to spot distractor answers.
The individuals who create the AP® Calculus questions commonly include distractor answers as options.
Contained in the list of answer alternatives is the correct or best alternative, which is the answer, and incorrect or inferior alternatives, known as distractors. Your job is to select the correct answer/alternative.
Oftentimes on math exams, distractors can be made to be very close in nature to the actual answer. One example of a common distracter is to use the correct numerical answer but to put a minus sign in front of it. Or, if the answer is supposed to be a negative number they will place a distractor with a positive sign in front of it.
Check out this Magoosh article for more examples and tips on how to spot distractors.
3. 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.
Start by taking a full-length AP® Calculus practice test (you can find four on Albert). As you complete the test, do the following three steps:
Keep track of all the questions you guessed on.
To do this, list all the questions on your AP® Calculus practice test that you guessed on– even if you ultimately answered it correctly. The questions you guess on are your weaknesses.
Keep track of all the questions you scored incorrectly on.
Keep another list of the questions you thought were correct, but were actually incorrect.
Reflect on your choices.
Once you’ve compiled your lists, get to the root of why you didn’t understand these questions. Did you simply read it wrong? Were you running out of time? Are you confused by a certain type of function? Write down why you got each question wrong and what you will do to prevent a similar mistake from happening again.
4. Know the different types of calculus problems.
One of the most common AP® Calculus questions 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. In 2016, the AP® Calculus exam started including some problems that could only be solved by using L’Hôpital’s rule. This is a great way of determining the limit of a function divided by another function.
The other types of problems found on the AP® Calculus exam are the following: continuity of functions, asymptotic functions, antidifferentiation, and the Fundamental Theorem of Calculus.
5. 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.
Here is a Khan Academy video going over how to apply the chain rule twice:
6. Understand and master the Product and Quotient Rules.
These are two more essential rules you should memorize during your AP® Calculus review sessions. The Product Rule explains how to differentiate expressions that are the product of two other expressions. The Quotient Rule is a formula for differentiating expressions that are the quotient of two other expressions.
Check out this link for more AP® Calculus review and practice on the Product Rule, and this link for more on the Quotient Rule.
7. Don’t round answers until the last step.
Don’t round off your answers at each step of the problem. If you start rounding off your answers too early, it will create a cumulative rounding error. This can affect the accuracy of your final answer, and give you a number that isn’t one of the multiple choice answer options.
Also, do not simplify a numerical answer. For example, if you get 1(1) + 2(3) + 5cos 45…….stop! You are finished. Students can often lose multiple points from this error.
8. Know what the graphs of basic functions look like.
You should be able to draw sine, cosine, tangent, ex, ln(x), 1/x, and simple polynomials easily and relatively accurately. It’s essential to have a firm understanding of functions for limits, derivatives, integrals and all the other main areas of calculus. You’ll also want to know the unit circle and how the trigonometric functions relate to it.
9. Understand the graphical implications of derivative and original.
Keep in mind these two fundamental concepts. 1- Remember that the derivative at a point is the instantaneous slope of the graph or the slope of the line tangent to the graph at that point. 2- Know that the derivative of an entire function is just another function where the y, or f'(x), values are simply the value of the slope of the original function at the given x value.
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AP® Calculus Free Response Tips
The following AP® Calculus FRQ 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 exercises. In essence, the FRQ is digging deeper in attempting to assess your more global understanding of Calculus.
1. Practice released AP® Calculus FRQs.
Visit AP® Central here for past AB examples and here for past BC examples of the AP® Calculus Free Response Questions. These will give you the best idea of what to expect on your exam.
You can also find the past Chief Reader reports at those same links. These are a great tool to help you understand how to complete FRQs.
Chief Reader Reports give an overview of each free-response question and how different students performed on the question.
They describe typical student errors, which can help you pinpoint where you went wrong if you made the same errors. It also includes general comments regarding the skills and content that students frequently struggle with most.
They also provide some suggestions for improving student preparation. These suggestions can help guide your studying.
When you’re starting your AP® Calculus free response review, first complete a past set of released questions, then grade yourself with the scoring guidelines.
See how long it took you to answer each question, how effective you were at answering the actual question posed, and where you lost points.
Review the Chief Reader Reports on those FRQs for help understanding any common errors you might have made.
After practicing this a few times, you’ll begin to be more mindful of what the graders look for in your responses.
2. Memorize and Identify task verbs.
FRQs for the AP® Calculus exam are essentially word problems that come in multiple parts. They are usually between 3-5 parts per FRQ. FRQs are designed to test your ability for an “extended chain of reasoning.”
The first thing you should do when reading through the FRQ is identify the task verbs. Underline or circle them in the problem.
Task verbs are terms the AP® Calculus exam uses to tell you what you need to do. Each Task Verb requires a different type of response.
Underlining the task verbs will help focus your thoughts and guide your next steps. The most common task verbs used on the AP® Calculus exam as provided by CollegeBoard are below:
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.
This is one of the more important AP® Calculus FRQ tips to master.
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.
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 revisit a particular question.
5. 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 sign chart 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 from the College Board:
6. Give units if required.
Don’t forget to do this if the question asks for it. 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. You don’t want to lose points unnecessarily for something like this.
7. Round off numerical answers properly.
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.
8. Memorize the key derivatives and integrals of common trigonometric and other functions.
This tip 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.
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Tips by AP® Calculus Teachers
The following are a collection of tips from teachers specifically designed to help you succeed on the AP® Calculus exam. These tips are broken in three sub-groups: Exam Day Tips, Content Tips, Preparation Tips.
AP® Calculus Exam Day Tips:
1. 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. Re-read the question to ensure you understood and answered all aspects of it.
2. Go through the multiple choice section three times.
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.
Then, 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 using mental energy early in the test on problems that are difficult.
3. Do not simplify a numerical answer.
For example, if you get 1(1) + 2(3) + 5cos 45…….stop! You are finished. Students can often lose multiple points from this error.
4. Don’t lose points for linkage errors and presentation errors. 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.
5. Answer only the question asked.
You don’t want to waste time doing something that isn’t needed or required for a problem.
6. Never leave a free response question blank.
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.
AP® Calculus Prep Tips:
1. 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 and do some problems together. Set up a recurring time to meet either virtually or in-person to complete practice problems together.
Most serious students do this to help take the stress out of the anticipation of taking such a high-level exam such as AP® Calculus.
2. 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.
3. Practice with a little less than the given time.
When taking AP® Calculus practice tests, 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.
4. Prepare yourself the night before.
There are a few things you can and should do the night before to prepare yourself for success on the exam day.
- Review all the formulas the night before and then go to sleep. Don’t try to cram and memorize anything new the night before. Focus on a quick AP® Calculus review of important formulas that you already know, and then get a good night’s sleep.
- Pack extra batteries for your calculator. It’s important to be prepared, just in case.
- Look up the directions to your testing center if it’s a new place. Give yourself at least 20 minutes extra time to get there.
5. Wear a watch.
Although there may be a clock on the wall in the room, there is no guarantee it will be in a clear position for you to see. If you bring your own watch, it’s easier for you to monitor your time and make sure you aren’t spending too long on any one section.
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AP® Calculus Content Tips:
1. Always remember to include +C in your answer for 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} ?
The correct answer is:
\int{Ydx}=\dfrac{{x}^{4}}{4}+\frac{{x}^{ 3}}{3}+C
2. Review past trigonometric concepts.
Many Calculus problems will use trig functions so it would be best for you to review trigonometric identities and common trigonometric formulas.
3. In doing derivatives, remember 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. Practice this by using different problems, such as the ones with n = e, a = pi.
4. Memorize the unit circle to calculate equations more quickly.
If you have it memorized, you can calculate the trig values of the angles without a calculator. This skill will save you precious time on the AP® Calculus exam.
5. Remember that 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.
6. The limit value does not depend on the actual value of f(a).
In this example graph below from Magoosh, the limit exists at x = 3 even though the function has a hole in the graph at that point.
7. 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.
8. Know the major theorems, both hypothesis and conclusion.
Know IVT, EVT, Rolle’s, MVT (derivatives), Fundamental Theorem of Calculus I (including its use for Net Change on a rate), Fundamental Theorem of Calculus 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.
9. The mean value theorem and its converse must be understood at all levels.
You also must be able to provide a simplistic physics example to explain it.
10. 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.
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Wrapping Things Up: The Ultimate List of AP® Calculus Tips
Studying for the AP® Calculus can be intimidating at first, but having a study plan, and a good grasp of what to expect on the exam will help you stay calm and focused.
We’ve covered a lot in this article. To recap, remember to:
- Practice as much as you can. Use Albert as a resource for detailed practice questions with explanations, as well as full-length AP® Calculus AB and AP® Calculus BC practice exams.
- Budget your time wisely. Plan to spend about two minutes on each multiple choice question, 15 minutes each FRQ.
- Show all your work. This is the best way to earn the maximum amount of points, and it will allow you to solve your problems more efficiently.
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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