# How to study for the AP^{®} Calculus AB Exam

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Written by Colin. When he is not tutoring for us, Colin is a full time mathematics teacher at a top ranked STEM school in the US. His students average over a 4.7 on the AP Calculus AB exam.

## Introduction

There are many guides to the Calculus AB exam, so why should you read this one?

One reason is that I’ve been teaching AP Calculus in high school since 2010, and my students have an average score of over 4.7 over the years. I’ve taught mathematics from Algebra 1 to Multivariable Calculus at high schools and generally focus on conceptual understanding and problem-solving skills that are not specific to a standardized test. That, followed by a detailed review and preparation in the weeks before the exam gets you ready to do your best.

More generally, it is important to assimilate information from various sources. Each guide you read will have things in common, expressed slightly differently, which will reinforce your memory and understanding of different concepts, and each guide may have unique insights or perhaps will present just one or two key ideas in a way that really speaks to you.

This guide does not propose to be a comprehensive review of what you need to know to pass the test; rather, it offers a general approach. It works well to have an overall plan for your review that should take place over the course of several weeks – slow and steady aces the test.

## The Outline

I often suggest that when preparing for any large test the first thing to do is to sit down with a blank sheet of paper and try to write an outline of the concepts of the course. I encourage you to do this before reading on. Start with unit names and try to get as many details as you can, possibly even to the level of sample problems that you might remember.

After you’ve exhausted what you can write down from memory, look over tests, notes, or books that you’ve used and try to fill in and edit your outline. You can find many other outlines online; compare yours with one of them and consider whether you need to refresh yourself about the topics you forgot to include.

Generally, the outline will be written in the order that you learned things in the course, with some exceptions. For example, you learn L’Hopital’s rule towards the end of the course, but I include it in the Limits part of the outline. Also some classes may introduce the derivatives of exponential, logarithmic, and inverse trig functions after having discussed the integral. So don’t think that the example below is the only way it can be written. Skipping details, the skeleton of your outline might look something like the following; can you fill in definitions, equations, examples?

- Limits
- Graphically
- Algebraically
- One-sided
- Ways a limit may not exist
- Continuity – definition and IVT
- L’Hopital’s Rule

- The Derivative
- Limit Definition
- Derivatives of Parent Functions
- More Advanced Rules – product, quotient, and chain rules [the chain rule also allows us to take implicit derivatives]
- Derivatives of Inverse Functions – including logarithmic and inverse trig
- Higher Order Derivatives

- Derivatives and the Graph of a Function
- Tangent Lines
- Finding local and global extrema – critical points and endpoints
- Mean Value Theorem
- Concavity and points of inflection

- More Applications of the Derivative
- Related Rates – an application of the chain rule
- Optimization – an application of finding extrema
- Rectilinear Motion – we should really wait until we have definite integrals to talk about this
- Differentials – an application of tangent line approximation, with tricky notation

- Intro to Integration
- Anti-Derivatives (a.k.a. indefinite integrals)
- The Area Problem – or more generally, integration as a measure of accumulation
- Riemann Sums – know that there are a lot of variants [left, right, midpoint, trapezoid rule; equal sub-intervals or not]
- The Definite Integral (as a limit of Riemann Sums)

- Fundamental Theorem of Calculus
- Calculating definite integrals using anti-derivatives
- Rate of change of a definite integral with respect to changes in the interval of integration

- Inverse and Logarithmic Functions
- General rule for the derivative of the inverse of a function
- Here is where I like to define as the integral and as its inverse
- Derivative of inverse trig functions, and recognizing them as anti-derivatives

- Further Applications of Integration
- Area between curves
- Volumes
- Separable Differential Equations

## Gather Your Resources

It will be useful to have a concise review sheet of useful definitions, key ideas, notations, and formulas. The official College Board* ^{®}* course description is comprehensive, though a bit long; here is the link!

A dense ‘cheat sheet’ will sometimes be enough to trigger your memory of the details, and we suggest writing one up as an exercise. We have a number of examples we can provide to our students.

You will want practice tests, both multi-choice (MCQ) and free-response (FRQ) type, both calculator allowed and no-calc sections. These, along with solution guides, are easily found from several sources. The College Board website has a great list of past FRQ’s, including scoring rubrics and sample responses. We also have a huge file of exams.

It’s been said that you don’t really understand something unless you can explain it to your grandmother. I will suggest that it is helpful to show and explain your solutions to a friend who is also studying for the test. It is also very effective to work with an expert who can help diagnose your mistakes, show some new problem solving skills, or better ways to set up your work.

## MCPractice

From the above link, take the multi-choice test with reasonably strict test-like conditions: no extra resources, but allow yourself extra time, if needed, to have a chance to think about each question.

Score yourself and see if there is a pattern to the topic of the questions that you missed, which will let you know what needs more review from the focused practice problems. Also, try to determine what kind of error you made – whether you were totally off-base or had a generally good approach but missed a small detail or had a miscalculation. The key is to be efficient and know where to look for targeted review of the skills you most need to practice.

## The Calculator

A small note about the calculator. You may notice that only about 1/3 of the points on the test are on “calculator allowed” parts of the test, and that only a small fraction of the questions on those parts actually require the use of the calculator. There are many questions that are conceptual in a way that a calculator cannot possibly help you, but some where it would be impossible to give a precise answer without a calculator, and even very challenging to estimate. There are relatively few where a calculator can be efficiently used to check an answer that could be found without one. Here are a couple scenarios when you can expect to use your calculator:

- Calculating a definite integral when there is no clear way to find an anti-derivative. Even if you think you can do it, when they want a decimal answer to a definite integral, it is a clear sign of what they expect you to do. For full credit in a Free Response question, you usually just have to write down the definite integral, with proper notation, and the value that you obtained from the calculator.
- Finding zeros of a difficult function (often to find critical points). It is a skill learned through experience to be able to gauge when an equation will be easy enough to solve by hand vs. when a decimal approximation from the calculator is preferred.
- Evaluate a complicated function as a specific value (e.g. to find an instantaneous rate of change by calculating the derivative at some point). Be clear and efficient with your written answers here. Usually the function will have a name, and it will be preferred (even expected) that you write down just something like with decimal values for and rather than writing out the full expression.

## Pre-requisite mistakes

Assess your work critically. How many of your mistakes actually came from a difficulty with Geometry [e.g. drawing diagrams, labeling variables, and using facts from geometry to find relations between those variables], Algebra [e.g. making mistakes with factoring, incorrectly simplifying rational expressions], or other pre-calc topics [e.g. rules of logs and exponents, understanding function notation and concepts]. It is not uncommon for one’s AP Calculus score to be held back not by a lack of understanding the main ideas of calculus, but by lack of fluency with more elementary topics, though I feel the College Board has gotten better in recent years at writing questions that minimize this risk. A Mathematics curriculum is built in a way that learning new skills require, but also reinforce, old ideas.

So, while there is no quick fix for this, taking each new problem as a chance for review is the right attitude. Also, when it comes to test-day strategy, don’t be shy to skip questions that make you uncomfortable with a plan to come back later when you know how much time you have. Realize that you only need about 2/3 of the available points to get a 5 on the test, so it is more important to focus first on getting right the questions that you are most confident in.

## Free Response Practice

The Free Response questions are where students can lose a lot of points by not presenting their work properly despite understanding ideas and being able to do the calculations. Sometimes writing down the correct final answer will only get you 1 point, with the rest for showing clearly where it came from. For example, you can lose a point by not stating the units of the answer, from being sloppy with variable usage or function notation, or from not using clear declarative sentences to show that you have properly justified a conclusion. Using stock phrases like “ is a point of inflection because changes from increasing to decreasing” is the key; you will lose a point if you say ‘it’ instead of or ‘the derivative’. It is essential to look at many prior test questions and know what they are looking for in an answer.

Also realize that while the test has only 6 questions worth 9 points each, that is a bit misleading. Each question usually has at least 3 parts, and the parts are often independent of each other. Don’t skip a question just because you’re not sure about part a!

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