Part II

About the AP Calculus Exams

AP Calculus is divided into two types: AB and BC. The former is supposed to be the equivalent of a semester of college calculus; the latter, a year. In truth, AB calculus covers closer to three quarters of a year of college calculus. In fact, the main difference between the two is that BC calculus tests some more theoretical aspects of calculus and it covers a few additional topics. In addition, BC calculus is harder than AB calculus. The AB exam usually tests straightforward problems in each topic. They’re not too tricky and they don’t vary very much. The BC exam asks harder questions. But neither exam is tricky in the sense that the SAT is. Nor do they test esoteric aspects of calculus. Rather, both tests tend to focus on testing whether you’ve learned the basics of differential and integral calculus. The tests are difficult because of the breadth of topics that they cover, not the depth. You will probably find that many of the problems in this book seem easier than the problems you’ve had in school. This is because your teacher is giving you problems that are harder than those on the AP.

STRUCTURE OF THE EXAMS

Now, some words about the test itself. The AP exam comes in two parts. First, there is a section of multiple-choice questions covering a variety of calculus topics. The multiple choice section has two parts. Part A consists of 28 questions; you are not permitted to use a calculator on this section. Part B consists of 17 questions; you are permitted to use a calculator on this part. These two parts comprise a total of 45 questions.

After this, there is a free response section consisting of six questions, each of which requires you to write out the solutions and the steps by which you solved it. You are permitted to use a calculator for the first two problems but not for the four other problems. Partial credit is given for various steps in the solution of each problem. You’ll usually be required to sketch a graph in one of the questions. The College Board does you a big favor here: You may use a graphing calculator. In fact, The College Board recommends it! And they allow you to use programs as well. But here’s the truth about calculus: Most of the time, you don’t need the calculator anyway. Remember: These are the people who bring you the SAT. Any gift from them should be regarded skeptically!

OVERVIEW OF CONTENT TOPICS

Topics in italics are BC Topics. This list is drawn from the topical outline for AP calculus furnished by the College Board. You might find that your teacher covers some additional topics, or omits some, in your course. Some of the topics are very broad, so we cannot guarantee that this book covers these topics exhaustively.

I. Functions, Graphs, and Limits

A. Analysis of Graphs

You should be able to analyze a graph based on “the interplay between geometric and analytic information.” The preceding phrase comes directly from the College Board. Don’t let it scare you. What the College Board really means is that you should have covered graphing in precalculus, and you should know (a) how to graph and (b) how to read a graph. This is a precalculus topic and we won’t cover it in this book.

B. Limits

You should be able to calculate limits algebraically, or to estimate them from a graph or from a table of data.
You do not need to find limits using the Delta-Epsilon definition of a limit.

C. Asymptotes

You should understand asymptotes graphically and be able to compare the growth rates of different types of functions (namely polynomial functions, logarithmic functions, and exponential functions). This is a topic that should have been covered in precalculus, and we won’t cover it in this book.
You should understand asymptotes in terms of limits involving infinity.

D. Continuity

You should be able to test the continuity of a function in terms of limits and you should understand continuous functions graphically.
You should understand the intermediate value theorem and the extreme value theorem.

E. Parametric, Polar, and Vector Functions

You should be able to analyze plane curves given in any of these three forms. Usually, you will be asked to convert from one of these three forms back to Rectangular Form (also known as Cartesian Form).

II. Differential Calculus

A. The Definition of the Derivative

You should be able to find a derivative by finding the limit of the difference quotient.

You should also know the relationship between differentiability and continuity. That is, if a function is differentiable at a point, it’s continuous there. But if a function is continuous at a point, it’s not necessarily differentiable there.

B. Derivative at a Point

You should know the Power Rule, the Product Rule, the Quotient Rule, and the Chain Rule.
You should be able to find the slope of a curve at a point, and the tangent and normal lines to a curve at a point.
You should also be able to use local linear approximation and differentials to estimate the tangent line to a curve at a point.
You should be able to find the instantaneous rate of change of a function using the derivative or the limit of the average rate of change of a function.
You should be able to approximate the rate of change of a function from a graph or from a table of values.
You should be able to find Higher-Order Derivatives and to use Implicit Differentiation.

C. Derivative of a Function

You should be able to relate the graph of a function to the graph of its derivative, and vice-versa.
You should know the relationship between the sign of a derivative and whether the function is increasing or decreasing (positive derivative means increasing; negative means decreasing).
You should know how to find relative and absolute maxima and minima.
You should know the mean value theorem for derivatives and Rolle’s theorem.

D. Second Derivative

You should be able to relate the graph of a function to the graph of its derivative and its second derivative, and vice-versa. This is tricky.
You should know the relationship between concavity and the sign of the second derivative (positive means concave up; negative means concave down).
You should know how to find points of inflection.

E. Applications of Derivatives

You should be able to sketch a curve using first and second derivatives and be able to analyze the critical points.
You should be able to solve Optimization problems (Max/Min problems), and Related Rates problems.
You should be able to find the derivative of the inverse of a function.
You should be able to solve Rectilinear Motion problems.

F. More Applications of Derivatives

You should be able to analyze planar curves in parametric, polar, and vector form, including velocity and acceleration vectors.
You should be able to use Euler’s Method to find numerical solutions of differential equations.
You should know L’Hôpital’s Rule.

G. Computation of Derivatives

You should be able to find the derivatives of Trig functions, Logarithmic functions, Exponential functions, and Inverse Trig functions.
BC students should be able to find the derivatives of parametric, polar, and vector functions.

III. Integral Calculus

A. Riemann Sums

You should be able to find the area under a curve using left, right, and midpoint evaluations and the Trapezoidal Rule.
You should know the fundamental theorem of calculus:

B. Applications of Integrals

You should be able to find the area of a region, the volume of a solid of known cross-section, the volume of a solid of revolution, and the average value of a function.
You should be able to solve acceleration, velocity, and position problems.
BC students should also be able to find the length of a curve (including a curve in parametric form) and the area of a region bounded by polar curves.

C. Fundamental Theorem of Calculus

You should know the first and second fundamental theorems of calculus and be able to use them to find the derivative of an integral, and in analytical and graphical analysis of functions.

D. Techniques of Antidifferentiation

You should be able to integrate using the power rule and U-Substitution.
You should be able to do Integration by Parts and simple Partial Fractions.
You should also be able to evaluate Improper Integrals as limits of definite integrals.

E. Applications of Antidifferentiation

You should be able to find specific antiderivatives using initial conditions.

You should be able to solve separable differential equations and logistic differential equations.

You should be able to interpret differential equations via slope fields. Don’t be intimidated. These look harder than they are.

IV. Polynomial Approximations and Series

A. The Concept of a Series

You should know that a series is a sequence of partial sums and that convergence is defined as the limit of the sequence of partial sums.

B. Series Concepts

You should understand and be able to solve problems involving Geometric Series, the Harmonic Series, Alternating Series, and P-Series.
You should know the Integral Test, the Ratio Test, and the Comparison Test, and how to use them to determine whether a series converges or diverges.

C. Taylor Series

You should know Taylor Polynomial Approximation; the general Taylor Series centered at x = a; and the Maclaurin Series for e^x, sin x, cos x, and .
You should know how to differentiate and antidifferentiate Taylor Series and how to form new series from known series.
You should know functions defined by power series and radius of convergence.
You should know the Lagrange error bound for Taylor Series.

GENERAL OVERVIEW OF THIS BOOK

The key to doing well on the exam is to memorize a variety of techniques for solving calculus problems, and to recognize when to use them. There’s so much to learn in AP calculus that it’s difficult to remember everything. Instead, you should be able to derive or figure out how to do certain things based on your mastery of a few essential techniques. In addition, you’ll be expected to remember a lot of the math that you did before calculus—particularly trigonometry. You should be able to graph functions, find zeros, derivatives, and integrals with the calculator.

Furthermore, if you can’t derive certain formulae, you should memorize them! A lot of students don’t bother to memorize the trigonometry special angles and formulae because they can do them on their calculators. This is a big mistake. You’ll be expected to be very good with these in calculus, and if you can’t recall them easily, you’ll be slowed down and the problems will seem much harder. Make sure that you’re also comfortable with analytic geometry. If you rely on your calculator to graph for you, you’ll get a lot of questions wrong because you won’t recognize the curves when you see them.

This advice is going to seem backward compared to what your teachers are telling you. In school you’re often yelled at for memorizing things. Teachers tell you to understand the concepts, not just memorize the answers. Well, things are different here. The understanding will come later, after you’re comfortable with the mechanics. In the meantime, you should learn techniques and practice them, and, through repetition, you will ingrain them in your memory.

Each chapter is divided into three types of problems: examples, solved problems, and practice problems. The first type is contained in the explanatory portion of the unit. The examples are designed to further your understanding of the subject and to show you how to get the problems right. Each step of the solution to the example is worked out, except for some simple algebraic and arithmetic steps that should come easily to you at this point.

The second type is solved problems. The solutions are worked out in approximately the same detail as the examples. Before you start work on each of these, cover the solution with an index card or something, then check the solution afterward. And you should read through the solution, not just assume that you knew what you were doing because your answer was correct.

The third type of problem is practice problems. Only the answers to them are given. We hope you’ll find that each chapter offers enough practice problems for you to be comfortable with the material. The topics that are emphasized on the exam have more problems; those that are de-emphasized have fewer. In other words, if a chapter has only a few practice problems, it’s not an important topic on the AP exam and you shouldn’t worry too much about it.

HOW AP EXAMS ARE USED

Different colleges use AP Exams in different ways, so it is important that you go to a particular college’s web site to determine how it uses AP Exams. The three items below represent the main ways in which AP Exam scores can be used:

College Credit. Some colleges will give you college credit if you score well on an AP Exam. These credits count towards your graduation requirements, meaning that you can take fewer courses while in college. Given the cost of college, this could be quite a benefit, indeed.

Satisfy Requirements. Some colleges will allow you to “place out” of certain requirements if you do well on an AP Exam, even if they do not give you actual college credits. For example, you might not need to take an introductory-level course, or perhaps you might not need to take a class in a certain discipline at all.

Admissions Plus. Even if your AP Exam will not result in college credit or even allow you to place out of certain courses, most colleges will respect your decision to push yourself by taking an AP Course or even an AP Exam outside of a course. A high score on an AP Exam shows mastery of more difficult content than is taught in many high school courses, and colleges may take that into account during the admissions process.

OTHER RESOURCES

There are many resources available to help you improve your score on the AP Calculus Exam, not the least of which are your teachers. If you are taking an AP class, you may be able to get extra attention from your teacher, such as obtaining feedback on your essays. If you are not in an AP course, reach out to a teacher who teaches calculus, and ask if the teacher will review your free response questions or otherwise help you with content.

Another wonderful resource is AP Central, the official site of the AP Exams. The scope of the information at this site is quite broad and includes:

Course Description, which includes details on what content is covered and sample questions
Sample test questions
Essay prompts from previous years

The AP Central home page address is: http://apcentral.collegeboard.com/apc/Controller.jpf.

The AP Calculus AB Exam Course home page address is: http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2178.html

The AP Calculus BC Exam Course home page address is: http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2118.html

Finally, The Princeton Review offers tutoring and small group instruction. Our expert instructors can help you refine your strategic approach and add to your content knowledge. For more information, call 1-800-2REVIEW.

DESIGNING YOUR STUDY PLAN

As part of the Introduction, you identified some areas of potential improvement. Let’s now delve further into your performance on Test 1, with the goal of developing a study plan appropriate to your needs and time commitment.

Read the answers and explanations associated with the Multiple Choice questions (starting at this page). After you have done so, respond to the following questions:

Review the Overview of Content Topics on this page. Next to each topic, indicate your rank of the topic as follows: “1” means “I need a lot of work on this,” “2” means “I need to beef up my knowledge,” and “3” means “I know this topic well.”
How many days/weeks/months away is your exam?
What time of day is your best, most focused study time?
How much time per day/week/month will you devote to preparing for your exam?
When will you do this preparation? (Be as specific as possible: Mondays & Wednesdays from 3 to 4 pm, for example.)
Based on the answers above, will you focus on strategy (Part Two) or content (Part Three) or both?
What are your overall goals in using this book?