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BARRON’S ESSENTIAL 5

As you review the content in this book to work toward earning that 5 on your AP CALCULUS AB exam, here are five things that you MUST know above everything else:

derivatives (p. 99) and antiderivatives (p. 191) of common functions;

the Product (p. 99), Quotient (p. 99), and Chain Rules (p. 100) for finding derivatives;

the midpoint, left and right rectangle, and trapezoid approximations for estimating definite integrals (p. 224);

finding antiderivatives by substitution (p. 202);

the important theorems: Rolle’s Theorem (p. 115), the Mean Value Theorem (p. 114), and especially the Fundamental Theorem of Calculus (p. 217).

use L’Hospital’s Rule to find limits of indeterminate forms (only and ) (p. 116);

find equations of tangent lines (p. 141);

determine where a function is increasing/decreasing (p. 143), is concave up/down (p. 144), or has maxima, minima, or points of inflection (pp. 145, 150);

analyze the speed, velocity, and acceleration of an object in motion (p. 160);

solve related rates problems (p. 168), using implicit differentiation (p. 111) when necessary.

the average value of a function (p. 237);

area (p. 255) and volume (p. 262);

the position of objects in motion and distance traveled (p. 307);

total amount when given the rate of accumulation (p. 313);

differential equations, including solutions and slope fields (p. 325).

answering all the multiple-choice questions;

knowing how and when to use your calculator, and when not to;

understanding what work you need to show;

knowing how to explain, interpret, and justify answers when a question requires that. (The free-response solutions in this book model such answers.)

BARRON’S ESSENTIAL 5

As you review the content in this book to work toward earning that 5 on your AP CALCULUS BC exam, here are five things that you MUST know above everything else:

Introduction

AP Calculus AB and AP Calculus BC are both full-year courses in the calculus of functions of a single variable. Both courses emphasize:

(1) understanding of concepts and applications of calculus over manipulation and memorization;

(2) developing the student’s ability to express functions, concepts, problems, and conclusions analytically, graphically, numerically, and verbally and to understand how these are related; and

(3) using a graphing calculator as a tool for mathematical investigations and for problem solving.

Both courses are intended for those students who have already studied college-preparatory mathematics: algebra, geometry, trigonometry, analytic geometry, and elementary functions (linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise).

Content Areas

The AP Calculus course topics can be arranged into four content areas:

1.Limits

2.Derivatives

3.Integrals and the Fundamental Theorem

4.Series

The AB exam tests content areas 1, 2, and 3. The BC exam tests all four content areas. There are BC only topics in content areas 2 and 3 (such as planar motion, Euler’s method, and logistic growth). Roughly, 40 percent of the points available for the BC exam are BC only topics.

To review all of the specific topics covered within each of these content areas, be sure to consult the review chapters throughout this book and visit the College Board website for any recent updates.

Exam Format

Both the AP Calculus AB exam and the AP Calculus BC exam follow the same format. Both exams are 3 hours and 15 minutes long, and the shared format of both exams is outlined in the table.

Scoring of the Exams

Each completed AP exam receives a grade according to the following five-point scale:

5Extremely well qualified

4Well qualified

3Qualified

2Possibly qualified

1No recommendation

Many colleges and universities accept a grade of 3 or better for credit or advanced placement or both (a score of 3 has been historically awarded for earning over 40 of 108 possible points). Be sure to check the AP credit policies on individual colleges’ websites.

The multiple-choice questions in Section I are scored by a machine. Note that the score will be the number of questions answered correctly. Since no points can be earned if answers are left blank, and there is no deduction for wrong answers, you should answer every question. For questions you cannot solve, try to eliminate as many of the choices as possible and then pick the best remaining answer.

The problems in Section II are graded by college and high school teachers called readers. The answers in any one exam booklet are evaluated by different readers, and for each reader, all scores given by preceding readers are concealed, as are the student’s name and school. Each free-response question in Section II is graded out of 9 points, and all problems in Section II are counted equally. Readers are provided with sample solutions for each problem, with detailed scoring scales and point distributions that allow partial credit for correct portions of a student’s answer.

When determining the overall grade for each exam, the two sections are given equal weight. The total raw score is then converted into one of the five grades listed previously. Do not think of these raw scores as percents in the usual sense of testing and grading. In general, you will not be expected to answer all the questions correctly in either Section I or Section II. For instance, if you average 6 out of 9 points on the Section II questions and perform similarly well on Section I’s multiple-choice questions, you may possibly earn a 5.

Great care is taken by all involved in the scoring and reading of exams to make certain that they are graded consistently and fairly so that a student’s overall AP grade reflects as accurately as possible the student’s achievement in calculus.

NOTE

Students who take the BC exam are given a Calculus BC grade and a Calculus AB subscore grade. The latter is based on the part of the BC exam that deals with topics in the AB syllabus.

Using Your Graphing Calculator on the AP Exam

Guidelines for Calculator Use

1. On the multiple-choice questions in Section I, Part B, you may use any feature or program on your calculator. Warning: Don’t rely on it too much! Only a few of these questions require the calculator, and in some cases using it may be too time-consuming or otherwise disadvantageous.

2. On the free-response questions in Section II, Part A:

(a) You may use the calculator to perform any of the four procedures listed below. When you do, you need only write the equation, derivative, or definite integral (called the setup) that will produce the solution and then write the calculator result to the required degree of accuracy (three places after the decimal point unless otherwise specified). Note that a setup must be presented in standard algebraic or calculus notation, not in calculator syntax. For example, you must include in your work the setup even if you use your calculator to evaluate the integral.

(b) For a solution for which you use a calculator capability other than the four listed below, you must write down the mathematical steps that yield the answer. A correct answer alone will not earn full credit and will likely earn no credit.

(c) You must provide mathematical reasoning to support your answer. Calculator results alone will not be sufficient.

The Four Calculator Procedures

Each student is expected to bring a graphing calculator to the AP exam. Different models of calculators vary in their features and capabilities; however, there are four procedures you must be able to perform on your calculator:

C1. Produce the graph of a function within an arbitrary viewing window.

C2. Solve an equation numerically.

C3. Compute the derivative of a function numerically.

C4. Compute definite integrals numerically.

The Procedures Explained

Here is more detailed guidance for the four allowed procedures.

C1. Produce the graph of a function within an arbitrary viewing window. More than likely, you will not have to produce a graph on the exam that will be graded. However, you must be able to graph a wide variety of functions, both simple and complex, and be able to analyze those graphs. Skills you need include, but are not limited to, typing complex functions correctly into your calculator including correct notation, which will ensure that the graph on the screen is what the question writer intended you to see, and finding a window that accurately represents the graph and its features. Note that on rare occasions you may wish to draw a graph in your exam booklet to justify an answer in the free-response section; such a graph must be clearly labeled as to what is being graphed, and there should be an accompanying sentence or two explaining why the graph you produced justifies the answer.

C2. Solve an equation numerically is equivalent to Find the zeros of a function or Find the point of intersection of two curves. Remember: you must first show your setup—write the equation out algebraically; then it is sufficient just to write down the calculator solution.

C3. Compute the derivative of a function numerically. When you seek the value of the derivative of a function at a specific point, you may use your calculator. First, indicate what you are finding—for example, f′(6)—then write the numerical answer obtained from your calculator. Note that if you need to find the derivative of the function, rather than its value at a particular point, you must write the derivative symbolically. Note that some calculators are able to perform symbolic operations.

C4. Compute definite integrals numerically. If, for example, you need to find the area under a curve, you must first show your setup. Write the complete integral, including the integrand in terms of a single variable, with the limits of integration. You may then simply write the calculator answer; you need not compute an antiderivative.

Accuracy

Calculator answers must be correct to three decimal places. To achieve this required accuracy, never type in decimal numbers unless they came from the original question. Do not round off numbers at intermediate steps, as this is likely to produce error accumulations that could result in a loss of credit. If necessary, store intermediate answers in the calculator’s memory. Do not copy them down on paper; storing is faster and avoids transcription errors. Round off, or truncate, only after your calculator produces the final answer.

Sample Solution for a Free-Response Question

The following example question illustrates proper use of your calculator on the AP exam. This example has been simplified (compared to an actual free-response question); it is designed to illustrate just the procedures (C1–C4) that you can use by supplying the setup and the value from your calculator.

Example

For 0 ≤ t ≤ 3, a particle moves along the x-axis. The velocity of the particle at time t is given by . The particle is at position x = 5 at time t = 2.

(a) Find the acceleration of the particle at t = 2.

Solution (a)

The acceleration is the derivative of the velocity—that connection must be made in your work. (C3) Since the velocity function is defined, you can use the derivative at a point function on your calculator to find v′(2). The calculator gives the value as v′(2) = –3.78661943164, which you may write as either v′(2) = –3.787 or v′(2) = –3.786 under the decimal presentation rules.

Your work should look like this:

(b) At what time(s) is the velocity of the particle equal to zero?

Solution (b)

(C2) You will need to solve the equation v (t) = 0. Some calculators have a solve function on the calculator/home screen, but sometimes they are a little difficult to work with. (C1) Our suggestion is to graph the function, v (t), and use the calculator’s root/zero function on the graph page. There is only one zero for v (t) on the interval 0 ≤ t ≤ 3, and it occurs at t = 2.64021.

Your work should look like this:

(c) Find the position of the particle at t = 1.

Solution (c)

(C4) You will use a definite integral by using the Fundamental Theorem of Calculus (FTC) to find the position. The form of the FTC you use is . Using this form of the FTC, you need to know a value of the function, f(a), and the rate of change of the function, f′(x). Here we know the position at t = 2 (i.e., x(2) = 5), and the rate of change of the position is the velocity, x′(t) = v(t). We want to find x(1), so our setup using the FTC is , and our calculator gives a value of 0.4064274888. Notice in the work below that we left the integrand as v(t); you may also do this on the AP Calculus exam since it is a defined function.

Your work should look like this:

A Note About Solutions in This Book

Students should be aware that in this book we sometimes do not observe the restrictions cited previously on the use of the calculator. When providing explanations for solutions to illustrative examples or to exercises, we often exploit the capabilities of the calculator to the fullest. Indeed, students are encouraged to do just that on any question in Section I, Part B of the AP exam for which they use a calculator. However, to avoid losing credit, you must carefully observe the restrictions imposed on when and how the calculator may be used when answering questions in Section II of the exam.

Additional Notes and Reminders

• SYNTAX. Learn the proper syntax for your calculator: the correct way to enter operations, functions, and other commands. Parentheses, commas, variables, or parameters that are missing or entered in the wrong order can produce error messages, waste time, or (worst of all) yield wrong answers.

• RADIANS. Keep your calculator set in radian mode. Almost all questions about angles and trigonometric functions use radians. If you ever need to change to degrees for a specific calculation, return the calculator to radian mode as soon as that calculation is complete.

• TRIGONOMETRIC FUNCTIONS. Many calculators do not have keys for the secant, cosecant, or cotangent functions. To obtain these functions, use their reciprocals.

For example, .

Evaluate inverse functions such as arcsin, arccos, and arctan on your calculator. Those function keys are usually denoted as sin–1, cos–1, and tan–1.

Don’t confuse reciprocal functions with inverse functions. For example:

• NUMERICAL DERIVATIVES. You may be misled by your calculator if you ask for the derivative of a function at a point where the function is not differentiable because the calculator evaluates numerical derivatives using the difference quotient (or the symmetric difference quotient). For example, if f(x) = |x|, then f′(0) does not exist. Yet the calculator may find the value of the derivative as