AP Calculus: DenverJDS

This class will prepare students for the AP Calculus AB exam. It will be taught in a similar style to precalculus, but at a quicker pace. There will be a major test every month or so, and tests/quizzes will account for about two thirds of your final grade. Homework will be collected and graded. Students are expected to put in considerable time outside of class, for homework, studying for exams and preparing for the AP exam. Limited extra credit opportunities will be available, but extra credit will not count for more than a third of a letter grade.

The following is information excerpted from the College Board's AP Calculus 2010/2011 Course Description

Before studying calculus, all students should complete four years of secondary

mathematics designed for college-bound students: courses in which they study

algebra, geometry, trigonometry, analytic geometry, and elementary functions. These

functions include linear, polynomial, rational, exponential, logarithmic, trigonometric,

inverse trigonometric, and piecewise-defi ned functions. In particular, before studying

calculus, students must be familiar with the properties of functions, the algebra of

functions, and the graphs of functions. Students must also understand the language

of functions (domain and ra nge, odd and even, periodic, symmetry, zeros, intercepts,

and so on) and know the values of the trigonometric functions at the numbers

0, pi/6 , pi/4 , pi/3 , pi/2 , and their multiples.

Topic Outline for Calculus AB

This topic outline is intended to indicate the scope of the course, but it is not

necessarily the order in which the topics need to be taught. Teachers may fi nd that

topics are best taught in different orders. (See AP Central [apcentral.collegeboard.

com] for sample syllabi.) Although the exam is based on the topics listed here,

teachers may wish to enrich their courses with additional topics.

I. Functions, Graphs, and Limits

Analysis of graphs With the aid of technology, graphs of functions are often

easy to produce. The emphasis is on the interplay between the geometric and

analytic information and on the use of calculus both to predict and to explain the

observed local and globa l behavior of a function.

Limits of functions (including one-sided limits)

• An intuitive understanding of the limiting process

• Calculating limits using algebra

• Estimating limits from graphs or tables of data

Asymptotic and unbounded behavior

• Understanding asymptotes in terms of graphical behavior

• Describing asymptotic behavior in terms of limits involving infinity

• Comparing relative magnitudes of functions and their rates of change (for

example, contrasting exponential growth, polynomial growth, and logarithmic

Continuity as a property of functions

• An intuitive understanding of continuity. (The function values can be made as

close as desired by taking sufficiently close values of the domain.)

• Understanding continuity in terms of limits

• Geometric understanding of graphs of continuous functions (Intermediate

Value Theorem and Extreme Value Theorem)

Concept of the derivative

• Derivative presented graphically, nume rically, and analytically

• Derivative interpreted as an instantaneous rate of change

• Derivative defined as the limit of the difference quotient

• Relationship between differentiability and continuity

Derivative at a point

• Slope of a curve at a point. Examples are emphasized, including points at which

there are vertical tangents and points at which there are no tangents.

• Tangent line to a curve at a point and local linear approximation

• Instantaneous rate of change as the limit of average rate of change

• Approximate rate of change from graphs and tables of values

Derivative as a function

• Corresponding characteristics of graphs of ƒ and ƒ'

• Relationship between the increasing and decreasing behavior of ƒ and the sign

• The Mean Value Theorem and its geometric interpretation

• Equations involving derivatives. Verbal descriptions are translated into

equations involving derivatives and vice versa.

• Corresponding characteri stics of the graphs of ƒ, ƒ', and ƒ''

• Relationship between the concavity of ƒ and the sign of ƒ'

• Points of inflection as places where concavity changes

• Analysis of curves, including the notions of monotonicity and concavity

• Optimization, both absolute (global) and relative (local) extrema

• Modeling rates of change, including related rates problems

• Use of implicit differentiation to find the derivative of an inverse function

• Interpretation of the derivative as a rate of change in varied applied contexts,

including velocity, speed, and acceleration

• Geometric interpretation of differential equations via slope fields and the

relationship between slope fields and solution curves for differential equations

Computation of derivatives

• Knowledge of derivatives of basic functions, including power, exponential,

logarithmic, trigonometric, and inverse trigonometric functions

• Derivative rules for sums, products, and quotients of functions

• Chain rule and i mplicit differentiation

Interpretations and properties of definite integrals

• Definite integral as a limit of Riemann sums

• Definite integral of the rate of change of a quantity over an interval interpreted

as the change of the quantity over the interval

• Basic properties of definite integrals (examples include additivity and linearity)

Applications of integrals Appropriate integrals are used in a variety of

applications to model physical, biological, or economic situations. Although only

a sampling of applications can be included in any specific course, students should

be able to adapt their knowledge and techniques to solve other similar application

problems. Whatever applications are chosen , the emphasis is on using the

method of setting up an approximating Riemann sum and representing its limit as

a definite integral. To provide a common foundation, specific applications should

include finding the area of a reg ion, the volume of a solid with kn own cross

sections, the average value of a function, the distance traveled by a particle along

a line, and accumulated change from a rate of change.

Fundamental Theorem of Calculus

• Use of the Fundamental Theorem to evaluate definite integrals

• Use of the Fundamental Theorem to represent a particular antiderivative, and

the analytical and graphical analysis of functions so defined

Techniques of antidifferentiation

• Antiderivatives following directly from derivatives of basic functions

• Antiderivativ es by substitution of variables (including change of limits for

• Finding specific antiderivatives using initial conditions, including applications to

• Solving separable differential equations and using them in modeling (including

the study of the equation y' = ky and exponential growth)

Numerical approximations to definite integrals Use of Riemann sums (using

left, right, and midpoint evaluation points) and trapezoidal su ms to approximate

definite integrals of functions represented algebraically, graphically, and by tables

Graphing Calculator Capabilities for the Exams

The committee develops exams based on the assumption that all students have access

to four basic calculator capabilities used extensively in calculus. A graphing calculator

appropriate for use on the exams is expected to have the built-in capability to:

1) plot the graph of a function within an arbitrary viewing window,

2) fi nd the zeros of functions (solve equations numerically),

3) numerically calculate the derivative of a function, and

4) numerically calculate the value of a defi nite integral.

One or more of these capabilities should provide the suffi cient computational tools

for successful development of a solution to any exam question that requires the use of

a calculator. Care is taken to ensure that the exam questions do not favor students

who use graphing calculators with more extensive built-in features.

Students are expected to bring a graphing calculator with the capabilities listed

above to the exams. AP teachers should check their own students’ calcu lators to

ensure that the required conditions are met. A list of acceptable calculators can be

found at AP Central. Teachers must contact the AP Program (609 771-7300) before

April 1 of the testing year to inquire whether a student can use a calculator that is not

on the list. If the calculator is approved, written permission will be given.

Technology Restrictions on the Exams

Computers, electronic writing pads, pocket organizers, nongraphing scientifi c

calculators, and calculator models with any of the following are not permitted for use

on the AP Calculus Exams: QWERTY keypads as part of hardware or software, peninput/

stylus/touch-screen capability, wireless or Bluetooth® capability, paper tapes,

“talking” or noise-making capability, need for an electrical outlet, ability to access the

Internet, cell phone capability or audio/video recording capability, digi tal audio/video

players, or camera or scanning capability. In addition, the use of hardware peripherals

with an approved graphing calculator is not permitted.

Test administrators are required to check calculators before the exam. Therefore, it

is important for each student to have an approved calculator. The student should be

thoroughly familiar with the operation of the calculator he or she plans to use.

Calculators may not be shared, and communication between calculators is prohibited

during the exam. Students may bring to the exam one or two (but no more than two)

graphing calculators from the approved list.

Calculator memories will not be cleared. Students are allowed to bring calculators

contain ing whatever programs they want. They are expected to bring calculators that

are set to radian mode.

Students must not use calculator memories to take test materials out of the room.

Students should be warned that their grades will be invalidated if they attempt to

remove test materials by any method.

Showing Work on the Free-Response Sections

An important goal of the free-response section of the AP Calculus Exams is to provide

students with an opportunity to communicate their knowledge of correct reasoning

and methods. Students are required to show their work so that AP Exam Readers can

assess the students’ methods and answers. To be eligible for partial credit, methods,

reasoning, and conclusions should be presented clearly. Answers without supporting

work will usually not receive credit. Students should use complete sentences in

responses that include explanations or justifi cations.

For resu lts obtained using one of the four required calculator capabilities listed on

page 13, students are required to write the mathematical setup that leads to the

solution along with the result produced by the calculator. These setups include the

equation being solved, the derivative being evaluated, or the defi nite integral being

evaluated. For example, if a problem involves fi nding the area of a region, and the area

is appropriately computed with a defi nite integral, students are expected to show the

defi nite integral—written in standard mathematical notation—and the answer. In

general, in a calculator-active problem that requires the value of a defi nite integral,

students may use a calculator to determine the value; they do not need to compute an

antiderivative as an intermediate step. Similarly, if a calculator-active problem requires

the value of a derivative of a given function at a specifi c point, students may use a

calculator to determine the value; they do not need to state the symbolic derivative

expression. For solutions obtained using a calculator capability other than one of the

four listed on page 13, students must show the mathematical steps necessary to

produce their results; a calculator result alone is not suffi cient. For example, if

students are asked to fi nd a relative minimum value of a function, they are expected to

use calculus and show the mathematical steps that lead to the answer. It is not

suffi cient to graph the function or use a calculator application that fi nds minimum

A graphing calculator is a powerful t ool for exploration, but students must be

cautioned that exploration is not a substitute for a mathematical solution. Exploration

with a graphing calculator can lead a student toward an analytical solution, and after a

solution is found, a graphing calculator can often be used to check the reasonableness

of the solution. Therefore, when students are ask ed to justify or explain an answer, the

just ifi cation must include mathematical, noncalculator reasons, not merely calculator

results. Also, within solutions and justifi cations of answers, any functions, graphs,

tables, or other objects that are used must be clearly labeled.

As on previous AP Calculus Exams, if a calculation is given as a decimal

approximation, it should be correct to three places after the decimal point unless

otherwise indicated in the problem. Students should be cautioned against rounding

values in intermediate steps before a fi nal calculation is made. Students should also be

aware that there are limitations inherent in graphing calculator technology. For

example, answers obtained by tracing along a graph to fi nd roots or points of

intersection might not produce the required accuracy.

Sign charts by themselves are not accepted as a suffi cient response when a freeresponse

problem requires a justifi cation for the existence of either a local or an

absolute extremum of a function at a particula r point in its domain. Rather, the

justifi cation must include a clear explanation about how the behavior of the derivative

and/or second derivative of the function indicates the particular extremum. For more

detailed information on this policy, read the article “On the Role of Sign Charts in AP

Calculus Exams for Justifying Local or Absolute Extrema,” which is available on the

Course Home Pages for Calculus AB and Calculus BC at AP Central.

For more information on the instructions for the free-response sections, read the

“Calculus FRQ Instruction Commentary,” which is available on the Course Home

Pages for Calculus AB and Calculus BC at AP Central.

The Calculus AB and BC Exams seek to assess how well a student has mastered the

concepts and techniques of the subject matter of the corresponding courses. Each

exam consists of two sections, as described below.

Section I: a multiple-choic e section testing profi ciency in a wide variety of topics

Section II: a free-response section requiring the student to demonstrate the ability to

solve problems involving a more extended chain of reasoning

The time allotted for each AP Calculus Exam is 3 hours and 15 minutes. The

multiple-choice section of each exam consists of 45 questions in 105 minutes. Part A of

the multiple-choice section (28 questions in 55 minutes) does not allow the use of a

calculator. Part B of the multiple-choice section (17 questions in 50 minutes) contains

some questions for which a graphing calculator is required.