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Contemplanda course syllabus

AP® Calculus AB

A Contemplanda online course.

This course follows the College Board's published AP Calculus AB framework.

AP® Calculus AB and AP Calculus BC are two College Board courses built on a single framework. The framework asks students to grasp the central ideas of calculus and put its methods to use, to move between a function's graph, a table of its values, its formula, and a description in words, and to back each conclusion with an argument that rests on definitions and theorems.

The College Board describes AP Calculus AB as the equivalent of a first-semester college calculus course in differential and integral calculus, and AP Calculus BC as the equivalent of both the first and the second semester of college calculus.

AP Calculus BC includes everything in AP Calculus AB and goes further. Within Units 6 to 8 it adds integration by parts, linear partial fractions, improper integrals, Euler's method, logistic models, and the arc length of a planar curve, and its two further units take up parametric equations, polar coordinates, and vector-valued functions (Unit 9) and infinite sequences and series (Unit 10).

The College Board's framework has ten units. AP Calculus AB covers Units 1 to 8, and AP Calculus BC covers all ten: Units 9 and 10, and topics 6.11, 6.12, 6.13, 7.5, 7.9, and 8.13, are part of BC only. The College Board presents the framework as helping to prepare students for college credit or placement.

This course teaches eight of the ten College Board units, with a lesson for every topic the AP Calculus AB Exam assesses.

Course at a glance

Course title
AP Calculus
Publisher
Contemplanda
Framework
Follows the College Board's published AP Calculus AB framework
Structure
8 teaching units over two semesters (1, 2, 3, 4, and 5 in semester 1; 6, 7, and 8 in semester 2), plus 3 units holding a single unit test (6A, 7A, and 8A) and 4 semester exams: 15 units in all
Lessons
98 lessons: 86 teaching lessons, 8 unit tests, and 4 semester exams
Credit
Not awarded by Contemplanda
Enrollment
Purchasing is not open yet
Course page
www.contemplanda.com/courses/ap-calculus

Credit and enrollment

Contemplanda does not award high school credit. A student who wants credit for the course should ask their own school whether it will accept it.

Under College Board rules, a course may carry the AP designation on a student's transcript only if it is authorized through the College Board's AP Course Audit.

Purchasing is not open yet. Questions about enrolling can go to info@contemplanda.com.

Students who are already enrolled sign in at direct.contemplanda.com.

How the course works

  • Each topic lesson combines one to two videos, written explanations with typeset mathematics, practice, and graded checks, and most lessons include graphs.
  • Practice comes one question at a time and continues until the student answers four in a row correctly. After a wrong answer, the practice keeps drawing questions.
  • Many questions have hints that open one step at a time. After a wrong answer, the next hint is offered when the question has one. Opening a hint costs nothing and never lowers a mastery level, but the top level, mastered, also requires at least one graded question answered correctly without opening a hint.
  • Answers take many forms: algebraic expressions checked for mathematical equivalence and for the form a question asks for; points placed on a graph; graphs sketched; Riemann rectangles set up on a graph; the region an integral measures, shaded on a graph; tables filled in; along with multiple-choice and numeric answers.
  • Most figures in the lessons are live graphs that the student can pan and zoom.
  • Every topic lesson has a graded check whose questions are drawn for each student from a pool.
  • Every figure has a written text description.

Recommended preparation

The College Board expects every calculus student to have completed the equivalent of four years of college-preparatory high school mathematics, including algebra, geometry, trigonometry, analytic geometry, and elementary functions, with a solid grounding in algebraic reasoning and algebraic structures.

For AP Calculus AB and BC, a working knowledge of

  • Linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions
  • The properties, composition, algebra, and graphs of functions
  • The language of functions: domain and range, odd and even, periodic, symmetry, zeros, intercepts, and increasing and decreasing
  • The sine and cosine functions as defined from the unit circle, and the values of the trigonometric functions at 0, π/6, π/4, π/3, and π/2 and at their multiples

For AP Calculus BC, also

  • Basic familiarity with sequences and series
  • Some exposure to parametric and polar equations

Units and lessons

Every published lesson of AP Calculus AB, by semester and unit. In the titles below, (AB) marks a lesson, unit test, or exam for AP Calculus AB only; every other lesson is also part of AP Calculus BC.

Semester 1

College Board unit 1: Limits and Continuity

Assessed on the AP Calculus AB and BC Exams

How change at an instant leads to the idea of a limit, then limits estimated from graphs and tables and found with algebraic properties, algebraic manipulation, and the squeeze theorem; types of discontinuity, continuity at a point and over an interval, infinite limits and limits at infinity with their asymptotes, and the Intermediate Value Theorem.

1
Limits and continuity
Topics 1.1 to 1.16
  1. 11.1 Introducing calculus: can change occur at an instant?
  2. 21.2 Defining limits and using limit notation
  3. 31.3 Estimating limit values from graphs
  4. 41.4 Estimating limit values from tables
  5. 51.5 Determining limits using algebraic properties of limits
  6. 61.6 Determining limits using algebraic manipulation
  7. 71.7 Selecting procedures for determining limits
  8. 81.8 Determining limits using the squeeze theorem
  9. 91.9 Connecting multiple representations of limits
  10. 101.10 Exploring types of discontinuities
  11. 111.11 Defining continuity at a point
  12. 121.12 Confirming continuity over an interval
  13. 131.13 Removing discontinuities
  14. 141.14 Connecting infinite limits and vertical asymptotes
  15. 151.15 Connecting limits at infinity and horizontal asymptotes
  16. 161.16 Working with the intermediate value theorem (IVT)
  17. 17Unit 1 testUnit test15 questions

College Board unit 2: Differentiation: Definition and Fundamental Properties

Assessed on the AP Calculus AB and BC Exams

Average and instantaneous rates of change, the definition of the derivative and its notation, estimating derivatives at a point, and how differentiability relates to continuity; then the basic rules: the power rule, the constant, sum, difference, and constant multiple rules, the derivatives of cosine, sine, e^x, and ln x, the product and quotient rules, and the derivatives of the other four trigonometric functions.

2
Differentiation: definition and fundamental properties
Topics 2.1 to 2.10
  1. 12.1 Defining average and instantaneous rates of change at a point
  2. 22.2 Defining the derivative of a function and using derivative notation
  3. 32.3 Estimating derivatives of a function at a point
  4. 42.4 Connecting differentiability and continuity: determining when derivatives do and do not exist
  5. 52.5 Applying the power rule
  6. 62.6 Derivative rules: constant, sum, difference, and constant multiple
  7. 72.7 Derivatives of cos x, sin x, e^x, and ln x
  8. 82.8 The product rule
  9. 92.9 The quotient rule
  10. 102.10 Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions
  11. 11Unit 2 testUnit test13 questions

College Board unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Assessed on the AP Calculus AB and BC Exams

The chain rule, implicit differentiation, the derivatives of inverse functions and of inverse trigonometric functions, choosing a procedure for calculating a derivative, and derivatives of higher order.

3
Differentiation: composite, implicit, and inverse functions
Topics 3.1 to 3.6
  1. 13.1 The chain rule
  2. 23.2 Implicit differentiation
  3. 33.3 Differentiating inverse functions
  4. 43.4 Differentiating inverse trigonometric functions
  5. 53.5 Selecting procedures for calculating derivatives
  6. 63.6 Calculating higher-order derivatives
  7. 7Unit 3 testUnit test12 questions
Exam

Semester 1 midterm

  1. Semester 1 midterm exam28 questions

Covers units 1 to 3.

College Board unit 4: Contextual Applications of Differentiation

Assessed on the AP Calculus AB and BC Exams

The derivative read in context: position, velocity, and acceleration in straight-line motion, rates of change in applied settings other than motion, related rates, approximation by local linearity and linearization, and L'Hospital's Rule for limits of indeterminate forms.

4
Contextual applications of differentiation
Topics 4.1 to 4.7
  1. 14.1 Interpreting the meaning of the derivative in context
  2. 24.2 Straight-line motion: connecting position, velocity, and acceleration
  3. 34.3 Rates of change in applied contexts other than motion
  4. 44.4 Introduction to related rates
  5. 54.5 Solving related rates problems
  6. 64.6 Approximating values of a function using local linearity and linearization
  7. 74.7 Using L'Hospital's rule for determining limits of indeterminate forms
  8. 8Unit 4 testUnit test12 questions

College Board unit 5: Analytical Applications of Differentiation

Assessed on the AP Calculus AB and BC Exams

The Mean Value Theorem and the Extreme Value Theorem, critical points, where a function increases and decreases, the first derivative test, the candidates test, and the second derivative test for extrema, concavity, sketching and connecting a function with its first and second derivatives, optimization problems, and the behavior of implicit relations.

5
Analytical applications of differentiation
Topics 5.1 to 5.12
  1. 15.1 Using the mean value theorem
  2. 25.2 Extreme value theorem, global versus local extrema, and critical points
  3. 35.3 Determining intervals on which a function is increasing or decreasing
  4. 45.4 Using the first derivative test to determine relative (local) extrema
  5. 55.5 Using the candidates test to determine absolute (global) extrema
  6. 65.6 Determining concavity of functions over their domains
  7. 75.7 Using the second derivative test to determine extrema
  8. 85.8 Sketching graphs of functions and their derivatives
  9. 95.9 Connecting a function, its first derivative, and its second derivative
  10. 105.10 Introduction to optimization problems
  11. 115.11 Solving optimization problems
  12. 125.12 Exploring behaviors of implicit relations
  13. 13Free-response workshop: justifying conclusions with theorems and derivatives
  14. 14Unit 5 testUnit test14 questions
Exam

Semester 1 final

  1. Semester 1 final exam32 questions

Covers all of semester 1, units 1 to 5.

Semester 2

College Board unit 6: Integration and Accumulation of Change

Assessed on the AP Calculus AB and BC Exams

Accumulation of change and its approximation by Riemann sums, definite integral notation, the Fundamental Theorem of Calculus and accumulation functions, properties of definite integrals, antiderivatives and indefinite integrals, integration by substitution, by long division, and by completing the square, and choosing a technique of antidifferentiation. Integration by parts, linear partial fractions, and improper integrals are part of AP Calculus BC only.

6
Integration and accumulation of change
  1. 16.1 Exploring accumulations of change
  2. 26.2 Approximating areas with Riemann sums
  3. 36.3 Riemann sums, summation notation, and definite integral notation
  4. 46.4 The fundamental theorem of calculus and accumulation functions
  5. 56.5 Interpreting the behavior of accumulation functions involving area
  6. 66.6 Applying properties of definite integrals
  7. 76.7 The fundamental theorem of calculus and definite integrals
  8. 86.8 Finding antiderivatives and indefinite integrals: basic rules and notation
  9. 96.9 Integrating using substitution
  10. 106.10 Integrating functions using long division and completing the square
  11. 116.14 Selecting techniques for antidifferentiation (AB)
  12. 12Free-response workshop: rates, tables, and accumulation
6A
Unit 6 test (AB)
  1. 1Unit 6 test (AB)Unit test15 questions

College Board unit 7: Differential Equations

Assessed on the AP Calculus AB and BC Exams

Differential equations as models of situations, verifying their solutions, sketching and reasoning with slope fields, general and particular solutions by separation of variables, and exponential models. Euler's method and logistic models are part of AP Calculus BC only.

7
Differential equations
  1. 17.1 Modeling situations with differential equations
  2. 27.2 Verifying solutions for differential equations
  3. 37.3 Sketching slope fields
  4. 47.4 Reasoning using slope fields
  5. 57.6 Finding general solutions using separation of variables
  6. 67.7 Finding particular solutions using initial conditions and separation of variables
  7. 77.8 Exponential models with differential equations
  8. 8Free-response workshop: differential equations
7A
Unit 7 test (AB)
  1. 1Unit 7 test (AB)Unit test12 questions
Exam

Semester 2 midterm (AB)

  1. Semester 2 midterm exam (AB)24 questions

College Board unit 8: Applications of Integration

Assessed on the AP Calculus AB and BC Exams

Definite integrals applied: the average value of a function, position, velocity, and acceleration, including displacement and the total distance traveled, accumulation in applied contexts, areas between curves, and the volumes of solids with known cross sections and of solids of revolution by the disc and washer methods. The arc length of a smooth planar curve is part of AP Calculus BC only.

8
Applications of integration
Topics 8.1 to 8.12
  1. 18.1 Finding the average value of a function on an interval
  2. 28.2 Connecting position, velocity, and acceleration of functions using integrals
  3. 38.3 Using accumulation functions and definite integrals in applied contexts
  4. 48.4 Finding the area between curves expressed as functions of x
  5. 58.5 Finding the area between curves expressed as functions of y
  6. 68.6 Finding the area between curves that intersect at more than two points
  7. 78.7 Volumes with cross sections: squares and rectangles
  8. 88.8 Volumes with cross sections: triangles and semicircles
  9. 98.9 Volume with disc method: revolving around the x- or y-axis
  10. 108.10 Volume with disc method: revolving around other axes
  11. 118.11 Volume with washer method: revolving around the x- or y-axis
  12. 128.12 Volume with washer method: revolving around other axes
  13. 13Free-response workshop: motion, area, and volume
  14. 14Free-response practice set: six questions
8A
Unit 8 test (AB)
  1. 1Unit 8 test (AB)Unit test14 questions
Exam

Semester 2 final (AB)

  1. Semester 2 final exam (AB)32 questions

Covers all of semester 2, units 6 to 8.

Mathematical practices

The College Board organizes the skills of AP Calculus AB and BC into four mathematical practices, which it expects students to develop throughout the course. Every task on the AP Exams draws on these skills, and Practice 4 is assessed only in the free-response section.

1Implementing Mathematical Processes

Finding expressions and values by carrying out mathematical procedures and rules.

1.A
Recognize what question is being asked or what problem needs solving (not assessed).
1.B
Pick out the information that matters for answering the question or solving the problem (not assessed).
1.C
Choose a suitable rule or procedure from the form of the expression at hand, such as the chain rule for a composite function.
1.D
Choose a suitable rule or procedure from how concepts or processes are related, such as rate of change and accumulation, or differentiation and antidifferentiation.
1.E
Carry out the appropriate rules or procedures, with and without technology.
1.F
Explain how an approximation relates to the value it approximates.

2Connecting Representations

Moving mathematical information within one representation or between several.

2.A
Recognize the common structure beneath problems set in different contexts.
2.B
Read mathematical information from graphical, numerical, analytical, and verbal representations.
2.C
Recognize the same mathematical information when it is expressed in another form.
2.D
Recognize how the characteristics or properties of a function appear in different representations.
2.E
Describe how representations of a function and of its derivatives relate to one another.

3Justification

Supporting reasoning and solutions with justified arguments.

3.A
Use technology to form claims and conjectures (not assessed).
3.B
Choose the definition, theorem, or test that applies.
3.C
Check that the hypotheses or conditions of the chosen definition, theorem, or test are met.
3.D
Apply the chosen definition, theorem, or test.
3.E
Give reasons for solutions and conclusions.
3.F
Explain what a mathematical solution means in its context.
3.G
Check that solutions are accurate and appropriate.

4Communication and Notation

Communicating results and solutions with correct notation, language, and mathematical conventions.

4.A
Express ideas in precise mathematical language.
4.B
Attach the correct units of measure.
4.C
Write with correct mathematical symbols and notation, including the notations for a derivative.
4.D
Draw graphs with appropriate technique.
4.E
Round results appropriately.

Three big ideas run through the College Board's framework: change, limits, and analysis of functions. Derivatives describe rates of change and definite integrals describe net change, and the Fundamental Theorem of Calculus ties the two together.

Assessment

Each topic lesson includes graded checks.

Units 1 to 5 each close with a unit test, and the unit tests of Units 6, 7, and 8 are in Units 6A, 7A, and 8A. A student takes 8 unit tests.

Each semester has a midterm exam and a final exam.

Unit tests and semester exams are graded automatically, and all graded work counts toward the student's mastery record, which shows each skill as attempted, familiar, proficient, or mastered.

Contemplanda publishes no grade weighting for this course. A school that decides to grant credit for it also decides how the work counts toward a grade.

Academic integrity

Work in the course must be the student's own.

Unit tests and semester exams are closed book: no notes, no lesson look-ups, and no outside help. Each question is worked on paper and the answer entered in the course.

Answer keys for graded questions are never sent to the student's device.

Graphing calculator

The unit tests and semester exams are written to be worked without a calculator.

The College Board treats a graphing calculator as an integral part of AP Calculus AB and BC, and one is required on two parts of each AP Exam. It also expects students to explore ideas with technology, a graphing calculator above all, across the whole course.

The College Board expects a calculator used on the exam to graph a function in any viewing window, find zeros of functions, and compute a derivative and a definite integral numerically. When a free-response answer uses one of those capabilities, the student must show the setup, such as the equation to be solved or the derivative or integral to be computed, together with the calculator's result; any other calculator feature needs the mathematical steps that produce the result.

A student preparing for the AP Exam should also practice with a graphing calculator, such as the Desmos graphing calculator, the kind built into the exam.

The AP Exam

The AP Calculus AB and AP Calculus BC Exams share one format. Each lasts 3 hours and 10 minutes and has two sections, with 42 multiple-choice questions and 6 free-response questions. The AB Exam assesses Units 1 to 8, and the BC Exam assesses all ten units.

Section I, Part A: multiple choice

Questions
29
Time
62 minutes
Share of score
35%
Calculator
Not permitted

Section I, Part B: multiple choice

Questions
13
Time
38 minutes
Share of score
15%
Calculator
Graphing calculator required

Section II, Part A: free response

Questions
2
Time
30 minutes
Share of score
16.7%
Calculator
Graphing calculator required

Section II, Part B: free response

Questions
4
Time
60 minutes
Share of score
33.3%
Calculator
Not permitted

On the exam, a Desmos graphing calculator is built into the testing app, available only in the parts that require a calculator, and students may also bring up to two handheld graphing calculators from the College Board's approved list.

Three of the six free-response questions are common to the AB and BC Exams and assess AB content, and each exam includes at least two questions set in a real-world context.

Students answer the multiple-choice questions and read the free-response questions in the College Board's digital testing app, and they handwrite their free-response answers in a paper booklet.

Scores are reported on a scale of 1 to 5. A student who takes the AP Calculus BC Exam also receives a Calculus AB subscore on the same scale, which reports how the student did on the AB material, about 60 percent of the exam.

This is the exam format in effect from the May 2027 exams.

The AP Exams are given in schools in May, and a student may not take both the AB and the BC Exam in the same year.

A student who takes the AP Exam at a school other than the one where they take the course, or who takes the exam without taking the course, gets a join code from the AP coordinator at the school where they will test and joins an exam-only section there.

Materials and technology

Where the course runs
In a web browser, through Contemplanda Direct
Graphing calculator
Not needed for the unit tests or semester exams; recommended for AP Exam practice
Textbook
None required. The course's own lessons are the text.
Course framework
apcentral.collegeboard.org/courses/ap-calculus-ab

Published at www.contemplanda.com/courses/ap-calculus-ab/syllabus

Units, lessons, and counts in this syllabus are generated from the published course.

Questions: info@contemplanda.com

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