Section outline

  • Course Information

    Go to section Course Information

    To see more about the College Board AP-Calculus BC exam, please see this link: https://apcentral.collegeboard.org/courses/ap-calculus-bc 

    Overview of AP Calculus AB and AP Calculus BC

    AP Calculus AB and AP Calculus BC emphasize a deep understanding of calculus concepts through hands-on experience with key methods and practical applications. The courses are centered around the big ideas of calculus—such as modeling change, understanding limits and approximations, and analyzing functions—making the curriculum a unified whole rather than a series of disconnected topics. Students are expected to use formal definitions and theorems to construct logical arguments and support their conclusions.

    A multi-representational approach is integral to both courses, so concepts, results, and problems are explored graphically, numerically, analytically, and verbally. Making connections among these various representations fosters a comprehensive understanding of how calculus uses limits to develop key ideas, definitions, formulas, and theorems. Consistent attention to clear communication—explaining methods, reasoning, justifications, and conclusions—is crucial throughout the program. Technology also plays a regular role, helping students visualize relationships among functions, verify work, experiment, and interpret results.

    AP Calculus BC Course Details

    AP Calculus BC is designed to be equivalent to both the first and second semesters of college-level calculus. Building on the foundation established in AP Calculus AB, this course extends concepts to parametrically defined curves, polar curves, and vector-valued functions. It introduces additional techniques and applications for integration and covers the topics of sequences and series.

    Prerequisites

    Students preparing for calculus should complete four years of secondary mathematics aimed at college-bound learners, equipping them with strong skills in algebraic reasoning and mastery of algebraic structures. Essential prior coursework includes algebra, geometry, trigonometry, analytic geometry, and elementary functions—such as linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions.

    Before enrolling, students must be comfortable with the properties and graphs of functions, as well as the concepts of domain and range, odd and even functions, periodicity, symmetry, zeros, intercepts, and descriptors like increasing or decreasing. They should also understand how sine and cosine are defined using the unit circle and know key trigonometric values at 0, π/6, π/4, π/3, π/2, and their multiples. For AP Calculus BC, some familiarity with sequences and series, parametric equations, and polar equations is also expected.

  • UNIT 1: Limits and Continuity

    Go to section UNIT 1: Limits and Continuity
    Developing Understanding

    Limits highlight the important difference between simply plugging in a value to a function and considering the value the function approaches as x nears a certain point. This subtlety lets us formally define concepts like asymptotes and holes in graphs, and lays the groundwork for understanding continuity. When introducing limits, it can be helpful to use rational functions as examples rather than starting with a comprehensive review of precalculus. Limits are essential for later topics—differentiation, integration, and infinite series in Unit 10 (BC-level only). They underpin key definitions, theorems, and methods used to solve real-world problems involving change and to justify mathematical conclusions.

    Building Mathematical Practices

    •    2.B
    •    2.C
    •    3.C
    •    3.D
    Mathematical information can be organized in various ways: graphically, numerically, analytically, or verbally. Mathematicians must communicate effectively across all these formats and move smoothly between them. Learning about limits helps students grow in these skills, showing them how to translate information from a table, equation, or text into a graph, and back again. Guidance should focus on matching and converting between analytical and verbal representations. Using graphing calculators to explore these connections is strongly encouraged.

    Mathematicians also build arguments and justify conclusions using precise definitions, theorems, and tests. A frequent student mistake is neglecting to state all relevant information before concluding a theorem. During Unit 1, it’s important to give students clear instruction and time to practice connecting the dots: they should first confirm that all conditions or hypotheses are met before making a conclusion.

    Preparing for the AP Exam

    This course should be a full-year journey towards mastering the material, culminating in the AP Exam. It’s important to develop both specific content knowledge and overall understanding. After completing Unit 1, students should be able to evaluate or estimate limits using graphs, tables, equations, or verbal descriptions.
    To avoid losing points on the AP Exam, students should regularly practice correct notation, show their setups, and properly round answers when using calculators. Remember, two test sections do not allow calculators, while the other two may require them. From the start of the course, stress the importance of stating and checking the hypotheses for theorems. Students should always make sure a theorem’s conditions are met before applying it, establishing sound mathematical habits early on.

    Activities: 20

  • UNIT 2: Differentiation: Definition and Fundamental Properties

    Go to section UNIT 2: Differentiation: Definition and Fundamental Properties
    Developing Understanding of Derivatives

    The concept of the derivative is fundamental for determining instantaneous rates of change. To help students grasp how the definition of a derivative uses limits to refine average rates of change, it is beneficial to create opportunities for them to investigate average rates over progressively smaller intervals. Activities such as graphing calculator explorations can demonstrate how different mathematical operations affect the slopes of tangent lines, aiding students in understanding basic rules and properties of differentiation. As students choose which differentiation rules to apply, they should be encouraged to follow the order of operations. Building differentiation skills will prepare students to model realistic instantaneous rates of change in Unit 4 and analyze graphical representations in Unit 5.

    Building Mathematical Practices

    Practices: 1.E, 2.B, 4.C
    Mathematicians recognize that a solution's accuracy depends on the precision of the procedure used. The difference between a correct and incorrect answer often stems from minor arithmetic or procedural errors. For students, applying mathematical procedures—especially differentiation rules—with care and precision can be challenging. Mistakes such as omitting necessary notation (like parentheses) or misapplying the product rule by differentiating each factor separately and multiplying the results are common. The material in Unit 2 serves as a foundational opportunity for students to practice applying procedures accurately and to develop habits of self-correction before errors occur.

    This unit also offers a chance to revisit and reinforce the skill of connecting various mathematical representations. Students will encounter derivatives expressed analytically, numerically, graphically, and verbally. Practice in extracting information about the original function f from a graph of its derivative f' can help prevent common misunderstandings, such as confusing the graph of the derivative with the graph of the original function itself.

    Preparing for the AP Exam

    Students should practice presenting clear mathematical structures that connect their work to definitions or theorems. For example, when asked to estimate the slope of a tangent line to a curve at a specific point using a table of values—as in the 2013 AP Exam Free-Response Question 3 Part A—they must explicitly present the difference quotient:
    •    C' (3.5) approximately = (C(4) - C (3)) / (4 - 3) = (12.8 - 11.2) / 1
    Failing to present this structure, even with a correct numerical answer, results in lost credit. Similarly, when evaluating the derivative of f(x) = u(x) * v(x) at x = 3, students should write out the product rule: as in f′(3) = u(3) * v′(3) + v(3) * u′(3)
    and substitute the appropriate values. It is also important for students to demonstrate how expressions are evaluated using the calculator and to apply specified rounding procedures, such as rounding or truncating to three decimal places. Developing the habit of storing intermediate results in the calculator can help prevent the accumulation of rounding errors throughout calculations.

    Activities: 12