AP Calculus BC

What the course covers

10 units, with exam weights

1. Unit 1

   Limits and Continuity

   5-10% of examFree
   Limits make it possible to talk about change at an instant. This unit estimates and computes limits from graphs, tables, and formulas, classifies discontinuities, connects limits to asymptotes, and uses the Intermediate Value Theorem, the first theorem that must be justified with verified hypotheses.
   16 topics

   1. 1.1Introducing Calculus: Can Change Occur at an Instant?
   2. 1.2Defining Limits and Using Limit Notation
   3. 1.3Estimating Limit Values from Graphs
   4. 1.4Estimating Limit Values from Tables
   5. 1.5Determining Limits Using Algebraic Properties of Limits
   6. 1.6Determining Limits Using Algebraic Manipulation
   7. 1.7Selecting Procedures for Determining Limits
   8. 1.8Determining Limits Using the Squeeze Theorem
   9. 1.9Connecting Multiple Representations of Limits
   10. 1.10Exploring Types of Discontinuities
   11. 1.11Defining Continuity at a Point
   12. 1.12Confirming Continuity over an Interval
   13. 1.13Removing Discontinuities
   14. 1.14Connecting Infinite Limits and Vertical Asymptotes
   15. 1.15Connecting Limits at Infinity and Horizontal Asymptotes
   16. 1.16Working with the Intermediate Value Theorem (IVT)

2. Unit 2

   Differentiation: Definition and Fundamental Properties

   5-10% of exam
   The derivative is defined as the limit of a difference quotient and then turned into efficient rules. This unit estimates derivatives from graphs and tables, connects differentiability to continuity, and builds the basic rules: power, sum, product, quotient, and the derivatives of trigonometric, exponential, and logarithmic functions.
   10 topics

   1. 2.1Defining Average and Instantaneous Rates of Change at a Point
   2. 2.2Defining the Derivative of a Function and Using Derivative Notation
   3. 2.3Estimating Derivatives of a Function at a Point
   4. 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
   5. 2.5Applying the Power Rule
   6. 2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple
   7. 2.7Derivatives of cos x, sin x, e^x, and ln x
   8. 2.8The Product Rule
   9. 2.9The Quotient Rule
   10. 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

3. Unit 3

   Differentiation: Composite, Implicit, and Inverse Functions

   5-10% of exam
   The chain rule unlocks the derivative of any composition, and from it come implicit differentiation, derivatives of inverse functions, and inverse trigonometric derivatives. The unit ends with choosing among rules efficiently and computing higher-order derivatives, including second derivatives of implicit relations.
   6 topics

   1. 3.1The Chain Rule
   2. 3.2Implicit Differentiation
   3. 3.3Differentiating Inverse Functions
   4. 3.4Differentiating Inverse Trigonometric Functions
   5. 3.5Selecting Procedures for Calculating Derivatives
   6. 3.6Calculating Higher-Order Derivatives

4. Unit 4

   Contextual Applications of Differentiation

   5-10% of exam
   Derivatives become rates in the real world. This unit interprets derivatives with units, analyzes straight-line motion, solves related rates problems, approximates values with tangent lines, and evaluates indeterminate limits with L'Hospital's Rule.
   7 topics

   1. 4.1Interpreting the Meaning of the Derivative in Context
   2. 4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
   3. 4.3Rates of Change in Applied Contexts Other Than Motion
   4. 4.4Introduction to Related Rates
   5. 4.5Solving Related Rates Problems
   6. 4.6Approximating Values of a Function Using Local Linearity and Linearization
   7. 4.7Using L'Hospital's Rule for Determining Limits of Indeterminate Forms

5. Unit 5

   Analytical Applications of Differentiation

   10-15% of exam
   Derivatives reveal the shape of a function. This unit applies the Mean Value and Extreme Value Theorems, locates relative and absolute extrema with the first derivative, second derivative, and candidates tests, analyzes concavity and inflection points, connects the graphs of f, f′, and f″, and solves optimization problems, all with written justifications.
   12 topics

   1. 5.1Using the Mean Value Theorem
   2. 5.2Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
   3. 5.3Determining Intervals on Which a Function Is Increasing or Decreasing
   4. 5.4Using the First Derivative Test to Determine Relative (Local) Extrema
   5. 5.5Using the Candidates Test to Determine Absolute (Global) Extrema
   6. 5.6Determining Concavity of Functions over Their Domains
   7. 5.7Using the Second Derivative Test to Determine Extrema
   8. 5.8Sketching Graphs of Functions and Their Derivatives
   9. 5.9Connecting a Function, Its First Derivative, and Its Second Derivative
   10. 5.10Introduction to Optimization Problems
   11. 5.11Solving Optimization Problems
   12. 5.12Exploring Behaviors of Implicit Relations

6. Unit 6

   Integration and Accumulation of Change

   15-20% of exam
   Integrals accumulate change. This unit approximates accumulation with Riemann and trapezoidal sums, defines the definite integral as a limit, connects it to derivatives through the Fundamental Theorem of Calculus, and builds every antidifferentiation technique on the exam, including the BC-only topics of integration by parts, linear partial fractions, and improper integrals (6.11 to 6.13).
   14 topics

   1. 6.1Exploring Accumulations of Change
   2. 6.2Approximating Areas with Riemann Sums
   3. 6.3Riemann Sums, Summation Notation, and Definite Integral Notation
   4. 6.4The Fundamental Theorem of Calculus and Accumulation Functions
   5. 6.5Interpreting the Behavior of Accumulation Functions Involving Area
   6. 6.6Applying Properties of Definite Integrals
   7. 6.7The Fundamental Theorem of Calculus and Definite Integrals
   8. 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
   9. 6.9Integrating Using Substitution
   10. 6.10Integrating Functions Using Long Division and Completing the Square
   11. 6.11Integrating Using Integration by Parts
   12. 6.12Integrating Using Linear Partial Fractions
   13. 6.13Evaluating Improper Integrals
   14. 6.14Selecting Techniques for Antidifferentiation

7. Unit 7

   Differential Equations

   5-10% of exam
   Differential equations describe a quantity through its rate of change. This unit models situations with differential equations, verifies solutions, reasons with slope fields, approximates solutions with Euler's method (BC only), solves separable equations with initial conditions, and analyzes exponential and logistic growth (logistic is BC only).
   9 topics

   1. 7.1Modeling Situations with Differential Equations
   2. 7.2Verifying Solutions for Differential Equations
   3. 7.3Sketching Slope Fields
   4. 7.4Reasoning Using Slope Fields
   5. 7.5Approximating Solutions Using Euler's Method
   6. 7.6Finding General Solutions Using Separation of Variables
   7. 7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
   8. 7.8Exponential Models with Differential Equations
   9. 7.9Logistic Models with Differential Equations

8. Unit 8

   Applications of Integration

   5-10% of exam
   Definite integrals answer geometric and applied questions. This unit finds average value, position and total distance, and accumulated amounts in context, then computes areas between curves, volumes by cross sections, discs, and washers about any horizontal or vertical line, and the arc length of a curve (BC only).
   13 topics

   1. 8.1Finding the Average Value of a Function on an Interval
   2. 8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
   3. 8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
   4. 8.4Finding the Area Between Curves Expressed as Functions of x
   5. 8.5Finding the Area Between Curves Expressed as Functions of y
   6. 8.6Finding the Area Between Curves That Intersect at More Than Two Points
   7. 8.7Volumes with Cross Sections: Squares and Rectangles
   8. 8.8Volumes with Cross Sections: Triangles and Semicircles
   9. 8.9Volume with Disc Method: Revolving Around the x- or y-Axis
   10. 8.10Volume with Disc Method: Revolving Around Other Axes
   11. 8.11Volume with Washer Method: Revolving Around the x- or y-Axis
   12. 8.12Volume with Washer Method: Revolving Around Other Axes
   13. 8.13The Arc Length of a Smooth, Planar Curve and Distance Traveled

9. Unit 9

   Parametric Equations, Polar Coordinates, and Vector-Valued Functions

   10-15% of exam
   BC only. Calculus extends to curves traced by parameters, by angles, and by vectors. This unit differentiates parametric and polar curves, computes arc length and speed, analyzes planar motion with vector-valued functions, and finds areas of regions bounded by polar curves. It anchors the BC-only calculator free-response question.
   9 topics

   1. 9.1Defining and Differentiating Parametric Equations
   2. 9.2Second Derivatives of Parametric Equations
   3. 9.3Finding Arc Lengths of Curves Given by Parametric Equations
   4. 9.4Defining and Differentiating Vector-Valued Functions
   5. 9.5Integrating Vector-Valued Functions
   6. 9.6Solving Motion Problems Using Parametric and Vector-Valued Functions
   7. 9.7Defining Polar Coordinates and Differentiating in Polar Form
   8. 9.8Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve
   9. 9.9Finding the Area of the Region Bounded by Two Polar Curves

10. Unit 10

    Infinite Sequences and Series

    15-20% of exam
    BC only. An infinite sum can have a finite value, and a function can be written as an infinite polynomial. This unit decides convergence with every test on the exam, bounds approximation error, builds Taylor polynomials and series, finds intervals of convergence, and manipulates known series. It anchors the final free-response question of every BC exam.
    15 topics

    1. 10.1Defining Convergent and Divergent Infinite Series
    2. 10.2Working with Geometric Series
    3. 10.3The nth Term Test for Divergence
    4. 10.4Integral Test for Convergence
    5. 10.5Harmonic Series and p-Series
    6. 10.6Comparison Tests for Convergence
    7. 10.7Alternating Series Test for Convergence
    8. 10.8Ratio Test for Convergence
    9. 10.9Determining Absolute or Conditional Convergence
    10. 10.10Alternating Series Error Bound
    11. 10.11Finding Taylor Polynomial Approximations of Functions
    12. 10.12Lagrange Error Bound
    13. 10.13Radius and Interval of Convergence of Power Series
    14. 10.14Finding Taylor or Maclaurin Series for a Function
    15. 10.15Representing Functions as Power Series

The exam, part by part

4 parts, 3 h 10 min in all.

• Section I, Part A: Multiple Choice (no calculator)

  Questions

  29

  Time

  1 h 2 min

  Weight

  35%

  No calculator

  Format details

  Four choices (A to D). Draw only items with calculator "not-allowed". About 2.1 minutes per question.

• Section I, Part B: Multiple Choice (graphing calculator required)

  Questions

  13

  Time

  38 min

  Weight

  15%

  Calculator allowed

  Format details

  Four choices (A to D). Draw only items with calculator "allowed"; some questions need the calculator, some do not. About 2.9 minutes per question.

• Section II, Part A: Free Response (graphing calculator required)

  Questions

  2

  Time

  30 min

  Weight

  16.7%

  Calculator allowed

  Format details

  Task types: rate-accumulation, parametric-vector-polar, area-volume, particle-motion

  Draw FRQs whose calculator field is "allowed". The real Part A is one AB-shared context question (usually rates and accumulation) plus one BC-only question (usually parametric, vector, or polar).

• Section II, Part B: Free Response (no calculator)

  Questions

  4

  Time

  1 h

  Weight

  33.3%

  No calculator

  Format details

  Task types: graph-analysis, differential-equations, series, analytic-function, rate-accumulation, area-volume, particle-motion, parametric-vector-polar

  Draw FRQs whose calculator field is "not-allowed". A realistic Part B always ends with a series question and usually includes a graph-analysis question and a differential equations question.

Skills the exam scores

• 1.CChoose a procedure from the form of an expression

  Practice 1, Implementing Mathematical Processes. Pick the right rule or procedure by classifying the expression, such as recognizing a composition that needs the chain rule.

• 1.DChoose a procedure from how concepts relate

  Practice 1, Implementing Mathematical Processes. Pick a rule or procedure from the relationship between ideas, such as rate of change and accumulation, or differentiation and antidifferentiation.

• 1.ECarry out rules and procedures

  Practice 1, Implementing Mathematical Processes. Execute derivative, integral, limit, and series procedures accurately, with and without a calculator.

• 1.FRelate an approximation to the actual value

  Practice 1, Implementing Mathematical Processes. Explain whether and why an estimate (tangent line, Riemann sum, Euler, Taylor) is too high or too low, and by how much at most.

• 2.ARecognize shared structure

  Practice 2, Connecting Representations. See the same mathematical structure underneath different contexts, such as related rates in cones and in shadows.

• 2.BRead information from a representation

  Practice 2, Connecting Representations. Pull the needed values and features from a graph, table, formula, or verbal description.

• 2.CRe-express information

  Practice 2, Connecting Representations. Rewrite given information in an equivalent form, such as a limit of a Riemann sum as a definite integral.

• 2.DLink properties across representations

  Practice 2, Connecting Representations. Connect a characteristic of a function in one representation to the same characteristic in another.

• 2.ERelate a function and its derivatives

  Practice 2, Connecting Representations. Describe how the graphs and behaviors of f, f′, and f″ determine one another.

• 3.BIdentify the theorem or test that applies

  Practice 3, Justification. Name the definition, theorem, or test (IVT, MVT, EVT, a derivative test, a series test) that fits the question.

• 3.CConfirm hypotheses

  Practice 3, Justification. Check that the conditions of a definition, theorem, or test hold before using it.

• 3.DApply a definition, theorem, or test

  Practice 3, Justification. Use the selected result correctly to reach a conclusion.

• 3.EGive reasons for conclusions

  Practice 3, Justification. Support an answer with the specific mathematical reason, such as a sign change of f′ at a point.

• 3.FExplain meaning in context

  Practice 3, Justification. Interpret a derivative, integral, or solution in the language of the problem, with units and the relevant time or interval.

• 3.GConfirm answers are accurate and appropriate

  Practice 3, Justification. Check that a solution is reasonable, satisfies the domain, and answers the question asked.

• 4.AUse precise mathematical language

  Practice 4, Communication and Notation (free response only). State conclusions with exact vocabulary, not vague words like "it" or "the graph".

• 4.BUse correct units

  Practice 4, Communication and Notation (free response only). Attach appropriate units to rates, accumulations, and averages.

• 4.CUse correct symbols and notation

  Practice 4, Communication and Notation (free response only). Write derivatives, integrals, limits, and series in standard notation, never calculator syntax.

• 4.DUse appropriate graphing techniques

  Practice 4, Communication and Notation (free response only). Sketch slope fields and graphs that show the required features.

• 4.ERound appropriately

  Practice 4, Communication and Notation (free response only). Report decimals accurate to three places and avoid rounding intermediate values early.