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Explore limits, derivatives, integration, differential equations, parametric and polar applications, and infinite sequences and series through complete explanations, labeled models, worked examples and original practice. Units 1–10 are available as review editions.
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Your AP Calculus BC learning path
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Your AP Calculus BC
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Choose a unit, expand an official topic, and open a focused lesson. Units 1–10 are available as review editions; independent teacher review and student testing remain pending.
10 units210 focused lessonsLearn · Explore · Practice · Review
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Review edition · Teacher review pending
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01 Limits and Continuity 24 lessons · Review edition
Build the BC foundation in limits and continuity through graphs, tables, algebra and careful explanations. 24 focused lessons cover the 16 official topics, with 48 explained questions, 72 recall cards and 24 written challenges. These foundations are shared with AB. An optional rotatable secant graph supports the complete 2D explanation. Independent teacher review and student testing remain pending.
1.1 Introducing Calculus: Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
1.4 Estimating Limit Values from Tables
1.5 Determining Limits Using Algebraic Properties of Limits
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
1.8 Determining Limits Using the Squeeze Theorem
1.9 Connecting Multiple Representations of Limits
1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
1.12 Confirming Continuity over an Interval
1.13 Removing Discontinuities
1.14 Connecting Infinite Limits and Vertical Asymptotes
1.15 Connecting Limits at Infinity and Horizontal Asymptotes
1.16 Working with the Intermediate Value Theorem (IVT)
02 Differentiation: Definition and Fundamental Properties 21 lessons · Review edition
Build derivative meaning before using derivative rules. 21 focused lessons cover ten official topics, with 42 explained questions, 63 recall cards and 21 written challenges. These foundations are shared with AB. An optional 3D comparison connects tangent slope with derivative height; complete 2D alternatives remain available. Independent teacher review and student testing remain pending.
2.1 Defining Average and Instantaneous Rates of Change at a Point
2.2 Defining the Derivative of a Function and Using Derivative Notation
2.3 Estimating Derivatives of a Function at a Point
2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
2.5 Applying the Power Rule
2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
2.7 Derivatives of cos x, sin x, eˣ, and ln x
2.8 The Product Rule
2.9 The Quotient Rule
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
03 Differentiation: Composite, Implicit, and Inverse Functions 18 lessons · Review edition
Connect the chain rule, implicit and inverse differentiation, inverse trigonometric functions, method selection and higher derivatives. 18 focused lessons cover six official topics, with 36 explained questions, 54 recall cards and 18 written challenges. These foundations are shared with AB. An optional 3D fold shows how swapping coordinates creates an inverse graph; full 2D explanations remain available. Independent teacher review and student testing remain pending.
3.1 The Chain Rule
3.2 Implicit Differentiation
3.3 Differentiating Inverse Functions
3.4 Differentiating Inverse Trigonometric Functions
3.5 Selecting Procedures for Calculating Derivatives
3.6 Calculating Higher-Order Derivatives
04 Contextual Applications of Differentiation 18 lessons · Review edition
Apply derivatives to motion, changing quantities, related rates, local approximation and eligible indeterminate limits. 18 focused lessons cover seven official topics, with 36 explained questions, 54 recall cards and 18 written challenges. These foundations are shared with AB. Explore a conical tank in optional 3D to connect water depth, radius and flow; full 2D explanations remain available. Independent teacher review and student testing remain pending.
4.1 Interpreting the Meaning of the Derivative in Context
4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
4.3 Rates of Change in Applied Contexts Other Than Motion
4.4 Introduction to Related Rates
4.5 Solving Related Rates Problems
4.6 Approximating Values of a Function Using Local Linearity and Linearization
4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms
05 Analytical Applications of Differentiation 20 lessons · Review edition
Connect the Mean Value Theorem, extrema, derivative signs, concavity, graph analysis, optimization and implicit relations. 20 focused lessons cover twelve official topics, with 40 explained questions, 60 recall cards and 20 written challenges. These foundations are shared with AB. Fold an open box in optional 3D to connect the cut size, net and completed volume; full 2D explanations remain available. Independent teacher review and student testing remain pending.
5.1 Using the Mean Value Theorem
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
5.6 Determining Concavity of Functions over Their Domains
5.7 Using the Second Derivative Test to Determine Extrema
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
06 Integration and Accumulation of Change 28 lessons · Review edition
Connect rates to signed accumulation, approximate with Riemann sums, apply the Fundamental Theorem and select BC integration methods. 56 explained questions, 84 recall cards and 28 written challenges. Optional 3D tank exploration with complete 2D alternatives. All fourteen official topics include parts, linear partial fractions and improper integrals. Independent teacher review and student testing remain pending.
6.1 Exploring Accumulations of Change
6.2 Approximating Areas with Riemann Sums
6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
6.4 The Fundamental Theorem of Calculus and Accumulation Functions
6.5 Interpreting the Behavior of Accumulation Functions Involving Area
6.6 Applying Properties of Definite Integrals
6.7 The Fundamental Theorem of Calculus and Definite Integrals
6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
6.9 Integrating Using Substitution
6.10 Integrating Functions Using Long Division and Completing the Square
6.11 Integrating Using Integration by Parts
6.12 Integrating Using Linear Partial Fractions
6.13 Evaluating Improper Integrals
6.14 Selecting Techniques for Antidifferentiation
07 Differential Equations 16 lessons · Review edition
Build a rate model, follow slope fields, compute Euler steps, separate variables and interpret exponential and logistic change. 32 explained questions, 48 recall cards and 16 written challenges. Optional 3D draining tank with complete 2D alternatives. All nine official topics; teacher review and student testing remain pending.
7.1 Modeling Situations with Differential Equations
7.2 Verifying Solutions for Differential Equations
7.3 Sketching Slope Fields
7.4 Reasoning Using Slope Fields
7.5 Approximating Solutions Using Euler’s Method
7.6 Finding General Solutions Using Separation of Variables
7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
7.8 Exponential Models with Differential Equations
7.9 Logistic Models with Differential Equations
08 Applications of Integration 23 lessons · Review edition
Connect integrals to average values, motion, stored amounts, areas and volumes. 46 explained original questions, 69 recall cards and 23 written challenges. Inspect square, rectangular, triangular and semicircular sections, disks and washers in optional 3D with complete 2D alternatives. BC topics 8.1–8.13 include chord approximations, dx/dy arc-length integrals and distance with retracing. Independent teacher review and student testing remain pending.
8.1 Finding the Average Value of a Function on an Interval
8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
8.4 Finding the Area Between Curves Expressed as Functions of x
8.5 Finding the Area Between Curves Expressed as Functions of y
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
8.7 Volumes with Cross Sections: Squares and Rectangles
8.8 Volumes with Cross Sections: Triangles and Semicircles
8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
8.10 Volume with Disc Method: Revolving Around Other Axes
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
8.12 Volume with Washer Method: Revolving Around Other Axes
8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
09 Parametric Equations, Polar Coordinates, and Vector-Valued Functions 18 lessons · Review edition
Follow a parameter through a plane curve, connect position, velocity and speed, and build polar areas from sectors. 36 explained original questions, 54 recall cards and 18 written challenges cover official topics 9.1–9.9. Explore tangents, concavity, path length, initial-value vectors, signed radii and overlapping polar regions. An optional 3D time-lift view separates repeated visits to the same point, with complete 2D alternatives. Independent teacher review and student testing remain pending.
9.1 Defining and Differentiating Parametric Equations
9.2 Second Derivatives of Parametric Equations
9.3 Finding Arc Lengths of Curves Given by Parametric Equations
9.4 Defining and Differentiating Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
9.7 Defining Polar Coordinates and Differentiating in Polar Form
9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
9.9 Finding the Area of the Region Bounded by Two Polar Curves
10 Infinite Sequences and Series 24 lessons · Review edition
Connect individual terms to running totals, choose and justify convergence tests, and build Taylor approximations with error guarantees. 48 explained original questions, 72 recall cards and 24 written challenges cover official topics 10.1–10.15. Explore partial sums, alternating brackets, convergence intervals and polynomial approximations. An optional 3D cube shows a geometric series through shrinking slabs, with complete 2D alternatives. Independent teacher review and student testing remain pending.
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power Series
10.14 Finding Taylor or Maclaurin Series for a Function
10.15 Representing Functions as Power Series
Curriculum sources & scope · AP Calculus BC
College Board
Shared AB/BC CED plus Fall 2026 corrections; ten BC units.
Complete official content is in the CED unit guides, including topic IDs, learning objectives, essential knowledge and course skills. The outline on this page is navigation only.
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Refresh Kid offers one-to-one online AP Calculus BC tutoring alongside a free Units 1–10 study guide. Use the guide to find a concept, then ask about live help with the reasoning, practice or study plan that needs attention.
Parametric, polar or series questions feel unfamiliar
Identify the representation first. Then practise choosing an appropriate method and explaining its conditions, instead of treating every problem as another formula to substitute into.
Earlier skills slow down the current unit
Bring an example where algebra, trigonometry or function notation interrupted the calculus. Separating that obstacle from the new idea makes the next practice goal more specific.
How is BC different from AB, and where should I begin?
BC extends the calculus studied in AB and includes work with parametric and polar representations and infinite series. Begin with your current class topic and prerequisites; do not assume that being in a later unit means every earlier skill is secure. Your school determines placement and course requirements.
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Course reference: College Board’s AP Calculus BC course overview.
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01 Limits, derivatives, and integrals
02 Advanced integration and differential equations
03 Parametric equations and polar coordinates
04 Sequences and infinite series
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