Course overview
AP Calculus AB develops core ideas of change and accumulation. You begin with limits and continuity, build derivative techniques and interpretations, then use integration and differential equations to model and solve problems.
Complete curriculum
Every unit and lesson
- 01
Unit 1 · 4 lessons
Limits and Continuity
Define limits numerically, graphically, and algebraically, and decide when a function is continuous at a point or over an interval.
Why it mattersLimits are the entry point to calculus because both derivatives and integrals are defined using limiting behavior.
By the end, you will be able to- Estimate and evaluate one-sided and two-sided limits.
- Determine continuity and classify removable, jump, and infinite discontinuities.
- Use limit laws and algebraic simplification to compute limits.
- 1.1Reading limits from graphs and tables
Interpret approaching behavior without relying on direct substitution.
- 1.2Limit laws and algebraic techniques
Apply limit laws, factoring, and rationalization to resolve indeterminate forms.
- 1.3Continuity at a point and on an interval
Check all continuity conditions and identify where they fail.
- 1.4Average and instantaneous change preview
Connect secant slopes to the idea of a tangent slope via limits.
- 02
Unit 2 · 4 lessons
Differentiation: Definition and Basic Rules
Build the derivative from first principles, interpret derivative values, and use core differentiation rules for common function forms.
Why it mattersThe derivative is the main tool for describing instantaneous rate of change and local linear behavior.
By the end, you will be able to- Compute derivatives from the limit definition in simple cases.
- Interpret derivative values and units in context.
- Differentiate polynomial, power, exponential, and trigonometric functions using basic rules.
- 2.1Derivative as a limit of difference quotients
Move from average rate of change to instantaneous rate of change.
- 2.2Notation and tangent line meaning
Use f'(x), dy/dx, and derivative values to describe local behavior.
- 2.3Power, sum, and constant multiple rules
Differentiate efficiently without rebuilding limits each time.
- 2.4Derivatives of trig and exponential functions
Apply standard derivatives and combine them in mixed expressions.
- 03
Unit 3 · 4 lessons
Composite, Implicit, and Inverse Functions
Differentiate composite functions with the chain rule, handle implicit relationships, and compute derivatives of inverse functions.
Why it mattersMany real formulas are nested or not solved for one variable, so these methods make derivatives practical in realistic settings.
By the end, you will be able to- Apply the chain rule to nested function structures.
- Use implicit differentiation to find dy/dx when x and y are linked.
- Find derivatives involving inverse functions and interpret restrictions.
- 3.1Chain rule for composite functions
Track outer and inner functions and multiply derivative layers correctly.
- 3.2Implicit differentiation workflow
Differentiate both sides with respect to x and isolate dy/dx.
- 3.3Derivatives of inverse functions
Use inverse relationships to connect derivatives at paired points.
- 3.4Logarithmic differentiation basics
Differentiate products and powers more efficiently by taking logs.
- 04
Unit 4 · 4 lessons
Contextual Applications of Differentiation
Use derivatives to model rates in context, including motion and related rates, with careful attention to units and interpretation.
Why it mattersCalculus is most useful when derivative values answer real questions about speed, flow, growth, and changing geometry.
By the end, you will be able to- Interpret derivative signs and magnitudes in verbal and physical contexts.
- Solve related-rates problems by linking variables and differentiating over time.
- Use position, velocity, and acceleration relationships in motion models.
- 4.1Rates in context and units
Translate derivative statements into precise real-world meaning.
- 4.2Related rates setup
Build a geometric or physical equation before differentiating with respect to time.
- 4.3Linearization and local approximation
Use tangent lines to estimate nearby values and changes.
- 4.4Motion along a line
Connect derivatives of position to velocity and acceleration decisions.
- 05
Unit 5 · 4 lessons
Analytical Applications of Differentiation
Analyze function behavior using first and second derivatives, then solve optimization and curve-analysis problems.
Why it mattersDerivative tests turn symbolic expressions into decisions about increase, decrease, extrema, and shape.
By the end, you will be able to- Find intervals of increase and decrease using first-derivative sign analysis.
- Classify local extrema and concavity with derivative tests.
- Solve optimization problems with clear constraints and interpretations.
- 5.1Critical points and first derivative test
Locate and classify potential extrema from derivative behavior.
- 5.2Concavity and second derivative test
Identify concavity changes and possible inflection points.
- 5.3Optimization from context
Write objective and constraint equations before differentiating.
- 5.4Curve sketching with derivative evidence
Combine intercepts, asymptotes, monotonicity, and concavity into one graph story.
- 06
Unit 6 · 4 lessons
Integration and Accumulation of Change
Introduce antiderivatives, Riemann sums, and definite integrals, then use the Fundamental Theorem of Calculus to connect accumulation with derivatives.
Why it mattersIntegration formalizes accumulated change, so you can recover total amounts from rate information.
By the end, you will be able to- Approximate area and accumulation with left, right, and midpoint Riemann sums.
- Compute antiderivatives and evaluate definite integrals.
- Apply both parts of the Fundamental Theorem of Calculus.
- 6.1Area accumulation and Riemann sums
Interpret sigma-style sums as approximations to accumulated quantity.
- 6.2Antiderivatives and indefinite integrals
Use reverse differentiation patterns to find families of functions.
- 6.3Definite integrals and net change
Evaluate integrals as signed accumulation over an interval.
- 6.4Substitution as reverse chain rule
Transform integrals into simpler variables and bounds.
- 07
Unit 7 · 4 lessons
Differential Equations
Model rates with differential equations, visualize slope fields, and solve separable equations with initial conditions.
Why it mattersDifferential equations describe systems where change depends on current state, including growth, cooling, and mixing behavior.
By the end, you will be able to- Interpret slope fields to compare families of possible solutions.
- Solve separable first-order differential equations.
- Use initial conditions to determine a specific solution curve.
- 7.1From verbal model to differential equation
Translate rate statements into equations involving derivatives.
- 7.2Slope fields and qualitative behavior
Use local slope patterns to predict long-run tendencies.
- 7.3Separable equations
Separate variables, integrate both sides, and apply constants correctly.
- 7.4Exponential growth and decay solutions
Use dy/dt = ky with initial value data to model changing quantities.
- 08
Unit 8 · 4 lessons
Applications of Integration
Apply definite integrals to area, accumulated change, and simple geometric or physical quantities derived from rates.
Why it mattersThese applications show how integration answers total-amount questions that derivatives alone cannot.
By the end, you will be able to- Compute net and total change from rate functions on intervals.
- Find area between curves using intersection bounds.
- Model accumulation in position, revenue, and other applied settings.
- 8.1Accumulation from rates
Convert a known rate function into total change over time.
- 8.2Area between two curves
Set up top-minus-bottom integrals with correct bounds.
- 8.3Volume from cross sections
Integrate known cross-sectional area formulas across an interval.
- 8.4Average value of a function
Use integral mean value to summarize behavior on an interval.