Unit 1 Limits and Continuity
You’ll start to explore how limits will allow you to solve problems involving change and to better understand mathematical reasoning about functions.
Asymptotes and limits at infinityDefinition and properties of limits in various representationsDefinitions of continuity of a function at a point and over a domainHow limits help us to handle change at an instantReasoning using the Squeeze theorem and the Intermediate Value Theorem
Unit 2 Differentiation: Definition and Fundamental Properties
You’ll apply limits to define the derivative, become skillful at determining derivatives, and continue to develop mathematical reasoning skills.
Applying differentiation rulesConnecting differentiability and continuityDefining the derivative of a function at a point and as a functionDetermining derivatives for elementary functions
Unit 3 Differentiation: Composite, Implicit, and Inverse Functions
You’ll master using the chain rule, develop new differentiation techniques, and be introduced to higher-order derivatives.
Determining higher-order derivatives of functionsDifferentiation of general and particular inverse functionsImplicit differentiationThe chain rule for differentiating composite functions
Unit 4 Contextual Applications of Differentiation
You’ll apply derivatives to set up and solve real-world problems involving instantaneous rates of change and use mathematical reasoning to determine limits of certain indeterminate forms.
Applying understandings of differentiation to problems involving motionGeneralizing understandings of motion problems to other situations involving rates of changeIdentifying relevant mathematical information in verbal representations of real-world problems involving rates of changeLocal linearity and approximationL’Hospital’s ruleSolving related rates problems
Unit 5 Analytical Applications of Differentiation
After exploring relationships among the graphs of a function and its derivatives, you'll learn to apply calculus to solve optimization problems.
Behaviors of Implicit relationsDerivatives and properties of functionsHow to solve optimization problemsHow to use the first derivative test, second derivative test, and candidates testMean Value Theorem and Extreme Value TheoremSketching graphs of functions and their derivatives
Unit 6 Integration and Accumulation of Change
You’ll learn to apply limits to define definite integrals and how the Fundamental Theorem connects integration and differentiation. You’ll apply properties of integrals and practice useful integration techniques.
Accumulation functions, the Fundamental Theorem of Calculus, and definite integralsAntiderivatives and indefinite integralsApproximating integrals using Riemann SumsProperties of integrals and integration techniquesUsing definite integrals to determine accumulated change over an interval
Unit 7 Differential Equations
You’ll learn how to solve certain differential equations and apply that knowledge to deepen your understanding of exponential growth and decay.
Deriving and applying a model for exponential growth and decayInterpreting verbal descriptions of change as separable differential equationsSketching slope fields and families of solution curvesSolving separable differential equations to find general and particular solutions
Unit 8 Applications of Integration
You’ll make mathematical connections that will allow you to solve a wide range of problems involving net change over an interval of time and to find areas of regions or volumes of solids defined using functions.
Determining the average value of a function using definite integralsDetermining volume with cross-sections, the disc method, and the washer methodFinding the area between curvesModeling particle motionSolving accumulation problems
