Course Content
Unit 1 Limits and Continuity
• How limits help us to handle change at an instant
• Definition and properties of limits in various representations
• Definitions of continuity of a function at a point and over a domain
• Asymptotes and limits at infinity
• Reasoning using the Squeeze theorem and the Intermediate Value Theorem
Unit 2 Differentiation- Definition and Fundamental Properties
• Defining the derivative of a function at a point and as a function
• Connecting differentiability and continuity
• Determining derivatives for elementary functions
• Applying differentiation rules
Unit 3 Differentiation- Composite, Implicit, and Inverse Functions
• The chain rule for differentiating composite functions functions
• Implicit differentiation
• Differentiation of general and particular inverse functions
• Determining higher-order derivatives of functions
Unit 4 Contextual Applications of Differentiation
• Identifying relevant mathematical information in verbal representations of real-world problems involving rates of change
• Applying understandings of differentiation to problems involving motion
• Generalizing understandings of motion problems to other situations involving rates of change
• Solving related rates problems
• Local linearity and approximation
Unit 5 Analytical Applications of Differentiation
• Mean Value Theorem and Extreme Value Theorem
• Derivatives and properties of functions
• How to use the first derivative test, second derivative test, and candidates test
• Sketching graphs of functions and their derivatives
• How to solve optimization problems
• Behaviors of Implicit relations
Unit 6 Integration and Accumulation of Change
• Using definite integrals to determine accumulated change over an interval
• Approximating integrals using Riemann Sums
• Accumulation functions, the Fundamental Theorem of Calculus, and definite integrals
• Antiderivatives and indefinite integrals
• Properties of integrals and integration techniques
Unit 7 Differential Equations
• Interpreting verbal descriptions of change as separable differential equations
• Sketching slope fields and families of solution curves
• Solving separable differential equations to find general and particular solutions
• Deriving and applying a model for exponential growth and decay
Unit 8 Applications of Integration
• Determining the average value of a function using definite integrals
• Modeling particle motion
• Solving accumulation problems
• Finding the area between curves
• Determining volume with cross-sections, the disc method, and the washer method