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Unit 1: Limits & 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

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Unit 2: Differentiation: Fundamentals

• 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

• Implicit differentiation

• Differentiation of general and particular inverse functions

• Determining higher-order derivatives of functions

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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

• L’Hospital’s rule

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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

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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

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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

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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