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Unit 1: Limits and Continuity (100)
• How limits help us to handle change at an instant (20)
• Definition and properties of limits in various representations (20)
• Definitions of continuity of a function at a point and over a domain (20)
• Asymptotes and limits at infinity (20)
• Reasoning using the Squeeze theorem and the Intermediate Value Theorem (20)
Unit 2: Differentiation: Definition and Fundamental Properties (80)
• Defining the derivative of a function at a point and as a function (20)
• Connecting differentiability and continuity (20)
• Determining derivatives for elementary functions (20)
• Applying differentiation rules (20)
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (80)
• The chain rule for differentiating composite functions (20)
• Implicit differentiation (20)
• Differentiation of general and particular inverse functions (20)
• Determining higher-order derivatives of functions (20)
Unit 4: Contextual Applications of Differentiation (120)
• Identifying relevant mathematical information in verbal representations of real-world problems involving rates of change (20)
• Applying understandings of differentiation to problems involving motion (20)
• Generalizing understandings of motion problems to other situations involving rates of change (20)
• Solving related rates problems (20)
• Local linearity and approximation (20)
• L’Hospital’s rule (20)
Unit 5: Analytical Applications of Differentiation (120)
• Mean Value Theorem and Extreme Value Theorem (20)
• Derivatives and properties of functions (20)
• How to use the first derivative test; second derivative test and candidates test (20)
• Sketching graphs of functions and their derivatives (20)
• How to solve optimization problems (20)
• Behaviors of Implicit relations (20)
Unit 6: Integration and Accumulation of Change (40)
• Using definite integrals to determine accumulated change over an interval (0)
• Approximating integrals with Riemann Sums (10)
• Accumulation functions; the Fundamental Theorem of Calculus and definite integrals (0)
• Antiderivatives and indefinite integrals (0)
• Properties of integrals and integration techniques; extended (20)
• Determining improper integrals (10)
Unit 7: Differential Equations (70)
• Interpreting verbal descriptions of change as separable differential equations (20)
• Sketching slope fields and families of solution curves (20)
• Using Euler’s method to approximate values on a particular solution curve (0)
• Solving separable differential equations to find general and particular solutions (20)
• Deriving and applying exponential and logistic models (10)
Unit 8: Applications of Integration (110)
• Determining the average value of a function using definite integrals (20)
• Modeling particle motion (20)
• Solving accumulation problems (20)
• Finding the area between curves (20)
• Determining volume with cross-sections; the disc method and the washer method (20)
• Determining the length of a planar curve using a definite integral (10)
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (60)
• Finding derivatives of parametric functions and vector-valued functions (10)
• Calculating the accumulation of change in length over an interval using a definite integral (10)
• Determining the position of a particle moving in a plane (10)
• Calculating velocity; speed and acceleration of a particle moving along a curve (10)
• Finding derivatives of functions written in polar coordinates (10)
• Finding the area of regions bounded by polar curves (10)
Unit 10: Infinite Sequences and Series (50)
• Applying limits to understand convergence of infinite series (10)
• Types of series: Geometric; harmonic and p-series (10)
• A test for divergence and several tests for convergence (0)
• Approximating sums of convergent infinite series and associated error bounds (10)
• Determining the radius and interval of convergence for a series (10)
• Representing a function as a Taylor series or a Maclaurin series on an appropriate interval (10)