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- 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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- 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
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- 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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- 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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- 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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- Using definite integrals to determine accumulated change over an interval
- Approximating integrals with Riemann Sums
- Accumulation functions, the Fundamental Theorem of Calculus, and definite integrals
- Antiderivatives and indefinite integrals
- Properties of integrals and integration techniques, extended
- Determining improper integrals
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- Interpreting verbal descriptions of change as separable differential equations
- Sketching slope fields and families of solution curves
- Using Euler’s method to approximate values on a particular solution curve
- Solving separable differential equations to find general and particular solutions
- Deriving and applying exponential and logistic models
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- 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
- Determining the length of a planar curve using a definite integral
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- Finding derivatives of parametric functions and vector-valued functions
- Calculating the accumulation of change in length over an interval using a definite integral
- Determining the position of a particle moving in a plane
- Calculating velocity, speed, and acceleration of a particle moving along a curve
- Finding derivatives of functions written in polar coordinates
- Finding the area of regions bounded by polar curves
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- Applying limits to understand convergence of infinite series
- Types of series: Geometric, harmonic, and p-series
- A test for divergence and several tests for convergence
- Approximating sums of convergent infinite series and associated error bounds
- Determining the radius and interval of convergence for a series
- Representing a function as a Taylor series or a Maclaurin series on an appropriate interval
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