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Course and Exam Description
This is the core document for the course. It clearly lays out the course content and describes the exam and AP Program in
general.
Learn the similarities and differences between these two courses and exams.
Course Content
Course Content
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.
Topics may include:
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
You'll apply limits to define the derivative, become skillful at determining derivatives, and continue to develop mathematical reasoning skills.
Topics may include:
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
You'll master using the chain rule, develop new differentiation techniques, and be introduced to higher-order derivatives.
Topics may include:
The chain rule for differentiating composite functions
Implicit differentiation
Differentiation of general and particular inverse functions
Determining higher-order derivatives of 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.
Topics may include:
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
LHospitars rule
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.
Topics may include:
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
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.
Topics may include:
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
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 and logistic models.
Topics may include:
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
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 lengths of curves, areas of regions, or volumes of solids defined using functions.
Topics may include:
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
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC only)
You'll solve parametrically defined functions, vector-valued functions, and polar curves using applied knowledge of differentiation and integration.
You'll also deepen your understanding of straight-line motion to solve problems involving curves.
Topics may include:
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
Unit 10: Infinite Sequences and Series (BC only)
You'll explore convergence and divergence behaviors of infinite series and learn how to represent familiar functions as infinite series. You'll also learn how to determine the largest possible error associated with certain approximations involving series.
Topics may include:
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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