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About the Course
Explore the concepts, methods, and applications of differential and integral calculus, including topics such as parametric, polar, and vector functions, and series. You’ll perform experiments and investigations and solve problems by applying your knowledge and skills.
Skills You'll Learn
Determining expressions and values using mathematical procedures and rules
Connecting representations
Justifying reasoning and solutions
Using correct notation, language, and mathematical conventions to communicate results or solutions
Equivalency and Prerequisites
College Course Equivalent
A first-semester college calculus course and the subsequent single-variable calculus course
Recommended Prerequisites
You should have successfully completed courses in which you studied algebra, geometry, trigonometry, analytic geometry, and elementary functions. In particular, you should understand the properties of linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions, as well as sequences, series, and polar equations. You should know how to graph these functions and solve equations involving them. You should also be familiar with algebraic transformations, combinations, compositions, and inverses for general functions.
Exam Date
About the Units
The course content outlined below is organized into commonly taught units of study that provide one possible sequence for the course. Your teacher may choose to organize the course content differently based on local priorities and preferences.
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
On The Exam
4%–7% of exam score
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
On The Exam
4%–7% of exam score
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
On The Exam
4%–7% of exam score
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
- L’Hospital’s rule
On The Exam
6%–9% of exam score
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
On The Exam
8%–11% of exam score
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
On The Exam
17%–20% of exam score
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
On The Exam
6%–9% of exam score
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
On The Exam
6%–9% of exam score
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
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
On The Exam
11%–12% of exam score
Unit 10: Infinite Sequences and Series
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
On The Exam
17%–18% of exam score
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