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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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