#### AP Calculus: Course Outline

**Unit 1**: The Cartesian Plane and Functions

- Real Numbers and the Real Line
- The Cartesian Plane
- Functions and Graphs
- Trigonometric Functions

**Unit 2**: Limits and their Properties

- Introduction to Limits
- Limit Power Point Presentations
- Properties for Limits
- Techniques for Evaluating Limits
- Evaluating Limits numerically and graphically. Students use the graph and the table to evaluate the limit.
- Continuity and one-sided limits
- Infinite Limits
- Limits as x→∞ Answers can be confirmed with the graphs and the numerical data.
- Asymptotic Behavior

**Unit 3**: Differentiation

- Limit Definition of the Derivative
- Differentiability and Continuity
- Differentiation Rules and Rates of Change
- Product, Quotient rules and Chain Rules
- Higher-Order Derivatives
- Applications to position, velocity, speed and acceleration
- The derivative graphically, numerically and analytically
- Implicit Differentiation
- Particle Motion
- Related Rates (Related Rate Project)

**Unit 4**: Applications of Derivatives

- Extrema on an Interval
- Rolle’s Theorem and Mean Value Theorem
- Increasing and decreasing functions and the First Derivative Test
- Concavity and the Second Derivative Test
- Analysis of the graph of a function
- Limits at Infinity
- Characteristics & relationships of the graphs of f, f ’, f “ (Families of Curves)
- Optimization and Modeling
- Tangent line to a curve and linear approximation

**Unit 5**: Integration

- Antiderivatives and Indefinite Integration
- Initial Condition
- Approximating areas
- Definite Integrals
- Riemann Sums
- Trapezoidal Rule
- Approximate definite integrals of functions using Riemann sums and Trapezoidal sums represented analytically, graphically, and tables of data
- Distance Traveled by a Particle Along a Line
- The Fundamental Theorem of Calculus
- The Second Fundamental Theorem of Calculus
- Integration by Substitution
- Change of Variables
- Mean Value Theorem for Integrals and the average value of a function

**Unit 6**: Transcendental Functions and Mathematical Modeling

- Natural Logarithmic Differentiation and Integration
- Derivative of an Inverse Function
- Derivatives and Integrals of Exponential Functions
- Bases other than e and applications
- Separating Variables: Solving Differential Equations
- Exponential Growth and Decay
- Slope Fields
- Initial Value
- Inverse Trigonometric Functions and their Derivatives
- Integrals Involving Inverse Trigonometric Functions

**Unit 7**: Applications of Integration

- The integral as an accumulator of rates of changes
- Area of a Region Between Two Curves
- Volumes of Solids of Revolution: Disc and Washer Method
- Volumes of Solids with known Cross Sections
- Applications of integration involving a particle moving along a line

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