### Calculus

**Course Overview:**

This course is equivalent to an introductory level university calculus course. This course meets 3 days a week for 42 minutes and 1 day a week for 87 minutes over a period of 180 days. Throughout this course, students will develop an understanding of the concepts of calculus and experience its methods and applications. The broad topics in the course include, but are not limited to, the following: derivatives, definite and indefinite integrals, limits, and approximations.

Each student is required to have a graphing calculator in class every day and one available to use at home for assignments. It is recommended that students have a TI-83 Plus or TI-84/Plus/Silver. We will use graphing calculators to discover and strengthen the concepts of calculus. During the first few weeks of class, students will be given extra time to familiarize themselves with their calculators.

**Evaluation**: Course grades will be determined by student performance within the following four areas:

Tests, Quizzes, and projects- 60%

Portfolio- 10%

Homework- 20%

Participation-10%

*Each student taking the AP exam will be required to come in one additional date outside of school time (date to be determined) to take a practice exam.

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**Unit 1: Pre-calculus Review (1 week)**

- Graphs
- Sketching
- Intercepts
- Symmetry
- Lines
- Slope as rate of change
- Equations of lines
- Parallel and perpendicular lines
- Function and graphs
- Functions
- Domain and range
- Transformations
- Families of function
- Composition of functions
- Even and odd function

Unit 2: Limits and Continuity (3 weeks)

- Introduction to Limits
- Finding graphically
- Finding numerically
- Properties of limits
- Continuity
- Continuous functions
- Discontinuous functions

i. Removable discontinuity

ii. Non-removable discontinuity

- Properties of continuity
- Intermediate Value Theorem
- Infinite limits
- Determining graphically
- Asymptotic behavior
- Properties of limits

Unit 3: Derivatives (6 weeks)

- Definition of the derivative

1. Graphically and analytically

- Slope of curve at a point
- Limit of the difference quotient
- Rates of change
- Applications to velocity and acceleration
- Differentiability

1. Relationship with continuity

- Differentiation Rules
- Higher-order derivatives
- Derivatives of Trigonometric Functions
- The Chain Rule
- Implicit Differentiation
- Related rates to solve real-life problems

Unit 4: Applications of Differentiation (6 weeks)

- Extreme Values

1. Local (relative) extrema

2. Global (absolute) extrema

- Using the derivative

1. Rolle’s Theorem

2. Mean Value Theorem

3. Increasing and Decreasing functions

- Analysis of graphs

1. Critical numbers

2. First derivative test

3. Concavity

4. Inflection points

5. Second derivative test

6. Horizontal Asymptotes

- Optimization Problems
- Local linear approximation (tangent line)

Unit 5: Integration (5 weeks)

- Definition of an antiderivative (indefinite integral)

1. Basic integration rules

2. Vertical motion applications

3. Solve differential equations

- Approximating areas

1. Riemann sums

2. Definite integral (including properties)

3. Common geometric figures

4. Trapezoidal rule

- The Fundamental Theorem

1. The Fundamental Theorem of Calculus (part 1)

2. Average Value Theorem

3. The Fundamental Theorem of Calculus (part 2)

- Integration using substitution

Unit 6: Logarithmic, Exponential, and Other Transcendental Functions (4 weeks)

- Logarithmic, trigonometric, power, exponential, and inverse trigonometric functions

1. Derivatives

2. Integration

Unit 7: Differential Equations (2 weeks)

- Finding a solution (initial condition)
- Slope fields
- Growth and decay
- Separation of variables

Unit 8: Applications of Integration (2 weeks)

- Area of a region between two curves
- Volumes

1. Disk method