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Unit I: Limits Unit IV: Integration Unit VII: Apps of Integration Unit IX: Infinite Series Unit XII: AP Exam Prep Final Exams Midterm Exam Semester 1 Semester 2

Curriculum Blueprint

Course Description

This course starts with four major ideas: limits, derivatives, indefinite integrals, and definite integrals. Then it delves into their applications. From there, it covers series and sequences, parametric and polar equations, vectors in a plane, and advanced integration techniques. You will learn how to work with functions represented in a variety of ways: graphical, numerical, analytical, and verbal.

The approach of the course is to develop skills that you apply to solving problems – skills that will become second nature because you will understand the why behind the major ideas. You’ll interact with your classmates and me through an online discussion board, posting solutions to problems. Each assignment will build on the previous ones, so success will depend on your completing the daily assignments.

You will start using your graphing calculator soon after the start of the class. Throughout the year, you will become comfortable using it as a tool to help solve problems, experiment, interpret results, and verify conclusions. However, it will not replace your pen and pencil.

Objectives & Proficiencies

THE SUCCESSFUL STUDENT WILL:

Exhibit a proficiency in the topics covered in this course.
Engage in logical and critical thinking.
Demonstrate solutions to problems by translating descriptions into mathematical terms.
Be prepared for the Advanced Placement Exam.
Be prepared for college calculus and courses utilizing calculus.

KEYS TO SUCCESS:

Keep up with daily homework.
Participate actively in the Discussion Board.
Read the textbook thoroughly.
Ask questions when you don’t understand concepts.
Study regularly, allocate preparation time, and don’t cram!

Grading Policy

50% — Tests at the end of each chapter (and mid-chapter).
10% — Daily interactive WebAssign homework.
10% — Quizzes using AP-Type questions for major topics.
10% — First Semester Exam in December.
10% — Final Exam in the format of an AP Exam in April.
5% — Contents of your Dropbox folder (scans of written work).
5% — Participation in the Discussion Board.

Semester 1: Differential & Integral Calculus

Limits and Their Properties (2 Weeks)

An intuitive understanding of the limiting process
Find limits graphically and numerically
Evaluate limits analytically
An intuitive understanding of continuity
Continuity and one-sided limits
Intermediate Value Theorem
Infinite limits and vertical asymptotes
Limits at Infinity and horizontal asymptotes

II.

Differentiation (2-3 Weeks)

The derivative and the tangent line problem
Differentiability and continuity concepts
Basic differentiation rules and rates of change (average and instantaneous)
Product and Quotient Rules and Higher Order derivatives
The Chain Rule
Implicit differentiation
Related Rates

III.

Applications of Differentiation (3 Weeks)

Extrema on an interval
Rolle’s Theorem and the Mean Value Theorem
Increasing and decreasing functions
The First Derivative Test
Concavity and points of inflection
The Second Derivative Test
Summary of Curve Sketching (including monotonicity)
Optimization and business problems
Newton’s Method
Differentials
Linear (or tangent line) approximations

IV.

Introduction to Integral Calculus (3 Weeks)

Antiderivatives and indefinite integration
Sigma Notation and concept of Area as the limit of a sum
Riemann sums (including left, right, and midpoint evaluation points)
Definite integrals: Properties and Solutions
The Fundamental Theorem of Calculus
The Mean Value Theorem for Integrals and Average Value of a Function
The Second Fundamental Theorem of Calculus
The Net Change Theorem
Integration using u-substitution
Numerical Integration and Trapezoidal Approximation

Transcendental Functions (2-3 Weeks)

The Natural Logarithmic Function and Differentiation
The Natural Logarithmic Function and Integration
Inverse Functions
Exponential Functions: Differentiation and Integration
Bases other than e and applications
Inverse trigonometric functions and Differentiation
Inverse trigonometric functions and Integration

VI.

Differential Equations (1-2 Weeks)

Differential equations: Slope fields and Euler’s Method
Differential equations: Growth and decay
Differential equations: Separation of variables and the Logistic Equation

VII.

Applications of Integration (2 Weeks)

Position, Velocity and Acceleration Functions
Net Change and Accumulation
Area of a region between two curves
Volume: Disk method
Volume: Washer method
Volume: Known cross-sections
Arc Length

Midterm Exam

The midterm exam includes problems from past AP exams that test the students’ abilities to connect concepts graphically, analytically, numerically, and verbally.

Semester 2: Advanced Topics & Series

VIII.

Integration Techniques (2-3 Weeks)

Basic integration techniques
Integration by parts
Partial fractions
Indeterminate forms and L’Hopital’s Rule
Improper integrals

IX.

Infinite Series (5 Weeks)

Sequences
Series and convergence
The Integral Test and p-Series
Comparison of series
Alternating series
The Ratio and Root Tests
Taylor Polynomials and Approximations
Power series
Representation of functions by power series
Taylor and Maclaurin Series

Parametric, Polar, and Calculus (2 Weeks)

Plane curves and parametric equations
Parametric equations and calculus
Polar coordinates and polar graphs
Area and Arc Length in polar coordinates

XI.

Vector-Valued Functions (1 Week)

Vectors in a plane
Vector-valued functions
Differentiation and integration of vector-valued functions
Velocity and acceleration

XII.

AP Exam Preparation (3 Weeks)

Mark Howell’s Be Prepared for the AP Calculus Exam
1997, 1998, 2003 and 2008 AP Exams
2004 to 2013 Free Response Questions

Final Exam

The final exam includes problems from past AP exams that test the students’ abilities to connect concepts graphically, analytically, numerically, and verbally.