ARS Mathematica Online                                                    AP Calculus BC
                                             AP Calculus BC Course Outline



Semester 1

P. Preparation for Calculus (.5 weeks)

  • Inverse Functions
  • Exponential and Logarithmic Functions

I. 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
  • Derivatives of Inverse Functions
  • Implicit differentiation
  • Related Rates
  • Newton's Method

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
  • Differentials
  • Linear (or tangent line) approximations

2nd Quarter

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
  • The natural logarithmic Function: Integration
  • Inverse Trigonometric Functions: Integration

V. Differential Equations (2 weeks)

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

VI. 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

VII. Integration Techniques (2-3 weeks)

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

VIII. 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
  • Lagrange Error Bound

IX. Parametric Equations, Polar Coordinates and Vectors (3 weeks)

  • Plane curves and parametric equations
  • Parametric equations and calculus
  • Polar coordinates and polar graphs
  • Area and Arc Length in polar coordinates
  • Vectors in a plane
  • Vector-valued functions
  • Differentiation and integration of vector-valued functions
  • Velocity and acceleration

4th Quarter

X. AP Exam Preparation (3 weeks)

  • Mark Howell’s Be Prepared for the AP Calculus Exam  
  • 1997, 1998, 2003, 2008 and 2012 Released AP Exams
  • 2013 to 2017 Audit AP Practice Exams

Final Exam:   The final exam is in the form of a past AP exam that tests the students’ abilities to connect concepts graphically, analytically, numerically, and verbally. (1 week)

Texts Required: Enhanced Web Assign with Calculus for AP 1st Edition eBook by Ron Larson, Paul Battaglia (2017) (available in mid-August through for $35.00)

Enhanced Web Assign helps you develop a deeper conceptual understanding of your subject matter and complete required online homework assignments. With Enhanced Web Assign, you receive access to a Premium eBook along with immediate feedback and tutorial content, which will include Watch It Videos, Animations, Practice It Questions, Master It Tutorials, Read It Text passages, and a Personal Study Plan to aid in the mastery of your course materials.

(Also available but not necessary) Calculu for AP 1st Edition Printed Version by Ron Larson, Paul Battaglia (2017) ISBN-13: 9781305674912.

Be Prepared for the AP Calculus Exam Third Edition by Mark Howell and Martha Montgomery Copyright © May, 2016 by Skylight Publishing ISBN 978-0-9972528-5-9.

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