AP Calculus (AB)

Mr. Jeff Sweigard Room 21

Course Description: AP Calculus offers students their first opportunity to attempt collegiate

level mathematics. This course has two distinct goals: First, to learn and appreciate Calculus as a

significant mathematical and scientific tool and as a human achievement. Second, to prepare for

the College Board’s Advanced Placement Examination in Calculus (AB) . Passing this examination may earn the student college credit when they enroll in a four year institution in the Fall of 2014.

Our textbook is Calculus of a Single Variable, by Ron Larson, Robert Hostetler, and Bruce

Edwards (Eighth Edition), 2006.

Supplemental materials include: Cracking the AP Calculus AB & BC Exams 2013 by David S. Kahn, The Princeton Review.

A graphing calculator (the TI-83 or TI-84 is particularly recommended) is required. Students

who are unable to obtain their own calculator may check out a calculator for the year in a manner

somewhat similar to that of checking out textbooks. We will use the calculator in a variety of ways including:

  • Graph functions within arbitrary windows.
  • Solve equations numerically
  • Analyze and interpret results.
  • Conduct explorations.
  • Justify and explain results of graphs and equations.

The “Rule of Four” will be emphasized in the approach and solving of problems. The four parts are:

  • Numerical analysis(where data points are known, but not an equation)
  • Graphical analysis(where a graph is known, but not an equation)
  • Analytic/algebraic analysis(traditional equation and variable manipulation)
  • Verbal/written method of representing problems(classic story problems as well as written justification of one’s thinking in solving a problem)

Throughout the course, students will be asked to explain how they solved a problem to the class. Student will regularly work in groups. I find that some students feel more comfortable asking fellow students for help. An atmosphere where students are comfortable asking questions is of great importance and will be emphasized. My hope is to build confidence and motivate students to do their best.

Student will have an opportunity after a test and some quizzes to get partial credit for missed questions. The student will write what they did wrong and what they should have done to get the correct answer. They can earn back up to half of the points lost.

Grading: Your homework, quizzes, notes, AP practice problems, and class participation are

40% of your grade. Tests are 50% of your grade. Citizenship will be 10% of your grade.

Grading uses this scale:

92 - 100 = A

84 - 91 = B

72 - 83 = C

60 - 71 = D

below 60 = F

Citizenship:

Every student is to be ready to learn: Bring book,pencil, paper, and notebook to class every day unless told otherwise. You will need a graphing calculator. Participate in all class activities.

This is a college level course offered in the high school. The highest standard of conduct and

integrity is required at all times. Disruption of instruction cannot be tolerated.

Topics of Study:

Summer Assignment: review topics in trigonometry and pre-calculus. (Approximately 5 days)

Unit 1: Limits and Continuity (Approximately 10 days)

  • What is a Limit?
  • Limits of Polynomials
  • Limits of Trigonometric Functions
  • The Definition of Continuity
  • Types of Discontinuities

Unit 2: Definition of the Derivative and Basic Differentiation (Approximately 15 days)

  • Deriving the Formula
  • The Slope of a Curve
  • The Secant and the Tangent
  • Differentiability
  • Notation
  • The Power Rule
  • Higher Order Derivatives
  • The Product Rule
  • The Quotient Rule
  • The Chain Rule
  • Derivatives of Trig Functions

Unit 3: Implicit Differentiation and Basic Applications of the Derivative (Approximately 10 days)

  • How to Do It
  • Second Derivatives
  • Equations of Tangent Lines and Normal Lines
  • The Mean Value Theorem for Derivatives
  • Rolle’s Theorem

Unit 4: Maxima and Minima, Motion (Approximately 5 days)

  • Applied Maxima and Minima Problems

** Sketch graphs and label all extrema, points of inflection, and asymptotes. Students will check their work using their graphic calculator.

  • Position, Velocity, and Acceleration

** A CBL experiment is conducted with students tossing a ball into the air. Students graph the height of the ball versus the time the ball is in the air. The calculator is used to find a quadratic equation to model the motion of the ball over time. Average velocities are calculated over different time intervals and students are asked to approximate instantaneous velocity. The tabular data and the regression equation are both used in these calculations. These velocities are graphed versus time on the same graph as the height versus time graph.

Unit 5: Exponential and Logarithmic Functions, Differential Calculus (Approximately 15 days)

  • The Derivative of ln x
  • The Derivative of ex
  • The Derivative of logax
  • The Derivative of ax
  • The Derivative of an Inverse Function
  • Derivatives of Parametric Functions
  • L’Hopital’s Rule
  • Differentials
  • Logarithmic Differentiation

Unit 6: The Integral (Approximately 10 days)

  • The Antiderivative
  • Integrals of Trig Functions
  • Addition and Subtraction
  • U-Substitution

Unit 7: Definite Integrals, Exponential and Logarithmic Functions (Approximately 16 days)

  • Area Under a Curve
  • Tabular Riemann Sums
  • The Fundamental Theorem of Calculus
  • The Trapezoid Rule
  • The Mean Value Theorems for Integrals
  • The Second Fundamental Theorem of Calculus
  • Integrals of Trig Functions
  • Integrating ex and ax

Unit 8: The Area Between Two Curves, The Volume of a Solid of Revolution (Approximately 10 days)

  • Vertical Slices
  • Horizontal Slices
  • Washer and Disks
  • Cylindrical Shells
  • Volumes of Solids with Known Cross-Sections

Unit 9: Integration by Parts, Trig Functions (Approximately 10 days)

  • The Formula
  • Inverse Trig Functions
  • Advanced Integration of Trig Functions

Unit 10: Differential Equations (Approximately 10 days)

  • Separation of Variables
  • Euler’s Method
  • Slope Fields

Review for AP Exam (Approximately 20 days)

Unit 11: Other Applications of the Integral (Approximately 10 days)

  • Length of a Curve
  • Parametric Functions
  • The Method of Partial Fractions
  • Improper Integrals
  • Calculus of Polar Curves

Unit 12: Infinite Series (Approximately 18 days)

  • Sequences and Series
  • Geometric Series
  • The Ratio Test
  • Alternating Series
  • Integral Test
  • Comparison Test
  • Power Series
  • Taylor Series and Polynomials