AP Calculus AB – Mr. Pugh
Somerville High School
2016 – 2017
Room 111
School Phone: 218-4108
Email:
The following is an outline of the material that will be covered in AP Calculus AB. Throughout the course, students will be expected to express mathematical ideas both orally and in written form on a daily basis. Particular attention will be paid to word-choice and the avoidance of pronouns. All concepts will be studied using an analytic and graphical approach. Computer technology will be used to develop ideas and enhance lessons when appropriate.
Through completion of this course students will have the ability to:
· Work with functions graphically, numerically, analytically, and verbally, as well as understand the connections between these representations
· Understand that the derivative represents a rate of change or a local linear approximation and will be able to apply derivatives to problem solving.
· Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and will be able to apply integrals to problem solving.
· Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus
· Communicate mathematics and explain solutions both verbally and in written form.
· Model a written description of a physical situation with a function, differential equation, or an integral.
· Use technology to help solve problems, experiment, interpret results, and justify conclusions.
· Verify the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
· Appreciate calculus as a coherent body of knowledge and as a human accomplishment.
Calculators
We will use a graphing calculator on a regular basis. I have a classroom set of TI-84s but it is strongly suggested that you obtain your own calculator for work outside of the classroom. A TI-84 or a TI-89 is recommended for this course. We will be using the calculator to perform the following tasks:
· Graph functions in arbitrary viewing windows
· Calculate the zeros of a function (i.e. its x-intercepts)
· Calculate the numerical derivative of a function
· Calculate the numerical integral of a function
· Interpret results and justify solutions obtained manually
Unit 1: Functions, Limits, and Continuity (16 days)
A. The Cartesian Plane and Functions
· Absolute Value
· Symmetry
· Even and Odd Functions
· Domain and Range
B. Limits and Their Properties
· One and two-sided limits
· Squeeze Theorem (look at proof of and confirm graphically)
· Calculate limits of polynomial and rational functions graphically, analytically, and by using a table of values
· Infinite Limits and Limits at Infinity
· Horizontal and Vertical Asymptotes of a Function
C. Continuity
· Develop definition of continuity
· Continuous and Discontinuous Functions
o Removable, Non-removable, Infinite, Jump Discontinuity (discuss y = int(x) )
· Intermediate Value Theorem
· Extreme Value Theorem (introduce and revisit in Unit 3)
Unit 2: Differentiation (21 days)
A. Rates of Change of a Function
· Average Rate of Change
· Tangent Line to a Curve
· Instantaneous Rate of Change
B. The Derivative
· Definition of the Derivative (difference quotient)
· Derivative at a Point
· One-sided derivatives
· Numerical Derivative of a Function (using nDeriv on the calculator)
· Graphing f`(x) using the graph of f(x)
· The Derivative as a Function
· Graphing the Derivative (explore using Y2 = nDeriv(Y1,X,X) on the calculator)
C. Differentiability
· Define differentiability
· Differentiability and Continuity
· Local Linearity
· Symmetric Difference Quotient
· Intermediate Value Theorem for Derivatives
D. Differentiation Rules
· Sum and Difference Rules
· Constant, Power, Product, and Quotient Rules
· Chain Rule
· Higher Order Derivatives
E. Applications of the Derivative
· Position, Velocity, Acceleration, and Jerk (show that vertical motion formulas from physics are related through differentiation)
· Particle Motion
F. Implicit Differentiation
· y` notation
· Expressing derivatives in terms of x and y.
G. Related Rates
Unit 3: Applications of Differentiation (20 days)
A. Extema and Related Theorems
· Absolute Extrema
· Extreme Value Theorem
· Relative Extrema
· Critical Values
· Rolle’s Theorem
· Mean Value Theorem
B. Determining Function Behavior
· Increasing and Decreasing Functions
· First Derivative Test to Locate Relative Extrema
· Concavity
· Using the Second Derivative to Locate Points of Inflection
· Second Derivative Test to Locate Relative Extrema
· The Relationship Between f(x), f`(x), and f``(x).
C. Optimization
D. Differentials
· Local Linearity
· Tangent Line Approximation
· Newton’s Method*
* Although this topic is not on the AP exam, it demonstrates a practical application of the linear approximation model which I believe will reinforce the concept with students.
Unit 4: Integration (22 days)
A. Antiderivatives
· Indefinite Integrals
· Initial Conditions and Particular Solutions
· Basic Integration Rules
B. Area Under a Curve
· RAM (Rectangle Approximation Method)
· Riemann Sums
· Left sums, right sums, midpoint sums
· Definite Integrals
C. The Fundamental Theorem of Calculus
· FTC Part 1
· Numerical Integral (using fnInt on the calculator)
· FTC Part 2
· Mean Value Theorem for Integrals
· Average Value of a Function
· Integration by Substitution
· Integrating with Respect to the x and y axes
D. Trapezoidal Rule
Unit 5: Transcendental Functions (15 days)
A. Trigonometric Functions
· Differentiation
· Integration
B. Inverse Trigonometric Functions
· Differentiation
· Integration
· General Rule for Derivative of an Inverse Function
C. Exponential and Logarithmic Functions
Unit 6: Differential Equations (10 days)
A. Slope Fields
B. Separable Differentiable Equations
C. Exponential Growth and Decay
Unit 7: Applications of Integration (20 days)
A. Net Change
· Distance vs. Displacement of a Particle in Motion
· Consumption over time
B. Area of a Region Between Two Curves
· Integrating with Respect to the x and y axes
C. Volume of 3-dimensional Regions
· Disk Method
· Shell Method
· Known Cross Sections
Unit 8: Review and AP Exam Preparation (15 days)
A. Multiple Choice Practice
B. Free Response Practice
· Rubric Review
· Self-scoring using rubric
· Peer evaluation of written responses
After the AP Exam
· L’Hôpital’s Rule
· Additional Integration Methods: Parts, Improper Integrals
· Projects related to the applications of the derivative and solids of revolution
· Investigation of Conic Sections and their Calculus
Textbook
Larson and Edwards. Calculus of a Single Variable. Ninth Edition. Brooks/Cole, 2010.
Supplemental Reference Materials
Finney, Demana, Waits, and Kennedy. Calculus – Graphical, Numerical, Algebraic. Second Edition. Pearson, Prentice Hall, 2006.
AP Calculus Workshop Handbook, College Board, 2008-2009.
Apcentral.collegeboard.com
Software
Texas Instruments TI-83 Flash Debugger
Texas Instruments TI-SmartView for TI-84
Geometer Sketchpad
Calculus in Motion.
GRADING
Tests 60% of your final grade
Quizzes 40% of your final grade
**Other Projects will be graded as I deem necessary
· If you have a question or concern about your grade, you must make an appointment with me after class, during lunch, or before or after school. DO NOT ASK ME ABOUT YOUR GRADE DURING CLASS TIME.
ABSENCES
· Students will have ONE day for each day you are absent to make up the work.
· If you miss a test or quiz, you must make it up the FIRST DAY you return to school.
PLAGIARISM / ACADEMIC DISHONESTY POLICY:
· Plagiarism and academic dishonesty are serious offenses. The academic work of a student is expected to be his/her own effort. Students must give the author(s) credit for any source material used. Students who commit any act of academic dishonesty will receive a failing grade in that portion of the course work. Acts of academic dishonesty will be reported to the administration.
EXPECTATIONS AND RULES
· I have very high expectations for this class. If you perform to the best of your ability, you will succeed. I look forward to helping you in any way during the school year and will be available if you need extra help. The following are specific rules of the classroom:
o FOLLOW THE RULES OF THE SCHOOL
o Be on time! You must be in your seat and ready to work when the bell rings.
o Be prepared! Bring daily-required materials to class every day.
o Be respectful! Show respect to everyone and everything!
o The teacher dismisses the class, not the bell!
DISCIPLINE POLICY FOR RULE VIOLATION
1st Offense: Verbal warning
2nd Offense: Teacher/student conference
3rd Offense: Phone call home
4th Offense: Detention before school, after school, or during lunch
5th Offense: Referral
I have read and understand the guidelines and procedures for Mr. Pugh’s AP Calculus AB class.
Student’s name: ______Period: ______
Student’s signature: ______
Thank you! I look forward to working with you this year!