Advanced Placement CalculusAB

Syllabus

Angie Avery

Note from the Instructor

It is my hope that each of you want to learn as much as you can about calculus. Mathematicians have been responsible for many great developments throughout history. Much of our understanding of the universe is a direct result of the contributions of mathematicians. I hope you learn to view math as more than just numbers, variables, processes, and algorithms. I hope you learn to apply your mathematical understandings to help you create a better understanding of the mathematical nature of our lives.

Course Objectives

By successfully completing this course, you will be able to do the following

  • Work with functions represented graphically, numerically, analytically, or verbally, and understand the connections among these representations.
  • Understand the meaning of a derivative in terms of a rate of change and local linear approximation and use derivatives to solve a variety of problems.
  • Understand the meaning of the definite integral both as a limit of Riemann sums and as a net accumulation of a rate of change, and understand the relationship between the derivative and integral.
  • Represent differential equations with slope fields and solve separable differential equations analytically.
  • Model problem situations with functions, differential equations, or integrals, and communicate both orally and in written form.
  • Use technology to help solve problems, interpret results, and verify conclusions.
  • Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
  • Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

Technology Requirement

A Texas Instruments 84 Plus graphing calculator will be used in class regularly. I would recommend purchasing a TI-84 Plus if you do not already own a graphing calculator. There will be a classroom set of TI-84 Plus calculators as well as some available for extended checkout from the media center.

Calculators will be used for the following:

  • Plotting the graph of a function within an arbitrary viewing window
  • Finding the zeros of functions (solving equations numerically)
  • Numerically calculating the derivative of a function
  • Numerically calculating the value of a definite integral
  • Viewing slope fields for differential equations

Textbook

Briggs, Cochran, and Gillett. Calculus – AP Edition. First edition. Pearson, 2014.

This textbook will be our primary resource. Students will benefit from reading each section. The text contains a number of interesting explorations and I encourage students to work through these together in order to construct understanding.

A Balanced Approach

This course will emphasize a “Rule of Four” technique: analytic/algebraic, numerical, graphical, and verbal methods of representing problems. Students must value each method, understand how they are connected to a given problem, and learn how to choose the most appropriate for a given problem.

With this balanced approach, students will be expected to do the following:

  • Obtain solutions algebraically or analytically
  • Support answers graphically or numerically with technology
  • Interpret the result in the original problem context

Course Outline – Tentative Timeline

Unit 1: Limits and Continuity (3-4 weeks)

  1. Rates of Change
  2. Average Speed
  3. Instantaneous Speed
  4. Limits at a Point
  5. One-sided Limits
  6. Two-sided Limits
  7. Sandwich Theorem
  8. In order to help tie the graphical implications of the Sandwich Theorem to the analytical applications of it, students will conduct an explorations where they graph the following equations: y1=x^2, y2=-x^2, and y3=sin(1/x). They then attempt to “see” the limit as x approaches 0 of x^2*sin(1/x).
  9. Limits involving Infinity
  10. Asymptotic behavior
  11. Horizontal
  12. Vertical
  13. End behavior models
  14. Properties of Limits – algebraic analysis
  15. Visualizing Limits – graphical analysis
  16. Continuity
  17. Continuity at a point
  18. Continuous functions
  19. Discontinuous functions
  20. Removable discontinuity
  21. Table groups conduct a tabular investigation of the limit as x approaches 1 of f(x) = (x^2 – 7x – 6)/(x – 1). Then the same group conducts an analytic investigation of the same function. Students in the group will then draw conclusions based on their observations. Finally, students will use their graphing calculators to graph the function and discuss, as a class, whether or not their table’s conclusions were verified or contradicted.
  22. Jump discontinuity
  23. Infinite discontinuity
  24. Rates of Change and Tangent Lines
  25. Average rate of change
  26. Tangent line to a curve
  27. Slope of a curve – algebraically and graphically
  28. Normal line to a curve – algebraically and graphically
  29. Instantaneous rate of change

Unit 2: The Derivative (5-6 weeks)

  1. Derivative of a Function
  2. Definition of the derivative
  3. Difference Quotient
  4. Derivative at a Point
  5. Relationships between the graphs of f and f’
  6. Graphing a derivative from data
  7. One-sided derivatives
  8. Differentiability
  9. Cases where f’(x) fail to exist
  10. Local Linearity
  11. Derivatives on the calculator
  12. Symmetric difference quotient
  13. Relationship between differentiability and continuity
  14. Intermediate Value Theorem for Derivatives
  15. Rules for Differentiation
  16. First order derivatives
  17. Constant, power, sum, difference, product, and quotient Rules
  18. Higher order derivatives
  19. Applications of the Derivative
  20. Position, velocity, acceleration, and jerk
  21. Particle motion
  22. Economics
  23. Marginal Cost
  24. Marginal Revenue
  25. Marginal Profit
  26. L’Hospital’s Rule
  27. Derivatives of Trigonometric Functions
  28. Chain Rule
  1. Implicit Differentiation
  2. Differential method
  3. y’ method
  4. Derivatives of Exponential and Logarithmic Functions

Unit 3: Applications of the Derivative (5-6 weeks)

  1. Extreme Values
  2. Relative Extrema
  3. Absolute Extrema
  4. Critical Points
  5. Implications of the Derivative
  6. Rolle’s Theorem
  7. Mean Value Theorem
  8. Increasing and Decreasing functions
  9. Connecting f’ and f’’ with the graph of f(x)
  10. First Derivative Test
  11. Second Derivative Test
  12. Concavity
  13. Inflection Points
  14. Relative Maximum/Minimum
  15. Students play a matching game with laminated cards which represent functions in 4 ways: a graph of the function, a graph of the derivative, a written description of the function, and a written description of the derivative.
  16. Optimization Problems
  17. Linearization Models
  18. Local Linearization
  19. Tangent Line Approximations
  20. Differentials
  21. Related Rates

Unit 4: The Definite Integral (3-4 weeks)

  1. Approximating Area
  2. Riemann Sums
  3. LRAM
  4. RRAM
  5. MRAM
  6. Trapezoidal sums
  7. Students are given a program to input into their graphing calculators which calculates the trapezoidal sums for trapezoids of equal width. Students are encouraged to think about altering the program to calculate rectangular sums.
  8. Definite Integrals
  9. Each student is asked to draw a linear function of their choosing. Then they are asked to use known geometric methods to calculate the definite integral over a given interval. Students then share their work with their tablemates.
  10. Properties of Definite Integrals
  11. Power Rule
  12. Mean Value Theorem for Definite Integrals
  13. The Fundamental Theorem of Calculus

Unit 5: Diffential Equations and Mathematical Modeling (4 weeks)

A. Slope Fields

  1. Antiderivatives
  2. Indefinite Integrals
  3. Power Formula
  4. Trigonometric Formulas
  5. Exponential and Logarithmic Formulas
  6. Separable Differential Equations
  7. Growth and Decay
  8. Slope Fields
  9. General Differential Equations
  10. Newton’s Law of Cooling
  11. Logistic Growth

Unit 6: Applications of Definite Integrals (3 weeks)

  1. Integral as net change
  2. Calculating distance traveled
  3. Consumption over time
  4. Net change from data
  5. Area Between Curves
  6. Area between a curve and an axis
  7. Integrating with respect to x
  8. Integrating with respect to y
  9. Area between intersecting curves
  10. Integrating with respect to x
  11. Integrating with respect to y
  12. Calculating Volume
  13. Cross Sections
  14. Disc Method
  15. Shell Method

Unit 7: Review/Test Preparation (4 weeks)

  1. Test taking strategies
  2. Multiple Choice
  3. Free Response
  1. Practice
  2. Multiple Choice
  3. Free Response
  4. Individual and Group practice
  5. Review rubrics for complete answers
  6. Collaborate to formulate responses
  7. Individual written responses

Unit 8: After the AP Exam (4-5 weeks) – scores will be treated as quizzes

  1. Projects designed to incorporate course curriculum
  2. Research projects on the historical development of mathematics
  3. Advanced Integration
  4. College placement exams/ College Math Requirements

Grading

Grades will include, but not limited to, the following: tests, quizzes, in-class activities, homework assignments, projects and classroom participation.

  • Homework (10%) – assigned almost daily and is due by the beginning of the next class period. Assignments will be completed and graded using

Homework Rubric – Homework will be graded by the percentage correct on each assignment. All problems can be reattempted for full credit.

Score / 4 / 3 / 2 / 1 / 0
Percentage Correct / 80-100 / 60-80 / 40-60 / 20-40 / 0-20 or NHI

Students will be given 2 free late passes (I will keep track of them) to turn in homework assignments late. Any other late assignments will have 1 point deducted for the score earned. All late assignments need to be turned in by the Chapter exam.

  • Quizzes (25%)–As long as we are covering new material, there will frequent quizzes, typically on Monday. Quiz questions may be selected directly from problems covered in class (warm-ups, notes, homework). The questions can come from any chapter, not just the chapter we are currently studying. Quizzes may be timed. If you are absent for a quiz, you will take a similar quiz the day you return to class. If the questions are not directly from questions covered in class, you will have the opportunity to reflect for up to ½ credit back (see reflection process below). If questions come directly from questions covered in class there will be no retakes or reflections. The final project after the AP exam will also count as a quiz. It will have a large impact on your grade as we will have fewer quizzes during the 2nd semester.
  • Tests (65%)– Tests will be divided into AP concepts covered in a particular chapter. This means that when you receive your results to a test, you will be able to identify how you performed on each concept separately. If you do not score well on a particular section, you may retake that section. Prior to retake, you will be requiredto complete a reflection on the entire section. The reflection needs to be stapled to the front of your test and will serve as your ticket into retake. No reflection, no retake. I will set a time frame for retakes (usually around 2 weeks). I will expect all students that receive less than a 70% to retake.

Reflection Process – Complete on a separate piece of paper and staple to the front of your original quiz. Use complete sentences and be specific (you will not receive full credit unless you do).

  1. Identify the concept missed for each incorrect problem.
  2. Explain what you misunderstood about the concept or what was incorrect with your original work.
  3. Redo the problem completely, showing ALL work.

Grading Scale

A: 93-100 A-: 90-92 B+: 87-89 B: 83-86 B-: 80-82 C+: 77-79 C: 73-76 C-: 70-72 D+: 67-69 D: 60-66 F: below 60

Important Dates

  • Tentative AP Review Study Sessions – These dates are subject to change. Sessions will be offered on Saturday mornings from 9-noon. If there is little interest, sessions may be canceled.

March: 4, 11, 18, 25 April: 15, 29May: 6

  • Mandatory AP Final Exam – As this is the final exam for our class, all students are expected to make arrangements to attend the scheduled exam.

Friday, April 21st from 12:30-4:30

  • 2017 AP Exam Date

Tuesday, May 9th