Saturday, March 26, 2011

Calculus Syllabus for the AP Audit.

This was enough to satisfy the review process wonks.  Feel free to use it verbatim as I can't claim to have made much up on my own. It is mostly a regurgitation of the table of contents.

I certainly don't feel that anyone should spend an iota of psychic energy on the syllabus - spend it making plans and working toward the best you can do for your students.

Case in point: "Definition by limit: epsilon-delta." was de-emphasized in AP years ago, is usually ignored by most college professors for MAT121 but emphasized by the Audit review process. I couldn't get the audit accomplished without lip service to it.

A change from this syllabus for me: I am changing from a school-hosted ePortfolio to a Moodle, blog and Wiki based on my personal server. There are several places to find free hosting for those. Additionally, I will be requiring certain homework to be done on WolframAlpha since no one can afford Mathematica.  I'd love to require an iPad but I'll have to win the lottery first!

Good luck in your setup.

Calculus Syllabus

Instructional Materials
  • Textbook: Calculus, 7th edition by Larson, Hostetler, Edwards; Houghton – Mifflin Publishing Company.
  • Technology: TI-83 / TI 84 graphing Calculator 
While materials can be borrowed from the school, it is highly recommended that students purchase both the book and the calculator. The book will be useful for many years and the utility and convenience of writing notes in the margins and highlighting passages cannot be minimized.
  • When appropriate, the class will make use of Geometer's Sketch Pad, excel, Google apps.
    • a gmail account would be most helpful.
  • E-Portfolios will be a part of the Course. See ____ for account and access information. Work will be done in groups, handed in separately (and included here). Typical Contents:
    • Your two sets of school-wide showcase materials
    • Chapter projects/labs assigned monthly.
    • Past years FRQ questions worked out and scored by each other.
    • Inter-curricular questions.

Introduction: What is Calculus About?

Functions Review
  • I. Sign Graph solutions
  • II. Coordinates, Slope
  • III. Linear Equations: [point slope, slope intercept, general]
    • A. Formulae [ distance, angle, perpendicular, bisectors]
  • IV. Polynomial functions – library of Functions
    • A. Domain, range, discontinuities, roots, synthetic division
    • B. Odd, even, Symmetry
    • C. composite functions g(f(x)), (g*f)(x), (gof)(x)
    • D. Slope = derivatives, differentiability,
    • E. "How to say it in Math." Introduction of proof techniques and format.
  • I. x =>a, one sided limits at discontinuity, peak, break in curve, inf. disc, etc.
  • II. [ graphically, algebraically, numerically]
  • III. definition of derivative as lim h =>0, delta, epsilon (refer back to "More or Less")
  • IV. Limits : add sub / mult / div
  • V. Infinite Limits
  • VI. Continuity, max-min theorem, Intermediate Value Theorem, Computer/calculator-aided exploration of situations in which lim f (x) x a D.N.E.
Derivatives of Polynomial Functions
  • I. Definition by limit: epsilon-delta.
  • II. Computer-aided exploration:
    • A. Using the definition of the derivative as a limit of slope
    • B. secants to plot f'(x)vs x. The relationship of the graph of f'(x)to f(x).
  • III. Power Rule, Constant Rule, Sum Rule, Product Rule, PowerRule2, Quotient Rule
  • IV. Implicit Differentiation
  • V. Chain Rule: composite functions
  • VI. "Explain" "Show that" "Why or Why Not" – how to justify an answer. Includes review of past free-response questions, and scoring methods, practice developing an appropriate answer with explanations. presentations by students
  • VII. Trigonometry
    • A. one-day review of basic functions, then Derivatives of trig functions
  • VIII. Linear
    • A. Tangents and Normals, Linear Approximations and Differentials – ZOOM and compare – analytic results vs. calc – relate back to epsilon/delta
    • B. Newton’s Method – writing a recursive program
    • C. Calculator computation of derivatives and graphs of derivatives.

Derivative Applications
  • I. Sign Graph analysis recap
  • II. Graphing
    • A. First derivative: increase, decrease,
    • B. Second derivative: concavity
  • III. Symmetry
  • IV. Asymptotes, limits, dominant terms, derivative of hyperbola vs parabola: behavior at extremes
  • V. Extrema Theory: Relative and Absolute Maxima and Minima;
    • A. 1st derivative test and the Extreme Value Theorem; Intervals,
    • B. Continuity;
      • 1. Why are the notions of continuity and limit crucial to the Calculus?
      • 2. Informal discussion of "function niceness."
      • 3. Definition of continuity theoretically compared to on the TI.
    • C. 2nd derivative test for maxima and minima
  • VI. Recap of "Explain" "Show that" "Why or Why Not" – justify an answer. "How does one know f(c) is minimum?" 1999 FRQ BC:4 Student Sample A and others.
  • VII. Maxima and minima: Problems
  • VIII. Related rates of change, physics, chemistry, Real Problems, dist-vel-acc-jerk, best solutions. Include here the instantaneous numerical calculations.
  • IX. Rolle's Theorem, Mean Value Theorem, Intermediate Value Theorem
  • X. Indeterminate forms and L'Hopital's Rule
  • XI. Fall School-wide Showcase: Presentations and Explanations (similar to typical portfolio except this is a chance for students to present some of their work to parents, faculty and members of the community.)
Indefinite Integrals
  • I. Indefinite Integrals - Primitives (antiderivatives) differential equations, Separation of variables, basic integration formulae, constant of integration.
  • II. Finding the constant of integration (C)
  • III. Substitution Method - U and dU
  • IV. Integrals of Trigonometric Functions
    • A. Basic Functions
    • B. Using Trig identities to solve integrals
  • V. The Area Under a Curve - Definite Integrals
    • A. Rectangles & Summation (Sigma notation)
    • B. Riemann Integral = Lim as number of rectangles incr. to infinity
  • VI. Calculating Areas by Summation
    • A. Rectangles
    • B. Trapezoids
  • VII. Fundamental Theorems of Calculus -
    • A. First F.T. of C. -- definition of Primitive F(x): derivative F(x) is f(x)
    • B. Second F.T. of C: definite integral is equal to F(b) - F(a)
  • VIII. Substitution
  • IX. Approximations of Definite Integrals
    • A. Trapezoidal Rule A=h/2 (1 2 2 2 2 ...... 2 1)
    • B. Simpson's Rule A= h/3 ( 1 4 2 4 .... 2 4 1 )
Integral Applications
  • I. Distance - velocity - acceleration - jerk
  • II. Areas: horizontal rectangles - dy, f(y), g(y); vertical rectangles - dx, f(x), g(x)
  • III. Volumes, by revolution
    • A. Creation: use potter’s wheel as example; zucchini-carving to show slicing; cardboard discs, wood tiles. Group–think: Given a lathe template for vase, find the volume
    • B. Volumes of revolution around x-axis and y-axis using Discs, Washers, Shells
  • IV. Known Cross Section
  • V. Average Value, average rate of change
  • VI. ** Optional extensions of integral theory
    • A. Length of Plane curves, Area of Surface of Revolution
    • B. Physics: Moment of Inertia, Center of mass, Centroids, Nonuniform mass distribution, Work done by variable forces (concurrent physics preferred)
Transcendental Functions
  • I. Inverse Functions and their derivatives
  • II. Inverse Trig functions
  • III. Derivatives of the Inverse trig functions and associated integrals
  • IV. Natural Log and Derivative
    • A. INT (1/cabin) d(cabin) = log (cabin) + C = houseboat
  • V. Exponential and logarithmic functions
    • A. derivatives and integrals
    • B. Applications [ rates of growth, decay, compound interest, Richter scale]
  • VI. Spring School-wide Showcase
Integration Methodology
  • I. Basic Forms
  • II. Integration by parts [Berger method]
  • III. Trig Functions, odd powers and even powers
  • IV. Trig Substitutions
  • V. Integrals involving .....
  • VI. Partial Fractions [Alfonso’s Method, Matrix solutions, Millitello cover]
  • VII. Numerical integration
** Post – Test Optional
  • I. Conic Sections – Definitions and uses of calculus – areas, etc.
  • II. Hyperbolic Functions –
    • A. Definitions, Derivatives and integrals
    • B. Cables, Catenaries (Gateway Arch and Power lines), and the suspension bridge.
    • C. Inverse Hyperbolics
  • III. Polar Coordinates
    • A. Graphing and common figures,
    • B. Derivatives and integrals in Polar coordinates
  • II. Infinite Sequences and series
    • A. e, sin, cos, pi expansions
    • B. Taylor, MacLaurin etc.

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