Are you teaching Calculus next year?

I've been doing it for how many years now? 27 years (I think) ... so much time, so many really cool kids. I guess one thing I really like about teaching calculus is that I can't recall a bad student.

Anyway, for those who are setting up a course for next year:

Text (by author):

• It will probably be Larson or Stewart. Stewart is the more common college text, but Larson is the more HS-friendly. I personally use Larson 7th. I've taught from this book since the third edition and I feel very comfortable with it.
• Hughes-Hallet-Gleason did things in an odd organization, IMNSHO. In a tutoring situation, I had this thing and it was oddly laid out, with integration happening before the end of differentials. You might like it, but I'm too used to the Larson / Stewart / Thomas Finney organization.
• Ostebee-Zora is dense for highschool students.
• Thomas and Finney was cool ten years ago but seems to have fallen out of favor. Don't know why.

If you want to stretch your boundaries and go 21st Century on your kids, there are many online texts.  I'd love to try this and I'm planning on putting selected sections on the Moodle. WolframAlpha will be required for some assignments.

• Neat place to start is the California site: http://www.clrn.org/home/
• http://www.whitman.edu/mathematics/calculus/
• MIT OC: http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm

Additional resources:

• http://www.math.temple.edu/~cow/
• http://archives.math.utk.edu/visual.calculus/
• and of course, Salman Khan on YouTube.

VERY IMPORTANT
AP insists you fulfill their audit process - mostly they need a syllabus, but you have to complete it if your school wants to put the "AP" label on it. Someone at your school may have already fulfilled that requirement -- make sure to check. Soon.  Here's my syllabus if you want to copy it.

The Summer Before
If you can, and can afford it or if your school with cover costs, hit up a summer AP seminar. The Summer AP Institute at St. Johnsbury VT is one of the best.
http://www.stjacademy.org/page.cfm?p=278
http://www.stjacademy.org/uploaded/documents/AP_Institute/AP_Institute_2011.pdf

The environment is as close as you can get to smart-people nirvana. One week surrounded by a bunch of AP teachers in an informal setting in the Vermont summer. Learn, work, relax, eat REALLY good food, play bocce on the lawn and drink free beer with a slew of brilliant people till dinner. Like you can get better than this? It's $1100 but worth it ... especially if your school will pony up. Skip the dorm and use the savings and a little extra of yours and take your wife -- a hotel isn't much more and it's a wonderful town.

Yeah, what the Hell, TI?

(from Randall at XKCD) And the marginally colored TI-Inspire doesn't get you off the hook, either.

Technology

• Every kid should have a TI-84 or the equivalent.  I don't require them to buy one but I'll warn them that the ones available in the room for borrowing are often changed and reset by the algebra I kids, making it difficult for the occasional user to get started. "Radians or degrees", language set to Deutsch, etc.  There is a new wrinkle, though:  The graphing calculator app for iPhone and Android devices.  It's got a better screen resolution and it only costs $3 for the phone they already own. 
• Excel is useful - Google apps works, too. Just be ready to reserve time occasionally in the computer room.
• Pencil and paper

To be continued.

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.

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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.

Limits

• 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.

Go read something.

Go read Robert Talbert's comments on Calculus books, calculus reform and the best classes for freshman.