On-line Math 21 Course Structure

1 Main sequence

1. Limits and Continuity

  (a) Toys:

    i. Two-piece function which allows user to re-position one half
      to make continuous, with read-out of formula.

    ii. Epsilon-delta chooser for specific functions (or choosable),
      for limit or continuity.

  (b) Limits, idea and examples. 

  (c) Methods of computing limits, limits at infinity, implications to
    curve-sketching.

  (d) Definitions of continuity, several versions, leading up to epsilon-delta.

  (e) Theory: IVT

  (f) Examples: Controls?? Margin of error, approximation errors.

2. The derivative

  (a) Toys: 

    i. Free-hand plotter that then plots slope

    ii. Fixed plot that has movable tangent line

    iii. secant-line approximation to tangent.

    iv. Animate some related rates problems

  (b) Definition as a limit

  (c) Differentiation rules -- include chain rule and inverses

  (d) Examples: Velocity/acceleration problems

  (e) Application: Linear approximation

  (f) Related-rates problems

3. The Zoo of functions and their relationship to calculus

  (a) Toys:

    i. Model EKG output as sum of trig functions

    ii. Live graphical ``fitter'' to write \cos (x) in terms of \sin (x-a) 

    iii. Exponential growth (population) modeler.

  (b) Polynomials, rational functions, algebraic functions

  (c) Trigonometric functions

    i. Finding the derivative of \sin (x)

    ii. Using trig to find derivatives of others, including inverses.

      A. Aside to trigonometric functions as integrals, definitions
        of \pi .

    iii. Why they are important; periodic phenomena

    iv. Inverse trig functions

  (d) Exponential functions

    i. Finding the derivative of e^{x}

    ii. Other exponentials, 

    iii. logarithm functions and their derivatives

      A. Aside to logarithms as integrals

    iv. Hyperbolic functions.

    v. Why they are important: 

      A. Exponential growth and decay

      B. Logarithmic response, sensations.

  (e) ``Elementary'' functions with examples of others.

4. Theory and applications

  (a) Toys:

    i. MVT ``finder'' 

    ii. Max-min locator

    iii. Animate some max-min word problems

      A. Little Red Riding Hood

      B. Box?

  (b) MVT

    i. Implications of MVT: Increasing/decreasing, anti-derivatives

  (c) Max-min problems

  (d) l'Hopital's rule

  (e) Curve sketching and interpretation.

5. The integral

  (a) Toys:

    i. F(x):=\int _{a}^{x}f(t)dt graphical output

    ii. Animator of volume problems

    iii. Riemann sum approximation integral computer

    iv. The mathematician's cornacopea.

  (b) Definition as limit of Riemann sums

    i. Summation formulas by induction and applications to integration

  (c) FTC

  (d) Rules for indefinite and definite integrals

  (e) Techniques of integration.

    i. u-substitution

    ii. parts

    iii. partial fractions.

  (f) Improper integrals.

6. Applications

  (a) Areas

  (b) Volume.

  (c) Average Value

  (d) Arclength

  (e) Work.

7. Approximate methods

  (a) Trapezoidal Rule

  (b) Simpson's Rule.

2 Review material

1. Algebra

  (a) Toys: 

    i. Moving terms around in formulas -- addition and multiplication

  (b) Factoring of polynomials

  (c) quadratic formula

    i. Aside: Cubic formula and quartic formula

    ii. Aside: Why there is no quintic formula

2. Basics on functions

  (a) What is an x?

  (b) Domain and Range (Image)

  (c) graphing.

  (d) Composition and Inverses.

3. Trigonometry

  (a) Right triangles and definitions

  (b) Unit circle and definitions

    i. radian measure

    ii. negative angles

    iii. periodicity,

  (c) Trigonometric identities

4. Exponential and logarithm functions

  (a) Bases and exponents

  (b) Differences begtween exponential and power functions

  (c) Logarithms.

  (d) Exponential growth and decay

  (e) Logarithms and senses. Decibels and lumens.

5. Hyperbolic functions and hyperbolic trigonometry

  (a) Nothing new

  (b) Analogies to trigonometry

  (c) Hyperbolas versus circles.

6. Curve sketching 

  (a) Asymptotes, intercepts

  (b) Qualitative behavior of functions.

3 Explorations

1. More on continuity and differentiability

  (a) Differentiability and continuity

  (b) Integrability and continuity

  (c) Uniform continuity

  (d) What to do when there is no formula for the derivative,

2. Sophisticated models

  (a) Logistic growth

  (b) Differential equations.

3. More integration theory

  (a) What is an integrable function

  (b) Impulses

4. Real-world applications

  (a) Simple circuits and differential equations

  (b) Basic economic models

  (c) Examples from physics.

5. History

  (a) Newton versus Leibnitz

  (b) l'Hopital, the Bernoullis, and the court mathematician

  (c) The advent of rigor and Cauchy

  (d) What did the Greeks know? How close were they to developing calculus?

  (e) Calculus from other cultures

  (f) Modern research 

    i. Measure theory

    ii. Limits and function spaces.

In addition to this, we need a comprehensive index/dictionary with
capsule definitions and links to the course, as well as strategically-placed
external links. Many examples will have changable