Web Links (compiled by Chris Wendl)

The web contains a plethora of interesting resources to help you visaulize important mathematical concepts -- the following is a selection of some of my favorites that pertain to 18.01 (and are also fun to play with). The list may be updated as I find more.

Most of these are Java applets, which run interactively as plug-ins in your web browser. To use them you must have Java enabled in your browser preferences -- by default, it probably is already.

• Some Java applets from MIT's OpenCourseWare page for 18.013A:
  • Operations on functions
  • Trigonometric functions
  • Slope of a line
  • Derivative and tangent line
  • Constant, linear, quadratic and cubic approximations
  • Numerical integration
  • Polar plotter
  • Curves in two dimensions
• Function and Derivative Animations, by Przemyslaw Bogacki and Gordon Melrose (.avi files, can play e.g. with Windows Media Player)
• Java applets of secant lines and tangent lines (from IES, Manipula Math):
  • a smooth function
  • one-sided derivatives at a point of non-differentiability
• Secant lines for a function with two non-differentiable points (applet by Daniel J. Heath)
• Animations of secant lines approaching (or not approaching) tangent lines (by Douglas N. Arnold):
  • at a point of differentiability
  • at a point of non-differentiability (the one-sided derivatives don't match)
  • at another point of non-differentiability (vertical tangent; the derivative is infinite)
• Constructing functions that are continuous but nowhere differentiable(!), applet from Maths Online
• Chain rule applet (from a multimedia calculus course by Scott Sarra)
• First and second derivatives applet (by Scott Sarra)
• More first and second derivatives, with parameters you can tweak (applet from Maths Online)
• Derivatives of a^x, sin x, cos x (applets by Daniel J. Heath)
• Converging to the number e (applet from IES, Manipula Math); note that the simulation doesn't let you go far enough to approach that close to e
• Zooming in on a tangent line (animation by Douglas N. Arnold)
• Linear approximation of sin x at 0 (applet from UBC Calculus Online)
• Finding a function's extremum, applet from Maths Online
• Rolle's theorem and the mean value theorem (applet from IES, Manipula Math)
• Some nice integral applets (by Daniel J. Heath):
  • Riemann sums
  • Definite integrals and the integral function 1
  • Definite integrals and the integral function 2
• Numerical Integration Simulation (by Joseph L. Zachary)
• Some applets on volumes of solids (from IES, Manipula Math):
  • Solid 1
  • Solid 2
  • Solid of revolution 1
• Direction field applet (by Scott Sarra)
• More direction field applets (from UBC Calculus Online)
• Yet another direction field applet (from IES, Manipula Math)
• Parametric equation applet (by Scott Sarra)
• Another parametric equation applet (from IES, Manipula Math)
• Cycloid animation (by Przemyslaw Bogacki and Gordon Melrose)
• Cycloid applet (from Maths Online)
• Computing arc length (animation by Przemyslaw Bogacki and Gordon Melrose)
• Approximating arc length (applet by Daniel J. Heath)
• Polar curve applet (from IES, Manipula Math)
• Several polar curve animations (by Przemyslaw Bogacki and Gordon Melrose):
  • limacons:
    • r = 1 + sin t
    • r = 3 - 2 cos t
    • r = 1 + 2 cos t
    • r = 3 - sin t
  • rose curves:
    • r = 3 cos(3t)
    • r = 5 sin(2t)
  • spiral: r = t
  • circle: r = cos t
• Converging and diverging series animations (by Przemyslaw Bogacki and Gordon Melrose):
  • 1 / N^(1/2)
  • 1 / N
  • 1 / N²
  • 1 / N¹⁵
• Numerical tools for computing sequences / series (from Maths Online)
• Taylor approximations (applet by Daniel J. Heath)
• Power series grapher

And just for the hell of it, there's the Quadratic Formula Song.