     Calculus Animations,Graphics and Lecture Notes Special Topics    Home About Kelly Liakos Sponsors Calculus 1- Limits and Derivatives Calculus 1 - The Second Derivative Calculus 1 and 3 Formula Sheets Chain Rule Computer Lab Assignments Conic Sections Differential Equations Directional Derivatives/Gradient Double Integrals Equillibrium Solutions 1st order DEs 1st order Diff Eqs -Motion Flux Integrals and Surface Integrals Infinite Sequences Infinie Series Level Curves and Level Surfaces Line Integrals Optimization and Related Rates Optimization for Functions of 2 Variables Parametric Equations 2-space Parametric Equations 3-space Partial Derivatives Polar Coordinate System Polar Coordinates- Derivatives and Integrals PreCalculus Riemann Sums and the Fundamental Theorem of Calculus 2d order Diff EQS-Motion 2d Partial Derivatives Supplemental Exercises and Solutions Tangent Planes/ Differential for f(x,y) Trigonometry Trigonometry Applications Triple Integrals Unit Tangent Vectors/Unit Normal Vectors Vectors in General Vector Valued Functions Visualizing Limits of Functions of 2 Variables Work Links Special Topics Scratch Paper 2 Poems The animations for the hypocycloids has been moved to the Parametric Equations 2 Space page for obvious reasons

Theoerm

Suppose f(x) is continuous on [0,1] and f(0) =f(1)

Using the Intermediate Value Theorem

Prove that for n > 1 there is at least one point c in [0,1-1/n]  such that

f(c+1/n) = f(c)

Proof Using IVT

The following are my notes on the application of Improper Integration applied to escape velocity and the Swarzchild Radius

Escape Velocity

A history professor asked me to develop animations to  help explain to his class the Michelson - Morley Experiment . I'll not get into a discussion of the Michelson-Morley experiment here-there are plenty of much better discussions than I could give. However, The following 2 animations are provided.

The first is what happens when light is sent through an interferometer in a labaratory at rest. Two beams are split and recombine in phase

The second is what was supposed to happen in a lab on earth as we move through the ether. The beams split and recombine out of phase.

Unfortunately,( or fortunately for science), this didn't happen. The results were the same as the lab at rest result. The same result occured regardles of the orientation of the interferometer.

The results of this "failed experiment"  had tremendous implications in modern physics including special relativity.

Lab at Rest--What happened

Lab in Motion- What was supposed to happen

In The following 3 downloads is a scheme for programming Mathcad to generate fractals from Julia Sets

Fractals

Dragon

Cloud

Generating Pythagorean Triples    The Smart Bunny-A very short story by Kurt Vonnegut Jr.  