Calculus Animations,Graphics and Lecture Notes

Special Topics

About Kelly Liakos
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
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 Applications
Triple Integrals
Unit Tangent Vectors/Unit Normal Vectors
Vectors in General
Vector Valued Functions
Visualizing Limits of Functions of 2 Variables
Special Topics
Scratch Paper
2 Poems

The animations for the hypocycloids has been moved to the Parametric Equations 2 Space page for obvious reasons


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




Generating Pythagorean Triples

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

©2008-2010 Kelly Liakos

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