Interesting Aspects of Calculus
Video Lectures
Lecture 1 Play Video 
The Fundamental Theorem of Calculus In this lecture, Lee explains the fundamental theorem of calculus. He believes that the understanding of this theorem has been declining and strongly feels that students should know exactly why they can find areas under the graph by the definite integral. NOTE: A(x) is one of the antiderivatives f(x) and F(x) is any antiderivatives of f(x). There are a few mistakes towards the end of the video. 
Lecture 2 Play Video 
The Fundamental Theorem of Calculus II
In this lecture, Lee continues his explanation of the fundamental theorem of calculus. He believes that the understanding of this theorem has been declining and strongly feels that students should know exactly why they can find areas under the graph by the definite integral. 
Lecture 3 Play Video 
Leibniz's Quest for Pi In this 3part video, we follow Leibniz in his pursuit to find the area of a quater circle of unit radius via integration, trigonometry and series expansion. 
Lecture 4 Play Video 
Leibniz's Quest for Pi II
In this 3part video, we follow Leibniz in his pursuit to find the area of a quater circle of unit radius via integration, trigonometry and series expansion. 
Lecture 5 Play Video 
Leibniz's Quest for Pi III
In this 3part video, we follow Leibniz in his pursuit to find the area of a quater circle of unit radius via integration, trigonometry and series expansion. 
Lecture 6 Play Video 
Wallis' Product I: Integrating sin Raised to the nth Power We shall take a journey through the mind of English mathematician John Wallis and see how he in 1656 discovered an intrinsic formula known as Wallis's Product. NOTE: Lee had to make this a 4part video without sacrificing the mathematical rigour. Watch all of them for a complete proof. 
Lecture 7 Play Video 
Wallis' Product II: Using the Reduction Formula
We shall take a journey through the mind of English mathematician John Wallis and see how he in 1656 discovered an intrinsic formula known as Wallis's Product. 
Lecture 8 Play Video 
Wallis' Product III: Taking the Limit
We shall take a journey through the mind of English mathematician John Wallis and see how he in 1656 discovered an intrinsic formula known as Wallis's Product.

Lecture 9 Play Video 
Wallis' Product IV: Taking the Limit
We shall take a journey through the mind of English mathematician John Wallis and see how he in 1656 discovered an intrinsic formula known as Wallis's Product.

Lecture 10 Play Video 
In Search for the Orbit This video is a short introduction on the development of planetary orbits by tracing the ideas of Ptolemy, Copernicus, Kepler and Newton. ERRATA: Copernicus, NOT Ptolemy, suggested that the Earth is in the centre and also published 'On the Revolution of Celestial Bodies'. 
Lecture 11 Play Video 
Kepler's Laws: Preliminaries I Before proving Kepler's Laws, we need to formulate some preliminaries namely setting up the coordinate axis. 
Lecture 12 Play Video 
Kepler's Laws: Preliminaries II Before proving Kepler's Laws, we need to formulate some preliminaries namely setting up the coordinate axis. 
Lecture 13 Play Video 
Kepler's First Law I In this video, Lee proves Kepler's First Law of planetary motion. This simple looking law actually turns out to be the hardest one to derive. The law states: A planet revolves in an elliptical orbit with the sun at one of its focus.

Lecture 14 Play Video 
Kepler's First Law II
In this video, Lee proves Kepler's First Law of planetary motion. This simple looking law actually turns out to be the hardest one to derive. The law states: A planet revolves in an elliptical orbit with the sun at one of its focus. 
Lecture 15 Play Video 
Kepler's Second Law In this video, Lee proves Kepler's Second Law: the area swept out by a line segment from the Earth to the Sun is equal at equal time intervals.

Lecture 16 Play Video 
Kepler's Third Law Lee concludes his explanation of Kepler's Laws of Planetary Motion by proving, with mathematical rigour as always, Kepler's Third Law  the square of the period of a planet moving around its elliptical orbit is proportional to the cube of the semimajor axis of that orbit.

Lecture 17 Play Video 
Hyperbolic Functions: Definitions and Graph of cosh(x) A short introduction to hyperbolic functions. Don't get mislead by their 'unpopularity' compared to trigonometric functions. Hyperbolic functions do have their uses. 
Lecture 18 Play Video 
Hyperbolic Functions: Graph of sinh(x) A short introduction to hyperbolic functions. Don't get mislead by their 'unpopularity' compared to trigonometric functions. Hyperbolic functions do have their uses. 
Lecture 19 Play Video 
Hyperbolic Functions: Derivatives Derivatives of hyperbolic functions leading up to the problem of the catenary. 
Lecture 20 Play Video 
The Catenary Problem In this video, Lee explains the classical problem of the catenary. This problem requires knowledge of hyperbolic functions to solve the integration part. 