Interesting Aspects of Calculus

Video Lectures

Displaying all 20 video lectures.
Lecture 1
The Fundamental Theorem of Calculus
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
The Fundamental Theorem of Calculus II
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.

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 3
Leibniz's Quest for Pi
Play Video
Leibniz's Quest for Pi
In this 3-part 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
Leibniz's Quest for Pi II
Play Video
Leibniz's Quest for Pi II

In this 3-part 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
Leibniz's Quest for Pi III
Play Video
Leibniz's Quest for Pi III

In this 3-part 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
Wallis' Product I: Integrating sin Raised to the nth Power
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 4-part video without sacrificing the mathematical rigour. Watch all of them for a complete proof.
Lecture 7
Wallis' Product II: Using the Reduction Formula
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.

NOTE: Lee had to make this a 4-part video without sacrificing the mathematical rigour. Watch all of them for a complete proof.

Lecture 8
Wallis' Product III: Taking the Limit
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.

NOTE: Lee had to make this a 4-part video without sacrificing the mathematical rigour. Watch all of them for a complete proof.

 

Lecture 9
Wallis' Product IV: Taking the Limit
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.

NOTE: Lee had to make this a 4-part video without sacrificing the mathematical rigour. Watch all of them for a complete proof.

 

Lecture 10
In Search for the Orbit
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
Kepler's Laws: Preliminaries I
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
Kepler's Laws: Preliminaries II
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
Kepler's First Law I
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
Kepler's First Law II
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
Kepler's Second Law
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
Kepler's Third Law
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 semi-major axis of that orbit.

Lecture 17
Hyperbolic Functions: Definitions and Graph of cosh(x)
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
Hyperbolic Functions: Graph of sinh(x)
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
Hyperbolic Functions: Derivatives
Play Video
Hyperbolic Functions: Derivatives
Derivatives of hyperbolic functions leading up to the problem of the catenary.
Lecture 20
The Catenary Problem
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.