  # Interesting Aspects of Calculus

## Video Lectures

Displaying all 20 video lectures.
 Lecture 1 Play Video The Fundamental Theorem of CalculusIn 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. 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 Play Video Leibniz's Quest for PiIn 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 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 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 Play Video Wallis' Product I: Integrating sin Raised to the nth PowerWe 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 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 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 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 Play Video In Search for the OrbitThis 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 IBefore proving Kepler's Laws, we need to formulate some preliminaries namely setting up the coordinate axis. Lecture 12 Play Video Kepler's Laws: Preliminaries IIBefore proving Kepler's Laws, we need to formulate some preliminaries namely setting up the coordinate axis. Lecture 13 Play Video Kepler's First Law IIn 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 LawIn 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 LawLee 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 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: DerivativesDerivatives of hyperbolic functions leading up to the problem of the catenary. Lecture 20 Play Video The Catenary ProblemIn this video, Lee explains the classical problem of the catenary. This problem requires knowledge of hyperbolic functions to solve the integration part.