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At IB Prep Academy, our mission is to provide an advanced education to students who aspire to better comprehend mathematics and physics, IB program , AP Calculus AB/BC, Mathematics courses for Machine Learning, Data Scientists and Math Olympiad.
02/13/2022
Calculus Volume methods: Shell Method.
Shell Method
Suppose we rotate the following area around the y-axis.
We get the volume in the following animation. The volume is generated by drawing reference rectangles parallel to the axis of revolution and spinning them around the axis to get “shells.” The shells are shown in blue in the animation.
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KEPLER’S LAWS FOR PLANETARY MOTION
THEOREMS
Kepler’s Laws of Planetary Motion
The path of any planet about the Sun is elliptical in shape, with the center of the Sun located at one focus of the ellipse (the law of ellipses).
A line drawn from the center of the Sun to the center of a planet sweeps out equal areas in equal time intervals (the law of equal areas) .
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of the lengths of their semimajor orbital axes (the law of harmonies).
Reference: https://openstax.org/books/calculus-volume-3/pages/3-4-motion-in-space
04/18/2021
How about defining an area between two functions in terms of double integration, means writing a single integration as double integration.
04/17/2021
Some of the problems are hard so be patient, Most of the inequality problems can be solved by Cauchy-Swartz Inequality.
04/17/2021
International Mathematical Olympiad 2020 Results:
Rank 1 : China
Rank 2 : Russian Federation
Rank 3 : USA Not #1 any more : Because our kids are no longer studying, they're controlled by "Love Bombs" from THE SIREN. The first and most ancient seducer of all is the Siren.
Why does Odysseus put wax in his men's ears? Refer: Homers Odyssey.
Ulysses wanted to hear the Sirens' song although he knew that doing so would render him incapable of rational thought. He put wax in his men's ears so that they could not hear and had them tie him to the mast so that he could not jump into the sea. We need someone to put wax to our kids ear to win the Math Olympiad.
Rank 4: Korea
Rank 5: Thailand
Rank 6: Tie Poland and Italy
04/17/2021
04/17/2021
Cauchy–Schwarz inequality many different forms and proofs. It is a very powerful result. Many Ph.D students used it in their thesis to get their doctorate degrees. One of my research paper based on this inequality too. You need to understand it is not just one single form, there are many variation of it in many branches of mathematics, physics and engineering. Of course in life science and finance, economics.
This article is about various forms of Cauchy–Schwarz inequality and proofs.
04/17/2021
If a²+b²+c²=16, x²+y²+z²=25 and ax+by+cz=20 then what is the value of (a+b+c)/(x+y+z)?
If a²+b²+c²=16, x²+y²+z²=36 and ax+by+cz=30 then what is the value of (a+b+c)/(x+y+z)?
How hard are these innocent looking problems? These problems can be solved by Cauchy–Schwarz inequality , still used to derive many new inequalities.
Augustin-Louis Cauchy(1789 - 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra.
A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy).Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics.
04/16/2021
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IB Prep Academy – IB Prep Academy Arduino High School Robotics Course This course focuses on learning how to teach programming, basic electronics, and additional STEM concepts using the Arduino Uno powered Parallax Shield-bot. You can earn a IB Prep Academy Robotics Academy Certification by completing various assignments throughout....
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