The tools of partial derivatives, the gradient, etc. Jun 15, 2016 i saw some projects simulating visual calculus techniques that simulated a presentation from a caltech professor, mamikon mnatsakanian new horizons in geometrydolciani mathematical expositions hardcover january 18, 20 by tom apostol author, mamikon mnatsakanian author. Apostol also provided academic content for an acclaimed video lecture series. New horizons in geometry dolciani mathematical expositions. The method of sweeping tangents the mathematical gazette. A visual approach to calculus problems, 2000, viewed online via caltech website, apr. Detailed instructions on implementing these ideas with various public domain, shareware and commercial software packages are provided. While an undergraduate at caltech in 1959, mamikon mnatsakanian devised a. Visual calculus by mamikon mnatsakanian is an approach to solving a variety of integral calculus problems. Worried about a required collegelevel or ap calculus class. How to arrive at 2670777 as an approximation to the square root of 2 and other rational approximations to other square roots, of course from first principles, using nothing fancy like continued fractions. Many problems which would otherwise seem quite difficult yield to the method with hardly a line of calculation, what martin gardner calls aha. The math forums internet math library is a comprehensive catalog of web sites and web pages relating to the study of mathematics.
Subtangentsan aid to visual calculus semantic scholar. Various plugins are needed to view some of the pages. Visual calculus by mamikon we will playfully discover amazing properties of natural shapes and their, mostly unknown, dynamic relationships. Lately i have been particularly interested in the geowall, a system which uses polarized filters to display stereo images in three dimensions. Kumaresan, department of mathematics and statistics, university of hyderabad, po. Mamikon s method easily and naturally prompts us to childish generalizations of these classical figures, with new features, amazing relationships. Short descriptions and examples for limits, derivatives, and integrals. Download it once and read it on your kindle device, pc, phones or tablets. It allows students to more readily visualize volumes and areas. Professor tom apostle, who taught calculus at cal tech for almost 50 years said, i was stunned to learn that many standard problems in calculus can be easily solved by an innovative visual approach that makes no use of formulas. Jan 30, 2016 multivariable calculus i took this course as a freshman in my chemical engineering program. Being able to describe the region, the upper and lower limits of integration.
As an undergraduate he specialized in the development of geometric methods for solving calculus problems by a visual. Now, let us go ahead and start talking about the other coordinate system. Maplesoft is a leading developer of interactive mathematical software such as maple. Mamikon mnatsakanian public figure, local business. For more information visit visual calculus by mamikon or read m. This really should be called visual calculus, but that would be a less intriguing title. Thoughtprovoking shockwave flash animations of geometric constructions, graphs, games, and much more. Mnatsakanian s 48 research works with 205 citations and 1,594 reads, including. As long as you understand, again, this is the important part. Mamikons method for the area of the cycloid wolfram. Keplers second law of motion is that the vector from a central sun to a moving planet sweeps out equal areas in equal times. Differentiation with respect to a parameter joseph wiener. He helped popularize the visual calculus devised by mamikon mnatsakanian with whom he also wrote a number of papers, many of which appeared in the american mathematical monthly.
In 1959, mamikon mnatsakanian, usually known as mamikon, devised an original method for solving problems in geometry. Now armenian mathematician mamikon mnatsakanian has devised a sweepingtangent theorem that accomplishes the same proof using intuition. In this workshop, we will see how mamikon s sweepingtangent theorem can be used to evaluate a wide range of integrals without relying on the first fundamental theorem of calculus. Throughout this course, simple examples and easytounderstand graphs and figures are presented to explore all of the main concepts of both differential and integral calculus. You can search for animation software tutorials on youtube to learn how to make videos. Mnatsakanians 48 research works with 205 citations and 1619 reads, including. The method was conceived in 1959 by a young undergraduate student at yerevan university in armenia named mamikon a. A new areal proof of keplers second law of planetary motion is presented, based on mamikon s sweepingtangent theorem.
The collaborative work of tom apostol and mamikon mnatsakanian has been lauded for its clarity and originality. The types of courses that a student should take prior to calculus vary according to whether the student is taking calculus in high school or in college. Revising mamikon s 195961 manuscript calculus without calculations. We will use specially designed handson activities and manipulatives, interactive computer games. These are very useful in practice, and to a large extent this is why people study multivariable calculus.
Traditional math education software has tended to be built using closed systems with a bare minimum of computation capabilities. Apostol and mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Jmulti was originally designed as a tool for certain econometric procedures in time series analysis that are especially difficult to use and that are not available in other packages, like impulse response analysis with bootstrapped confidence intervals for varvec modelling. Its suite of products harnesses the power of mathematics to provide industry, government and academia with technical productivity tools.
He is the coauthor of a recently published geometry work that provides fresh and powerful insights to that field 5. How can the area under a curve be calculated without using. Here are solutions to some problems from the 2007 and 2008 exams which i think might be most useful. Math softwarefor school, university teachers and students. Officially called calculus ii, the course description is as follows. Mamikon a visual approach to calculus problems california. Citeseerx document details isaac councill, lee giles, pradeep teregowda. But with its immensely powerful computational engine and unique interface creation technology, mathematica redefines what is possible, and makes it realistic even for individual students and educators to create rich.
Mamikon mnatsakanian is a former soviet doctor of sciences in. You need to have a strong foundation in algebra and pre calculus. Visualization of 3d mathematics university of minnesota. Visual calculus development and tools springerlink. Apostol and mamikon provide fresh and powerful insights into geometry that requires only a. New horizons in geometry dolciani mathematical expositions kindle edition by tom apostol, mamikon mnatsakanian. An aid to visual calculus, generalized cyclogons, sums of squares of distances, the generalized problem of counterfeit coins, and more. With the help of a few mathematicians in sacramento and at uc davis, he was a visual approach to calculus problems by tom m.
In this demonstration, an example is presented based on work by ujjwal rane 2, where the area of the cycloid is obtained without recourse to the methods of calculus. The most interesting software are pspp, qtiplot, r, mathcast, rstudio. It serves as a capstone to an amazing collaboration. Statistics software to implement data analysis, multivariate statistical analysis and 2d, 3d, 4d data visualization. Mnatsakanian was professor of astrophysics at yerevan state university and director of the mathematical modeling center of physical processes, armenian academy of sciences. Fit data with linear, nonlinear curve models, surface models, volume models and polynomial models. The arch encloses an area three times that of the circle, a result commonly proven using calculus. His richly illustrated conference talk brought archimedes. Visual calculus is an easytouse calculus grapher for graphing limit, derivative function, integral, 3d vector, double integral, triple integral, series, ode etc. Surprisingly, the area can be obtained more easily without calculus, regardless of the bikes path, using a dynamic visual approach called the method of sweeping tangents that does not require equations for the curves. Everyday low prices and free delivery on eligible orders. Mnatsakanian, the astrophysicist, sounds like a clever and interesting fellow. Mnatsakanians research works california institute of. New horizons in geometry represents the fruits of 15 years of work in geometry by a remarkable team of prizewinning authorstom apostol and mamikon mnatsakanian.
Visual calculus by mamikon mnatsakanian known as mamikon is an approach to solving a variety of integral calculus problems. What does the area under a curve represent, exactly. Mamikon came to california about a decade ago to work on an earthquakepreparedness program for armenia, and when the soviet government collapsed, he was stranded in the united states without a visa. Applications of multivariable derivatives khan academy. Mnatsakanian has developed various techniques for solving integration problems visually. New horizons in geometry dolciani mathematical expositions ebook. In this volume the authors present an impressive collection of geometric results that reveal surprising connections between lengths, areas and volumes in various shapes, and allow one to compute difficult integrals, all using intuitive visual calculations. Now armenian mathematician mamikon mnatsakanian has devised a sweepingtangent theorem. I also saw cool applications of teachers teaching trigonometry. Other than the obvious visual space of the graph, it usually means how much do we have after some time period. While an undergraduate at caltech in 1959, mamikon mnatsakanian devised a novel solution to this problem dubbed visual calculus, which led to another proof of the pythagorean theorem. Mnatsakanian in 1, but this paper seems to have escaped notice, probably because it appeared in russian in an armenian journal. Visualization of 3d mathematics an ongoing project to provide visualization tools to students in multivariable calculus has led to a broader interest in the visualization of mathematics. Visual calculus academic dictionaries and encyclopedias.
Nov 21, 2014 as a circle rolls along a line, a point on its circumference traces an arch called a cycloid. This really should be called visual calculus, but that would be a less intriguing title, though more accurate. New horizons in geometry by tom apostol and mamikon mnatsakanian. I saw some projects simulating visual calculus techniques that simulated a presentation from a caltech professor, mamikon mnatsakanian new horizons in geometrydolciani mathematical expositions hardcover january 18, 20 by tom apostol author, mamikon mnatsakanian author. This is an application developed for an ap calculus course. If you have access to dynamic geometry software, you can illustrate the. Areas and volumes of many classical regions can be determined by an intuitive geometric method that does not require integral calculus or even equations for the boundaries. Free software for windows can be found in the mathematics category. New insight into cycloidal areas mathematical association of. Chris rorres e ditor archimedes in the 21st century. The method and its many applications was first announced by mamikon a. Celebrating 10 years of math horizons, washington, dc.
New horizons in geometry represents the fruits of 15 years of work in geometry by a remarkable team of prizewinning authors tom apostol and mamikon mnatsakanian. Visual calculus, invented by mamikon mnatsakanian known as mamikon, is an approach to solving a variety of integral calculus problems. Originally, this collection was designed for instructors to give some ideas how technology, in particular, computers, can be used in the teaching of calculus. Mamikon mnatsakanian perhaps one of the greatest living and relatively unknown scientists is mamikon mnatsakanian. Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation, often reminiscent of what martin gardner calls aha. The college mathematics journal may 2001 mathematical.
Use features like bookmarks, note taking and highlighting while reading new horizons in geometry dolciani mathematical expositions. As an undergraduate he invented visual calculus, more fully described in this book. These modules were written as an aid for teachers who wanted to use these software packages to demonstrate concepts from calculus. New horizons in geometry mathematical association of america. Start with this course, and get a graphical introduction to calculus. Wrens solution for cycloidal length fully relies on the fact that the involute of cycloid is a cycloid of the same size, as in picture, which is known from calculus, or differential geometry, prior to arc lengths, as a more basic. Mamikon, annular rings of equal area, math horizons, 5 3, 1997 pp. Mamikon mnatsakanian is the developer of visual calculus, an ingenious approach to solving many problems in geometry and integral calculus. Use of mathematical software for teaching and learning.
Curriculum vitae california institute of technology. This application allows the user to either enter their own functions, or to choose from an extensive list of textbook questions and previous ap exam questions. Mamikon s method works easily also for parts of all these figures indefinite integrals, extends to finding higher moments gravity centers etc, and even solves problems unsolved by calculus. If you have this software available, it is not really much of a problem. If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. But rather than teach calculus, i would like him to come to noosa and think about how to quantify the effects of the moon and planets on earths weather and climate.
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