Citeseerx document details isaac councill, lee giles, pradeep teregowda. Mnatsakanian has developed various techniques for solving integration problems visually. Visual calculus by mamikon mnatsakanian known as mamikon is an approach to solving a variety of integral calculus problems. Mnatsakanian was professor of astrophysics at yerevan state university and director of the mathematical modeling center of physical processes, armenian academy of sciences. Various plugins are needed to view some of the pages. New horizons in geometry dolciani mathematical expositions kindle edition by tom apostol, mamikon mnatsakanian. 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. While an undergraduate at caltech in 1959, mamikon mnatsakanian devised a. 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.
Applications of multivariable derivatives khan academy. Apostol and mamikon provide fresh and powerful insights into geometry that requires only a. The arch encloses an area three times that of the circle, a result commonly proven using calculus. This is an application developed for an ap calculus course. 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. Statistics software to implement data analysis, multivariate statistical analysis and 2d, 3d, 4d data visualization. Here are solutions to some problems from the 2007 and 2008 exams which i think might be most useful. A new areal proof of keplers second law of planetary motion is presented, based on mamikon s sweepingtangent theorem. I also saw cool applications of teachers teaching trigonometry. Use features like bookmarks, note taking and highlighting while reading new horizons in geometry dolciani mathematical expositions. Its suite of products harnesses the power of mathematics to provide industry, government and academia with technical productivity tools. Mnatsakanian in 1, but this paper seems to have escaped notice, probably because it appeared in russian in an armenian journal. As an undergraduate he invented visual calculus, more fully described in this book.
Thoughtprovoking shockwave flash animations of geometric constructions, graphs, games, and much more. 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. If you have access to dynamic geometry software, you can illustrate the. Start with this course, and get a graphical introduction to calculus. Now armenian mathematician mamikon mnatsakanian has devised a sweepingtangent theorem that accomplishes the same proof using intuition. The college mathematics journal may 2001 mathematical. Revising mamikon s 195961 manuscript calculus without calculations. Visual calculus is an easytouse calculus grapher for graphing limit, derivative function, integral, 3d vector, double integral, triple integral, series, ode etc. Download it once and read it on your kindle device, pc, phones or tablets. If you have this software available, it is not really much of a problem.
Short descriptions and examples for limits, derivatives, and integrals. Officially called calculus ii, the course description is as follows. Traditional math education software has tended to be built using closed systems with a bare minimum of computation capabilities. The visual calculus project started out as a collection of modules that provided detailed instructions for the use of a variety of software packages to illustrate various concepts in the first year calculus sequence. Now, let us go ahead and start talking about the other coordinate system. Everyday low prices and free delivery on eligible orders. Visualization of 3d mathematics university of minnesota.
As an undergraduate he specialized in the development of geometric methods for solving calculus problems by a visual. If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. The collaborative work of tom apostol and mamikon mnatsakanian has been lauded for its clarity and originality. Mamikon, annular rings of equal area, math horizons, 5 3, 1997 pp. 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. It allows students to more readily visualize volumes and areas. 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. His richly illustrated conference talk brought archimedes. Now armenian mathematician mamikon mnatsakanian has devised a sweepingtangent theorem. Mamikon mnatsakanian perhaps one of the greatest living and relatively unknown scientists is mamikon mnatsakanian. Math softwarefor school, university teachers and students. Differentiation with respect to a parameter joseph wiener. For more information visit visual calculus by mamikon or read m. Jan 30, 2016 multivariable calculus i took this course as a freshman in my chemical engineering program.
Subtangentsan aid to visual calculus semantic scholar. Curriculum vitae california institute of technology. He is the coauthor of a recently published geometry work that provides fresh and powerful insights to that field 5. 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. You need to have a strong foundation in algebra and pre calculus. New insight into cycloidal areas mathematical association of. It serves as a capstone to an amazing collaboration. 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. 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. 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. Visual calculus development and tools springerlink. Being able to describe the region, the upper and lower limits of integration. The method of sweeping tangents the mathematical gazette.
The tools of partial derivatives, the gradient, etc. 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. 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. Nov 21, 2014 as a circle rolls along a line, a point on its circumference traces an arch called a cycloid. Apostol and mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Free software for windows can be found in the mathematics category. Maplesoft is a leading developer of interactive mathematical software such as maple.
Mamikon mnatsakanian public figure, local business. 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. We will use specially designed handson activities and manipulatives, interactive computer games. The method and its many applications was first announced by mamikon a. What does the area under a curve represent, exactly. With the help of a few mathematicians in sacramento and at uc davis, he was a visual approach to calculus problems by tom m. 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. 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. The math forums internet math library is a comprehensive catalog of web sites and web pages relating to the study of mathematics. Mamikon mnatsakanian is a former soviet doctor of sciences in.
This really should be called visual calculus, but that would be a less intriguing title. 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. Lately i have been particularly interested in the geowall, a system which uses polarized filters to display stereo images in three dimensions. An aid to visual calculus, generalized cyclogons, sums of squares of distances, the generalized problem of counterfeit coins, and more. How can the area under a curve be calculated without using. Visual calculus academic dictionaries and encyclopedias. Visual calculus by mamikon we will playfully discover amazing properties of natural shapes and their, mostly unknown, dynamic relationships. Buy visual calculus by jesse russell, ronald cohn isbn. Mnatsakanians research works california institute of. Detailed instructions on implementing these ideas with various public domain, shareware and commercial software packages are provided. New horizons in geometry mathematical association of america. Mamikons method for the area of the cycloid wolfram. Kumaresan, department of mathematics and statistics, university of hyderabad, po.
A visual approach to calculus problems, 2000, viewed online via caltech website, apr. The method was conceived in 1959 by a young undergraduate student at yerevan university in armenia named mamikon a. New horizons in geometry dolciani mathematical expositions. In 1959, mamikon mnatsakanian, usually known as mamikon, devised an original method for solving problems in geometry. The most interesting software are pspp, qtiplot, r, mathcast, rstudio. Use of mathematical software for teaching and learning. New horizons in geometry by tom apostol and mamikon mnatsakanian. Celebrating 10 years of math horizons, washington, dc.
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. 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. Keplers second law of motion is that the vector from a central sun to a moving planet sweeps out equal areas in equal times. 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. Visual calculus, invented by mamikon mnatsakanian known as mamikon, is an approach to solving a variety of integral calculus problems. Um ihnen ein besseres nutzererlebnis zu bieten, verwenden wir cookies. Originally, this collection was designed for instructors to give some ideas how technology, in particular, computers, can be used in the teaching of calculus.
Fit data with linear, nonlinear curve models, surface models, volume models and polynomial models. Mnatsakanians 48 research works with 205 citations and 1619 reads, including. As long as you understand, again, this is the important part. Mamikon s method easily and naturally prompts us to childish generalizations of these classical figures, with new features, amazing relationships.
Mnatsakanian s 48 research works with 205 citations and 1,594 reads, including. These are very useful in practice, and to a large extent this is why people study multivariable calculus. 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. New horizons in geometry dolciani mathematical expositions ebook.
Mamikon a visual approach to calculus problems california. Many problems which would otherwise seem quite difficult yield to the method with hardly a line of calculation, what martin gardner calls aha. Other than the obvious visual space of the graph, it usually means how much do we have after some time period. Mamikon mnatsakanian is the developer of visual calculus, an ingenious approach to solving many problems in geometry and integral calculus. 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. Worried about a required collegelevel or ap calculus class. 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. You can search for animation software tutorials on youtube to learn how to make videos. Mnatsakanian, the astrophysicist, sounds like a clever and interesting fellow.
Visual calculus by mamikon mnatsakanian is an approach to solving a variety of integral calculus problems. 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. These modules were written as an aid for teachers who wanted to use these software packages to demonstrate concepts from calculus. 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. Apostol also provided academic content for an acclaimed video lecture series. Chris rorres e ditor archimedes in the 21st century. This really should be called visual calculus, but that would be a less intriguing title, though more accurate.
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