Related Topics:

Mamikon even takes this simple observation about curves to establish a new relationship between the tractrix and exponential curves (Apostol & Mamikon 2002).
Mamikon's visual understanding and explanation of calculus is not limited to two-diemnsional curves, nor does he concern himself only with new insights into mathematical relationships. In another paper, again published with Apostol, Mamikon established new proofs for Archimedes' discoveries concerning polyhedrons and their circumscribing prisms (Apostol & Mamikon 2004). Again, his explanation abounds with visual examples, clearly shaded in various tones to correlate areas and volumes for an easy understanding of the relationships Mamikon is describing. The mathematical formula are present too, of course, but they are far more easily understood for most students when accompanied with visual examples.

In sharing these and other visual learning techniques with students, I would start (as Mamikon does) with examples familiar to their daily lives -- the curve made...

Handouts, prepared in advance, would illustrate these activities along with more diagrammatic illustrations. Once the relationship of the tangent to the area of a curve is established (for example), its applications in more strictly mathematical settings (e.g. measuring the area of a graphed curve) can be examined. For less visual thinkers, mathematical formulas could be included on each page. This will help to establish and reaffirm the relationship between illustration and formula for all students.

References

Apostol, T. & Mamikon, M. (2002). "Subtangents -- An Aid to Visual Calculus." The American Mathematical Monthly, 109(6), pp. 525-33.

Apostol, T. & Mamikon, M. (2004). "A Fresh Look at the Method of Archimedes." The American Mathematical Monthly, 11(6), pp. 496-508.

Mamikon. (2000) "Bicycle Puzzle." Visual Calculus by Mamikon. CalTehc ITS website. Accessed 7 July 2009. http://www.its.caltech.edu/~mamikon/calculus.html

Apostol, T. & Mamikon, M. (2002). "Subtangents -- An Aid to Visual Calculus." The American Mathematical Monthly, 109(6), pp. 525-33.

Apostol, T. & Mamikon, M. (2004). "A Fresh Look at the Method of Archimedes." The American Mathematical Monthly, 11(6), pp. 496-508.

Mamikon. (2000) "Bicycle Puzzle." Visual Calculus by Mamikon. CalTehc ITS website. Accessed 7 July 2009. http://www.its.caltech.edu/~mamikon/calculus.html

PAGES 3
WORDS 824

Being able to "crunch the numbers" is an essential part of the manager's role. Too often managers feel uncomfortable working with numbers because of their limited mathematical background. This reduces their usefulness, however. Strong managers are not intimidated by the numbers, but rather view them as an essential component of the job. Therefore, part of the process of studying business management is to build the set of tools that

PAGES 3
WORDS 1264

(Hilton, 26) in general, no mathematician would be willing to accept the solution to a problem without some sort of proof, and in the same way, no student of calculus would be ready to accept the resolution of a problem without the necessary proof. (Cadena; Travis; Norman, 77) It must be stated that Newton's mathematics that involved 'fluxions' was one of the first forms of the area defined as 'differential

PAGES 5
WORDS 1309

Calculus and Definitions of Its Concepts Indefinite integration Indefinite integration is the act of reversing any process of differentiation. It is the process of obtaining a function from its derivative. It is also called anti-derivative of f. A function F. is an anti-derivative of f on an interval I, if F'(x) = f (x) for all x in I. A function of F (x) for which F'(x)=f (x), this means that for

PAGES 10
WORDS 3205

Nevertheless, an individual may prefer to have this type of calculus removed for other reasons or otherwise as part of a long-term treatment regimen. For example, Bennett and Mccrochan note that, "When the American Dental Association later approved Warner-Lambert's mouthwash, Listerine, by stating that 'Listerine Antiseptic has been shown to help prevent and reduce supragingival plaque accumulation and gingivitis. . ., ' sales rose significantly" (1993:398). It remains unclear,

PAGES 2
WORDS 452

The semi-minor and semi-major axis are easily determined, and can then be subbed into the standard equation for an ellipse. Taking the square root of y will result in a plus/minus, and discarding the minus erases the lower half of the ellipse. The long axis extends horizontally, and the short axis extends vertically. The x and y axis bisects the ellipse already, so both a and B. are available:

PAGES 2
WORDS 688

Derivatives and Definite Integrals Word Count (excluding title and works cited page): 628 Calculus pioneers of the seventeenth century such as Leibniz, Newton, Barrow, Fermat, Pascal, Cavelieri, and Wallis sought to find solutions to puzzling mathematical problems. Specifically, they expressed the functions for derivatives and definite integrals. Their areas of interest involved discussions on tangents, velocity and acceleration, maximums and minimums, and area. This introductory paper shall briefly introduce four specific questions