Dans' Mathematica

Finite mathematics & applied calculus: everything! Wolfram Mathworld The Principles of Mathematics - Bertrand Russell MIT Open Courseware Khan Academy Wolfram Alpha College Algebra (vids) - Prof Richard Delaware Calculus I (vids) - Prof Richard Delaware Highlights of Calculus (vids) Calculus - Open Courseware Consortium

Paul's Online Notes

"Functions are fundamental to the study of calculus, and here we review what they are, their graphs, how they are combined and transformed, and various ways they are classified. A function can be represented by an equation, by a numerical table, by a graph, or by verbal descriptions. The graph of a function is a particularly useful visualization of its features and overall behavior, and we review several ways for obtaining a graph, including the use of graphing calculators and computer graphing s/w. We look at the main types of functions that occur in calculus, with special emphasis (in this chapter) on the exponential functions and their inverses, the logarithmic functions. Trig functions are summarized in App B, along with several other basic topics, including the real number system, Cartesian coordinates in the plane, straight lines, parabolas, and circles." (George Thomas)

"The concept of a limit is a central idea that distinguishes calculus from algebra & trig. It is fundamental to finding the tangent to a curve or the velocity of an object. We develop the limit, first intuitively, and then formally. We use limits to describe the way a function f varies. Some functions vary continuously; small changes in "x" produce only small changes in f(x). Other functions can have values that jump or vary erratically. The notion of a limit gives a precise definition to distinguish between these behaviors. The geometric application of using limits to define the tangent to a curve leads at once to the important concept of the derivative of a function. The derivative quantifies the way a function's values change." (George Thomas)

"This chapter studies some of the important applications of derivatives. We learn how derivatives are used to find extreme values of functions, to determine and analyze the shapes of graphs, to calculate limits of fractions whose numerators and denominators both approach zero or infinity, and to find numerically where a function equals zero. We also consider the process of recovering a function from its derivative. The key to many of these accomplishments is the Mean Value Theorem , a theorem whose corollaries provide the gateway to integral calculus." (George Thomas) In calculus, the mean value theorem states, roughly, that given an arc of a smooth differentiable (continuous) curve, there is at least one point on that arc at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the secant chord connecting the endpoints of the arc. The theorem i...

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Khan Academy

They're just showing off! MATH 32500. Algebra I. 100 Units. MATH 32500 deals with rings, fields, algebras, ideals, maximal ideals,zero divisors and nilpotent elements, idempotents. PIDs, UFDs, Euclidan rings. Also included are the Chinese remainder theorem, PID implies UFD, polynomial rings and Gauss' lemma, spectrum of an element of an algebra, structure of finite dimensional commutative C*-algebras without nilpotent elements, group algebra of a group, duality and Fourier transform for finite abelian groups. Also included are modules: simple, semisimple, cyclic, finitely generated, and free modules. Topics may also include Schur’s lemma, Wedderburn theory, Jacobson density theorem, structure theory of finitely generated modules over PIDs, and applications to finitely generated abelian groups and to linear algebra U of Chicago Course Catalog

In calculus, the squeeze theorem (known also as the pinching theorem, the sandwich theorem, the sandwich rule and sometimes the squeeze lemma) is a theorem regarding the limit of a function. The squeeze theorem is a technical result that is very important in proofs in calculus and mathematical analysis. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Gauss. In Italy, Russia and France, the squeeze theorem is also known as the two carabinieri theorem, two militsioner theorem, two gendarmes theorem, or two policemen and a drunk theorem. The story is that if two policemen are escorting a drunk prisoner between them, and both officers go to a cell, then (regardless of the path taken, and the fact that the prisoner may be wobbling about between the policemen) t...

The Area under a function

The area shaded in red stripes can be estimated as h times ƒ(x). Alternatively, if the function A(x) were known , it could be computed as A(x + h) − A(x). These two values are approximately equal, particularly for small h. Computing the derivative of a function and “finding the area” under its curve are "opposite" operations More exactly, the theorem deals with definite integration with variable upper limit and arbitrarily selected lower limit. This particular kind of definite integration allows us to compute one of the infinitely many antiderivatives of a function (except for those which do not have a zero). Hence, it is almost equivalent to indefinite integration, defined by most authors as an operation which yields any one of the possible antiderivatives of a function, including those without a zero. This is a good starting point. Yes, it's all new but he provides the big picture, "the Forest," so to speak, not just the trees. Gets you acquainted w...