The following sections of this BookRags Literature Study Guide is offprint from Gale's For Students Series: Presenting Analysis, Context, and Criticism on Commonly Studied Works: Introduction, Author Biography, Plot Summary, Characters, Themes, Style, Historical Context, Critical Overview, Criticism and Critical Essays, Media Adaptations, Topics for Further Study, Compare & Contrast, What Do I Read Next?, For Further Study, and Sources.
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The following sections, if they exist, are offprint from Beacham's Encyclopedia of Popular Fiction: "Social Concerns", "Thematic Overview", "Techniques", "Literary Precedents", "Key Questions", "Related Titles", "Adaptations", "Related Web Sites". (c)1994-2005, by Walton Beacham.
The following sections, if they exist, are offprint from Beacham's Guide to Literature for Young Adults: "About the Author", "Overview", "Setting", "Literary Qualities", "Social Sensitivity", "Topics for Discussion", "Ideas for Reports and Papers". (c)1994-2005, by Walton Beacham.
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One of the surprises of calculus is the fact that the derivatives and integrals of some functions look very different from the original functions. This is usually the case for inverse trigonometric and logarithm functions. These functions are "transcendental," in the sense that they cannot be written as polynomials, quotients of polynomials, or roots of polynomials. Yet their derivatives are rational functions or roots of rational functions. The simplest illustrations are:
A common feature of all of these examples is the fact that the transcendental function is the inverse of a function that satisfies a simple first-order differential equation. The natural logarithm, for example, is the inverse of the exponential function, which satisfies the equation dy/dx = y. In each case, the formula for the derivative of the inverse function follows easily from the corresponding differential equation and from the inverse function theorem.
For the student of calculus, it is also important to realize that each of these derivative formulas leads to a corresponding integral formula, and hence that the integral of an innocuous-appearing rational function may be a transcendental function.