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.
(c)1998-2002; (c)2002 by Gale. Gale is an imprint of The Gale Group, Inc., a division of Thomson Learning, Inc. Gale and Design and Thomson Learning are trademarks used herein under license.
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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Rolle's theorem implies that if we have a function f that is continuous on an interval [a, b] and f(a) = f(b), then there always exists at least one critical point of the function in (a, b). The theorem is usually written symbolically as: Assume f is a function that is continuous on [a, b] and differentiable on (a, b). If f(a) = f(b) then there is a number c in (a, b) such that the derivative of f at c, f'(c) = 0. From Rolle's theorem it follows that c is at least one critical point of f in (a, b). This means that at some point (c, f(c)) on the graph of f that the slope of the tangent line is 0. There may be more than one such point c but the theorem provides that there exist at least one. The line tangent to point (c, f(c)) is parallel to the line joining (a, f(a)) to (b, f(b)).
Rolle's theorem is named after the French mathematician Michel Rolle who developed the theorem during the 17th century. Rolle's theorem is very closely related to the mean value theorem. In the mean value theorem f(a) does not have to equal f(b) but there is still at lest one point in between these two on f where the slope of the tangent line is parallel to that of the line connecting (a, f(a)) and (b, f(b)). The mean value theorem is a generalized form of Rolle's theorem that is applicable to a wider variety of situations but has a similar meaning.