{"id":388,"date":"2016-10-22T15:10:35","date_gmt":"2016-10-22T19:10:35","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=388"},"modified":"2017-08-22T17:01:07","modified_gmt":"2017-08-22T21:01:07","slug":"historical-applications-and-formulations-of-intermediate-value-theorem","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2016\/10\/22\/historical-applications-and-formulations-of-intermediate-value-theorem\/","title":{"rendered":"Historical Applications and Formulations of Intermediate Value Theorem"},"content":{"rendered":"

Formulations of Intermediate Value Theorem<\/strong><\/h4>\n

What is Intermediate Value Theorem? Let\u2019s start with an interesting story I found online that can illustrate the result for Intermediate Value Theorem.<\/p>\n

One day a monk leaves at sunrise to climb up a mountain. He walks at a leisurely pace, sometimes stopping to enjoy the view, even retracing his path to look again at a pretty flower. He arrives at the summit at sundown, spends the night meditating, and starts home down the same path the next day at sunrise, arriving home at sunset. The question is this: Was there a time of day when he was exactly at the same point on the trail on the two days?<\/p>\n

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\"Edison3\"<\/a><\/p>\n

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We can think of two graphs for those two days with distance from the bottom of his track on the y-axis, and time on the x-axis. Position is a continuous function with time here. Then we can see that the two curves intersect at a point which indicates there was a time of day when the monk was exactly at the same point on the trail on the two days.\u00a0
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\"Edison4\"<\/a><\/p>\n

This result can also be proven by the Intermediate Value Theorem easily. The Intermediate Value Theorem states:<\/p>\n

If a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, <\/strong>for every d<\/em> between f(a)<\/em> and f(b)<\/em>, there exists a c<\/em> between a<\/em> and b<\/em> so that f(c)<\/em> = d<\/em>.<\/strong><\/p>\n

The Intermediate Value Theorem was first proven by Bernard Bolzano in 1817, who was, in fact, a monk, and the French mathematician Augustin-Louis Cauchy provided a proof in 1821. Bernard Bolzano published a paper with the full title that \u201cPurely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least on real root of the equation<\/em>\u201d. He stated:<\/p>\n

If two functions of x, \"fx\" and \"\phi, vary according to the law of continuity either for all values x or only for those which lie between \"\alpha\" and \"\beta\", and if \"f\alpha and \"f\beta, then there is always a certain value of x between \"\alpha\" and \"\beta\" for which \"fx=\phi<\/strong>.<\/p>\n

Bolzano defined the Intermediate Value Theorem different from the modern definition by using two functions of x. I\u00a0attach Bolzano’s paper in 1817 below, and you can discover more about his contribution in the rigorous foundation of analysis in this paper.<\/p>\n

http:\/\/www.sciencedirect.com\/science\/article\/pii\/0315086080900361<\/a><\/p>\n

This becomes the Intermediate Value Theorem. This theorem seems obvious, and before Bolzano proved it in 1817, it was freely used by mathematicians in the 18th<\/sup> century. Mathematicians made assumptions that this basic theorem does not need justification. However, Bolzano had different opinions towards this theorem. Bolzano wants to give a justification instead of confirmation to this theorem. Even if the theorem is obvious, he provides a proof of the Intermediate Value Theorem in order to support the foundation of analysis.<\/p>\n

The French mathematician Cauchy stated the Intermediate Value Theorem in a slightly different way:<\/p>\n

If the function f(x) is continuous with respect to the variable x between the limits \"x=x_0\" and \"x=X\" and if b designates a quantity between \"f(x_0)\" and \"f(X)\", one can always satisfy the equation \"f(x)=b\" for one or several real values of x between \"x_0\" and \"X\".<\/strong><\/p>\n

This statement is still recognizable today, even if the statement uses unusual notation and wordings. Cauchy actually provided two proofs for this theorem. The first proof only used continuity to show that a continuous function is an unbroken curve, and used geometry instead of algebra to prove it. The second proof used\u00a0Newton’s method <\/strong>(for approximation of roots of a function; see the tutorial video below)\u00a0which is more popularly known to the public readers.<\/p>\n

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