Transcription of Intermediate Value Theorem, Rolle’s Theorem and Mean …
{{id}} {{{paragraph}}}
Example: air traffic controller
Intermediate Value Theorem, Rolle’s Theorem and Mean …
Intermediate Value Theorem , rolle s Theoremand Mean Value TheoremFebruary 21, 2014In many problems, you are asked to show that something exists, but are notrequired to give a specific example or formula for the answer. Often in this sortof problem, trying to produce a formula or specific example will be following three theorems are all powerful because they guarantee theexistence of certain numbers without giving specific 1( Intermediate Value Thoerem).Iffis a continuous function onthe closed interval[a,b], and ifdis betweenf(a)andf(b), then there is a numberc [a,b]withf(c) = an example, letf(x) = cos(x) x. Sincef(0) = 1 andf( ) = 1 ,there must be a numbertbetween 0 and withf(t) = 0 (sotsatisfies cos(t) =t).
11. (MCMC 2004 II.1) Suppose fis a continuous real-valued function on the interval [0;1]. Show that Z 1 0 x2f(x)dx= 1 3 f(˘) for some ˘2[0;1]. Solution. Because fis continuous, it attains its minimum and maximum at points aand b, both in [0;1], giving f(a) Z 1 0 x2dx Z 1 0 x2f(x)dx f(b) Z 1 0 x2 dx or f(a) 3 Z 1 0 x2f(x)dx f(b):
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: