apcalcIntermediate+Value+Theorem

//Intermediate Value Theorem//

﻿ Definition:
==Let //f// (//x//) be a continuous function on the interval [//a//, //b//]. If d is a number between f(a) and f(b), then there is a number **// c //** inside the interval [a,b], such that //f// (//c// ) = //d//. ==



Note: 1. There can be more than or only one number that in x-axis than f(c) is equal d. 2.A nice use of the Intermediate Value Theorem is to prove the existence of roots of equations. 3.It’s also important to note that the Intermediate Value Theorem only says that the function will take on the value of //M// somewhere between //a// and //b//. It doesn’t say just what that value will be. It only says that it exists.

Show that [[image:http://tutorial.math.lamar.edu/Classes/CalcI/Continuity_files/eq0026M.gif width="164" height="17" align="baseline"]]has a root somewhere in the interval [-1,2].
What we’re really asking here is whether or not the function will take on the value somewhere between -1 and 2. In other words, we want to show that there is a number //c// such that   and. However if we define and acknowledge that  and  we can see that these two condition on //c// are exactly the conclusions of the Intermediate Value Theorem. Because p(-1)=-8 and p(2)=-19 So we have: ﻿ -19=p(2)<0<p(-1)=8 Therefore is between  and  and since  is a polynomial it’s continuous everywhere and so in particular it’s continuous on the interval [-1,2]. So by the Intermediate Value Theorem there must be a number -1<c<2,so that, Therefore the polynomial does have a root between -1 and 2.
 * //Solution//**

Question:   ﻿ ﻿1.If possible, determine if  takes the following values in the interval [0,5]. a)Does ? b)Does  ? 2.Show that the function has a solution between 2 and 3.

History:
The Intermediate Value Theorem was found by [|Bernard Bolzano](1781-1848) in 1817. The intermediate Value Theorem is a theoretical tool in our calculus class room today.(I am still working on how it attain its role in nineteenth-century). Earlier authors held the result to be intuitively obvious, and requiring no proof.

Works Cited: Intermediate Value Theorem -- from Wolfram MathWorld." //Wolfram MathWorld: The Web's Most Extensive Mathematics Resource//. Web. 05 June 2011. . "The Intermediate Value Theorem." //Oregon State University//. Web. 05 June 2011. . "Intermediate Value Therem." //Calculus Applets//. Web. 05 June 2011. .