Tutorial 7 - Intermediate Value Theorem Dr C Sean Bohun Limits and Continuity Tutorial 07 Page 1 Intermediate Value Theorem To begin with lets start

Tutorial 7 - Intermediate Value Theorem Dr C Sean Bohun...

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Intermediate Value Theorem: Dr. C. Sean Bohun Limits and Continuity, Tutorial 07 Page 1 Intermediate Value Theorem To begin with, let’s start with the basic statement of the theorem. Theorem: If f ( x ) is continuous on a closed interval [ a, b ] and N is any number f ( a ) < N < f ( b ) then there exists a value c ( a, b ) such f ( c ) = N . The illustration corresponding to the theo- rem is to the right and indicates that there may be more than one possible value for c . The important restrictions are that f ( x ) be continuous and the interval [ a, b ] is closed. The primary purpose of this theorem is to in- dicate when numbers with various properties exist. a c b f ( b ) N f ( a ) For example, suppose we have the function g ( x ) = x 2 - 4 x and we wish to show there is a number x * such that g ( x * ) = 1. Step 1) Notice that since g ( x ) is continuous, we can use the intermediate value theorem. Step 2) Make a new function f ( x ) = x 2 - 4 x - 1 so that f ( x ) = 0 when we have the correct x * . Note that replacing the function in this manner always makes the N in the theorem equal to zero. Step 3) We need to find an a and a b so that either f ( a ) > 0 and f ( b ) < 0 or f ( a ) < 0 and f ( b ) > 0. The point here is that we need a change in sign. Choosing

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