## - The Intermediate Value Theorem

Once we know that a root lies within a particular interval we need a process to help us to determine the value of the root or an approximation to the root. To do this we use ITERATIVE processes. An ITERATIVE process is a repeated calculation which allows us get closer to the desired result with each iteration.

If you had to guess a value for one of the roots of the equation what would you guess? That's quite a challenge. It may be easier to try to guess a range of values in which you think the root occurs.

That's where the Intermediate Value Theorem comes in. This Theorem allows us to determine if a root occurs within a specified range.

 THE INTERMEDIATE VALUE THOREM If is a continuous function on the closed interval and the product then there exists in such that

EXAMPLE 1

Use the Intermediate Value Theorem to show that has a root between 1 and 2.

 SOLUTION is a polynomial and therefore continuous on the interval . By the Intermediate Value Theorem there must be some such that . Therefore, there is a root between 1 and 2.

EXAMPLE 2

Use the Intermediate Value Theorem to verify that there is a root of the equation between 1.1 and 1.2.

 SOLUTIONFirstly we need to rewrite the equation so that we have .Therefore, NB: is NOT a polynomial so we do not make that assertion in our statement for the INTERMEDIATE VALUE THEOREM. is a continuous on the interval . By the Intermediate Value Theorem there must be some such that . Therefore, there is a root between 1.1 and 1.2.