Expert: Socrates - 8/6/2006

Question
i wanna ask about wht is "IVT"? wht is this theorem for? what is the most important things we need to know? thanks


Answer
The Intermediate Value Theorem tells you about the values of a continuous funtion. If a continuous function is 5 when x is 0 and 10 when x is 1 , then you can pick any number between 5 and 10 , say 7 , and then you know that the function value will be 7 for some x between 0 and 1.
In function notation , if f(0) = 5 and f(1) = 10 , the IVT tells you that there will be some number t between 0 and 1 with f(t) = 7.
Another example , suppose f(4) = 41 and f(6) = 100 . You can choose any number between 41 and 100 and you know the function will have that value somewhere between 4 and 6.
You could choose 75 as the number between 41 and 100. The IVT says there will be some number , call it t , between 4 and 6 so that f(t) = 75.
The IVT doesn't tell you what the number t will be , just that there must be a number , t , between 4 and 6 with
f(t) = 75

The IVT gives you information about the function values between points that you know. It sort of tells what the function is doing between known points

The IVT has many useful applications, for example , approximating zeros of polynomials.

Let f(x) = x^3 + x - 20

f(2) = -10

f(3) = 10

Since 0 is between -10 and 10 , the IVT tells you that the polynomial will be zero for some value for x between 2 and 3 . In other words, the polynomial has a root between 2 and 3 . We don't know exactly what it is, just that it is some number between 2 and 3.

I hope this helps