intermediate value theorem?

Question

can you please the intermediate value theorem in words, thanks.

Answer 1

If f is a continuous function on the interval [a,b] then f takes on all values between f(a) and f(b) (the function values at the endpoints of the interval [a,b]). This becomes very intuitive if you plot of an arbitrary continuous function.

Another formulation is that if f is continuous on the interval [a,b] and f(a) and f(b) have the opposite sign then there must a point in between where the function value is 0.

Answer 2

The intermediate theorum basically defines a continuous function.

Let's look at something simple in two dimensions. Pick some limits on the x axis. They usually use a and b, let's use 3 and 7. Now draw a curve which is continuous on the interval between 3 and 7. A straight line will work or any curve. Make sure the curve (line) has no breaks and doesn't do anything nuts like go off the graph to infinity in the interval from 3 to 7. This called a continuous function -- it CONTINUES through the interval with no breaks and the function is defined by some equation..

The intermediate value theorum says that if you pick a number between a and b (3 and 7) and plug it into the equation, the result will fall on the curve. It's as simple as that.

Answer 3

The Intermediate Value Theorem says that a continuous function on an interval [a,b] takes all the values between f(a) and f[b] (Note the function might take more values, the theorem says that at least it takes all the values between f(a) and f(b).) Here f(0) is positive and and f(-1) is negative, so there is (at least) a c in the open interval (-1,0) so that f(c)=0.

Answer 4

Hi,

Basically the theorem is saying that if a function is continuous (on some range), there is a point where the value of the function exists for every input (in that range).

Example. graph y=x for the range 0
now for every value of y(x) (between and including y(0) and y(10)) there exists some x that makes the y(x) that value.

Please note that there could be more than one, for example y=x^2

Hope that helps.

Answer 5

In a nutshell:

If a function is continuous on a closed interval [a,b], then, within (a,b), it assumes every value between its values at the endpoints.

This is often used to prove that an equation of the form f(x) = 0 has a solution in (a,b). If it is continuous on the [a,b], and its values at the endpoints have opposite signs, then, within (a,b), f(x) must assume every value in between, including zero. Therefore the equation f(x) = 0 must have a solution in (a,b).

Answer 6

Ted and James L have both given correct mathematical statements of the theorem. In non-mathematical terms, it amounts to saying that if you are moving around on the number line (think of the y-axis), then you can't get from one point to another without passing through every point in between. (You aren't allowed to "jump" - that would violate the "continuity" requirement in the theorem.)

Another way of looking at it that is a little more general: if you are moving around in a plane that has an infinite horizontal straight line in it, and you are on one side of the line, then you can't get to the other side of the line without getting on the line at least once. Again, jumping not allowed because of the continuity requirement.

Answer 7

states that for value A(a)B(b) the value of a-b is within the points A-B and that the value of (ab) falls on or within/between the values set by AB simply put A and B are barriers that contain (ab) and all permutations set by actual function elements . good for calculating differentials and spaces and defining the right computational line for a variable set.