# Intermediate Value Theorem Word Problem

• harrietstowe
In summary, the conversation discusses the problem of showing that there is at least one point on a tangled string that has the same distances from two walls as a straight string that lies on the floor. The solution involves parameterizing the strings by arc length and looking at the difference between the x coordinates.
harrietstowe

## Homework Statement

The image can be seen at:
http://s1130.photobucket.com/albums/m521/harrietstowe/?action=view&current=photo1.jpg

The rectangle in the picture represents the floor of a room and AB a straight piece of string lying on the floor whose ends touch the opposite walls w1 and w2. The blob is the same string tangled up. I need to show that there is at least one point of the tangled string whose distances from the two walls are exactly the same as they were before.

## The Attempt at a Solution

I tried to graph the situation by saying the regular string is f(x) and the tangled string is g(x) and then graphing h(x)=f(x)-g(x) but I am now stuck.
Thank You

Think about the straight string of length L with x coordinate parameterized by arc length s so its x coordinate is f(s) = s, 0 ≤ s ≤ L. Then think of the tangled string still parameterized by s so

R(s) = < x(s), y(s) >, 0 ≤ s ≤ L

Then look at f(s) - x(s) and see what you come up with.

## 1. What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a mathematical theorem that states that if a continuous function has a different sign at two points, then it must have at least one root (or solution) between those points.

## 2. How is the Intermediate Value Theorem used to solve word problems?

The Intermediate Value Theorem can be used to solve word problems by first identifying a continuous function that represents the situation in the problem. Then, it can be applied to find a root (or solution) that satisfies the given conditions.

## 3. Can the Intermediate Value Theorem be applied to all types of functions?

No, the Intermediate Value Theorem can only be applied to continuous functions. This means that the function must have no breaks or gaps in its graph.

## 4. What is the importance of the Intermediate Value Theorem in mathematics?

The Intermediate Value Theorem is important because it guarantees the existence of solutions for certain types of problems. It also provides a method for finding these solutions, which can be applied to a wide range of mathematical and scientific scenarios.

## 5. Are there any limitations to the Intermediate Value Theorem?

Yes, the Intermediate Value Theorem has some limitations. It can only be used to find a solution within a given interval and it does not provide any information about the uniqueness or multiplicity of the solution. Additionally, it assumes that the function is continuous, which may not always be the case in real-world scenarios.

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