Exploring Limits: Applications of the Intermediate Value Theorem

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In summary, the conversation discusses questions 6, 7, and 8 related to functions, fixed points, and the Intermediate Value Theorem. Question 6 asks to show the existence of a fixed point for a continuous function on a closed interval, while question 7 asks to prove that stretching a rubber band will result in at least one point returning to its original position. The summary also mentions the use of composite functions and the need to apply the Intermediate Value Theorem for question 7. The conversation ends with a question about how to approach question 8, which involves the use of the Intermediate Value Theorem as well. There is also a mention of using composite functions, but the person is unsure how to proceed.
  • #1
Latios1314
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Have a couple of questions that I've been stuck with for some time. Would greatly appreciate the help. No idea how I should approach these questions and how do I start.

6. Suppose that a function f is continuous on the closed interval [0,1]and that 0≤f(x)≤1 for every x in [0,1].Show that there must exist a number c in [0,1] such that f(c)=c(c is called a fixed point of f).

(7) Is it true that if you stretch a rubber band by moving one end to the right and the other end to the left, some point of the band will end up in its original position? Give reasons for your answer.

(8) Is there any reason to believe that there is always a pair of antipodal (diametrically opposite) points on Earth’s equator where the temperatures are the same? Explain.
 
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  • #2
Latios1314 said:
6. Suppose that a function f is continuous on the closed interval [0,1]and that 0≤f(x)≤1 for every x in [0,1].Show that there must exist a number c in [0,1] such that f(c)=c(c is called a fixed point of f).

Consider g(x) = f(x) - x. What property does this function have? Try to think what else you could do with g(x).
 
  • #3
From where do I derive the equation g(x)=f(x)-x?
 
  • #4
I reckon that I should perhaps use composite functions to do this but I have no idea from where can i get the function g(x) from.
 
  • #5
Managed to get both questions 6 and 7 down.

Not sure whether my working for 7 is correct though.

Let x1 be the new position of the left end of the elongated rubber band.
let d(x) be the displacement from the original position.

d(x) = x - x1 <0

Let x2 be the new position of the right end of the elongated rubber band.
d(x) = x2 - x >0

Therefore by Intermediate Value Theorem, d(x) = 0 must exist. Hnece some part of the elongated rubber band must be at the oringal position.
 
  • #6
how do i go about question 8 though?

I don't really get the question. Know that i must make use of intermediate value theorem somehow but i have no idea how I'm supposed to do it.
 

FAQ: Exploring Limits: Applications of the Intermediate Value Theorem

What is a limit in mathematics?

A limit in mathematics is a value that a function or sequence approaches as its input or index approaches a certain value. It is used to describe the behavior of a function or sequence near a specific point.

How do you find the limit of a function?

To find the limit of a function, you can evaluate the function at values approaching the specific point from both the left and right sides. If the values approach the same value, then that value is the limit. If the values approach different values, or the function is undefined at that point, the limit does not exist.

Can a limit be infinity?

Yes, a limit can be infinity. This means that the function approaches positive infinity, negative infinity, or both as the input approaches the specific value. This can happen if the function has a vertical asymptote or if the function increases or decreases without bound as the input approaches the specific value.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the values approaching the specific point from one direction, either the left or the right. A two-sided limit considers the values approaching the specific point from both the left and right sides and requires that the values approach the same value for the limit to exist.

Why are limits important in mathematics?

Limits are important in mathematics because they allow us to understand the behavior of functions and sequences near specific points. They are also used in calculus to analyze rates of change and to calculate derivatives and integrals. Additionally, limits are essential in the study of continuity and differentiability of functions.

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