A chord at the edges of a graph

In summary: You can see that triangle as being made up of two smaller right angled triangles. The length of the chord is the square root of the sum of the squares of the lengths of the sides. The sum of the squares of the lengths of the sides is the value of the function f at a minus the value of the function f at b all squared plus the length of the interval a to b all squared. This is how the formula in post#2 arose.In summary, the conversation discusses an exercise involving a continuous function f with domain [0,1] and f(0)=f(1)=0. There are two parts to the exercise: A) proving the existence of a chord with length 1/2, and B) proving the
  • #1
Kostas Tzim
94
1
Greetings, i found an interesting exercise from my perspective, it's not about HW, i just want to see different approaches than the Greek math forum i posted yesterday, so we have:

If $$ f $$ is a function, then a chord is a straight portion whose edges belong to $$ C_f $$
f is a continuous function. its domain is $$ [0,1] $$ and $$ f(0)=f(1)=0 $$


A) Prove that a chord with length $$ \tfrac{1}{2} $$ exists
B) Prove that a chord with length $$ \tfrac{1}{n} $$ exists where n=1,2,3..

ps: (
sorry for the ugly latex appearance), i also think that A) question is a result of B)
 
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  • #2
Kostas Tzim said:
Greetings, i found an interesting exercise from my perspective, it's not about HW, i just want to see different approaches than the Greek math forum i posted yesterday, so we have:

If [itex]f [/itex] is a function, then a chord is a straight portion whose edges belong to [/itex]C_f[/itex]
f is a continuous function. its domain is [itex][0,1][/itex] and [itex]f(0)=f(1)=0[/itex]


A) Prove that a chord with length [itex] \tfrac{1}{2} [/itex] exists
B) Prove that a chord with length [itex]tfrac{1}{n}[/itex] exists where n=1,2,3..

ps: (
sorry for the ugly latex appearance), i also think that A) question is a result of B)
On this board, use [ itex ] and [ /itex ] (without the spaces) rather than the dollar signs to get "in line" Latex.
Does "[itex]C_f[/itex]" mean the graph of f? I don't believe that is standard notation. Let g(x) be the length of the chord from (0, 0) to (x, f(x)). Show that g is a continuous function. Therefore g takes on all values between 0 (when x= 0) and 1 (when x= 1).
 
  • #3
I think the translation is confusing: a chord is normally defined as a line segment joining two points on the graph of a function. The words "portion" and "boundary" are not useful here.

Assuming this is what you mean, the more general result that for any function f continuous in the interval [a, b] chords exist of all lengths ## l; 0 < l \le \sqrt {(f(b)-f(a))^2 + (b-a)^2} ## can be proved from the definition of a continuous function. Can you see how?
 
  • #4
Thanks for your answer..Yes the term "portion" is bad sorry :/ . The solution i got from the other forum was different : Assume the function [itex] F_n(x)=f(x+\dfrac{1}{n})-f(x) [/itex] and then i use the Bolzano theorem if [itex] Fn(a)F_n(b)<0 [/itex], (sorry i find it extremely hard to translate some greek terms we use) for the second case we follow an inequallity method

Could you explain me your method? i can't see very clearly how you ended up with this specific inequallity
 
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  • #5
#https://www.physicsforums.com/members/hallsofivy.331/ you are right the [itex]C_f[/itex] is a symbol we use in greece, it means the graph of [itex]f[/itex]
 
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  • #6
Kostas Tzim said:
Could you explain me your method? i can't see very clearly how you ended up with this specific inequallity
How long is the chord joining ## f(a) ## and ## f(b) ##? Note that you can also use two # (hash) characters as a short cut to bracket inline ## \LaTeX ## on this forum.
 
  • #7
I didnt understand this..what do you mean by how long? sometimes my brains completely stops working :P
 
  • #8
Kostas Tzim said:
I didnt understand this..what do you mean by how long? sometimes my brains completely stops working :P
"How long" here means "what is the length of". The chord is the hypotenuse of a right angled triangle.
 

Related to A chord at the edges of a graph

1. What is a chord at the edges of a graph?

A chord at the edges of a graph refers to a line segment connecting two vertices that are located at the outermost points of the graph.

2. How is a chord at the edges of a graph different from a regular chord?

A chord at the edges of a graph is different from a regular chord because it connects two vertices that are located at the outermost points of the graph, while a regular chord can connect any two vertices within the graph.

3. What is the importance of studying chords at the edges of a graph?

The study of chords at the edges of a graph can provide valuable information about the connectivity and structure of the graph. It can also help identify any isolated or disconnected parts of the graph.

4. Are there any practical applications of chords at the edges of a graph?

Yes, chords at the edges of a graph have practical applications in fields such as computer science, transportation planning, and social network analysis. They can be used to optimize network efficiency and identify key nodes in a graph.

5. Can a chord at the edges of a graph be part of a cycle?

Yes, a chord at the edges of a graph can be part of a cycle. In fact, a chord at the edge of a cycle is known as a bridge and is an important concept in graph theory.

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