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Does this function have a name?

by guysensei1
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guysensei1
#1
Jul21-14, 08:36 AM
P: 22
A function f(x) where f(x)=length of the graph curve/line from 0 to x

Can this function be expressed in algebraic form or some other form?

Does it have a name?
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HallsofIvy
#2
Jul21-14, 09:07 AM
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You are talking about a particular way of getting a function, not a specific function so, no, it does not have a name. As one learns in Calculus, the length of the graph of y= f(x), from 0 to x is given by [itex]\int_0^x \sqrt{1+ (f'(t))^2} dt[/itex].
1MileCrash
#3
Jul21-14, 09:27 AM
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Quote Quote by HallsofIvy View Post
You are talking about a particular way of getting a function, not a specific function so, no, it does not have a name. As one learns in Calculus, the length of the graph of y= f(x), from 0 to x is given by [itex]\int_0^x \sqrt{1+ (f'(t))^2} dt[/itex].
It's without a doubt a specific function, one whose domain is the Cartesian product of x and the function space. I don't think that this is a good reason for it to not have a name.

MrAnchovy
#4
Jul21-14, 03:20 PM
P: 545
Does this function have a name?

Quote Quote by guysensei1 View Post
Can this function be expressed in algebraic form or some other form?
Not in general, but the solutions for a straight line and for a circle should be fairly obvious. Closed form solutions do exist for a few more complex curves e.g. parabola, catenery and cycloid but not for most others, even the humble ellipse.

Quote Quote by guysensei1 View Post
Does it have a name?
The calculation of the length of a curve between two points is called rectification, or simply calculating arc length.
guysensei1
#5
Jul21-14, 10:44 PM
P: 22
Quote Quote by MrAnchovy View Post
Not in general, but the solutions for a straight line and for a circle should be fairly obvious. Closed form solutions do exist for a few more complex curves e.g. parabola, catenery and cycloid but not for most others, even the humble ellipse.



The calculation of the length of a curve between two points is called rectification, or simply calculating arc length.

What I was looking for is a function that gives itself when the length of curve function is applied.
MrAnchovy
#6
Jul22-14, 04:37 AM
P: 545
Quote Quote by guysensei1 View Post
What I was looking for is a function that gives itself when the length of curve function is applied.
Try looking for this function. Clearly f(0) = 0. Let y = f(1). The length of the arc between (0, 0) and (1, y) is given by ## \sqrt{1 + y^2} ## so we have ## y = \sqrt{1 + y^2} ## or ## y^2 = 1 + y^2 ## which has no solution - the function you are looking for does not exist (over any non-zero domain).
skiller
#7
Jul22-14, 09:30 AM
P: 235
Quote Quote by MrAnchovy View Post
Try looking for this function. Clearly f(0) = 0. Let y = f(1). The length of the arc between (0, 0) and (1, y) is given by ## \sqrt{1 + y^2} ##
Why is that? That surely just gives the length of the straight line from (0, 0) to (1, y). The curve you are looking for is not going to be of that form.
Nugatory
#8
Jul22-14, 11:08 AM
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Quote Quote by skiller View Post
Why is that? That surely just gives the length of the straight line from (0, 0) to (1, y). The curve you are looking for is not going to be of that form.
No, but its length has to be strictly greater than the length of the straight line between the two points.
MrAnchovy
#9
Jul22-14, 11:08 AM
P: 545
Quote Quote by skiller View Post
Why is that? That surely just gives the length of the straight line from (0, 0) to (1, y). The curve you are looking for is not going to be of that form.
Oh, I thought one thing and wrote something slightly different, let's try again.

Quote Quote by guysensei1 View Post
What I was looking for is a function that gives itself when the length of curve function is applied.
Try looking for this function. Clearly f(0) = 0. Let y = f(1). The shortest arc between (0, 0) and (1, y) is simply the diagonal of length ## \sqrt{1 + y^2} ##, and so y must be at least as large as that. So we have ## y \ge \sqrt{1 + y^2} ## or ## y^2 \ge 1 + y^2 ## which has no real solution. The function you are looking for does not exist (over any non-zero domain).


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