General equation for fractional error

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Homework Help Overview

The discussion revolves around deriving a general expression for fractional error, Δq/q, using the error propagation rule for a function of a single variable, specifically q(x) = x^n, where n is an integer. Participants are exploring the relationship between the variables involved, including n, x, and Δx.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of the error propagation rule, questioning how to express fractional uncertainty and the implications of having dx in the denominator. There is a consideration of whether to apply the propagation rule for multiplication and how many multiplications are involved with the variable x.

Discussion Status

The discussion is active, with participants offering various interpretations of the problem and questioning the definitions and relationships between the variables. Some guidance has been provided regarding the application of the propagation rule, but there is no explicit consensus on the final expression for fractional error.

Contextual Notes

Participants are navigating the constraints of the problem, particularly focusing on the implications of using a single independent variable and the nature of the uncertainties involved. There is an ongoing exploration of how to incorporate the integer n into the final equation.

benji55545
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Homework Statement


Using the error propagation rule for functions of a single variable, derive a general expression for the fractional error, Δq/q, where q(x)=x^n and n is an integer. Explain your answer in terms of n, x, and Δx.

Homework Equations


The uncertainty of a function of one variable will be Δq=abs(dq/dx)Δx

The Attempt at a Solution


Okay, so I figured I could divide both sides of the equation above by dq, which will give a fractional uncertainty. This seems okay, but having dx in the denominator doesn't seem like a good idea. Any ideas on where to begin? Thanks.
 
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benji55545 said:

Homework Statement


Using the error propagation rule for functions of a single variable, derive a general expression for the fractional error, Δq/q, where q(x)=x^n and n is an integer. Explain your answer in terms of n, x, and Δx.


Homework Equations


The uncertainty of a function of one variable will be Δq=abs(dq/dx)Δx


The Attempt at a Solution


Okay, so I figured I could divide both sides of the equation above by dq, which will give a fractional uncertainty. This seems okay, but having dx in the denominator doesn't seem like a good idea. Any ideas on where to begin? Thanks.

Isn't what they are asking is for you to apply the propagation rule for multiplication? Addition and subtraction are the sum of the absolute errors and multiplication and division are the sum of the relative (fractional) uncertainties. So xn results in how many multiplications?
 
LowlyPion said:
Isn't what they are asking is for you to apply the propagation rule for multiplication? Addition and subtraction are the sum of the absolute errors and multiplication and division are the sum of the relative (fractional) uncertainties. So xn results in how many multiplications?

Right, the propagation rule for multiplication says Δq/q=sqrt[(Δx/x)2+...(Δz/z)2]
But if it's only for one variable, it reduces to Δq/q=sqrt[(Δx/x)2] ---> Δq/q=Δx/x right?

xn results in n multiplications... of what, though, beside x?

Thanks.
 
benji55545 said:
Right, the propagation rule for multiplication says Δq/q=sqrt[(Δx/x)2+...(Δz/z)2]
But if it's only for one variable, it reduces to Δq/q=sqrt[(Δx/x)2] ---> Δq/q=Δx/x right?

xn results in n multiplications... of what, though, beside x?

Thanks.

X is the only independent variable it says.
 
Well yeah. So the original question asked for a general equation for fractional uncertainty where q(x)=x^n. But that's not the answer obviously. If you just take the reduced form of the propagation of uncertainty, you get Δq/q=Δx/x. So...
q(x)=(Δx/x)1. That doesn't seem right. Maybe I need to set xn equal to Δx/x, then the result of that is my q(x)?
 
benji55545 said:
Well yeah. So the original question asked for a general equation for fractional uncertainty where q(x)=x^n. But that's not the answer obviously. If you just take the reduced form of the propagation of uncertainty, you get Δq/q=Δx/x. So...
q(x)=(Δx/x)1. That doesn't seem right. Maybe I need to set xn equal to Δx/x, then the result of that is my q(x)?

Just wondering why you are avoiding saying Δq/q = n*(Δx/x)
 
I'm afraid I don't see why that's true...
What is n representing in this case?
 
benji55545 said:
I'm afraid I don't see why that's true...
What is n representing in this case?

Isn't your function q = xn ?

Δq/q = Δx/x + Δx/x + Δx/x ... Δx/x

n times?

Δq/q = n*(Δx/x)
 
Isn't your function q = xn ?

Δq/q = Δx/x + Δx/x + Δx/x ... Δx/x

n times?

Δq/q = n*(Δx/x)
Oooh okay. I guess I was getting caught up with incorporating the exponential n in the final equation.
Δx/x + Δx/x + Δx/x ... Δx/x is all the uncertainties added together, each which is dependent only on x. I think I got it, thanks for the help.
 
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