General equation for fractional error

AI Thread Summary
The discussion focuses on deriving a general expression for fractional error, Δq/q, for the function q(x) = x^n using the error propagation rule. Participants clarify that the uncertainty for a single variable can be expressed as Δq = abs(dq/dx)Δx, leading to the fractional uncertainty Δq/q = Δx/x. It is noted that for the function x^n, the fractional error can be generalized to Δq/q = n*(Δx/x), as the uncertainty is summed n times due to the multiplicative nature of the function. The conversation emphasizes understanding the role of the exponent n in relation to the uncertainty. Ultimately, the correct expression for fractional error is established as Δq/q = n*(Δx/x).
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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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