General equation for fractional error

[benji55545]

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.

 

[LowlyPion]

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?

 

[benji55545]

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.

 

[LowlyPion]

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.

 

[benji55545]

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)?

 

[LowlyPion]

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)

 

[benji55545]

I'm afraid I don't see why that's true...
What is n representing in this case?

 

[LowlyPion]

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)

 

[benji55545]

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.