# General equation for fractional error

1. Oct 1, 2008

### benji55545

1. The problem statement, all variables and given/known data
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.

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

3. 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.

2. Oct 1, 2008

### LowlyPion

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?

3. Oct 1, 2008

### benji55545

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.

4. Oct 1, 2008

### LowlyPion

X is the only independent variable it says.

5. Oct 1, 2008

### 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)?

6. Oct 1, 2008

### LowlyPion

Just wondering why you are avoiding saying Δq/q = n*(Δx/x)

7. Oct 1, 2008

### benji55545

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

8. Oct 1, 2008

### LowlyPion

Isn't your function q = xn ?

Δq/q = Δx/x + Δx/x + Δx/x ... Δx/x

n times?

Δq/q = n*(Δx/x)

9. Oct 1, 2008

### benji55545

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.

Last edited: Oct 1, 2008