Absolute and Relatiave Uncertainty (partial derivatives)

In summary, to calculate the given expressions with uncertainties, we need to use partial derivatives. This involves holding certain variables constant while differentiating. It is assumed that the reader knows how to take derivatives.
  • #1
3ephemeralwnd
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Homework Statement



Calculate the following, expressing all results with uncertainties both in absolute and relative (percentage) form:

a) A + B

b) A x B

c) Asin(theta)

d) A^2 / Bcos(theta)


The relevant formula for the absolute uncertainty is below, but i have no idea how to use it! The teacher just sort of gave us the assignment without much explanation.. I know that it uses something called partial derivatives, could someone explain how to find that?
 

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  • #2
Assuming you know how to take derivatives, a partial derivative is just holding all the other variables as constant.

So if we had z=xy and we wanted to get ∂z/∂x, we would hold 'y' constant and differentiate like normal to get ∂z/∂x = y.

For this exercise, you should at least know how to take derivatives.
 

FAQ: Absolute and Relatiave Uncertainty (partial derivatives)

1. What is the difference between absolute and relative uncertainty?

Absolute uncertainty refers to the actual amount of uncertainty in a measurement, while relative uncertainty is the ratio of the absolute uncertainty to the measured value. In other words, absolute uncertainty tells us the range of possible values for a measurement, while relative uncertainty tells us how uncertain that measurement is in relation to its value.

2. How are partial derivatives used to calculate uncertainty?

Partial derivatives are used to calculate uncertainty in a multi-variable function by determining the sensitivity of the function to changes in each variable. This allows us to determine the effect of each variable on the overall uncertainty in the function.

3. Can partial derivatives be negative?

Yes, partial derivatives can be negative. A negative partial derivative means that the function is decreasing as the corresponding variable increases. This can be useful in determining the direction of change in a measurement and its effect on overall uncertainty.

4. How do you interpret the results of a partial derivative?

The result of a partial derivative indicates the rate of change of the function with respect to a specific variable. A larger absolute value of the partial derivative indicates a greater sensitivity of the function to changes in that variable, and therefore a larger impact on the overall uncertainty.

5. Can partial derivatives account for all sources of uncertainty in a measurement?

No, partial derivatives can only account for the uncertainty associated with the variables in the function. Other sources of uncertainty, such as measurement errors or instrument limitations, must be considered separately in the overall uncertainty calculation.

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