Propagation Of Uncertainty Calculator With Solution

Propagation Of Uncertainty:

\[ \sigma_f = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2\sigma_x^2 + \left(\frac{\partial f}{\partial y}\right)^2\sigma_y^2 + \cdots} \]

Uncertainty Result:

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1. What Is Propagation Of Uncertainty?

Propagation of uncertainty (or error propagation) is the effect of variables' uncertainties on the uncertainty of a function based on those variables. It's essential in experimental sciences and engineering for quantifying measurement errors.

2. How The Calculator Works

The calculator uses the general formula for uncertainty propagation:

\[ \sigma_f = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2\sigma_x^2 + \left(\frac{\partial f}{\partial y}\right)^2\sigma_y^2 + \left(\frac{\partial f}{\partial z}\right)^2\sigma_z^2 + \cdots} \]

Where:

\( \sigma_f \) — Uncertainty in the function result
\( \frac{\partial f}{\partial x} \) — Partial derivative of f with respect to x
\( \sigma_x \) — Uncertainty in variable x
Similarly for other variables y, z, etc.

3. Step-by-Step Partial Derivatives

Process: The calculator computes partial derivatives for each variable, squares them, multiplies by the squared uncertainty, sums all terms, and takes the square root of the sum.

4. Using The Calculator

Instructions: Enter the mathematical function, list variables separated by commas, provide their measured values, and specify their respective uncertainties.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can be analyzed?
A: The calculator handles most mathematical functions including polynomials, trigonometric, exponential, and logarithmic functions.

Q2: How are partial derivatives calculated?
A: The calculator uses symbolic differentiation to compute partial derivatives with respect to each variable.

Q3: What if variables are correlated?
A: This calculator assumes uncorrelated variables. For correlated variables, additional covariance terms are needed.

Q4: Can I use this for complex functions?
A: Yes, the calculator can handle multi-variable functions of various complexity levels.

Q5: How accurate are the results?
A: Results are mathematically exact based on the uncertainty propagation formula, assuming the first-order Taylor approximation is valid.