Question

A fractional error in $ x $, $ y $, and $ z $ are 0.1, 0.2, and 0.5 respectively. Find the maximum fractional error in $ x^{-2} y^{3/2} z^{-2/5} $.

• A. 0.2
• B. 0.7
• C. 0.6
• D. 0.3

Hint:

To calculate the maximum fractional error in a product or quotient, add the fractional errors of each term. For powers, multiply the fractional error by the power.

Answer 1

The Correct Option is B

Step 1: Understand the formula for fractional error.
For a function \( f(x, y, z) \) of multiple variables, the maximum fractional error in \( f \) is given by: \[ \frac{\Delta f}{f} = \left| \frac{\partial f}{\partial x} \cdot \frac{\Delta x}{x} \right| + \left| \frac{\partial f}{\partial y} \cdot \frac{\Delta y}{y} \right| + \left| \frac{\partial f}{\partial z} \cdot \frac{\Delta z}{z} \right| \] For the function \( f(x, y, z) = x^{-2} y^{3/2} z^{-2/5} \), the fractional errors are propagated as follows.
Step 2: Calculate the partial derivatives.
We take the logarithmic derivative of the function to find the fractional errors: \[ \frac{\Delta f}{f} = \left| -2 \cdot \frac{\Delta x}{x} \right| + \left| \frac{3}{2} \cdot \frac{\Delta y}{y} \right| + \left| -\frac{2}{5} \cdot \frac{\Delta z}{z} \right| \]
Step 3: Substitute the given fractional errors.
We are given: \[ \frac{\Delta x}{x} = 0.1, \quad \frac{\Delta y}{y} = 0.2, \quad \frac{\Delta z}{z} = 0.5 \] Substitute these into the error propagation formula: \[ \frac{\Delta f}{f} = \left| -2 \cdot 0.1 \right| + \left| \frac{3}{2} \cdot 0.2 \right| + \left| -\frac{2}{5} \cdot 0.5 \right| \] \[ \frac{\Delta f}{f} = 0.2 + 0.3 + 0.2 = 0.7 \]
Step 4: Conclusion.
The maximum fractional error in the expression \( x^{-2} y^{3/2} z^{-2/5} \) is \( 0.7 \).