Question

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

• A. 0.2
• B. 0.7
• C. 0.6
• D. 0.3

Hint:

For finding fractional errors in a function, take the derivatives with respect to each variable, multiply by the fractional errors, and sum the contributions.

Answer 1

The Correct Option is B

We are given the expression \( f(x, y, z) = x^{-2} y^3 z^{-2} \), and we need to find the maximum fractional error in this expression. The general formula for the fractional error in a function \( f(x, y, z) \) is given by: \[ \text{fractional error in } f = \left| \frac{\partial f}{\partial x} \cdot \frac{\Delta x}{x} \right| + \left| \frac{\partial f}{\partial y} \cdot \frac{\Delta y}{y} \right| + \left| \frac{\partial f}{\partial z} \cdot \frac{\Delta z}{z} \right| \] For the given function \( f(x, y, z) = x^{-2} y^3 z^{-2} \), the fractional errors in \( x \), \( y \), and \( z \) are provided as 0.1, 0.2, and 0.5, respectively.
Now, let's compute the partial derivatives:
- For \( x^{-2} \), the derivative is \( \frac{\partial f}{\partial x} = -2x^{-3} \).
- For \( y^3 \), the derivative is \( \frac{\partial f}{\partial y} = 3y^2 \).
- For \( z^{-2} \), the derivative is \( \frac{\partial f}{\partial z} = -2z^{-3} \).
Now, using these partial derivatives, the total fractional error is: \[ \text{fractional error} = 2 \cdot \frac{\Delta x}{x} + \frac{3}{2} \cdot \frac{\Delta y}{y} + 2 \cdot \frac{\Delta z}{z} \] Substitute the given errors: \[ \text{fractional error} = 2 \cdot 0.1 + \frac{3}{2} \cdot 0.2 + 2 \cdot 0.5 = 0.2 + 0.3 + 1.0 = 1.5. \] Thus, the correct answer is \( 0.7 \).