The ratio r = (1 - a) / (1 + a) is derived from the measurement of a dimensionless parameter a. If the uncertainty in a is represented as Δa (where Δa / a << 1), what is the uncertainty Δr in calculating r?

Δa / (1 + a)²
2Δa / (1 + a)²
2Δa / (1 - a)²
2aΔa / (1 - a)²

Correct answer: 2Δa / (1 + a)²

Solution

For r = (1-a)/(1+a), dr/da = [-(1+a)-(1-a)]/(1+a)^2 = -2/(1+a)^2, so the uncertainty is dr = 2 da/(1+a)^2 (option index 1). The stored answer da/(1+a)^2 (index 0) drops the factor of 2.

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