Perform the following operations and give the result with...

Perform the following operations and give the result with standard deviation. The standard deviations of quantities are indicated by $\pm$; they are independent of each other. (a) $15.000 /(5.0 \pm 0.1)$ (b) $(30.0 \pm 0.9) /(5.0 \pm 0.2)$ (c) $\log _{10}(1000 \pm 2)$ (d) $(20.0 \pm 0.3) \exp [-(2.00 \pm 0.01)]$

Perform the following operations and give the result with standard deviation. The standard deviations of quantities are indicated by $\pm$; they are independent of each other.
(a) $15.000 /(5.0 \pm 0.1)$
(b) $(30.0 \pm 0.9) /(5.0 \pm 0.2)$
(c) $\log _{10}(1000 \pm 2)$
(d) $(20.0 \pm 0.3) \exp [-(2.00 \pm 0.01)]$

Key Concepts

Exponential Function Uncertainty Propagation

For functions involving the exponential, the uncertainty is usually propagated through the derivative of the exponential function. If an exponent has an uncertainty, then error propagation can be done by differentiating the exponential, making use of the fact that the derivative of exp(x) is exp(x). This allows one to quantify the uncertainty in expressions like (20.0 ± 0.3) exp[-(2.00 ± 0.01)].

Error Propagation

Error propagation is a method used to determine the uncertainty in a result calculated from measurements that have their own uncertainties. When combining measurements through functions such as addition, subtraction, multiplication, division, or more complex functions like exponential or logarithmic functions, the uncertainties are combined using appropriate formulas, often involving derivatives of the functions with respect to the variables.

Multiplication and Division Uncertainty

When quantities are multiplied or divided, their relative (or fractional) uncertainties are combined. For independent quantities, the relative uncertainty of the product or quotient is given by the square root of the sum of the squares of the individual relative uncertainties. This method allows for the computation of the uncertainty for operations like evaluating 15.000/(5.0 ± 0.1) or (30.0 ± 0.9)/(5.0 ± 0.2).

Logarithmic Function Uncertainty Propagation

For functions involving logarithms, the uncertainty in the logarithm can be approximated by using the derivative of the logarithm function. Specifically, if x has an uncertainty ?x, then the uncertainty in log10(x) is approximately ?(log10 x) = ?x / (x ln(10)). This approach helps in estimating the uncertainty for operations like log10(1000 ± 2).

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