Question

Show that the approximate relative error (df)/f of a product $f=gh$ is the sum of the approximate relative errors of the factors.

Short answer

The relative error of a product is the sum of the relative errors of its factors: $\frac{df}{f}=\frac{dg}{g}+\frac{dh}{h}$.

Step-by-step solution

Write the definition of approximate relative error

The approximate relative error for any function can be defined as the absolute error divided by the true value. For a function $f$, the relative error is given by $\frac{df}{f}$.

Express the function as a product

Given $f=gh$, where $g$ and $h$ are functions of some variable(s).

Differentiate the product

Differentiate $f$ with respect to the variable(s): $df=hdg+gdh$. This follows from the product rule of differentiation.

Divide by $f$

Divide both sides of the differentiated equation $df=hdg+gdh$ by $f=gh$: $\frac{df}{f}=\frac{hdg+gdh}{gh}$.

Simplify the expression

Simplify the right-hand side of the equation $\frac{df}{f}=\frac{hdg+gdh}{gh}$ to get $\frac{df}{f}=\frac{dg}{g}+\frac{dh}{h}$.

Interpret the result

The relative error of the product function $f=gh$ is the sum of the relative errors of its factors, $g$ and $h$. Therefore, we have shown that $\frac{df}{f}=\frac{dg}{g}+\frac{dh}{h}$.

Key concepts

Approximate Relative Error

Relative error is a measure of the uncertainty in a measurement compared to the size of the measurement itself. It gives a ratio of the absolute error to the true value. For a function or quantity, the relative error is represented as $\frac{df}{f}$. This ratio demonstrates how significant the error is in comparison to the function value.

To understand this better, if you have a product of two functions, say $f=gh$, one might want to know how the errors in $g$ and $h$ propagate to the error in $f$. By calculating the approximate relative error, we can get a sense of how the individual errors contribute to the overall error of the product.

Product Rule of Differentiation

Differentiating products of functions is crucial in understanding how relative errors propagate. The product rule of differentiation states that for two functions $g$ and $h$, the derivative of their product is given by:
$$f=gh$$ $$df=gdh+hdg$$
This means you differentiate one function while keeping the other constant, and then do the reverse. This rule is essential in deriving the relationship between the relative errors of the product function and its factors.

Error Propagation

Error propagation is the process of determining the uncertainty or error in a calculated result based on the errors in the individual inputs. When dealing with products, the error in the product is affected by the errors in the individual terms.
Using the product rule and dividing by the product itself, we can find the relationship between the approximate relative errors:
$$\frac{df}{f}=\frac{dg}{g}+\frac{dh}{h}$$
This equation shows that the relative error of a product is the sum of the relative errors of the individual components. This is an important concept because it illustrates how even small errors in the factors can add up and potentially lead to larger errors in the result. Understanding this can help in accurately predicting and managing error in complex calculations.