Question

assume the measurement of x is accurate within 3%. use a linear approximation to determine the error δf in the calculation of f and find the percentage error 100δf/f. the quantities f(x) and the true value of x are given. f(x)=5x³, x = 1.8 the percentage error 100δf/f is □%.

Explanation:

Step1: Find the derivative of \(f(x)\)

Given \(f(x) = 5x^{3}\), using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(f^\prime(x)=15x^{2}\).

Step2: Use the linear - approximation formula for relative error

The linear - approximation formula for \(\Delta f\) is \(\Delta f\approx f^\prime(x)\Delta x\). Then the relative error \(\frac{\Delta f}{f}\approx\frac{f^\prime(x)\Delta x}{f(x)}\).
Substitute \(f(x) = 5x^{3}\) and \(f^\prime(x)=15x^{2}\) into \(\frac{\Delta f}{f}\approx\frac{f^\prime(x)\Delta x}{f(x)}\), we get \(\frac{\Delta f}{f}\approx\frac{15x^{2}\Delta x}{5x^{3}}=\frac{3\Delta x}{x}\).

Step3: Calculate the percentage error

The percentage error is \(100\frac{\Delta f}{f}\). Since \(\frac{\Delta x}{x}=0.03\) (because the measurement of \(x\) is accurate within \(3\%\)), then \(100\frac{\Delta f}{f}=100\times\frac{3\Delta x}{x}\).
Substitute \(\frac{\Delta x}{x}=0.03\) into the formula, we have \(100\frac{\Delta f}{f}=100\times3\times0.03 = 9\).

Answer:

\(9\)