The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$. Find the approximation error $| \Delta f - d f |$. Round your answer, if appropriate.
-$$f ( x ) = \frac { 1 } { x } , x _ { 0 } = 3 , d x = 0.4$$
A) 0.16732
B) 0.08366
C) 0.34379
D) 0.00522

Question

Quizplus · Accepted Answer

The approximation error $| \Delta f - d f |$ can be found by comparing the actual change in the function, $\Delta f$, to the differential, $df$. For $f(x) = \frac{1}{x}$, the differential $df$ can be calculated using the derivative $f'(x) = -\frac{1}{x^2}$ and the formula $df = f'(x_0)dx$. Given $x_0 = 3$ and $dx = 0.4$, we find:$$df = -\frac{1}{x_0^2}dx = -\frac{1}{3^2}(0.4) = -\frac{1}{9}(0.4) = -\frac{4}{90} = -\frac{2}{45}.$$The actual change $\Delta f = f(x_0 + dx) - f(x_0) = f(3.4) - f(3) = \frac{1}{3.4} - \frac{1}{3}.$Calculating these values:$$df = -\frac{2}{45} \approx -0.04444,$$$$\Delta f = \frac{1}{3.4} - \frac{1}{3} = \frac{1}{3.4} - \frac{1}{3} \approx 0.2941 - 0.3333 = -0.0392.$$The approximation error is:$$| \Delta f - df | = |-0.0392 - (-0.04444)| \approx |0.00524|.$$Rounding to five decimal places gives an approximation error of approximately 0.00522, which corresponds to choice D.

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The Function $f ( x )$ Changes Value When $x$ Changes From $x _ { 0 }$