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.
-$$f ( x ) = x ^ { 2 } - x , x _ { 0 } = 5 , d x = 0.04$$
A) 0.0032
B) 0.0016
C) 0.2016
D) 0.2

Question

Quizplus · Accepted Answer

Using the formula for the approximation error:
$$|\Delta f - d f| = |f(x_0 + d x) - f(x_0) - d f| = |f'(x_0)d x - \frac{1}{2}f''(x_0)(d x)^2|$$
where $f'(x)$ and $f''(x)$ denote the first and second derivatives of $f(x)$, respectively.

Plugging in the given values, we have:
$$|\Delta f - d f| = |(10)(0.04) - \frac{1}{2}(2)(0.04)^2| = 0.0016$$
Rounding to four significant digits, we get $\boxed{\textbf{(B) }0.0016}$ as the answer.

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