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.
-$$f ( x ) = x + x ^ { 2 } , x _ { 0 } = 3 , d x = 0.04$$
A) 0.11264
B) 0.0016
C) 0.05632
D) 0.0032

Question

Quizplus · Accepted Answer

The approximation error $| \Delta f - d f |$ can be found by first calculating the actual change in $f$, $\Delta f = f(x_0 + dx) - f(x_0)$, and the differential $df = f'(x_0)dx$, then finding the absolute difference between these two values. For $f(x) = x + x^2$, $f'(x) = 1 + 2x$. At $x_0 = 3$, $f'(3) = 1 + 2(3) = 7$. So, $df = 7dx = 7(0.04) = 0.28$. Calculating $\Delta f$, $f(3.04) = 3.04 + (3.04)^2 = 3.04 + 9.2416 = 12.2816$, and $f(3) = 3 + 9 = 12$, so $\Delta f = 12.2816 - 12 = 0.2816$. The approximation error is $| \Delta f - d f | = |0.2816 - 0.28| = 0.0016$.

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