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.
-$$f ( x ) = x ^ { 2 } , x _ { 0 } = 5 , d x = 0.06$$
A) 0.0036
B) 0.5436
C) 0.0018
D) 0.0072

Question

Quizplus · Accepted Answer

Using the formula for linear approximation, we have:
$$\Delta f \approx f'(x_0)dx = 2x_0dx = 2(5)(0.06) = 0.6$$
$$df \approx f(x_0 + dx)-f(x_0) = (5+0.06)^2 - 5^2 = 0.6 + 0.06^2 \approx 0.636$$
Therefore, the approximation error is:
$$\Delta f - df \approx 0.6 - 0.636 = -0.036$$
Rounding to the nearest hundredth, we get an error of 0.00 (since the absolute value of -0.036 is less than 0.005). Thus, the best choice is A.

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