Chapt. 1 Quantities, Units, Conversions, Measurement, Significant Figures, and Uncertainties
13
sufficiently small (
x
−
x
0
). The smaller (
x
−
x
0
) is the less important the higher order terms. In
physics (and other fields no doubt), one is often interested in the behavior near some important
point
x
0
, and so truncates the Taylor’s series to find an simple approximate expression for the
neighborhood of
x
0
. For example
f
(
x
) =
f
(
x
0
)
,
0th order or
constant approximation;
f
(
x
0
) + (
x
−
x
0
)
f
′
(
x
0
)
,
1st order or
linear approximation;
f
(
x
0
) + (
x
−
x
0
)
f
′
(
x
0
) +
(
x
−
x
0
)
2
2
f
′′
(
x
0
)
,
2nd order or
quadratic approximation.
The 0th order expansion, approximates the function as a constant which is the function value at
x
0
; the 1st order expansion approximates the function by a line that is tangent to the function
at
x
0
; the 2nd order expansion approximates the function by a quadratic that is tangent to the
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 Fall '06
 Buchler
 Physics, Derivative, Leonhard Euler, Euler's formula, Taylor expand

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