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Unformatted text preview: Chapt. 1 Quantities, Units, Conversions, Measurement, Signiﬁcant Figures, and Uncertainties 13
suﬃciently small (x − x0 ). The smaller (x − x0 ) 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 x0 , and so truncates the Taylor’s series to ﬁnd an simple approximate expression for the
neighborhood of x0 . For example f (x) = f (x0 ) , f (x0 ) + (x − x0 )f ′ (x0 ) , 0th order or
1st order or
linear approximation; 2 f (x0 ) + (x − x0 )f ′ (x0 ) + (x − x0 ) f ′′ (x0 ) , 2 2nd order or
quadratic approximation. The 0th order expansion, approximates the function as a constant which is the function value at
x0 ; the 1st order expansion approximates the function by a line that is tangent to the function
at x0 ; the 2nd order expansion approximates the function by a quadratic that is tangent to the
function x0 .
a) Draw a general function on the Cartesian plane and at a general point x0 schematically
show how the 0th, 1st, and 2nd order approximations to the function behave.
b) Taylor expand
f (x) = 1
1+x to 2nd order about x0 = 0. (Here x − x0 is just x, of course.)
c) Taylor expand √ 1+x f (x) = √ 1
1+x f (x) =
to 2nd order about x = 0.
d) Taylor expand to 2nd order about x = 0.
e) Taylor expand
f (x) = sin(x)
to 3rd order about x = 0. Note that x must be in radians for the Taylor’s expansion to
work for the trigonometric functions. This is because the derivatives of these functions are
derived using radians.
f) Taylor expand
f (x) = cos(x)
to 2nd order about x = 0. ...
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- Fall '06