Taylor Polynomials
A graphical introduction
Approximating f ( x)  ( x  1) e
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Best first order (linear)
approximation at x=0.
OZ calls this straight line
function P1(x).
Note: f (0)=P1(0) and
f’ (0) = P’1(0).
 x2
Approximating f ( x)  ( x  1) e
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Best second order
(quadratic) approximation
at x=0.
OZ calls this quadratic
function P2(x).
Note: f (0)=P2(0),
f ’ (0) = P’2 (0), and
f ‘’ (0) = P’’2 (0).
 x2
Approximating f ( x)  ( x  1) e
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Best third order (cubic)
approximation at x=0.
OZ calls this cubic
function P3(x).
Note: f (0)=P3(0),
f ’ (0) = P’3 (0),
f ’ (0) = P’’3 (0), and
f ‘’’ (0) = P’'’3 (0).
 x2
Approximating f ( x)  ( x  1) e
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Best sixth order
approximation at x=0.
OZ calls this function
P6(x).
P6 “matches” the value
of f and its first six
derivatives at x = 0.
 x2
Approximating f ( x)  ( x  1) e
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Best eighth order
approximation at x=0.
OZ calls this function
P8(x).
P8 “matches” the value
of f and its first eight
derivatives at x = 0.
 x2
Approximating f ( x)  ( x  1) e
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Best tenth order
approximation at x=0.
This function is P10(x).
 x2
Approximating f ( x)  ( x  1) e
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Best hundredth order
approximation at x=0.
This function is P100(x).
Notice that we cannot
see any difference
between f and P100 on
the interval [-3,3].
 x2
Approximating f ( x)  ( x  1) e
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Best hundredth order
approximation at x=0.
This function is P100(x).
But what about [-6,6]?
 x2
Approximating f ( x)  ( x  1) e
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Best hundredth order
approximation at x=0.
This function is P100(x).
But what about [-6,6]?
 x2
Approximating f ( x)  ( x  1) e
 x2
Compare Different Centers
Third order approximation at x=0
Third order approximation at x = -1
Taylor Polynomial
Approximations
Three Themes
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Derivative matching as a means to good and
better approximation.
Can we find (Taylor) polynomials that do what
we want?
Approximating closely related functions by
similarly related polynomials.