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Macla= urin Expansions
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Given a "nice" function f(x) (e= .g. y =3D sin(x)) and x =3D a, we want to find a numerical method in calculating f(x)
for x around a.  
Macla= urin Expansions
------=_NextPart_01C97CE6.91C30810 Content-Location: file:///C:/23664A33/30MacLaurinExpansions_files/slide0211.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Given a "nice" function f(x) (e= .g. y =3D sin(x)) and x =3D a, we want to find a numerical method in calculating f(x)
for x around a.  
Macla= urin Expansions
It turns out that being "nice" in this case means that = the derivative exists. 
------=_NextPart_01C97CE6.91C30810 Content-Location: file:///C:/23664A33/30MacLaurinExpansions_files/slide0212.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Given a "nice" function f(x) (e= .g. y =3D sin(x)) and x =3D a, we want to find a numerical method in calculating f(x)
for x around a.  
Macla= urin Expansions
It turns out that being "nice" in this case means that the derivative exists.  The higher the derivative we <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>can find, the nicer the function is, the better the approximation we may achieve.  
------=_NextPart_01C97CE6.91C30810 Content-Location: file:///C:/23664A33/30MacLaurinExpansions_files/slide0213.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Given a "nice" function f(x) (e= .g. y =3D sin(x)) and x =3D a, we want to find a numerical method in calculating f(x)
for x around a.  
Macla= urin Expansions
It turns out that being "nice" in this case means that the derivative exists.  The higher the derivative we <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>can find, the nicer the function is, the better the approximation we may achieve.  
We will assume our functions are of the b= est kind,
i.e. they're infinitely differentiable an= d we will approximate the points around x =3D 0.
------=_NextPart_01C97CE6.91C30810 Content-Location: file:///C:/23664A33/30MacLaurinExpansions_files/slide0214.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Given a "nice" function f(x) (e= .g. y =3D sin(x)) and x =3D a, we want to find a numerical method in calculating f(x)
for x around a.  
Macla= urin Expansions
It turns out that being "nice" in this case means that the derivative exists.  The higher the derivative we <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>can find, the nicer the function is, the better the approximation we may achieve.  
We will assume our functions are of the b= est kind,
i.e. they're infinitely differentiable an= d we will approximate the points around x =3D 0.
Given such a nice functio= n, we'll try to find polynomials of higher and higher degree that give better and better approximation. <= /span>
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Given a "nice" function f(x) (e= .g. y =3D sin(x)) and x =3D a, we want to find a numerical method in calculating f(x)
for x around a.  
Macla= urin Expansions
It turns out that being "nice" in this case means that the derivative exists.  The higher the derivative we <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>can find, the nicer the function is, the better the approximation we may achieve.  
We will assume our functions are of the b= est kind,
i.e. they're infinitely differentiable an= d we will approximate the points around x =3D 0.
Given such a nice functio= n, we'll try to find polynomials of higher and higher degree that give better and better approximation.  We'll find the power series that is the same as the function in some cases.
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
=3D
pn= (x)
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
=3D
f(0)
pn= (x)
f(= 0)(0)
0!
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
f '(0= )x
=3D
f(0)
+
pn= (x)
f(= 0)(0)
0!
f(= 1)(0)
1!
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
f '(0= )x
f(= 2)(0)
+
2!
=3D
f(0)
+
x2=
pn= (x)
f(= 0)(0)
0!
f(= 1)(0)
1!
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
f '(0= )x
f(= 2)(0)
+
2!
=3D
f(0)
+
x2=
pn= (x)
f(= 0)(0)
0!
f(= 3)(0)
+
3!
x3= ..
f(= 1)(0)
1!
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
f '(0= )x
f(= 2)(0)
+
2!
=3D
f(0)
+
x2=
pn= (x)
f(= 0)(0)
0!
f(= 3)(0)
+
3!
x3= ..
f(= n)(0)
+
n!
xn=
f(= 1)(0)
1!
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
f '(0= )x
f(= 2)(0)
+
2!
=3D
f(0)
+
x2=
pn= (x)
f(= 0)(0)
0!
f(= 3)(0)
+
3!
x3= ..
f(= n)(0)
+
n!
xn=
f(= 1)(0)
1!
or  
pn(x)= =3D
Σ
k=3D0
n
xk=
k!
f(= k)(0)
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
f '(0= )x
f(= 2)(0)
+
2!
=3D
f(0)
+
x2=
pn= (x)
f(= 0)(0)
0!
f(= 3)(0)
+
3!
x3= ..
f(= n)(0)
+
n!
xn=
f(= 1)(0)
1!
or  
pn(x)= =3D
Σ
k=3D0
n
xk=
k!
f(= k)(0)
Thi= s is called the n'th (degree) Maclaurin polynomial (Mac-poly) for f(x). 
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
f '(0= )x
f(= 2)(0)
+
2!
=3D
f(0)
+
x2=
pn= (x)
f(= 0)(0)
0!
f(= 3)(0)
+
3!
x3= ..
f(= n)(0)
+
n!
xn=
f(= 1)(0)
1!
or  
pn(x)= =3D
Σ
k=3D0
n
xk=
k!
f(= k)(0)
Thi= s is called the n'th (degree) Maclaurin polynomial (Mac-poly) for f(x).  If we set n to be    , we get the <= /span>= Maclaurin series (Mac-series):<= /span>
∞
P(x) =3D
Σ
k=3D0
xk= .
k!
f(= k)(0)
∞
Macla= urine Expansions
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Given a function f(x) that is infinitely differentiable at
x =3D 0, we define the polynomial =
f '(0= )x
f(= 2)(0)
+
2!
=3D
f(0)
+
x2=
pn= (x)
f(= 0)(0)
0!
f(= 3)(0)
+
3!
x3= ..
f(= n)(0)
+
n!
xn=
f(= 1)(0)
1!
or  
pn(x)= =3D
Σ
k=3D0
n
xk=
k!
f(= k)(0)
Thi= s is called the n'th (degree) Maclaurin polynomial (Mac-poly) for f(x).  If we set n to be    , we get the <= /span>= Maclaurin series (Mac-series):<= /span>
∞
P(x) =3D
Σ
k=3D0
xk= .
k!
f(= k)(0)
∞
Macla= urine Expansions
We cal= l them the Mac-expansions of f(x).
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
Macla= urine Expansions
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
=3D
f(0)
pn= (0)
In other words,
Macla= urine Expansions
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
f  (0)
=3D
f(0)
+ #x = + #x2= + ..#xn-1|<= /span>x= =3D0 
pn= (0)
In other words,
=3D
pn= (0)
(1)<= /b>
(1)<= /b>
Macla= urine Expansions
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
f  (0)
=3D
f(0)
+ #x = + #x2= + ..#xn-1|<= /span>x= =3D0 =3D= f  (0)
pn= (0)
In other words,
=3D
pn= (0)
(1)<= /b>
(1)<= /b>
(1)=
Macla= urine Expansions
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
f  (0)
=3D
f(0)
+ #x = + #x2= + ..#xn-1|<= /span>x= =3D0 =3D= f  (0)
pn= (0)
In other words,
=3D
pn= (0)
(1)<= /b>
f  (0)
+ #x = + #x2= + ..#xn-2|<= /span>x= =3D0 
=3D
pn= (0)
(2)<= /b>
(2)<= /b>
(1)<= /b>
(1)=
Macla= urine Expansions
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
f  (0)
=3D
f(0)
+ #x = + #x2= + ..#xn-1|<= /span>x= =3D0 =3D= f  (0)
pn= (0)
In other words,
=3D
pn= (0)
(1)<= /b>
f  (0)
+ #x = + #x2= + ..#xn-2|<= /span>x= =3D0 =3D= f  (0)
=3D
pn= (0)
(2)<= /b>
(2)<= /b>
(2)=
(1)<= /b>
(1)=
Macla= urine Expansions
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
f  (0)
=3D
f(0)
+ #x = + #x2= + ..#xn-1|<= /span>x= =3D0 =3D= f  (0)
pn= (0)
In other words,
=3D
pn= (0)
(1)<= /b>
f  (0)
+ #x = + #x2= + ..#xn-2|<= /span>x= =3D0 =3D= f  (0)
=3D
pn= (0)
(2)<= /b>
(2)<= /b>
(2)=
(1)<= /b>
(1)=
and so= on, up to pn(0) =3D f  (0).
(n)=
(n)=
Macla= urine Expansions
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
f  (0)
=3D
f(0)
+ #x = + #x2= + ..#xn-1|<= /span>x= =3D0 =3D= f  (0)
pn= (0)
In other words,
=3D
pn= (0)
(1)<= /b>
f  (0)
+ #x = + #x2= + ..#xn-2|<= /span>x= =3D0 =3D= f  (0)
=3D
pn= (0)
(2)<= /b>
(2)<= /b>
(2)=
(1)<= /b>
(1)=
and so= on, up to pn(0) =3D f  (0).
(n)=
(n)=
In fact, pn(x) is the only degree n (or less) polyno= mial who= se values of the first n'th derivatives agree with those of f(x) at x =3D 0.
Macla= urine Expansions
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pn(x)= is a polynomial whose values of the first n'th = derivatives at x=3D0 are the sa= me as f(x).
f  (0)
=3D
f(0)
+ #x = + #x2= + ..#xn-1|<= /span>x= =3D0 =3D= f  (0)
pn= (0)
In other words,
=3D
pn= (0)
(1)<= /b>
f  (0)
+ #x = + #x2= + ..#xn-2|<= /span>x= =3D0 =3D= f  (0)
=3D
pn= (0)
(2)<= /b>
(2)<= /b>
(2)=
(1)<= /b>
(1)=
and so= on, up to pn(0) =3D f  (0).
(n)=
(n)=
In fact, pn(x) is the only degree n (or less) polyno= mial who= se values of the first n'th derivatives agree with those of f(x) at x =3D 0. In si= milar arguments, the Mac-series is the only power series that has all derivatives = agree with those of f(x) at x = =3D 0.
Macla= urine Expansions
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The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition. 
Macla= urine Expansions
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The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition.  Later, we will use these expansio= ns to calculate the expansions of oth= er functions (in stead of using the definition).
Macla= urine Expansions
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The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition.  Later, we will use these expansio= ns to calculate the expansions of oth= er functions (in stead of using the definition).
Example: A.  Find the Mac-expansion of
f(x) =3D 1 + x + x2 + x3 + x4 around x =3D 0.
Macla= urine Expansions
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The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition.  Later, we will use these expansio= ns to calculate the expansions of oth= er functions (in stead of using the definition).
Example: A.  Find the Mac-expansion of
f(x) =3D 1 + x + x2 + x3 + x4 around x =3D 0.
We need the derivatives of f(x):
Macla= urine Expansions
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The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition.  Later, we will use these expansio= ns to calculate the expansions of oth= er functions (in stead of using the definition).
Example: A.  Find the Mac-expansion of
f(x) =3D 1 + x + x2 + x3 + x4 around x =3D 0.
We need the derivatives of f(x):
f  (x) =3D 1 + 2x + 3x2 + 4x3  = 3;
(1)
Macla= urine Expansions
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The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition.  Later, we will use these expansio= ns to calculate the expansions of oth= er functions (in stead of using the definition).
Example: A.  Find the Mac-expansion of
f(x) =3D 1 + x + x2 + x3 + x4 around x =3D 0.
We need the derivatives of f(x):
f  (x) =3D 1 + 2x + 3x2 + 4x3 à f  (0) =3D 1. =
(1)
(1)
Macla= urine Expansions
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The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition.  Later, we will use these expansio= ns to calculate the expansions of oth= er functions (in stead of using the definition).
Example: A.  Find the Mac-expansion of
f(x) =3D 1 + x + x2 + x3 + x4 around x =3D 0.
We need the derivatives of f(x):
f  (x) =3D 1 + 2x + 3x2 + 4x3 à f  (0) =3D 1. =
(1)
(1)
f  (x) =3D 2 + 3*2x + 4*3x2 = 3;
(2)
Macla= urine Expansions
------=_NextPart_01C97CE6.91C30810 Content-Location: file:///C:/23664A33/30MacLaurinExpansions_files/slide0238.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition.  Later, we will use these expansio= ns to calculate the expansions of oth= er functions (in stead of using the definition).
Example: A.  Find the Mac-expansion of
f(x) =3D 1 + x + x2 + x3 + x4 around x =3D 0.
We need the derivatives of f(x):
f  (x) =3D 1 + 2x + 3x2 + 4x3 à f  (0) =3D 1. =
(1)
(1)
f  (x) =3D 2 + 3*2x + 4*3x2 à f  (0) =3D 2! =
(2)
(2)
Macla= urine Expansions
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The following are some examples of the Mac-polys and Mac-series of basic functio= ns calculated via the definition.  Later, we will use these expansio= ns to calculate the expansions of oth= er functions (in stead of using the definition).
Example: A.  Find the Mac-expansion of
f(x) =3D 1 + x + x2 + x3 + x4 around x =3D 0.
We need the derivatives of f(x):
f  (x) =3D 1 + 2x + 3x2 + 4x3 à f  (0) =3D 1. =
(1)
(1)
f  (x) =3D 2 + 3*2x + 4*3x2 à f  (0) =3D 2! =
(2)
(2)
f  (x) =3D 3*2 + 4*3*2x  = 3;
(3)
Macla= urine Expansions
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