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{SECT 0 {PARA 257 "" 0 "" {TEXT -1 49 "A Quadratic Approximation for t
he Cosine Function" }}{PARA 258 "" 0 "" {TEXT -1 18 "Jim Herod, Retire
d" }}{PARA 259 "" 0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 260 "" 0 ""
{TEXT -1 21 "herod@math.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 212 " This worksheet is concerned with g
etting a quadratic approximations for the cosine function on the inter
val [-/2, /2]. After all, the graph of the cosine function does look l
ike a quadratic on that interval." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 27 "plot(cos(x),x=-Pi/2..Pi/2);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1
{PARA 3 "" 0 "" {TEXT -1 29 "Method 1: Choose three points" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 207 " One way to
think of getting a quadratic approximation for the cosine function is
to choose three points and find the quadratic that exactly goes throu
gh those three points. We choose the three points [-" }{XPPEDIT 18 0 "
Pi;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT 273 20 "/2, 0], [0,1], and [" }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT 274 60 "/2,0]. All
three of these lie on the graph of the quadratic." }{TEXT 258 31 "\n \+
We seek the quadratic -- " }{XPPEDIT 18 0 "Q(x)=a*x^2+b*x+c" "6#/-
%\"QG6#%\"xG,(*&%\"aG\"\"\"*$F'\"\"#F+F+*&%\"bGF+F'F+F+%\"cGF+" }
{TEXT -1 22 " -- determined by the " }{TEXT 259 26 "conditions that Q(
0)=1, Q(" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT 275 14
"/2)=0, and Q(-" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT
276 65 "/2)=0. We only need to solve three equations with three unknow
ns:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 18 "Q:=x->a*x^2+b*x+c;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 45 "solve(\{Q(0)=1,Q(Pi/2)=0,Q(-Pi/2)=0\},\{a,b,c\});" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{PARA 0 ""
0 "" {TEXT -1 123 " We name this first of the five quadratics we h
ave computed and superimpose its graph with that of the cosine functio
n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Q1:=x->a*x^2+b*x+c;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(\{cos(t),Q1(t)\},t=-P
i/2..Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a:='a': b:='
b': c:='c':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1
{PARA 3 "" 0 "" {TEXT -1 28 "Method 2: Choose five points" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 256 " We present \+
an alternate method to that above for determing a quadratic approximat
ion for cos(x). Choose more than three points on the graph of the cos
ine function and find the quadratic by getting the \"least squared\" s
olution. We know the cosine of -" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 0 "" }{TEXT 277 7 "/4 and " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 0 "" }{TEXT 278 115 "/4. We add these two points to the thre
e points in Method 1. This will make five points to determine a quadra
tic.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "d1:=-Pi/2: d2:=-Pi
/4: d3:= 0: d4:=Pi/4: d5:=Pi/2:" }}}{PARA 0 "" 0 "" {TEXT 261 136 "We \+
get the least squares solution for these five equations with three unk
nowns. The statistics package contains a least squares routine." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
12 "with(stats):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "xcord:=
[d1,d2,d3,d4,d5];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ycord:
=[cos(d1),cos(d2),cos(d3),cos(d4),cos(d5)];" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 53 "fit[leastsquare[[x,y],y=a*x^2+b*x+c]]([xcord,ycord
]);" }}}{PARA 0 "" 0 "" {TEXT -1 96 " We assign the result of the \+
fit to Q2 and compare the graphs of Q2 and the cosine function." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Q2:=unapply(rhs(%),x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(\{cos(t),Q2(t)\},t=-Pi/
2..Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA
3 "" 0 "" {TEXT -1 31 "Method 3: A close hug at x = 0." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 338 " The classical
quadratic approximation for the cosine function is one that fits the \+
graph of the cosine function best at x = 0. This method is the \"natur
al\" one from the perspective of differential calculus: this quadratic
has the same value, the same first derivative, and the same second de
rivative at 0 as the cosine function does." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "Q3:=t->a*t^2+b*t+c;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 87 "solve(\{Q3(0)=cos(0), D(Q3)(0)=D(cos)(0),\n (D@
@2)(Q3)(0)=(D@@2)(cos)(0)\},\{a,b,c\});" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 46 "assign(%);\nplot(\{cos(t),Q3(t)\},t=-Pi/2..Pi/2);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a:='a'; b:='b'; c:='c';" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 66 "Method 4: The area between the two graphs is as small as \+
possible." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
5 " " }{TEXT 263 98 "In Method 4, we ask that the area between the
curves should be small. We ask that the integral of " }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " " }
{XPPEDIT 18 0 "(cos(s)-(a*x^2+b*x+c))^2" "6#*$,&-%$cosG6#%\"sG\"\"\",(
*&%\"aGF)*$%\"xG\"\"#F)F)*&%\"bGF)F.F)F)%\"cGF)!\"\"F/" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 105 "should be as small
as possible. We compute this integral, and recognize it as a function
of a, b, and c.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "h:=(a,
b,c)->int((cos(x) - (a*x^2 + b*x +c))^2,x=-Pi/2..Pi/2);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "h(a,b,c);" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 265 183 "The probl
em is to find a, b, and c that make this value as small as possible. \+
To find such a point, we take the derivatives with respect to a, b, an
d c and find where these are zero." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 80 "solve(\{diff(h(a,b,c),a)=0,diff(h(a,b,c),b)=0,\n \+
diff(h(a,b,c),c)=0\},\{a,b,c\});" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
19 "Q4:=x->a*x^2+b*x+c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "
plot(\{Q4(x),cos(x)\},x=-Pi/2..Pi/2);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 23 "a:='a': b:='b': c:='c':" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3
"" 0 "" {TEXT -1 80 "Method 5: The quadratic has the same first three \+
moments as the cosine function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 266 159 " In this last method, we find the q
uadratic which has the first three moments the same as the first three
moments of the cosine function on the interval [-" }{XPPEDIT 18 0 "Pi
;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT 279 4 "/2, " }{XPPEDIT 18 0 "Pi;" "
6#%#PiG" }{TEXT -1 0 "" }{TEXT 280 4 "/2]." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 223 "solve(\{int(cos(x),x=-Pi/2..Pi/2) =\n int(a*x^2+
b*x+c,x=-Pi/2..Pi/2),\nint(x*cos(x),x=-Pi/2..Pi/2) = \n int(x*(a*x^2
+b*x+c),x=-Pi/2..Pi/2),\nint(x^2*cos(x),x=-Pi/2..Pi/2) = \n int(x^2*
(a*x^2+b*x+c),x=-Pi/2..Pi/2)\},\{a,b,c\});" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 55 "Q5:=x->a*x^2+b*x+c;\nplot(\{Q5(x),cos(x)\},x=-Pi/2..Pi/2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "Remembering the Hausdor
f Moment Problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 89 " The Method V for finding an approximation for the co
sine function reminds us of the " }{TEXT 270 23 "Hausdorf Moment Probl
em" }{TEXT -1 58 ". This classical problem addresses the following re
quest:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1
31 "Characterize those sequences \{ " }{XPPEDIT 18 0 "c[n]" "6#&%\"cG6
#%\"nG" }{TEXT -1 71 " \} for which there is one and only one continuo
us function p(x) so that" }}{PARA 0 "" 0 "" {TEXT -1 50 " \+
" }{XPPEDIT 18 0 "c[n]" "6#&%\"cG
6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(x^n*p(x),x=a..b)" "6#-%
$intG6$*&)%\"xG%\"nG\"\"\"-%\"pG6#F(F*/F(;%\"aG%\"bG" }{TEXT -1 1 "."
}}{PARA 0 "" 0 "" {TEXT -1 72 " A reference for these ideas could \+
be found in Shohat and Tamarkin, " }{TEXT 272 18 "The Moment Problem"
}{TEXT -1 2 ", " }{TEXT 271 20 "Mathematical Surveys" }{TEXT -1 95 ", \+
AMS Publications, 1943, 1950 (revised), and 1963. (Library of Congres
s Listing: QA295.S55)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 323 " In what follows, we compute the moments for th
e cosine function over the interval [0, 4 ] and approximate this cosin
e function on that interval with a polynomial of degree N. The polynom
ial will have the same N+1 moments as the cosine function. To give per
spective, we compare the Taylor polynomial of the same degree." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 ""
{TEXT -1 32 "We use a polynomial of degree N." }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 5 "N:=6;" }}}{PARA 0 "" 0 "" {TEXT -1 51 "The moment
s are computed over the interval [0, 4 ]." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 57 "for i from 0 to N do\nc||i:=int(x^i*cos(x),x=0..4*Pi)
;\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 48 "A polynomial p(x) of degree N \+
with coefficients " }{XPPEDIT 18 0 "b[n]" "6#&%\"bG6#%\"nG" }{TEXT -1
12 " is defined." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p:=x->su
m('b||n'*x^n,n=0..N);" }}}{PARA 0 "" 0 "" {TEXT -1 136 "The requiremen
t that this polynomial should have the same N + 1 moments as the cosin
e function enables us to compute the coefficients. " }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 71 "fsolve(\{seq(c||m=int(x^m*p(x),x=0..4*Pi)
,m=0..N)\},\n\{seq(b||i,i=0..N)\});" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 10 "assign(%);" }}}{PARA 0 "" 0 "" {TEXT -1 66 "For compa
rison, we compute the Taylor polynomial for cosine at 2 ." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "series(cos(x),x=2*Pi,N+1);\nconvert
(%,polynom);\nT:=unapply(%,x);" }}}{PARA 0 "" 0 "" {TEXT -1 102 "Final
ly, we compare the three graphs to see how well the polynomial made fr
om the moments makes a fit." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
43 "plot(\{T(x),cos(x),p(x)\},x=0..4*Pi,y=-2..2);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT
1 {PARA 3 "" 0 "" {TEXT -1 30 "Trying to Help Taylor's Series" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 332 " One
might complain in that last graph that the Taylor's series was expand
ed about the wrong point -- that if the series were expanded about som
e point other than the midpoint 2 , then the graph of the Taylor polyn
omial might make a better approximation of the interval [0, 4 ] than t
he Hausdorf polynomial. We investigate this." }}{PARA 0 "" 0 "" {TEXT
-1 16 " We measure " }{TEXT 267 9 "closeness" }{TEXT -1 132 " usin
g the integral of the square of the difference between the two functio
ns. Thus, we expand the Taylor series about an arbitrary " }{XPPEDIT
18 0 "a" "6#%\"aG" }{TEXT -1 12 " and choose " }{XPPEDIT 18 0 "a" "6#%
\"aG" }{TEXT -1 42 " so that integral is as small as possible." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "series(cos(x),x=a,N+1):\ncon
vert(%,polynom):\nT:=unapply(%,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 169 " Surely the output of the next comp
utation is so long that we learn nothing from reading it. Take the out
put to define the measure of the closeness of T and call it " }
{XPPEDIT 18 0 "mcT" "6#%$mcTG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 31 "int((cos(x)-T(x))^2,x=0..4*Pi):" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "mcT:=unapply(%,a):" }}}{PARA 0 ""
0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 47 " We would lik
e to know where this function " }{XPPEDIT 18 0 "mcT" "6#%$mcTG" }
{TEXT -1 95 " takes on its minimum value. The usual idea is to take th
e derivative and set it equal to zero." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 15 "diff(mcT(a),a):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 142 " It surely will not be easy to see w
here this derivative is zero. We draw a graph and determine a numerica
l approximation to the solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 23 "plot(mcT(a),a=0..4*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 30 "plot(mcT(a),a=5..6,y=0..1000);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 36 "fsolve(diff(mcT(a),a)=0,a,5.4..5.6);" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " This last numbe
r may be rational multiple of Pi." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 12 "evalf(%/Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}
{PARA 0 "" 0 "" {TEXT -1 97 " For comparison with the Hausdorf pol
ynomial, we evaluate the Taylor polynomial about 7 / 4." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "series(cos(x),x=7*Pi/4,N+1):\nconve
rt(%,polynom):\nT:=unapply(%,x);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 30 "plot(\{cos(x),T(x)\},x=0..4*Pi);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT
1 {PARA 3 "" 0 "" {TEXT -1 8 "Summary:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 169 "This work sheet has reviewed a colle
ction of approximations to the cosine function. Also, it has served as
a reminder, or introduction, to the mystery and wonder of the " }
{TEXT 269 23 "Hausdorf Moment Problem" }{TEXT 268 1 "." }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Exercise for
the student." }}{PARA 0 "" 0 "" {TEXT -1 81 "Find three different cub
ic approximations to the sine function on the interval [ " }{XPPEDIT
18 0 "-Pi/2" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 3 ", " }
{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 " ]." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
3 "?||" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0"
0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }