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{SECT 0 {PARA 3 "" 0 "" {TEXT 259 55 "Approximation of functions using
 partitions of unity II" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone
, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 26.3.
2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}{PARA 0 "" 0 "" {TEXT 262 153 "In this worksheet we consider som
e more examples of families of functions which form partitions of unit
y, and which can be used to approximate functions." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Approximation usi
ng narrow \"bell shaped\" pulses II " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 268 1 "
n" }{TEXT -1 45 " be a positive integer and, for each integer " }
{TEXT 269 1 "k" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0<=k" "6#1\"\"!%
\"kG" }{XPPEDIT 18 0 "``<=n" "6#1%!G%\"nG" }{TEXT -1 20 ", define a fu
nction " }{XPPEDIT 18 0 "H[k,n];" "6#&%\"HG6$%\"kG%\"nG" }{TEXT -1 4 "
 by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "H[k,n](x) = P
IECEWISE([2*(1+n*x-k)^2/(1+(1+2*n*x-2*k)^2), (k-1)/n < x and x < k/n],
[2*(1-n*x+k)^2/(1+(1-2*n*x+2*k)^2), k/n < x and x < (k+1)/n],[0, other
wise]);" "6#/-&%\"HG6$%\"kG%\"nG6#%\"xG-%*PIECEWISEG6%7$*(\"\"#\"\"\"*
$,(F2F2*&F)F2F+F2F2F(!\"\"F1F2,&F2F2*$,(F2F2*(F1F2F)F2F+F2F2*&F1F2F(F2
F6F1F2F632*&,&F(F2F2F6F2F)F6F+2F+*&F(F2F)F67$*(F1F2*$,(F2F2*&F)F2F+F2F
6F(F2F1F2,&F2F2*$,(F2F2*(F1F2F)F2F+F2F6*&F1F2F(F2F2F1F2F632*&F(F2F)F6F
+2F+*&,&F(F2F2F2F2F)F67$\"\"!%*otherwiseG" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "n :
= 'n':\nH := (k,n,x) -> piecewise(x>(k-1)/n and x<=k/n,2*(1+n*x-k)^2/(
1+(1+2*n*x-2*k)^2),\nx>k/n and x<(k+1)/n,2*(1-n*x+k)^2/(1+(1-2*n*x+2*k
)^2),0):\n`H(k,n,x)`=H(k,n,x);\nplot(H(3,5,x),x=0..1,color=brown,thick
ness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)H(k,n,x)G-%*PIECEWISEG6
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1,&F0F,*&F1F,F/F2F2F>7$,$*(F+F,,(F,F,F.F2F1F,F+,&F,F,*$),(F,F,*(F+F,F/
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1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Suppose that we are given
 a real valued " }{TEXT 260 10 "continuous" }{TEXT -1 34 " function de
fined on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "Then the function " }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "g[n](x) = Sum(f(k/n)
*H[k,n](x),k = 0 .. n);" "6#/-&%\"gG6#%\"nG6#%\"xG-%$SumG6$*&-%\"fG6#*
&%\"kG\"\"\"F(!\"\"F4-&%\"HG6$F3F(6#F*F4/F3;\"\"!F(" }{TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 43 "provides an approximation for the functi
on " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the int
erval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 16 "Example \+
1:  n=10" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 39 "The graph below compares the graphs of " }
{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y = g[n](x);" "6#/%\"yG-&%\"gG6#%\"nG6#%\"xG" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 134 "f := x -> exp(-x^2);\nn := 10;\ng := (n,x) -> Sum(f(
k/n)*H(k,n,x),k=0..n);\nplot([f(x),g(n,x)],x=0..1,thickness=[1,2],colo
r=[black,red]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6
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11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "
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l#F^`l7$$\"3e&[[)f2Og**Fgy$\"3!**\\6`#)yPG#F^`l7$$\"3Q!**)*)RQdt**Fgy$
\"3QqAd!eQ.p\"F^`l7$$\"3yUU#*z.=!)**Fgy$\"3WPF0hllA8F^`l7$$\"3?&\\\\*>
py')**Fgy$\"31JB\"=Bn+:*F07$$\"3gZZ(*fMR$***Fgy$\"3zR1\"ozS`s%F07$$\"
\"\"F)F(-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F
bjal-%%VIEWG6$;F(Feial%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 50 "Approximation using wider \"bell shaped\" pulses II " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 58 "In the previous example a maximum of two terms in the sum " }
{XPPEDIT 18 0 "Sum(f(k/n)*H[k,n](x),k = 0 .. n);" "6#-%$SumG6$*&-%\"fG
6#*&%\"kG\"\"\"%\"nG!\"\"F,-&%\"HG6$F+F-6#%\"xGF,/F+;\"\"!F-" }{TEXT 
-1 32 " are non-zero for each value of " }{TEXT 270 1 "x" }{TEXT -1 
17 " in the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 130 "In this example we cons
truct \"bell shaped\" pulses which are more \"spread out\" so that mor
e terms can be nonzero for each value of " }{TEXT 271 1 "x" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "
Let n be a positive integer and, for each integer " }{TEXT 272 1 "k" }
{TEXT -1 6 " with " }{XPPEDIT 18 0 "-1 <= k;" "6#1,$\"\"\"!\"\"%\"kG" 
}{XPPEDIT 18 0 "`` <= n+1;" "6#1%!G,&%\"nG\"\"\"F'F'" }{TEXT -1 20 ", \+
define a function " }{XPPEDIT 18 0 "H[k,n];" "6#&%\"HG6$%\"kG%\"nG" }
{TEXT -1 4 " by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "H
[k,n](x) = PIECEWISE([(2+n*x-k)^2/(4+4*(1+n*x-k)^2), (k-2)/n < x and x
 < k/n],[(2-n*x+k)^2/(4+4*(1-n*x+k)^2), k/n < x and x < (k+2)/n],[0, o
therwise]);" "6#/-&%\"HG6$%\"kG%\"nG6#%\"xG-%*PIECEWISEG6%7$*&,(\"\"#
\"\"\"*&F)F3F+F3F3F(!\"\"F2,&\"\"%F3*&F7F3*$,(F3F3*&F)F3F+F3F3F(F5F2F3
F3F532*&,&F(F3F2F5F3F)F5F+2F+*&F(F3F)F57$*&,(F2F3*&F)F3F+F3F5F(F3F2,&F
7F3*&F7F3*$,(F3F3*&F)F3F+F3F5F(F3F2F3F3F532*&F(F3F)F5F+2F+*&,&F(F3F2F3
F3F)F57$\"\"!%*otherwiseG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Now define " }}{PARA 256 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "K[k,n] = PIECEWISE([H[-1,n](x)+H[0,n
](x), k = 0],[H[k,n](x), 1 <= k and k <= n-1],[H[n,n](x)+H[n+1,n](x), \+
k = n]);" "6#/&%\"KG6$%\"kG%\"nG-%*PIECEWISEG6%7$,&-&%\"HG6$,$\"\"\"!
\"\"F(6#%\"xGF3-&F06$\"\"!F(6#F6F3/F'F:7$-&F06$F'F(6#F631F3F'1F',&F(F3
F3F47$,&-&F06$F(F(6#F6F3-&F06$,&F(F3F3F3F(6#F6F3/F'F(" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "n := 'n':\nH := (k,n,x) -> \+
piecewise(x>(k-2)/n and x<=k/n,(2+n*x-k)^2/(4+4*(1+n*x-k)^2),\nx>k/n a
nd x<(k+2)/n,(2-n*x+k)^2/(4+4*(1-n*x+k)^2),0):\n`H(k,n,x)`=H(k,n,x);\n
\nK := proc(k,n,x)\n   if k>0 and k<n then H(k,n,x)\n   elif k=0 then \+
H(-1,n,x)+H(0,n,x)\n   elif k=n then H(n,n,x)+H(n+1,n,x)\n   else FAIL
 end if\nend proc;\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)H(k,n,x)G
-%*PIECEWISEG6%7$*&,(\"\"#\"\"\"*&%\"nGF,%\"xGF,F,%\"kG!\"\"F+,&\"\"%F
,*&F3F,),(F,F,F-F,F0F1F+F,F,F132,&*&,&F0F,F+F1F,F.F1F,F/F1\"\"!1,&F/F,
*&F0F,F.F1F1F<7$*&,(F+F,F-F1F0F,F+,&F3F,*&F3F,),(F,F,F-F1F0F,F+F,F,F13
2,&F/F1F?F,F<2,&F/F,*&,&F0F,F+F,F,F.F1F1F<7$F<%*otherwiseG" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%\"KGf*6%%\"kG%\"nG%\"xG6\"F*F*@)32\"\"!9$2
F/9%-%\"HG6%F/F19&/F/F.,&-F36%!\"\"F1F5\"\"\"-F36%F.F1F5F;/F/F1,&-F36%
F1F1F5F;-F36%,&F1F;F;F;F1F5F;%%FAILGF*F*F*" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The set of functions " }
{XPPEDIT 18 0 "\{E[0,n], E[1,n], ` . . . `, E[n,n]\};" "6#<&&%\"EG6$\"
\"!%\"nG&F%6$\"\"\"F(%(~.~.~.~G&F%6$F(F(" }{TEXT -1 45 " can be plotte
d together in the same picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "n := 8;\nplot([seq(K(k,n,x),
k=0..n)],x=0..1,\n  color=[seq(COLOR(HUE,k/(n+1)),k=0..n)],thickness=2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\")" }}{PARA 13 "" 1 "" 
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "For each \+
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}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Suppose that we are given
 a real valued " }{TEXT 260 10 "continuous" }{TEXT -1 34 " function de
fined on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "Then the function " }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "g[n](x) = Sum(f(k/n)
*K[k,n](x),k = 0 .. n);" "6#/-&%\"gG6#%\"nG6#%\"xG-%$SumG6$*&-%\"fG6#*
&%\"kG\"\"\"F(!\"\"F4-&%\"KG6$F3F(6#F*F4/F3;\"\"!F(" }{TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 43 "provides an approximation for the functi
on " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the int
erval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 16 "Example \+
1:  n=10" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 39 "The graph below compares the graphs of " }
{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y = g[n](x);" "6#/%\"yG-&%\"gG6#%\"nG6#%\"xG" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 136 "f := x -> exp(-x^2);\nn := 10;\ng := (n,x) -> Sum(f(
k/n)*'K(k,n,x)',k=0..n);\nplot([f(x),g(n,x)],x=0..1,thickness=[1,2],co
lor=[black,red]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6
\"6$%)operatorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%$SumG6$
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)em'yKH*F,7$$\"3\"30NJ()z&>\"*Ff\\l$!3uJ$*Q\"\\?a4\"F37$$\"30t>L%y#zD
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es1'=#Q\"*Ff\\l$!3/,Y#3B!>'R\"F37$$\"3yRF#z^JW9*Ff\\l$!3)))*4x,]o99F37
$$\"3#4m>\"HWk]\"*Ff\\l$!3ORjB&='H!R\"F37$$\"3;$e;.Mdo:*Ff\\l$!3nCLS%
\\MfK\"F37$$\"3+#QE&GP)*o\"*Ff\\l$!3/QtQ**>(*46F37$$\"3#3=On656=*Ff\\l
$!3rCSvN8S:%)F,7$$\"3mzf%\\]OK>*Ff\\l$!3Y)*eh'eT3L'F,7$$\"3\\yd:$*GO0#
*Ff\\l$!3;JFqB5_$o&F,7$$\"3s1k!f-3&=#*Ff\\l$!3)G5!)z\"fNcMF,7$$\"3&\\.
d'eJlJ#*Ff\\l$!3Jub#e>%\\.SFjjq7$$\"3=jwS\"H)zW#*Ff\\l$\"3*GJk61_V.#F,
7$$\"3U\"HeTUVzD*Ff\\l$\"3\\<2hoEQ;HF,7$$\"3'yvE(=wJp#*Ff\\l$\"3]2`e/O
9Q@F,7$$\"3UD_H8=p!G*Ff\\l$\"3#G0Ll5EZ[#Fiar7$$\"3=f%z0\"*yjG*Ff\\l$!3
6/ff!>#GH9F,7$$\"3'Hpjy+m?H*Ff\\l$!3?jC!\\*GGoIF,7$$\"3uEz90Jv(H*Ff\\l
$!3eD=mtkO@[F,7$$\"3_g@V--W.$*Ff\\l$!3*os#fxp'Hh'F,7$$\"3*el=//s*4$*Ff
\\l$!3K\"3k%3R]=')F,7$$\"3E^^SyQ];$*Ff\\l$!37`&Quz'*f/\"F37$$\"3_X;R;d
.B$*Ff\\l$!3K\"\\2]ezL?\"F37$$\"3*39yVbn&H$*Ff\\l$!3W83*)pzBD8F37$$\"3
EOYO#R*4O$*Ff\\l$!3#eeV*4J(\\S\"F37$$\"3_I6NI7jU$*Ff\\l$!3l_@zx+kQ9F37
$$\"3!fiP$oI;\\$*Ff\\l$!3Z=jyViMD9F37$$\"3E@TK1\\pb$*Ff\\l$!3)='HqX*Qv
O\"F37$$\"3!\\%)=&[a(zO*Ff\\l$!3eNP^t_*=;\"F37$$\"3_oNr!*fD!Q*Ff\\l$!3
kl_3#*)>P&*)F,7$$\"39#H3H`ODR*Ff\\l$!3LI?mO$Q;!oF,7$$\"3z:I5vq\"[S*Ff
\\l$!3+BFQ#))HE8'F,7$$\"3#)QEl+!zpT*Ff\\l$!34RB<dT.<UF,7$$\"3&=E-i#49H
%*Ff\\l$!3\"HmEg_h#Q9F,7$$\"3*[)=v^GIT%*Ff\\l$\"3jJF\"R*[A;5F,7$$\"3/4
:IxZY`%*Ff\\l$\"3?.m9#e4%>BF,7$$\"3)*pPvxjjl%*Ff\\l$\"3Gr/!oxDf0#F,7$$
\"3%4.1#yz!yZ*Ff\\l$\"3Wk%pUR>K=#Fjjq7$$\"3'>;K%yP*Q[*Ff\\l$!3#zYZ5'[r
$>\"F,7$$\"3)=He'y&z**[*Ff\\l$!3))eOMa6!o%GF,7$$\"3!=U%))y`1'\\*Ff\\l$
!3uU9'G&)4\"fYF,7$$\"3&Gb5\"z6:-&*Ff\\l$!3&oIqnz26a'F,7$$\"3<NY65$e$3&
*Ff\\l$!3l>!*HwBEO%)F,7$$\"3g=(=6WlX^*Ff\\l$!31;23j>1@5F37$$\"3/-G7sDx
?&*Ff\\l$!3;:)G`e(Qx6F37$$\"3O%)o7.(zp_*Ff\\l$!3)[p\\tq2ZI\"F37$$\"3om
48Mo=L&*Ff\\l$!3=m:E?!4nR\"F37$$\"36]]8lRRR&*Ff\\l$!3Q*pE_`e!\\9F37$$
\"3$=4P1`(\\U&*Ff\\l$!3Ep\"*)*>ajf9F37$$\"3aL\"Rh4,ca*Ff\\l$!3/#)>%3NR
(f9F37$$\"3;u6khYq[&*Ff\\l$!3Qlu3EH_\\9F37$$\"3)e@VrA3=b*Ff\\l$!35!Qb_
[6$H9F37$$\"3u_2!p8n\\c*Ff\\l$!3#****=oI*\\Z7F37$$\"3q!Hem/E\"y&*Ff\\l
$!3T+EHzmfG(*F,7$$\"3mGeTc\\G\"f*Ff\\l$!3)yCx[p5fJ(F,7$$\"3lmL<mQW/'*F
f\\l$!3.;o2JNXLlF,7$$\"30bh&yH4lh*Ff\\l$!3m)\\5xDjOr%F,7$$\"3bW*Q&HZdG
'*Ff\\l$!3XkB/Up;1?F,7$$\"31M<Ah,kS'*Ff\\l$\"3/(G!o5Pj_VFjjq7$$\"3YAX!
Hf0Fl*Ff\\l$\"3kO/56dl'z\"F,7$$\"3+G1DBdhk'*Ff\\l$\"3A1!*)pk`vm\"F,7$$
\"3cLnf`e_w'*Ff\\l$\"30!z7/#G;MTFiar7$$\"3CSG%R)fV)o*Ff\\l$!3Yj;(\\fw-
z#F,7$$\"3yX*)G9hM+(*Ff\\l$!3k$**o\\XOtH'F,7$$\"3215&4)>r1(*Ff\\l$!3VY
YWGM;B#)F,7$$\"3OmIhZy28(*Ff\\l$!36z*Ho*G505F37$$\"3mE^F9PW>(*Ff\\l$!3
c/$R!*G)Ho6F37$$\"3&p=P4e4es*Ff\\l$!3Q![X*ej@.8F37$$\"3CZ#*fZa<K(*Ff\\
l$!3RhI*QD)y-9F37$$\"3a28E98aQ(*Ff\\l$!3k/guS%=?Y\"F37$$\"3BQBfZUsT(*F
f\\l$!3'3C(o;#3cZ\"F37$$\"3$yOB4=2\\u*Ff\\l$!3%oQ\"Ht@Py9F37$$\"3U(RaU
6!4[(*Ff\\l$!3#*4&*>$)yUq9F37$$\"37GaeZIF^(*Ff\\l$!3#=6:n`z?X\"F37$$\"
3eW*Q]1xRw*Ff\\l$!3K>TyS!HzG\"F37$$\"39iC\\#3\"ow(*Ff\\l$!3])GX1*zeJ5F
37$$\"3szf%**4&Q*y*Ff\\l$!3p<-V1@p0zF,7$$\"3H(\\*R<\"*3-)*Ff\\l$!3tVow
jR,GqF,7$$\"3!3cCx(z23)*Ff\\l$!3#H8v;!Q%\\\"oF,7$$\"3@B'\\!Qo19)*Ff\\l
$!3_QI=:c,DlF,7$$\"3i&ou$)pb+#)*Ff\\l$!3?Va\"=gD)ejF,7$$\"38\\(*peX/E)
*Ff\\l$!3,ASRT*)[0lF,7$$\"3a6[->M.K)*Ff\\l$!3.(y[yTe!RrF,7$$\"3%R()\\$
zA-Q)*Ff\\l$!38lxJ(HkaT)F,7$$\"3YP\\nR6,W)*Ff\\l$!3=pDBNv+Z5F37$$\"3))
************\\)*Ff\\l$!3N`v1w[fT8F3-%'COLOURG6&%$RGBGF(F($\"*++++\"!\"
)-%+AXESLABELSG6$Q\"x6\"Q!Fb_^l-%%VIEWG6$;F($\"$&)*!\"$%(DEFAULTG" 1 
2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 4 
"Let " }{XPPEDIT 18 0 "f(x) = 1/2-abs(x-1/2)" "6#/-%\"fG6#%\"xG,&*&\"
\"\"F*\"\"#!\"\"F*-%$absG6#,&F'F**&F*F*F+F,F,F," }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "(a) Use \+
the1st \"bell-shaped\" family of functions described above to provide \+
an approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"
\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 117 "(b) Use th
e 2nd wider \"bell-shaped\" family of functions described above to pro
vide an approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"
fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#
7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "_______
________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 
"f(x) = abs(abs(abs(x-1/4)-1/2)-1/4);" "6#/-%\"fG6#%\"xG-%$absG6#,&-F)
6#,&-F)6#,&F'\"\"\"*&F2F2\"\"%!\"\"F5F2*&F2F2\"\"#F5F5F2*&F2F2F4F5F5" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 110 "(a) Use the1st \"bell-shaped\" family of functions descr
ibed above to provide an approximation for the function " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }
{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 117 "(b) Use the 2nd wider \"bell-shaped\" family of fun
ctions described above to provide an approximation for the function " 
}{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval
 " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 31 "_______________________________" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 31 "__
_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "
" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x*sqrt(1-x^2);" "6#/-%\"fG6
#%\"xG*&F'\"\"\"-%%sqrtG6#,&F)F)*$F'\"\"#!\"\"F)" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "(a) Use \+
the1st \"bell-shaped\" family of functions described above to provide \+
an approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"
\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 117 "(b) Use th
e 2nd wider \"bell-shaped\" family of functions described above to pro
vide an approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"
fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#
7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "_______
________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 \+
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }