{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 258 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 259 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "C ourier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "Couri er" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 30 "MLC Lab Visit - Lab 07 \+ - Maple" }}{PARA 0 "" 0 "" {TEXT 256 46 "Mth 355 (a.k.a. Mth 399) Feb \+ 19, 2003 Maple 7" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }{TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 260 0 "" }} {PARA 264 "" 0 "" {TEXT 261 22 "petersen@math.orst.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 259 181 "There are 5 prob lems below. Problem solutions are due Feb 26, 2003. Email your solutio ns to me as Maple worksheet attachments. Your worksheet must execute c orrectly for full credit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 74 "In this week's lab we investigate a few randomly c hosen features of Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Interpolation Polynomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Maple provides a builti n command interp() for computing interpolation polynomials. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "poly1:=interp([1,3,4,2],[2,1,3,1],z);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 "The first parameter we pass to interp() is the list of (distinct) abscissas, the second is t he list of ordinates and the third is a name, the name for the variabl e to be used in the polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 110 "If you want a polynomial function rather than a polynomial expression in some variable, you can use unapply(): " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "poly1fun:=unapply(interp([1,3,4,2],[2,1,3,1],z),z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Let's check that it worked:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "poly1fun(1); poly1fun(3); po ly1fun(4); poly1fun(2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 263 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "If you have a list of points you want to interpolate you can extract the abscissas and ordinates b y using the sequence comamnd seq()." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "data2:=[ [1,2], [2,-1], [3 ,-2], [-1,1], [-2,7], [8,6], [7,5] ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "XX2:=[seq(data2[k][1],k=1..nops(data2))];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "YY2:=[seq(data2[k][2],k=1..nops(dat a2))];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Let's construct the interpolation polynomial and plot it \+ together with the data:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "poly2:=interp(XX2,YY2,z);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "dataplot2:=plot(data2,style= point,symbol=circle,symbolsize=16,color=blue):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "polyplot2:=plot(poly2,z=1..7,color=red):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "He re's the plot:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "display([dataplot2,polyplot2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 9 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "A convenient way to construct an interpolation polynomial for a f unction is to use the map() command to evaluate the function at each a bscissa. Let's consider the function 1/(1+16x^2) on [-1,1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "X X3:=[seq(-1+k/6,k=0..12)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "YY3:=map(x->(1/(1+16*x^2)),XX3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Note the use of an anonymous fu nction in the argument of map()." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "We will need a list of points to plot the original data:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "data3:=[seq([XX3[k],YY3[k]],k=1..nops(XX3))]; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "poly3:=evalf(interp(XX3 ,YY3,t),6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "dataplot3:=p lot(data3,style=point,symbol=circle,symbolsize=16,color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "polyplot3:=plot(poly3,t=-1.. 1,color=red,thickness=2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "Note the previous example shows one way o f plotting two functions on one graph." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "display(\{polyplot3,dat aplot3\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 "Notice how the polynomial oscillates too much in some se nse. For example, the original function is positive on [-1,1] but the \+ interpolation polynomial is far from positive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Interpolation Splines" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 186 "Maple computes splines of all degree s. Here we will look only at (natural) cubic splines. The parameters a re much the same as for interp(), but the abscissas must be in increas ing order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "XX:=[seq(-1+k/6,k=0..12)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "YY:=map(x->(1/(1+16*x^2)),XX);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sp:=evalf(spline(XX,YY,x,cubic),6);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "pts:=[seq([XX[k],YY[k]],k=1. .nops(XX))];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "dataplot:=p lot(pts,style=point,symbol=circle,symbolsize=16,color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "splnplot:=plot(sp,x=-1..1,co lor=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(\{data plot,splnplot\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 103 "Notice how much better the interpolation spline t racks the data than the interpolation polynomial does." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Planar Graphs" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(networks):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 421 "We discussed planar graphs in cla ss. According to Kuratowski (circa 1930) a graph is planar if and only if it contains no subgraph that is homeomorphic to K5 or K3_3. Not e the term \"homeomorphic\" is used here in a sense peculiar to graph \+ theory: two graphs are homeomorphic if they can be obtained from the s ame graph by \"bisecting\" some of its edges. This is a beautiful resu lt though I have no idea how useful it is." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Maple does have a builtin plana rity test, isplane()." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G1:=icosahedron():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isplanar(G1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(G1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "It is hard to believe that G1 is planar!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "G2:=petersen():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isplanar(G2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(G2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "K3_3:=c omplete(3,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "isplanar(K 3_3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "draw(K3_3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "K5:=complete(5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isplanar(K5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(K5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The Maple default labeling of the \+ edges of K5 is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "edges(K5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "Once we know the labels we can remove an edge. Note the edge is removed from K5 directly - the new g raph replaces the old, so there is no point in assigning the result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "delete(e1,K5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dr aw(K5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "G3:=cube(3):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isplanar(G3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(G3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "G4:=cube(4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isplanar(G4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(G4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G5:=com plete(2,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isplanar(G5) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(G5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "Numeric Derivative Estimates" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 102 "Here is a silly example, but it illustra tes a general procedure (method of undetermined coefficients)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Suppose \+ we wish to have an estimate of the second derivative of f at a in \+ terms of f(a), f(a+h) and f(a+3h):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g:=h->A1*f(a)+A2*f(a+h)+A3 *f(a+3*h);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We expand g in a Taylor polynomial with center at h=0 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "expr:=taylor(g(h),h=0,4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "We want the first and sec ond coefficients here to be 0 and the third to be 1." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "for k fro m 0 to 3 do coef[k]:=coeff(expr,h,k); od;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "soln:=solve(\{coef[0]=0,coef[1]=0,coef[2]=(D@@2)(f) (a)\},\{A1,A2,A3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "exp r2:=subs(soln,expr);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "To seperate out the second derivative we need t o divide by h^2. So here's our expression" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "est:=subs(soln,g(h)/ h^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can estimate the error" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "taylor((D@@2)(f)(a)-est ,h=0,4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "So, only first order (unless the third derivative at a is zero)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Problems" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 9 "Pr oblem 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "The doudecahedron graph was discussed in clas. It has 30 edges an d 20 vertices each of degree 3. Enter the definition of the doudecahed ron into Maple and verify its (obvious of course) planarity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 266 9 "Problem 2" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "If we remove an edge from the complet e graph K5 is the resulting graph planar?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 9 "Problem 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "If we remove an edge from the bipartite graph K3_3 = complete(3,3) is the resulting graph planar?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 268 9 "Problem 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 160 "Find a numeric estimate for the first derivative \+ of f at a in terms of f(a-h), f(a) and f(a+h) of order 2. What p articularly nice property do you observe?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 9 "Problem 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "Find a numeric estimate for the third derivative of f \+ at a, of as high order as possible, in terms of f(a-h), f(a), f(a+ h), f(a+2h) and f(a+3h). What is the order?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }