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0 "" {TEXT -1 23 "Topic 1 : Calculating " }{XPPEDIT 18 0 "Pi" "6#% #PiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "We will use the fact that " }{XPPEDIT 18 0 "2 *Int(sqrt(1-x^2),x=-1..1)) " "6#*&\"\"#\"\"\"-%$I ntG6$-%%sqrtG6#,&F%F%*$%\"xGF$!\"\"/F.;,$F%F/F%F%" }{TEXT -1 47 " is \+ the area of a circle of radius 1 which is " }{XPPEDIT 18 0 "Pi" "6#%#P iG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "f:= x-> sqrt(1-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\"\"\"F0*$) 9$\"\"#F0!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plo t(f(x), x=-1..1, scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 363 363 363 {PLOTDATA 2 "6&-%'CURVESG6$7_o7$$!\"\"\"\"!$F*F*7$$!3-n;Hd Nvs**!#=$\"3aIk@d'**oP(!#>7$$!3/MLe9r]X**F/$\"3^xD,?$RD/\"F/7$$!3/,](= ng#=**F/$\"3W*\\#GuA(fF\"F/7$$!3%pmm\"HU,\"*)*F/$\"3=cuB4yNs9F/7$$!3() ***\\PM@l$)*F/$\"3gMpy'*3z+=F/7$$!3!RLL$e%G?y*F/$\"3iA8`Ih^w?F/7$$!3u* 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-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;F(F``l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 19 "Assignment Part 1: " }{TEXT 262 116 " Comput e the left-hand sums and right-hand sums with 10, 100, 1000 subinterva ls. Do not forget the student package" }{TEXT 261 1 "." }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 17 "Assignment Part 2" } {TEXT -1 141 ": What do you notice? Can you explain what you noticed ? You may want to graph some left-hand sums and right-hand sums to exp lain your answe" }{TEXT 263 2 "r." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "As a result we do not get \+ underestimates and overestimates for " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 29 " using the previous commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 17 "Assignment Part 3" }{TEXT -1 46 ": Find underestimates and overestimates for " }{XPPEDIT -1 0 "Pi ;" "6#%#PiG" }{TEXT -1 64 ", using left-hand sums and right-hand sums \+ with 5, 50, 500, 5000" }{TEXT 257 1 " " }{TEXT -1 33 "on the intervals [-1,0] and [0,1]" }{TEXT 264 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We can actually predict beforehand how large " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 73 " should be so that our underestimate s and overestimates are within, say, " }{XPPEDIT 18 0 "epsilon;" "6#%( epsilonG" }{TEXT -1 31 ". Recall that on the interval [" }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b;" "6#%\"bG" } {TEXT -1 24 "], the difference |LHS(" }{XPPEDIT 18 0 "n;" "6#%\"nG" } {TEXT -1 6 ")-RHS(" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 14 ")|=|f( b)-f(a)|" }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{XPPEDIT 18 0 "x;" "6# %\"xG" }{TEXT -1 50 " for a monotone function. If we split [-1,0] into " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 42 " subintervals the error is less than " }{XPPEDIT 18 0 "(f(0)-f(-1))/n = 1/n;" "6#/*&,&-% \"fG6#\"\"!\"\"\"-F'6#,$F*!\"\"F.F*%\"nGF.*&F*F*F/F." }{TEXT -1 11 ". \+ since " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{XPPEDIT 18 0 "x;" "6 #%\"xG" }{TEXT -1 7 "=(b-a)/" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 48 ". On the interval [0,1], if we split into " }{XPPEDIT 18 0 " n;" "6#%\"nG" }{TEXT -1 42 " subintervals, we have an error less than \+ " }{XPPEDIT 18 0 "(f(0)-f(1))/n = 1/n;" "6#/*&,&-%\"fG6#\"\"!\"\"\"-F' 6#F*!\"\"F*%\"nGF-*&F*F*F.F-" }{TEXT -1 58 ". So in the whole interva l [-1,1] the error is less than " }{XPPEDIT 18 0 "2/n;" "6#*&\"\"#\"\" \"%\"nG!\"\"" }{TEXT -1 12 " and, since " }{XPPEDIT 18 0 "Pi;" "6#%#Pi G" }{TEXT -1 48 " is twice the integral, the error in estimating " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 14 " is less than " }{XPPEDIT 18 0 "4/n;" "6#*&\"\"%\"\"\"%\"nG!\"\"" }{TEXT -1 67 ". If we want the error to be, say less than 0.001, we need to make " }{XPPEDIT 18 0 "4 /n < .1e-2;" "6#2*&\"\"%\"\"\"%\"nG!\"\"-%&FloatG6$F&!\"$" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 "4/.1e-2 < n;" "6#2*&\"\"%\"\"\"-%& FloatG6$F&!\"$!\"\"%\"nG" }{TEXT -1 100 ", i.e., n>4000. This explains why with 5000 subintervals we had gotten the first 3 decimals correct ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 270 18 "Assignment Part 4:" }{TEXT 267 32 " How large do you need to take " }{XPPEDIT 257 0 "n;" "6#%\"n G" }{TEXT 268 26 ", so that you can compute " }{XPPEDIT 259 0 "Pi;" "6 #%#PiG" }{TEXT 269 35 " with an error less than 0.0000001?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Unfortun ately, this number of subintervals is too large to work out with Maple . So we resort to the trapezoid rule and the midpoint rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 18 "Assignment Par t 5:" }{TEXT -1 15 " Compute TRAP(" }{XPPEDIT -1 0 "n;" "6#%\"nG" } {TEXT -1 7 "), MID(" }{XPPEDIT -1 0 "n;" "6#%\"nG" }{TEXT -1 8 "), wit h " }{XPPEDIT -1 0 "n;" "6#%\"nG" }{TEXT -1 50 "=10, 100, 1000 on the \+ whole interval [-1,1]. Which" }{TEXT 256 1 " " }{TEXT -1 62 "ones give underestimates and which ones give overestimates of " }{XPPEDIT 262 0 "Pi;" "6#%#PiG" }{TEXT -1 37 "? Why? What is the relation of TRAP( " }{XPPEDIT -1 0 "n;" "6#%\"nG" }{TEXT -1 11 ") with LHS(" }{XPPEDIT -1 0 "n;" "6#%\"nG" }{TEXT -1 2 ")?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 " To compute more accurately, we can use Simpson's rule: SIM(" } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 4 ")= " }{XPPEDIT 18 0 "(TRAP( n)+2*MID(n))/3;" "6#*&,&-%%TRAPG6#%\"nG\"\"\"*&\"\"#F)-%$MIDG6#F(F)F)F )\"\"$!\"\"" }{TEXT -1 49 " . There is a Maple command for Simpson's rule:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 65 "simpson(f(x), x=lowerlimit..upperlimit, 2*number of subintervals) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 265 20 "As signment Part 6 :" }{TEXT -1 2 " " }{TEXT 266 8 "Compute " } {XPPEDIT 271 0 "Pi;" "6#%#PiG" }{TEXT 272 27 " using Simpson's rule wi th " }{XPPEDIT 273 0 "n;" "6#%\"nG" }{TEXT 274 35 "=10, 100, 1000,1000 0 subintervals. " }}{PARA 258 "" 0 "" {TEXT 286 22 "Optional: Verify \+ your" }{TEXT -1 1 " " }{TEXT 275 45 "answers using the formula for Sim pson's rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 276 33 "Topic 2 : Substitution in Maple" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "Although Maple can often evaluate integrals directly it is useful to be able to use \+ the substitution technique." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Example: Compute " }{XPPEDIT 19 1 "int((x^2+1) ^4*x,x)" "6#-%$intG6$*&,&*$%\"xG\"\"#\"\"\"F+F+\"\"%F)F+F)" }}{PARA 0 "" 0 "" {TEXT -1 18 "using substitution" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E:=Int((x^2 +1)^4*x, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%$IntG6$*&),&*$ )%\"xG\"\"#\"\"\"F/F/F/\"\"%F/F-F/F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The first step is to substitute " }{XPPEDIT 18 0 "u =x^2 +1" "6 #/%\"uG,&*$%\"xG\"\"#\"\"\"F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F:=changeva r(u=x^2+1, E, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%$IntG6$,$ *&\"\"#!\"\"%\"uG\"\"%\"\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The second step is to evaluate this integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "V:=value(F) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*&\"#5!\"\"%\"uG\"\"&\"\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "The last step is to substit ute x back into the answer" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "S:=subs(u=x^2+1, V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,$*& \"#5!\"\",&*$)%\"xG\"\"#\"\"\"F.F.F.\"\"&F." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 37 "For comparison, we evaluate directly " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(E);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"\"\"\"#5F&*$)%\"xGF'F&F&F&*&#F&\"\"#F&*$)F*\" \")F&F&F&*$)F*\"\"'F&F&*$)F*\"\"%F&F&*&F,F&*$)F*F-F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "This seems different from S but they are \+ not - we expand S to check." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "expand(S, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&#\"\"\"\"#5F& *$)%\"xGF'F&F&F&*&#F&\"\"#F&*$)F*\"\")F&F&F&*$)F*\"\"'F&F&*$)F*\"\"%F& F&*&F,F&*$)F*F-F&F&F&F%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "We s till notice that there is a difference: the constant " }{XPPEDIT 18 0 "1/10;" "6#*&\"\"\"F$\"#5!\"\"" }{TEXT -1 146 ". This is so, because i n integration Maple does not care about constants. The most general an tiderivative is any of the previous expressions +C." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "One problem is that M aple does not tell us which substitution to use. So the choice is ours and this is where our work lies." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 25 "Assignment Part 7 (A) " }{TEXT 279 4 "Find" }{TEXT -1 1 " " }{XPPEDIT 19 1 "int((x^2+1)^4,x)" "6#-%$intG6 $*$,&*$%\"xG\"\"#\"\"\"F+F+\"\"%F)" }{TEXT 277 26 " directly by expa nding " }{XPPEDIT 18 0 "(x^2+1)^4 " "6#*$,&*$%\"xG\"\"#\"\"\"F(F(\"\" %" }{TEXT 280 101 " (There is a Maple command called expand that you c an use.) and then just integrating the terms. " }}{PARA 258 "" 0 " " {TEXT -1 3 "(B)" }{TEXT 281 9 " Find " }{XPPEDIT 19 1 "int((x^2+1 )^4,x);" "6#-%$intG6$*$,&*$%\"xG\"\"#\"\"\"F+F+\"\"%F)" }{TEXT 278 28 " using the substitution " }{XPPEDIT 256 0 "u = x^2+1;" "6#/%\"uG, &*$%\"xG\"\"#\"\"\"F)F)" }}{PARA 0 "" 0 "" {TEXT -1 21 "What do you co nclude?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 19 "Assignment Part 8 " }{TEXT -1 10 " Find " }{XPPEDIT 18 0 "Int(cos(x)^2*sin(x),x);" "6# -%$IntG6$*&-%$cosG6#%\"xG\"\"#-%$sinG6#F*\"\"\"F*" }{TEXT -1 67 " us ing a substitution. Check your answer with the value command.\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 19 "Assignment Part 9 " }{TEXT -1 10 " Find " } {XPPEDIT 18 0 "Int(exp(-x)*tan(exp(-x)),x);" "6#-%$IntG6$*&-%$expG6#,$ %\"xG!\"\"\"\"\"-%$tanG6#-F(6#,$F+F,F-F+" }{TEXT -1 67 " using a sub stitution. Check your answer with the value command.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "36 \+ 0 0" 18 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }