GSP4QThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.$_;Lpoly` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` ArialArial Narrow$    VVV  ~g   '''$'$'$'$'$'$'$'$'$'$' UUU$' $' %'&'8'8'8'9':') nQ\aAS[2]@|@4(@ 4`.2U@1: &  nQ\aAx6(n 6\.B@: &n  nQ\aAy='n   n &)ʢDֿA  & 6?B "=rd?interval=0e@ @.@   @ .{Gz?1   ,,c>>c/  /    .(\?a/3   $<0>f(nQ\aAm[1]v0 3nr !<0>e@t@ R0 wFĉ;6  ` nQ\aAaI;n     >nH@x@/@  @ /HzG?I $ tic mark ht    ?0.#  ?1 ?2.# ?3.# ?4.# ?5.# ?-1 .#! ?-2 .#" ?-3 .## ?-4 .#$ ?-5 .#%  - & L  -  (M ')N @Ђ@/@ + @, /)\(?O +-"? slider ht  . - /0 0. - 10 2. - 30 4. - 50. 0 . 2 . 4 . 6  7 .(\?b 8 .ףp= ?c 9 .)\(?d : . ףp= ?e - 71 - 81 - 91 - :1/36B>  <0>x1>nQ\aAm[5]r/3n4A=  <0>x1>nQ\aAm[4]r/3n0?;  O<0>x1>nQ\aAm[2]r/3n2@<  |<0>x1>nQ\aAm[3]rn*+-.#G+-.#H+-.#I+-.#> ,  ;0<2=4>6   &nQ\aA a,b,c,d,e = 0 -n *%n L'0 3nr -<0> R0 wFĉ;6  ` nQ\aAa0 3nEr '<0>e@t@ R0 wFĉ;6  ` nQ\aAb0 3nFr .<0> R0 wFĉ;6  ` nQ\aAc0 3nDr .<0>e@t@ R0 wFĉ;6  ` nQ\aAd0 3nCr .<<0> R0 wFĉ;6  ` nQ\aAeG 9nNOPQR  <0> j0DwFĉ;6 `````fH: S  4P!^uj?> ꨋI?y=f(x),  Q  Q !0.>>'0.- >>- T!0.f>>f4! !AI;, UNVOWPXQYRB>>>>>>>>>>>>>>>>>>>>>>m> FE[1]`r@=@@F[1] @=@@ \]/@ ]^ @_ /Xm?ht]/ `_/ \^ - abG[1] \c c` `]$$&5c`]\@@ ,@   A3   !<0>x nQ\aAA 0 3n Sjb ><0> B0wFĉ;6 p` nQ\aAm[14] UUUnLL 9>  .#0 3nr R-<0>T1c, R0 !7=ĉ;6  ` nQ\aAa 0 3nEr QW<0>T1c, R0 7=ĉ;6  ` nQ\aAb 0 3nFr P<0>T1c, R0 7=ĉ;6  ` nQ\aAc 0 3nDr Q<0>T1c, R0 7=ĉ;6  ` nQ\aAd 0 3nCr Q<0>T1c, R0  7=ĉ;6  ` nQ\aAe n0j@P@@Z[1]c@p@A3@   Lk<0>x nQ\aAB 0 3n mtnopqj ><0> 0G7=ĉ;6 ```````````````````` nQ\aAf(x) dx nA[2]dK@px@@B[2]dH@py@@ vw@E[2]dO@w@@F[2]dS@w@Q C@zyv vw$r<f@Q xw$r<f@Q yw$r<f@Q zw$r<f@Q C {w$r<f@Q0 3 Stb ^<0> B07=ĉ;6 p` nQ\aAf(B)  On^ b<0> >0v??Y@??nQ\aAnby0 3nb <0> B06wFĉ;6 `nQ\aAm[13] nljIW><->>`IOn42 inm VE8njk"(x,f(A)) V[3]Ax@X@@W[3]Ay@X@@ @X[3]Ax@8@"@ 4@4'"4 lj AW>  Ac.#$"&>&+9$">$"R>R+e$">1E8t"(x,f(B))  %0   w<0>x  V3  #<0>x  0 0 b v3<0>T1or B0ĉ;6`` 0 3 b %3<0>T1or B0 ĉ;6``   -  x[1]B A3   g<0>x  0 3 Sb Vf<0>T1or B0ĉ;6p`  E8f(x1)    5  5  mdpt1 A3   <0>x  0 3 Sb r<0>T1or B0ĉ;6p`  E8"f(mdpt1)    5f 50 3 kb <0>T1or B0ĉ;6`` 0 3 b <0>T1or B0ĉ;6``  J left) @P@@@0 !$ new J left J rt  @P@@@ J mdpte P@@@ J trape @P@@@@0 0 0 !$ new J mdpt A3   w<0>x nQ\aAm[24] A3n   <0>x nQ\aAm[26] A3n   <0>x nQ\aAm[28] A3n   <0>x nQ\aAm[30] 0 3n b <0>T1or B0<ĉ;6`` 0 3nkn <<0>T1or N0 <ĉ;6 `` ` !"new J rtt !$ new J trap Y@< jku I7@MA  MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MAf MA MA MA MA MA MA MA MA MA MA X(terminal J left X& terminal J rt X(terminal J mdpt X(terminal J trap A3   mu<0>nQ\aAm[23] A3n   q<0>nQ\aAm[25] A3n   <0>nQ\aAm[27] A3n   <0>nQ\aAm[29] 0 3}nn @<0>T1or N0 <ĉ;6 ``  ` "nQ\aALeft SumJ0 3nn +<0>T1or N0 <ĉ;6 ``  ` "nQ\aA Right Sum0 3fnn <0>T1or N0 <ĉ;6 ``  ` &nQ\aA Midpoint Suma0 3nn -E<0>T1or N0 <ĉ;6 ``  ` "nQ\aA Trap. Sum On^ <0> >0?YY@??nQ\aAt[1]  0 3njn q<0>T1or N0 oKĉ;6`  E8n near Au A3   a<0> 0 3 Sb <0>T1or B0oKĉ;6p`  E8M[5]A  Y@$ 4Z(MA MA MA ljeV>qP e;     > f<    O(*nQ\aAShow Mdpt. Rects.> n(    O*nQ\aAShow Right Rects.> n$   O*nQ\aAShow Left Rects.> n(    OQ(nQ\aAShow Trapezoids n )&u?4r4 shaded locus under curve A3   <0>Q 0 3 Sb +<0>T1or B0*ĉ;6p`  E86f(4 shaded locus under curve)  #+    ?K>    ;.nQ\aAHide Integral Shading> n$>@@@F   q*nQ\aAn+1 nonstop, n<80> n$>C@@F   ,nQ\aAn+10 nonstop, n<80n\TnDjku {nQ\aAH[22]n1y<[>>>>1|0 3nkj 9<0>T1or J0Q+NgR `` `0 3 b c<0>T1or B0Q+NgR``I; -s><[>>>>>    d*nQ\aAShow RS-LS Info.  On` !<0> @0rh?YY@??"nQ\aAA targetS!!4 On^ 2<0> >0rh?YY@??"nQ\aAB targetS2E On^ &<0> >0rh?YY@??nQ\aAzerog&.E8n"A targetSE8"B targetS>  !   J(nQ\aAMove to targets n`r>><0><0>>>{2} `_.#> n`W[kCqpDFonEmK-.   Z*nQ\aAShow all details nQ[7]a @`g@@R[7]a.m|a@`g@@ %&?o.#S[7]a @ c@@ %(?p.#U[7]aP@ c@@ (*?q.#X[7]aP@ b@@ *,?r.#Y[7]aa@ c@@ &.?s.#Z[7]aV@ c@@ .0?t.#A[8]aV@ b@@ 02?u.#B[8]a@X@ b@@ 24?v.#C[8]aR@`@@ 46?w.#D[8]aK@ b@@ ,8?x.# 86?y.#0 &.02468,*(%.#-#QThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.$4Lln` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` Arial $    VVV  ~g  '''$'$'$'$'$'$'$'$'$'$' UUU$' $' %'&'8'8'8'9':') nQ\aAS[2]@Pw@4(@ 4`.2U@1: &  nQ\aAx6(n 6\.B@: &n  nQ\aAy='n   n &(\?A  &Gz@B "=rd?interval=0e@ @.@   @ .{Gz?1   ,,c>>>c/  /    .(\?a/3   $<0>f(nQ\aAm[1]v0 3nr !<0>T1c, R0 wFĉ;6  ` nQ\aAaI;n     >nHv@ @/@  @ /HzG?I $ tic mark ht    ?0.#  ?1 ?2.# ?3.# ?4.# ?5.# ?-1 .#! ?-2 .#" ?-3 .## ?-4 .#$ ?-5 .#%  - & L  -  (M ')Nv@x@/@ + @, /)\(?O +-"? slider ht  . - /0 0. - 10 2. - 30 4. - 50. 0 . 2 . 4 . 6  7 .{Gz?b 8 .?c 9 .?d - 71 - 81 - 91 - :1/34@=  <0>x1>nQ\aAm[4]r/3n0>;  O<0>x1>nQ\aAm[2]r/3n2?<  |<0>x1>nQ\aAm[3]rn*+-.#E+-.#F+-.#G+- - *% I'0 3r <0>T1c, R0 wFĉ;6  ` nQ\aAa0 3nCr <0>T1c, R0 wFĉ;6  ` nQ\aAb0 3nDr <0>T1c, R0 wFĉ;6  ` nQ\aAc0 3nBr <0>T1c, R0 wFĉ;6  ` nQ\aAdG 9nKLMNv  <0>T1c, V0 DwFĉ;6 ` ` ` `fH: O  Es?-sZ@> ꨋI?y=f(x)h  V Vj V >Vj"Y "Y@m mZ >mZnm>FE[1])`r@=@@F[1]) @=@@ VW/@ WX @Y /Xm?ht]/ ZY/ VX - [\G[1]) V] ]Z ZW$$&5]ZWV@@ ,@   A3   %<0>)]nQ\aAA 0 3n Odb ><0>T1or B0wFĉ;6 p` nQ\aAm[14] UUUnLL 9>  .#0 3nr R-<0>T1c, R0 !7=ĉ;6  ` nQ\aAa 0 3nCr QW<0>T1c, R0 7=ĉ;6  ` nQ\aAb 0 3nDr P<0>T1c, R0 7=ĉ;6  ` nQ\aAc 0 3nBr Q<0>T1c, R0 7=ĉ;6  ` nQ\aAd n0j@P@@Z[1]c@p@A3@   R<0>)]nQ\aAB nA[2]M@Pz@@B[2]J@P{@@ no@E[2]P@y@@F[2]T@py@Q C@rqn no$r<f@tt po$r<f@tt qo$r<f@tt ro$r<f@tt C so$r<f@tt0 3 Omb ^<0>T1or B07=ĉ;6 p` nQ\aAf(B)  On^ b<0> >0v??Y@??nQ\aAnby0 3nzb <0>T1or B06wFĉ;6 `nQ\aAm[13] nljIW><->> IOn42 inm  VE8nde"(x,f(A)) ~V[3]Ax@@@W[3]Ay@@@ @X[3]Ax@h@"@ 4@4'"4 lj >W>  >`.#$"&> &+9$"> $"R> R+e$"> 1E8my"(x,f(B))  %0   w<0>)] V3  #<0>)] 0 0 zb v3<0>T1or B0ĉ;6`` 0 3 zb %3<0>T1or B0 ĉ;6``   -  x[1]B A3   g<0>)] 0 3 Ob Vf<0>T1or B0ĉ;6p`  E8f(x1)  ~ ~ 5~  5  mdpt1 A3   <0>)] 0 3 Ob r<0>T1or B0ĉ;6p`  E8"f(mdpt1)    5f ~5~0 3 eb <0>T1or B0ĉ;6`` 0 3 b <0>T1or B0ĉ;6``  J left) @P@@@0 !$ new J left J rt  @P@@@ J mdpte P@@@ J trape @P@@@@0 0 0 !$ new J mdpt A3   w<0>)]nQ\aAm[24] A3n   <0>)]nQ\aAm[26] A3n   <0>)]nQ\aAm[28] A3n   <0>)]nQ\aAm[30] 0 3n b <0>T1or B0<ĉ;6`` 0 3nen <<0>T1or N0 <ĉ;6 `` ` !"new J rtt !$ new J trap  O^ <0> >0?YY@??nQ\aAt[1]  0 3ndn q<0>T1or N0 oKĉ;6`  E8n near Ar A3   a<0>)] 0 3 Ob <0>T1or B0oKĉ;6p`  E8M[5]A  Y@${ 4Z(MA MA MA ljeV> e;   )&u?4r4 shaded locus under curve A3   <0> 0 3 Ob +<0>T1or B0*ĉ;6p`  E86f(4 shaded locus under curve)  #+   ?K>    <.nQ\aAHide Integral Shading> nz$>@@@F   q*nQ\aAn+1 nonstop, n<80> nz$>C@@F   ,nQ\aAn+10 nonstop, n<80n n: . ףp= ?e> ,  ;0<2=46   &nQ\aA a,b,c,d,e = 0/3n6A  <0>nQ\aAm[5],0 3nr Q<0>T1c, R0  7=ĉ;6  ` nQ\aAe 0 3ngmihjd B<0>PC 0;7=ĉ;6 ` ``  ` ``  `` ``` ``  ` ``  `` `` nQ\aAf(x) dxY@n<{ de~ I7@MA ~MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MAf MA MA MA MA X(terminal J left  A3   mu<0>nQ\aAm[23] 0 3}n{n 8<0>T1or N0 <ĉ;6 ``  ` "nQ\aALeft SumJ> n$   H*nQ\aAShow Left Rects.MAn MAn MAn MAn Xn(terminal J mdpt  A3   <0>nQ\aAm[27] 0 3fn{n <0>T1or N0 <ĉ;6 ``  ` &nQ\aA Midpoint Sump> fn<   H!*nQ\aAShow Mdpt. Rects.MAn Xn& terminal J rt  A3   q<0>!nQ\aAm[25] 0 3n {n %<0>T1or N0 <ĉ;6 ``  ` "nQ\aA Right Sum> n(    H*nQ\aAShow Right Rects.MAn Xn (terminal J trap  A3   <0>nQ\aAm[29] 0 3n{n )<<0>T1or N0 <ĉ;6 ``  ` "nQ\aA Trap. Sum> n(    HH(nQ\aAShow Trapezoids\TnDde {nQ\aAH[23]MAn  > nPSUey|}  jBDihCg{-.    e&nQ\aA Show Objects0 3nr _<0>T1c, R0 wFĉ;6  ` nQ\aAeI;n$QKRLSMTNC>>>>>>>>>>>>>2r>><0><0>>>{2} 2.# Ob <0>T1or B0?YY@?? "nQ\aAA target On^ <0> >0?YY@??"nQ\aAB target!n"A target!"B target>    (nQ\aAMove to targets nnQ\aAK@`g@@J0m|a@`g@@nQ\aAI@ c@@nQ\aAHO@ c@@nQ\aAGO@ b@@Fa@ c@@EU@ c@@DU@ b@@CX@ b@@BR@`@@AJ@ b@@ .#@  .#@  !.#@ !".#@ #.#@ #$.#@ $%.#@ %&.#@ &'.#@ "(.#@ ('.#@0 #$%&'("! .#-#HQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.5$6Lsine` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0594/Aug 28 20060#` Arial $    VVV  ~g  '''$'$'$'$'$'$'$'$'$'$' UUU$' $' %'&'8'8'8'9':') nQ\aAS[2]@v@4(@ 4\.D@1: &  nQ\aAx6(n 6\.B@: &n  nQ\aAy='n   n &&9߿A  &)I?B "=rd?interval=0e@@.@   @ .{Gz?1   ,,c>>>c/  /    .333333?a/3   $<0>f(nQ\aAm[1]v0 3nr !<0>T1c, R0 wFĉ;6  ` nQ\aAaI;n     >nHv@ @/@  @ /HzG?I $ tic mark ht    ?0.#  ?1 ?2.# ?3.# ?4.# ?5.# ?-1 .#! ?-2 .#" ?-3 .## ?-4 .#$ ?-5 .#%  - & L  -  (M ')Nv@x@/@ + @, /)\(?O +-"? slider ht  . - /0 0. - 10 2. - 30 4. - 50. 0 . 2 . 4 . 6  7 .{Gz?b 8 .?c 9 .?d - 71 - 81 - 91 - :1/34@=  <0>x1>nQ\aAm[4]r/3n0>;  O<0>x1>nQ\aAm[2]r/3n2?<  |<0>x1>nQ\aAm[3]rn*+-.#E+-.#F+-.#G+- - *% I'0 3r <0>T1c, R0 wFĉ;6  ` nQ\aAa0 3nCr <0>T1c, R0 wFĉ;6  ` nQ\aAb0 3nDr <0>T1c, R0 wFĉ;6  ` nQ\aAc0 3nBr <0>T1c, R0 wFĉ;6  ` nQ\aAdG 9nKLMNv  <0>T1c, V0 DwFĉ;6 ` ` ` `fH: O  +H ? Ԁ@> ꨋI?y=f(x)h  V Vw V Vw"Y "YNz mZ mZ{m>FE[1])`r@=@@F[1]) @=@@ VW/@ WX @Y /Xm?ht]/ ZY/ VX - [\G[1]) V] ]Z ZW$$&5]ZWV@@ ,@  A3   .<0>)]nQ\aAA 0 3n Odb ><0>T1or B0wFĉ;6 p` nQ\aAm[14] UUUnLL 9>  .#0 3nr R-<0>T1c, R0 !7=ĉ;6  ` nQ\aAa 0 3nCr QW<0>T1c, R0 7=ĉ;6  ` nQ\aAb 0 3nDr P<0>T1c, R0 7=ĉ;6  ` nQ\aAc 0 3nBr Q<0>T1c, R0 7=ĉ;6  ` nQ\aAd n0j@P@@Z[1]c@p@A3@   [~<0>)]nQ\aAB nA[2]P@y@@B[2]N@z@@ no@E[2]S@x@@F[2]@V@x@Q C@rqn no$r<f@tt po$r<f@tt qo$r<f@tt ro$r<f@tt C so$r<f@tt0 3 Omb ^<0>T1or B07=ĉ;6 p` nQ\aAf(B)  On^ b<0> >0v??Y@??nQ\aAnby0 3nzb <0>T1or B06wFĉ;6 `nQ\aAm[13] nljIW><->>` IOn42 inm  VE8nde"(x,f(A)) ~V[3]Ax@@@W[3]Ay@@@ @X[3]Ax@h@"@ 4@4'"4 lj <W>` <^.#$"&> &+9$"> $"R> R+e$"> 1E8my"(x,f(B))  %0   w<0>)] V3  #<0>)] 0 0 zb v3<0>T1or B0ĉ;6`` 0 3 zb %3<0>T1or B0 ĉ;6``   -  x[1]B A3   g<0>)] 0 3 Ob Vf<0>T1or B0ĉ;6p`  E8f(x1)  ~ ~ 5~  5  mdpt1 A3   <0>)] 0 3 Ob r<0>T1or B0ĉ;6p`  E8"f(mdpt1)    5f ~5~0 3 eb <0>T1or B0ĉ;6`` 0 3 b <0>T1or B0ĉ;6``  J left) @P@@@0 !$ new J left J rt  @P@@@ J mdpte P@@@ J trape @P@@@@0 0 0 !$ new J mdpt A3   w<0>)]nQ\aAm[24] A3n   <0>)]nQ\aAm[26] A3n   <0>)]nQ\aAm[28] A3n   <0>)]nQ\aAm[30] 0 3n b <0>T1or B0<ĉ;6`` 0 3nen <<0>T1or N0 <ĉ;6 `` ` !"new J rtt !$ new J trap  O^ <0> >0?YY@??nQ\aAt[1]  0 3ndn q<0>T1or N0 oKĉ;6`  E8n near Ar A3   a<0>)] 0 3 Ob <0>T1or B0oKĉ;6p`  E8M[5]A  Y@${ 4Z(MA MA MA ljeV>qPe;   )&u?4r4 shaded locus under curve A3   <0> 0 3 Ob +<0>T1or B0*ĉ;6p`  E86f(4 shaded locus under curve)  #+   ?K>    <.nQ\aAHide Integral Shading> nz$>@@@F   q*nQ\aAn+1 nonstop, n<80> nz$>C@@F   ,nQ\aAn+10 nonstop, n<80n n: . ףp= ?e> ,  ;0<2=46   &nQ\aA a,b,c,d,e = 0 /3n6A  <0>nQ\aAm[5],0 3nr Q<0>T1c, R0  7=ĉ;6  ` nQ\aAe 0 3nghmijd D<0> 0$7=ĉ;6 `` ` `` ```` ` `` `` nQ\aAf(x) dxY@n<{ de~ I7@MA ~MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MAf MA MA MA MA X(terminal J left  A3   mu<0>nQ\aAm[23] 0 3}n{n C<0>T1or N0 <ĉ;6 ``  ` "nQ\aALeft SumJ> n$   U*nQ\aAShow Left Rects.MAn MAn MAn MAn Xn(terminal J mdpt  A3   <0>nQ\aAm[27] 0 3fn{n <0>T1or N0 <ĉ;6 ``  ` &nQ\aA Midpoint Sump> fn<   U!*nQ\aAShow Mdpt. Rects.MAn Xn& terminal J rt  A3   q<0>!nQ\aAm[25] 0 3n {n .<0>T1or N0 <ĉ;6 ``  ` "nQ\aA Right Sum> n(    U*nQ\aAShow Right Rects.MAn Xn (terminal J trap  A3   <0>nQ\aAm[29] 0 3n{n 2<<0>T1or N0 <ĉ;6 ``  ` "nQ\aA Trap. Sum> n(    UH(nQ\aAShow Trapezoids\TnDde {nQ\aAH[23]MAn  > nPSUey|}  jBDihCg{-.    e&nQ\aA Show Objects 0 3nr _<0>T1c, R0 wFĉ;6  ` nQ\aAeI;n$QKRLSMTND>>>>>>>>>>>>>C8C!3|C8C!3|@dr>><0><0>>>{2}d Q.# Oz $<0>T1c, Z0?YY@??   "nQ\aAB target$7 On| <0>T1c, \0?YY@??   "nQ\aAA target#%!n"A target !" B target >    <(nQ\aAMove to targetsnnQ\aAK@`g@@nQ\aAJ0m|a@`g@@nQ\aAI@ c@@nQ\aAHO@ c@@nQ\aAGO@ b@@nQ\aAFa@ c@@nQ\aAEU@ c@@nQ\aADU@ b@@nQ\aACX@ b@@nQ\aABR@`@@nQ\aAAJ@ b@@  !.#@  ".#@ "#.#@ #$.#@ !%.#@ %&.#@ &'.#@ '(.#@ ().#@ $*.#@ *).#@0 !%&'()*$#" .#-#