ࡱ> 02/b %(jbjb LG,PPPPT$,J~RRRRR4(4666666,[RbiDRRDDb(RR(((DRR4(4 "D4((( P>(( 0(G(G((,,$ P,,PMath438 ACTIVITY 14: More on the simplex algorithm and why it works WHY: The stopping condition for the simplex algorithm (see theorem 3.3.7) is based on the effects our manipulations have on the bottom row of the simplex tableau, but the discussion of these effects requires the identification of several parts of the tableau these parts are also necessary for discussion of later work (sensitivity analysis). One condition for use of the simplex algorithm is that the bottom row (like other rows) must always have 0 entries below the pivot 1s of the tableau this may require one of more pivot operations before the use of the simplex algorithm. LEARNING OBJECTIVES: 1. Understand how to form the initial tableau for a problem in which one or more columns of the identity submatrix do not correspond to slack variables. 2. Discover how to label the initial tableau (using D, B, B-1, CB, CD ) and the current tableau (using B-1D , CBTB-1b, CBTB-1D - CD T CBTB-1b ) based on a particular basis matrix for further work with the simplex algorithm. 3. Understand why the optimality test (in theorem 3.3.7) works and what the bottom row of the simplex tableau is showing. 3. Gain experience with the simplex algorithm. CRITERIA: 1. Quality of the answers to the Critical Thinking Questions. 2. Success with the task using Maple. RESOURCES: 1. Section 3.3: Strategic Mathematics, especially examples 3.3.4 3.3.6. 2. 40 minutes 3. The Maple worksheet Activity14.mw on the Public server PLAN: 1. Look over the model 2. Complete the Exercise, using the Maple Worksheet 3. Answer the Critical Thinking Questions MODEL: Standard form: minimize 2x1 + x2 + x3 - x4 minimize 2x1 + x2 + x3 - x4 subj to: 6x1 - x2 + x3 d" 10 subj to: 6x1 - x2 + x3 + x5 = 10 x1 + x2 - x3 d" 4 x1 + x2 - x3 + x6 = 4 x1 + 5x2-2x3+ x4 = 5 x1 + 5x2 -2x3 + x4 = 5 x1 e" 0, x2 e" 0, x3 e" 0, x4 e" 0 xi e" 0 for all i A =  EMBED "Equation" "Word Object1" \* mergeformat  , b = EMBED "Equation" \* mergeformat , basic variables: x5, x6, x4, nonbasic variables: x1, x2, x3, We did not have to add slack variables to all the constraints, but the third constraint is the only one that contains x4 , so when we form the tableau we do have a pivot 1 in each row:  EMBED "Equation" "Word Object5" \* mergeformat  The initial basic feasible solution is [0 0 0 5 10 4 ]T but our tableau does not correctly express z in terms of the non-basic variables.. We see this in the fact that in the bottom row (the coefficient row) there are non-zero entries below some of the 1's [in the basic variable columns] of the identity matrix. We must pivot to make all bottom row entries under the basis columns ( 5, 6 and 4 ) zero. If we do not do this, we will not be able to make a correct choice of entering variable and will not locate the correct entry for the first pivot operation.. Clearing under x4 gives:  EMBED "Equation" \* mergeformat  This will be our initial tableau from which we begin the simplex algorithm. Our basic variables at this step are x5, x6, x4 in that order (to get a true identity matrix) so the basis matrix at this step is  EMBED Equation.3  (so  EMBED Equation.3  ) and  EMBED Equation.3  (order (x5, x6, x4) remember) so  EMBED Equation.3  - so that when we subtract the objective function coefficients we get  EMBED Equation.3  - the first part of the bottom row of the tableau. We also see that  EMBED Equation.3  - the last entry in the bottom row, and we can see that the value of the objective function for this basic feasible solution is 5. After this adjustment, we can start the simplex process: There is a 1 in the bottom row at the x3 column. The minimum positive ratio is 10/1 (row 1) so we pivot on a13 to get  EMBED Equation.3  (x5 leaves the basis because row 1 was the x5 row) Now the bottom row is d" 0, so we are finished. The basic variables [we read from the top row row down] are, x3 (=10), x6 (=14), x4 (=25) , nonbasic variables: x1, x2, x5. The final basic feasible solution is [0 0 10 25 0 14]T. According to our theorem, the optimal solution is x1= 0, x2= 0, x3= 10 ,x4 = 25, and the minimum value for the objective is -15. With this, our final basic feasible solution, we have our final basis (x3, x6, x4) , and so B =  EMBED "Equation" \* mergeformat , B-1 =  EMBED "Equation" "Word Object8" \* mergeformat , CBT = [1 0 -1],  EMBED Equation.3  - so that when we subtract the objective function coefficients we get  EMBED Equation.3  - the bottom row of the tableau [up to the z entry]. We also see that  EMBED Equation.3  - the last entry in the bottom row, and we can see that the value of the objective function for this basic feasible solution is 5. To illustrate the proof of theorem 3.3.7, we only need to identify the columns with the non-basic variables [not in final basis, that is] (representing matrix D) - in this case x1, x2, x5 Then we have D =  EMBED "Equation" \* mergeformat , CD T= [2 1 0] and we find that B-1D =  EMBED "Equation" \* mergeformat  , B-1b = [10 14 25 ]T , CBTB-1D - CD T= [-9 -5 -0] and CBTB-1b = -15 - all as they should be for Theorem 3.3.4. - that is, the final tableau can be written  EMBED "Equation" "Word Object2" \* mergeformat  . EXERCISE: 1. Put into standard form and write out the initial tableau for the following problem: minimize x1 + x2 + x3 + x4 subject to: 8x2 + 4x4 d" 8 x1 + 2x2 + x3 + 4x4 = 19 x1  4x2 + 2x4 d" 5 x1 e" 0, x2 e" 0, x3 e" 0, x4 e" 0 2. Call up Maple , and use the file Activity14..mw (in Public>Courses>Math>Math438) to solve the problem. 3. Identify (from the initial tableau for simplex after the clean-up{ pivot operation) the initial basic variables and initial basis matrix Binit and calculate for this basis B1, CBT, CBTB-1A CT, B-1b and CBTB-1b - show that they match the corresponding pieces of the initial tableau (thats after the bottom row is adjusted) 4. For each step (each tableau) do the same calculations as in 3 you will be using a different basis matrix each time. 5. Use the final tableau to identify the basic and non-basic variables and the final basis matrix B, and identify D, B, b, CB, and CD. Rearrange the variables to write the tableau in the form  EMBED "Equation" "Word Object1" \* mergeformat  used in theorem 3.3.7 6. Calculate B-1D, B-1b, and CBTB-1D-CDT based on the final tableau and verify that if we rewrite the final tableau with the columns in the order given in #5, we obtain  EMBED "Equation" "Word Object2" \* mergeformat  (as claimed in the proof of 3.3..7). CRITICAL THINKING QUESTIONS: 1. How do we use the final simplex tableau to find B ? 2. Why are the entries in the bottom row of the new tableau that correspond to the new non-basic variables obtained by computing CBTB-1D - CD T? If we extend this formula to the basic columns, why does this require us to force all the entries in the bottom row under the Identity columns to become zero? 3. Why is it true that the last column of the new tableau is obtained by computing B-1b and that the value of the objective function in the lower right hand corner is CBTB-1b ? 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BJ4!)p ThJ4!) ;)hn 6xVMT~imDPA(.^;9څBd1zorIrogRх . څA+c<~Crz >X>0_7_Ca| ?BAGCDEFIO^JKLMNPUQRSTV\WXYZ[_]`zabdcefghikjlmnoupqrstvwx|y{}~Root Entry F@P!-3@Data '4dWordDocumentLObjectPool9@P!-@P!-_1159001632 F@P!-@P!-Ole PIC TCompObjR  !"#$%'(+,/1456789:;<=>?@ABCDEFGHIJNQSTUVWXYZ[\]^_`abcdefghijklmnopqrtwxyz{|}~To*FMicrosoft EquationDNQE Equation.3TCIP; 4-` @@hdxpr  h, SymObjInfo OlePres000Ole10Native &Ole10FmtProgID  ) bol P.( "h currentpoint ",Times(P*6)p-,Times),1)1)0)r1)v0((1)1)-)+1)r0)t0)r1(@(1)5)-)+2)r1)v0)t0.(H (J ( *Q *P (H (J ( *Q *P O/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3072 div 1440 3 -1 roll exch div scale currentpoint translate 64 48 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Roman f1 (6) 105 272 sh (1) 728 272 sh (1) 1247 272 sh (0) 1799 272 sh (1) 2255 272 sh (0) 2727 272 sh (1) 97 752 sh (1) 632 752 sh (1) 1343 752 sh (0) 1799 752 sh (0) 2263 752 sh (1) 2719 752 sh (1) 97 1232 sh (5) 648 1232 sh (2) 1335 1232 sh (1) 1791 1232 sh (0) 2263 1232 sh (0) 2727 1232 sh 320 /Symbol f1 (-) 553 272 sh (-) 1168 752 sh (-) 1160 1232 sh (\351) -32 240 sh (\353) -32 1280 sh (\352) -32 544 sh (\352) -32 848 sh (\352) -32 1152 sh (\371) 2880 240 sh (\373) 2880 1280 sh (\372) 2880 544 sh (\372) 2880 848 sh (\372) 2880 1152 sh end MTsave restore dMATHt 6-1101011-100115-2100[]} 6-11010150001150100[] Equation 6-1101011-100115-2100[]TN*Equation Native *_1159001639F@P!-@P!-Ole -PIC .TCompObj0RObjInfo2OlePres0003Ole10NativeK5FMicrosoft EquationDNQE Equation.3TCIP4- @@hdxpr  h, Symbol P.( "h currentpoint ",Times(P"10+x4+x5.(H (J ( *Q *P (Hp (Jp (p *Q *P )/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 608 div 1440 3 -1 roll exch div scale currentpoint translate 64 48 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Roman f1 (10) 73 272 sh (4) 161 752 sh (5) 177 1232 sh 320 /Symbol f1 (\351) -32 240 sh (\353) -32 1280 sh (\352) -32 544 sh (\352) -32 848 sh (\352) -32 1152 sh (\371) 386 240 sh (\373) 386 1280 sh (\372) 386 544 sh (\372) 386 848 sh (\372) 386 1152 sh end MTsave restore d=MATH1  1045[]1 1045[] Equation1 1045[]Ole10FmtProgID L Equation Native MM_1159001622F@P!-@P!-Ole OPIC PTPICT RCompObjsRObjInfouT xFdxpr  F"F currentpoint ",2Times2 .+ 6, Symbol)-2)1)1)0)1)0)0)10( 1)5)0)0)0)1)0)4(. 1)1)0)1)0)0)0)5(@-2)2)-2)1)-2)1)1)0)0)1)0.(  *6 ( * * (  *6 ( * * 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5248 div 2240 3 -1 roll exch div scale currentpoint translate 64 49 translate 260 293 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (6) show 930 293 moveto 384 /Symbol f1 (-) show 1140 293 moveto 384 /Times-Roman f1 (1) show 1757 293 moveto (1) show 2407 293 moveto (0) show 2963 293 moveto (1) show 3521 293 moveto (0) show 4078 293 moveto (0) show 4602 293 moveto (10) show 260 869 moveto (1) show 1022 869 moveto (5) show 1758 869 moveto (0) show 2407 869 moveto (0) show 2964 869 moveto (0) show 3520 869 moveto (1) show 4078 869 moveto (0) show 4718 869 moveto (4) show 260 1445 moveto (1) show 1016 1445 moveto (1) show 1758 1445 moveto (0) show 2406 1445 moveto (1) show 2964 1445 moveto (0) show 3521 1445 moveto (0) show 4078 1445 moveto (0) show 4720 1445 moveto (5) show 158 2021 moveto 384 /Symbol f1 (-) show 368 2021 moveto 384 /Times-Roman f1 (2) show 930 2021 moveto 384 /Symbol f1 (-) show 1140 2021 moveto 384 /Times-Roman f1 (1) show 1671 2021 moveto 384 /Symbol f1 (-) show 1881 2021 moveto 384 /Times-Roman f1 (1) show 2406 2021 moveto (1) show 2964 2021 moveto (0) show 3521 2021 moveto (0) show 4077 2021 moveto (1) show 4715 2021 moveto (0) show 0 355 moveto 384 /Symbol f1 (\351) show 0 2106 moveto (\353) show 0 724 moveto (\352) show 0 1093 moveto (\352) show 0 1462 moveto (\352) show 0 1831 moveto (\352) show 5001 355 moveto (\371) show 5001 2106 moveto (\373) show 5001 724 moveto (\372) show 5001 1093 moveto (\372) show 5001 1462 moveto (\372) show 5001 1831 moveto (\372) show end FMicrosoft EquationDNQE Equation.3TCIPx" ; @@0dxpr  0, SymOlePres000v Ole10NativeOle10FmtProgID  Equation Native bol P.( "0 currentpoint ",Times(L>6)-,Times)+1)1)0)r1)v0)t0)l10(<1)1)-),1)r0)t0)r1)v0)4(<<1)5)-),2)r1)v0)t0)t0)5(&-),2)p-)+1)n-),1)p1)v0)t0)r1)0.(D ( ( *Q *Q *Q (D ( ( *Q *Q *Q /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4288 div 1888 3 -1 roll exch div scale currentpoint translate 64 32 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Roman f1 (6) 184 272 sh (1) 887 272 sh (1) 1406 272 sh (0) 1958 272 sh (1) 2414 272 sh (0) 2886 272 sh (0) 3350 272 sh (10) 3782 272 sh (1) 176 752 sh (1) 791 752 sh (1) 1502 752 sh (0) 1958 752 sh (0) 2422 752 sh (1) 2878 752 sh (0) 3350 752 sh (4) 3870 752 sh (1) 176 1232 sh (5) 807 1232 sh (2) 1494 1232 sh (1) 1950 1232 sh (0) 2422 1232 sh (0) 2886 1232 sh (0) 3350 1232 sh (5) 3886 1232 sh (2) 264 1712 sh (1) 887 1712 sh (1) 1502 1712 sh (1) 1950 1712 sh (0) 2422 1712 sh (0) 2886 1712 sh (1) 3342 1712 sh (0) 3878 1712 sh 320 /Symbol f1 (-) 712 272 sh (-) 1327 752 sh (-) 1319 1232 sh (-) 89 1712 sh (-) 712 1712 sh (-) 1327 1712 sh (\351) -32 240 sh (\353) -32 1760 sh (\352) -32 544 sh (\352) -32 848 sh (\352) -32 1152 sh (\352) -32 1456 sh (\371) 4095 240 sh (\373) 4095 1760 sh (\372) 4095 544 sh (\372) 4095 848 sh (\372) 4095 1152 sh (\372) 4095 1456 sh end MTsave restore dMATH 6-1101001011-10010415-210005-2-1-110010[] 6-110100101500010411010005-2-1-110010[] Equation 6-1101001011-10010415-210005-2-1-110010[]_1159001651/$F@P!-@P!-Ole PIC !#TPICT T xFdxpr  F"F currentpoint ",2Times2 .+ 6, Symbol)-2)1)1)0)1)0)0)10(1)5)0)0)0)1)0)4(.1)1)0)1)0)0)0)5(@4)-2)2)0)1)1)0)1)10.(  *6 ( * * (  *6 ( * * 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4896 div 2240 3 -1 roll exch div scale currentpoint translate 64 49 translate 154 293 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (6) show 735 293 moveto 384 /Symbol f1 (-) show 945 293 moveto 384 /Times-Roman f1 (1) show 1485 293 moveto (1) show 2043 293 moveto (0) show 2599 293 moveto (1) show 3157 293 moveto (0) show 3714 293 moveto (0) show 4238 293 moveto (10) show 154 869 moveto (1) show 827 869 moveto (5) show 1486 869 moveto (0) show 2043 869 moveto (0) show 2600 869 moveto (0) show 3156 869 moveto (1) show 3714 869 moveto (0) show 4354 869 moveto (4) show 154 1445 moveto (1) show 821 1445 moveto (1) show 1486 1445 moveto (0) show 2042 1445 moveto (1) show 2600 1445 moveto (0) show 3157 1445 moveto (0) show 3714 1445 moveto (0) show 4356 1445 moveto (5) show 158 2021 moveto (4) show 719 2021 moveto 384 /Symbol f1 (-) show 929 2021 moveto 384 /Times-Roman f1 (2) show 1486 2021 moveto (0) show 2042 2021 moveto (1) show 2599 2021 moveto (1) show 3157 2021 moveto (0) show 3713 2021 moveto (1) show 4238 2021 moveto (10) show 0 355 moveto 384 /Symbol f1 (\351) show 0 2106 moveto (\353) show 0 724 moveto (\352) show 0 1093 moveto (\352) show 0 1462 moveto (\352) show 0 1831 moveto (\352) show 4637 355 moveto (\371) show 4637 2106 moveto (\373) show 4637 724 moveto (\372) show 4637 1093 moveto (\372) show 4637 1462 moveto (\372) show 4637 1831 moveto (\372) show end FMicrosoft EquationDNQE Equation.3TCIP" ; @@8dxpr  8, Symbol P.( "8 currentpoiCompObj"&RObjInfoOlePres000%( Ole10Nativent ",Times(L<6)-,Times)+1)1)0)r1)v0)t0)n10(:1)1)-),1)r0)t0)r1)v0)4(<:1)5)-),2)r1)v0)t0)t0)5(&-),3)l-)+6)1)0)t0)t0)r1)r-),5.(D ( ( *Q *Q *Q (D ( ( *Q *Q *Q /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4320 div 1888 3 -1 roll exch div scale currentpoint translate 64 32 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Roman f1 (6) 176 272 sh (1) 879 272 sh (1) 1406 272 sh (0) 1958 272 sh (1) 2414 272 sh (0) 2886 272 sh (0) 3350 272 sh (10) 3789 272 sh (1) 168 752 sh (1) 783 752 sh (1) 1502 752 sh (0) 1958 752 sh (0) 2422 752 sh (1) 2878 752 sh (0) 3350 752 sh (4) 3877 752 sh (1) 168 1232 sh (5) 799 1232 sh (2) 1494 1232 sh (1) 1950 1232 sh (0) 2422 1232 sh (0) 2886 1232 sh (0) 3350 1232 sh (5) 3893 1232 sh (3) 264 1712 sh (6) 871 1712 sh (1) 1406 1712 sh (0) 1958 1712 sh (0) 2422 1712 sh (0) 2886 1712 sh (1) 3342 1712 sh (5) 3973 1712 sh 320 /Symbol f1 (-) 704 272 sh (-) 1327 752 sh (-) 1319 1232 sh (-) 89 1712 sh (-) 696 1712 sh (-) 3798 1712 sh (\351) -32 240 sh (\353) -32 1760 sh (\352) -32 544 sh (\352) -32 848 sh (\352) -32 1152 sh (\352) -32 1456 sh (\371) 4110 240 sh (\373) 4110 1760 sh (\372) 4110 544 sh (\372) 4110 848 sh (\372) 4110 1152 sh (\372) 4110 1456 sh end MTsave restore dMATH 6-1101001011-10010415-210005-3-610001-5[] 6-1101001015000104110100054-20110110[] Equation 6-1101001011-10010415-210005-3-610001-5[]Ole10FmtProgID ') Equation Native _1159001661,F@P!-@P!-Ole CompObj+-NObjInfo.Equation Native {_1159001668*41F@P!-@P!-   !$'*-./2456789:;<=>?@ABCDEGJKLMNOPQRSTUVWXYZ[\]^_`abcdfilnopqrstuvwxyz{|}~FMicrosoft EquationDNQE Equation.3_ B i =100010001[]FMicrosoft EquationDNQE Equation.3Ole CompObj02NObjInfo3Equation Native  f B i-1 =100010001[]FMicrosoft EquationDNQE Equation.3_11590016766F@P!-@P!-Ole  CompObj57 NObjInfo8Equation Native q_1159001683 M;F@P!-@P!-Ole CompObj:<NU C B i T =00-1[]FMicrosoft EquationDNQE Equation.3 C B i T B i-1 A=00-1ObjInfo=Equation Native _1159001693@F@P!-@P!-Ole []A=-1-51000[]FMicrosoft EquationDNQE Equation.3 C B i T B i-1 A-C T =CompObj?ANObjInfoBEquation Native _1159001701>HEF@P!-@P!-00-1[]A=-3-61000[]FMicrosoft EquationDNQE Equation.3b C B i T B i-1 b=0+0+-5=-5Ole "CompObjDF#NObjInfoG%Equation Native &~_1159001709JF@P!-@P!-Ole (CompObjIK)RObjInfoL+FMicrosoft EquationDNQE Equation.3 6-110100107000110141330120025-9-500-101-15[]Equation Native ,_1159001724CkQF@P!-@P!-Ole 0PIC NP1TT33--dxpr  --, Palatino .*- "-- currentpoint "( 0)1)0(0)0)1((1)0)0, Symbol.(  * ( *  *  ( ( * (( *  *  $30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1440 PICT 3CompObjOSFRObjInfoHOlePres000RUIdiv 1440 3 -1 roll exch div scale currentpoint translate 64 32 translate 132 288 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 320 /Palatino-Roman f1 (0) show 597 288 moveto (1) show 1050 288 moveto (0) show 132 768 moveto (0) show 591 768 moveto (0) show 1056 768 moveto (1) show 138 1248 moveto (1) show 591 1248 moveto (0) show 1050 1248 moveto (0) show 0 296 moveto 320 /Symbol f1 (\351) show 0 1345 moveto (\353) show 0 601 moveto (\352) show 0 906 moveto (\352) show 0 1211 moveto (\352) show 1224 296 moveto (\371) show 1224 1345 moveto (\373) show 1224 601 moveto (\372) show 1224 906 moveto (\372) show 1224 1211 moveto (\372) show end dXMATHL 010001100[] dFMicrosoft EquationDNQE Equation.3TCIP4-1 @@hdxpr  h, Symbol P.( "h currentpoint ",Times(P<1)0)t0((-,Times),1)p1)v0(@&-),2)t0)r1.(H (J ( *Q *P (H` (J` (` *Q *P /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1568 div 1440 3 -1 roll exch div scale currentpoint translate 64 48 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Roman f1 (1) 176 272 sh (0) 728 272 sh (0) 1192 272 sh (1) 272 752 sh (1) 720 752 sh (0) 1192 752 sh (2) 264 1232 sh (0) 728 1232 sh (1) 1184 1232 sh 320 /Symbol f1 (-) 97 752 sh (-) 89 1232 sh (\351) -32 240 sh (\353) -32 1280 sh (\352) -32 544 sh (\352) -32 848 sh (\352) -32 1152 sh (\371) 1345 240 sh (\373) 1345 1280 sh (\372) 1345 544 sh (\372) 1345 848 sh (\372) 1345 1152 sh end MTsave restore d^MATHR  100-110-201[]L 010001100[] EquationOle10NativeePOle10FmtProgID TVg Equation Native hn_1159001732[F@P!-@P!-R 100-110-201[]T33--dxpr  --, Palatino .*- "-- currentpoint "( 0)0)1(1)0)0((0)1)0, Symbol.(Ole jPIC XZkTPICT mCompObjY]R  * ( *  *  ( ( * (( *  *  $30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1440 div 1440 3 -1 roll exch div scale currentpoint translate 64 32 translate 132 288 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 320 /Palatino-Roman f1 (0) show 591 288 moveto (0) show 1056 288 moveto (1) show 138 768 moveto (1) show 591 768 moveto (0) show 1050 768 moveto (0) show 132 1248 moveto (0) show 597 1248 moveto (1) show 1050 1248 moveto (0) show 0 296 moveto 320 /Symbol f1 (\351) show 0 1345 moveto (\353) show 0 601 moveto (\352) show 0 906 moveto (\352) show 0 1211 moveto (\352) show 1224 296 moveto (\371) show 1224 1345 moveto (\373) show 1224 601 moveto (\372) show 1224 906 moveto (\372) show 1224 1211 moveto (\372) show end dXMATHL 001100010[] dFMicrosoft EquationDNQE Equation.3TCIP4RR-, @@h`dxpr  h`, Symbol P.( "h` currentpoint ",Times(P(1)v0)t0((1)t1)v0(@*2)t0)r1.(H (J ObjInfoOlePres000\_zOle10NativePOle10FmtProgID ^` ( *Q *P (H8 (J8 (8 *Q *P /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1408 div 1440 3 -1 roll exch div scale currentpoint translate 64 48 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Roman f1 (1) 97 272 sh (0) 569 272 sh (0) 1033 272 sh (1) 97 752 sh (1) 561 752 sh (0) 1033 752 sh (2) 105 1232 sh (0) 569 1232 sh (1) 1025 1232 sh 320 /Symbol f1 (\351) -32 240 sh (\353) -32 1280 sh (\352) -32 544 sh (\352) -32 848 sh (\352) -32 1152 sh (\371) 1186 240 sh (\373) 1186 1280 sh (\372) 1186 544 sh (\372) 1186 848 sh (\372) 1186 1152 sh end MTsave restore dXMATHL 100110201[]0L 001100010[] EquationEquation Native h_1159001742WfcF@P!-@P!-Ole CompObjbdNL 100110201[]FMicrosoft EquationDNQE Equation.3 C BT B -1 A=-10-1[]AObjInfoeEquation Native _1159001750hF@P!-@P!-Ole =-7-41-1-10[]FMicrosoft EquationDNQE Equation.3 C BT B -1 A-C T =-10CompObjgiNObjInfojEquation Native _1159001758azmF@P!-@P!--1[]A=-9-500-10[]FMicrosoft EquationDNQE Equation.3[ C BT B -1 b=-10+0+-5=-15Ole CompObjlnNObjInfooEquation Native w_1159001766tF@P!-@P!-Ole PIC qsTPICT \T!*\4;dxpr  4;"4; currentpoint ",2Times2 .+ 6, Symbol)-2)1)1(1)5)0(/1)1)0.(  *$ ( * ( 5 *$ (5 * ^30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1888 div 1664 3 -1 roll exch div scale currentpoint translate 64 39 translate 149 313 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (6) show 708 313 moveto 384 /Symbol f1 (-) show 918 313 moveto 384 /Times-Roman f1 (1) show 1443 313 moveto (1) show 149 889 moveto (1) show 800 889 moveto (5) show 1444 889 moveto (0) show 149 1465 moveto (1) show 794 1465 moveto (1) show 1444 1465 moveto (0) show 0 355 moveto 384 /Symbol f1 (\351) show 0 1550 moveto (\353) show 0 724 moveto (\352) show 0 1093 moveto (\352) show 0 1462 moveto (\352) show 1651 355 moveto (\371) show 1651 1550 moveto (\373) show 1651 724 moveto (\372) show 1651 1093 moveto (\372) show 1651 1462 moveto (\372) show end FMicrosoft EquationDNQE Equation.3CompObjrvRObjInfoOlePres000uxOle10NativeSTCIP4-0 @@hdxpr  h, Symbol P.( "h currentpoint ",Times(P*6)p-,Times),1)n1((1)1)0(@(1)5)0.(H (J ( *Q *P (H\ (J\ (\ *Q *P /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1536 div 1440 3 -1 roll exch div scale currentpoint translate 64 48 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Roman f1 (6) 105 272 sh (1) 728 272 sh (1) 1168 272 sh (1) 97 752 sh (1) 632 752 sh (0) 1176 752 sh (1) 97 1232 sh (5) 648 1232 sh (0) 1176 1232 sh 320 /Symbol f1 (-) 553 272 sh (\351) -32 240 sh (\353) -32 1280 sh (\352) -32 544 sh (\352) -32 848 sh (\352) -32 1152 sh (\371) 1329 240 sh (\373) 1329 1280 sh (\372) 1329 544 sh (\372) 1329 848 sh (\372) 1329 1152 sh end MTsave restore d[MATHO n 6-11110150[]pO 6-11150110[] EquationO 6-11110150[]Ole10FmtProgID wy Equation Native k_1159001777p~F@P!-@P!-Ole T3-2dxpr  -2, Palatino .*- "-2 currentpoint "( 1)1)0(6, Symbol)-)1) 1((1)5)0.(  * ( *  *  ( - * (- *  *  b30 dict begin currentpoint 3 -1 roll sub neg 3 1 rollPIC {}TPICT CompObj|RObjInfo     !"#$&),./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ\_`abcdefghijklmnopqrstuvwxyz{|}~ sub 1600 div 1440 3 -1 roll exch div scale currentpoint translate 64 32 translate 138 288 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 320 /Palatino-Roman f1 (1) show 677 288 moveto (1) show 1209 288 moveto (0) show 131 768 moveto (6) show 597 768 moveto 320 /Symbol f1 (-) show 772 768 moveto 320 /Palatino-Roman f1 (1) show 1215 768 moveto (1) show 138 1248 moveto (1) show 674 1248 moveto (5) show 1209 1248 moveto (0) show 0 296 moveto 320 /Symbol f1 (\351) show 0 1345 moveto (\353) show 0 601 moveto (\352) show 0 906 moveto (\352) show 0 1211 moveto (\352) show 1383 296 moveto (\371) show 1383 1345 moveto (\373) show 1383 601 moveto (\372) show 1383 906 moveto (\372) show 1383 1211 moveto (\372) show end d[MATHO d 1106-11150[]v FMicrosoft EquationDNQE Equation.3TCIP+4-4 @@hdxpr  h, Symbol P.( "h currentpoint ",Times(P86)~-,Times),1)n1(87)0)1(@"13)3)2.(H (J ( *Q *P (Hx (JOlePres000 Ole10Native%SOle10FmtProgID ' Equation Native (nx (x *Q *P /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1664 div 1440 3 -1 roll exch div scale currentpoint translate 64 48 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Roman f1 (6) 161 272 sh (1) 840 272 sh (1) 1280 272 sh (7) 161 752 sh (0) 752 752 sh (1) 1280 752 sh (13) 73 1232 sh (3) 760 1232 sh (2) 1288 1232 sh 320 /Symbol f1 (-) 665 272 sh (\351) -32 240 sh (\353) -32 1280 sh (\352) -32 544 sh (\352) -32 848 sh (\352) -32 1152 sh (\371) 1441 240 sh (\373) 1441 1280 sh (\372) 1441 544 sh (\372) 1441 848 sh (\372) 1441 1152 sh end MTsave restore d^MATHR 1 6-117011332[]O 1106-11150[] EquationR 6-117011332[]TX  R!dxpr  !, Palatino .*! ) "! currentpoint "( B, Symbol (-)1 +D)$B _1159001788F@P!-@P!-Ole *PIC +TPICT -R (Q-)1 +B)0)B (-)1 +b(C +B( T +B (-)1 +D) -)C +D(:T +0)1) C +B(~T +B (-)1 +b "F"d"r.(  * ( (  * ( 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5056 div 1056 3 -1 roll exch div scale currentpoint translate 64 56 translate 720 336 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 320 /Palatino-Bold f1 (B) show 945 194 moveto 224 /Symbol f1 (-) show 1067 194 moveto 224 /Palatino-Roman f1 (1) show 1178 336 moveto 320 /Palatino-Bold f1 (D) show 2327 336 moveto (B) show 2552 194 moveto 224 /Symbol f1 (-) show 2674 194 moveto 224 /Palatino-Roman f1 (1) show 2783 336 moveto 320 /Palatino-Bold f1 (B) show 3288 336 moveto (0) show 3953 336 moveto (B) show 4178 194 moveto 224 /Symbol f1 (-) show 4300 194 moveto 224 /Palatino-Roman f1 (1) show 4415 336 moveto 320 /Palatino-Bold f1 (b) show 125 816 moveto (C) show 372 896 moveto 224 ns (B) show 375 674 moveto (T) show 542 816 moveto 320 ns (B) show 767 674 moveto 224 /Symbol f1 (-) show 889 674 moveto 224 /Palatino-Roman f1 (1) show 1000 816 moveto 320 /Palatino-Bold f1 (D) show 1327 816 moveto 320 /Symbol f1 (-) show 1562 816 moveto 320 /Palatino-Bold f1 (C) show 1811 896 moveto 224 ns (D) show 1812 674 moveto (T) show 2578 816 moveto 320 ns (0) show 3296 816 moveto 320 /Palatino-Roman f1 (1) show 3742 816 moveto 320 /Palatino-Bold f1 (C) show 3989 896 moveto 224 ns (B) show 3992 674 moveto (T) show 4159 816 moveto 320 ns (B) show 4384 674 moveto 224 /Symbol f1 (-) show 4506 674 moveto 224 /Palatino-Roman f1 (1) show 4621 816 moveto 320 /Palatino-Bold f1 (b) show /sol { [] 0 setdash } def /dsh { [ exch 3 div dup .6 mul ] 0 setdash } def /dot { [ thick 3 -1 roll 5 div ] 0 setdash } def sol /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 141 435 moveto 4659 0 rlineto stroke sol 2186 33 moveto 0 873 rlineto stroke sol 3146 33 moveto 0 873 rlineto stroke sol 3606 33 moveto 0 873 rlineto stroke sol 0 296 moveto 320 /Symbol f1 (\351) show 0 913 moveto (\353) show 0 601 moveto (\352) show 0 858 moveto (\352) show 4826 296 moveto (\371) show 4826 913 moveto H(\373) show 4826 601 moveto (\372) show 4826 858 moveto (\372) show end dMATH TB -1 DB -1 B0B -1 bC BT B -1 D-C DT 01C BT B -1 b[]foFMicrosoft EquationDNQE Equation.3TCIPfF  @@dxpr  , SymCompObj[RObjInfo]OlePres000^< Ole10Nativebol P.(8 " currentpoint ",Times(ZB 0(6-,Times)1 P+$D(Z(B 0(6\-)1 P+$B)0)B 0(6-)1 P+$b(&C 0+<B(bT P+($B 0(-)1 P+$D)D-)8C 0+<D(T P+$0)1)rC 0+<B(T P+($B 0(-)1 P+$b r*rx   ^^.(F * ( (F{ * ({ D/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4736 div 992 3 -1 roll exch div scale currentpoint translate 64 40 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def /sol { [] 0 setdash } def /dsh { [ exch 3 div dup .6 mul ] 0 setdash } def /dot { [ thick 3 -1 roll 5 div ] 0 setdash } def sol 16 th 105 424 rndmvto 4411 0 rlineto stroke sol 1994 32 rndmvto 0 848 rlineto stroke sol 2931 32 rndmvto 0 848 rlineto stroke sol 3395 32 rndmvto 0 848 rlineto stroke sol /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Bold f1 (B) 649 320 sh (D) 1074 320 sh (B) 2146 320 sh (B) 2571 320 sh (0) 3083 320 sh (B) 3739 320 sh (b) 4164 320 sh (C) 89 800 sh (B) 489 800 sh (D) 914 800 sh (C) 1410 800 sh (0) 2382 800 sh (C) 3531 800 sh (B) 3931 800 sh (b) 4356 800 sh 192 ns (B) 329 880 sh (T) 329 656 sh (D) 1650 880 sh (T) 1650 656 sh (B) 3771 880 sh (T) 3771 656 sh 192 /Symbol f1 (-) 857 176 sh (-) 2354 176 sh (-) 3947 176 sh (-) 697 656 sh (-) 4139 656 sh 320 ns (-) 1186 800 sh (\351) -32 240 sh (\353) -32 848 sh (\352) -32 544 sh (\371) 4525 240 sh (\373) 4525 848 sh (\372) 4525 544 sh 192 /Times-Roman f1 (1) 962 176 sh (1) 2459 176 sh (1) 4052 176 sh (1) 802 656 sh (1) 4244 656 sh 320 ns (1) 3075 800 sh end MTsave restore dMATH TB -1 DB -1 B0B -1 bC BT B -1 D-C DT 01C BT B -1 b[] TB -1 DB -1 B0B -1 bC BT B -1 D-C DT 01C BT B -1 b[] EquationOle10FmtProgID  Equation Native _1159001804F@P!-@P!-Ole  TB -1 DB -1 B0B -1 bC BT B -1 D-C DT 01C BT B -1 b[]T "XPIC TPICT CompObjRObjInfo[dxpr  [, Palatino .* "[ currentpoint "( D)B)0)b, Symbol(-)C +D(T +-)C +B(1T +1)0" N""<"J.(  * ( ( V * (V 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2912 div 960 3 -1 roll exch div scale currentpoint translate 64 60 translate 332 264 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 320 /Palatino-Bold f1 (D) show 1304 264 moveto (B) show 2023 264 moveto (0) show 2488 264 moveto (b) show 138 744 moveto 320 /Symbol f1 (-) show 325 744 moveto 320 /Palatino-Bold f1 (C) show 574 824 moveto 224 ns (D) show 575 602 moveto (T) show 1098 744 moveto 320 /Symbol f1 (-) show 1285 744 moveto 320 /Palatino-Bold f1 (C) show 1532 824 moveto 224 ns (B) show 1535 602 moveto (T) show 2025 744 moveto 320 ns (1) show 2500 744 moveto (0) show /sol { [] 0 setdash } def /dsh { [ exch 3 div dup .6 mul ] 0 setdash } def /dot { [ thick 3 -1 roll 5 div ] 0 setdash } def sol /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 141 363 moveto 2526 0 rlineto stroke sol 949 33 moveto 0 801 rlineto stroke sol 1881 33 moveto 0 801 rlineto stroke sol 2341 33 moveto 0 801 rlineto stroke sol 0 296 moveto 320 /Symbol f1 (\351) show 0 841 moveto (\353) show 0 601 moveto (\352) show 2693 296 moveto (\371) show 2693 841 moveto (\373) show 2693 601 moveto (\372) show end d|MATHp  TDB0b-C DT -C BT 10[]60FMicrosoft EquationDNQE Equation.3TCIP XXU @OlePres000Ole10NativetOle10FmtProgID  Equation Native @dxpr  , Symbol P.(( " currentpoint ",Times(LWD)B)0)tb(&-,Times)*C 0+<D(T P+x$-)*C 0+<B(jT P+x$1)v0 d*d}   ...(D * ( (D * ( /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2720 div 928 3 -1 roll exch div scale currentpoint translate 64 32 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def /sol { [] 0 setdash } def /dsh { [ exch 3 div dup .6 mul ] 0 setdash } def /dot { [ thick 3 -1 roll 5 div ] 0 setdash } def sol 16 th 105 376 rndmvto 2386 0 rlineto stroke sol 842 32 rndmvto 0 800 rlineto stroke sol 1715 32 rndmvto 0 800 rlineto stroke sol 2179 32 rndmvto 0 800 rlineto stroke sol /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Bold f1 (D) 285 272 sh (B) 1174 272 sh (0) 1867 272 sh (b) 2331 272 sh (C) 258 752 sh (C) 1147 752 sh (1) 1867 752 sh (0) 2339 752 sh 192 ns (D) 498 832 sh (T) 498 608 sh (B) 1387 832 sh (T) 1387 608 sh 320 /Symbol f1 (-) 89 752 sh (-) 978 752 sh (\351) -32 240 sh (\353) -32 800 sh (\352) -32 544 sh (\371) 2500 240 sh (\373) 2500 800 sh (\372) 2500 544 sh end MTsave restore d|MATHp  TDB0b-C DT -C BT 10[]p TDB0b-C DT -C BT 10[] Equationp TDB0b-C DT -C BT 10[]_1159001820F@P!-@P!-Ole PIC TPICT R TX  R!dxpr  !, Palatino .*! ) "! currentpoint "( B, Symbol (-)1 +D)$B (Q-)1 +B)0)B (-)1 +b(C +B( T +B (-)1 +D) -)C +D(:T +0)1) C +B(      !"#$%&'()*+,-./0123456789:;<=>?ABCFGHIKLMNOPQSTUVX~T +B (-)1 +b "F"d"r.(  * ( (  * ( 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5056 div 1056 3 -1 roll exch div scale currentpoint translate 64 56 translate 720 336 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 320 /Palatino-Bold f1 (B) show 945 194 moveto 224 /Symbol f1 (-) show 1067 194 moveto 224 /Palatino-Roman f1 (1) show 1178 336 moveto 320 /Palatino-Bold f1 (D) show 2327 336 moveto (B) show 2552 194 moveto 224 /Symbol f1 (-) show 2674 194 moveto 224 /Palatino-Roman f1 (1) show 2783 336 moveto 320 /Palatino-Bold f1 (B) show 3288 336 moveto (0) show 3953 336 moveto (B) show 4178 194 moveto 224 /Symbol f1 (-) show 4300 194 moveto 224 /Palatino-Roman f1 (1) show 4415 336 moveto 320 /Palatino-Bold f1 (b) show 125 816 moveto (C) show 372 896 moveto 224 ns (B) show 375 674 moveto (T) show 542 816 moveto 320 ns (B) show 767 674 moveto 224 /Symbol f1 (-) show 889 674 moveto 224 /Palatino-Roman f1 (1) show 1000 816 moveto 320 /Palatino-Bold f1 (D) show 1327 816 moveto 320 /Symbol f1 (-) show 1562 816 moveto 320 /Palatino-Bold f1 (C) show 1811 896 moveto 224 ns (D) show 1812 674 moveto (T) show 2578 816 moveto 320 ns (0) show 3296 816 moveto 320 /Palatino-Roman f1 (1) show 3742 816 moveto 320 /Palatino-Bold f1 (C) show 3989 896 moveto 224 ns (B) show 3992 674 moveto (T) show 4159 816 moveto 320 ns (B) show 4384 674 moveto 224 /Symbol f1 (-) show 4506 674 moveto 224 /Palatino-Roman f1 (1) show 4621 816 moveto 320 /Palatino-Bold f1 (b) show /sol { [] 0 setdash } def /dsh { [ exch 3 div dup .6 mul ] 0 setdash } def /dot { [ thick 3 -1 roll 5 div ] 0 setdash } def sol /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 141 435 moveto 4659 0 rlineto stroke sol 2186 33 moveto 0 873 rlineto stroke sol 3146 33 moveto 0 873 rlineto stroke sol 3606 33 moveto 0 873 rlineto stroke sol 0 296 moveto 320 /Symbol f1 (\351) show 0 913 moveto (\353) show 0 601 moveto (\352) show 0 858 moveto (\352) show 4826 296 moveto (\371) show 4826 913 moveto H(\373) show 4826 601 moveto (\372) show 4826 858 moveto (\372) show end dMATH TB -1 DB -1 B0B -1 bC BT B -1 D-C DT 01C BT B -1 b[]foCompObjRObjInfoOlePres000< Ole10Native@FMicrosoft EquationDNQE Equation.3TCIPfF  @@dxpr  , Symbol P.(8 " currentpoint ",Times(ZB 0(6-,Times)1 P+$D(Z(B 0(6\-)1 P+$B)0)B 0(6-)1 P+$b(&C 0+<B(bT P+($B 0(-)1 P+$D)D-)8C 0+<D(T P+$0)1)rC 0+<B(T P+($B 0(-)1 P+$b r*rx   ^^.(F * ( (F{ * ({ D/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4736 div 992 3 -1 roll exch div scale currentpoint translate 64 40 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def /sol { [] 0 setdash } def /dsh { [ exch 3 div dup .6 mul ] 0 setdash } def /dot { [ thick 3 -1 roll 5 div ] 0 setdash } def sol 16 th 105 424 rndmvto 4411 0 rlineto stroke sol 1994 32 rndmvto 0 848 rlineto stroke sol 2931 32 rndmvto 0 848 rlineto stroke sol 3395 32 rndmvto 0 848 rlineto stroke sol /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 320 /Times-Bold f1 (B) 649 320 sh (D) 1074 320 sh (B) 2146 320 sh (B) 2571 320 sh (0) 3083 320 sh (B) 3739 320 sh (b) 4164 320 sh (C) 89 800 sh (B) 489 800 sh (D) 914 800 sh (C) 1410 800 sh (0) 2382 800 sh (C) 3531 800 sh (B) 3931 800 sh (b) 4356 800 sh 192 ns (B) 329 880 sh (T) 329 656 sh (D) 1650 880 sh (T) 1650 656 sh (B) 3771 880 sh (T) 3771 656 sh 192 /Symbol f1 (-) 857 176 sh (-) 2354 176 sh (-) 3947 176 sh (-) 697 656 sh (-) 4139 656 sh 320 ns (-) 1186 800 sh (\351) -32 240 sh (\353) -32 848 sh (\352) -32 544 sh (\371) 4525 240 sh (\373) 4525 848 sh (\372) 4525 544 sh 192 /Times-Roman f1 (1) 962 176 sh (1) 2459 176 sh (1) 4052 176 sh (1) 802 656 sh (1) 4244 656 sh 320 ns (1) 3075 800 sh end MTsave restore dMATH TB -1 DB -1 B0B -1 bC BT B -1 D-C DT 01C BT B -1 b[] TB -1 DB -1 B0B -1 bC BT B -1 D-C DT 01C BT B -1 b[] Equation TB -1 DB -1 B0B -1 bC BT B -1 D-C DTOle10FmtProgID D Equation Native E1TableGSummaryInformation(J 01C BT B -1 b[] Oh+'0 $ D P \ ht|'338 ACTIVITY 13:Charles F. 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