ࡱ> vxu#` bjbj\.\. .>D>D/B_-~~~~222 t$}}}8}T.~-ڪ ....5777777$hr[92[~~..Y "~.2.5 5 p2ɢ. `˒}UL0ڪm\T0ɢ2ɢ [[ ڪ---O}---}F | ~~~~~~  Finding the area under a curve with random probability. (The Monte Carlo method for estimating area under curve) Summary: Initially, students will graph a curve whose area can be found using Geometry methods using the Monte Carlo method that uses random points and probability to estimate the area under the curve. Students also calculate the area geometrically to prove that the method provides a reasonable estimate of area. Students then use the method to estimate the area under non-geometric curves whose exact area is found using calculus methods. Key Words: Monte Carlo, Area Under Curve, Probability Ratios Background Knowledge: Geometry area formulas Key strokes for random number, list, store, window setting Ratios and percentages Calculation of mean OACS Standards: Mathematical Process Measurement Standard G: Write clearly and coherently about mathematical thinking and ideas. Measurement Standard Standard C - Grades 8-10: Apply indirect measurement techniques, tools and formulas, as appropriate to find perimeter, circumference and area of circles, triangles, quadrilaterals and composite shapes and to find volume of prisms, cylinders and pyramids Standard C - Grades 11-12: Estimate and compute areas and volume in increasingly complex problem situations Learning Objectives: The student will find the area under a curve using geometry formulas Students will apply the Monte Carlo method to estimate the area under a curve on a given interval Students will make comparisons between the estimated area and the actual area Materials: Graphing calculator Copy of inquiry based activity Suggested Procedures: Use soft dart board and set it on wall to simulate probability of hitting area of target (Attention Getter) Group students in threes so students can compare individual results and create a group average for increased accuracy Assessment: Collect activities from each group Monitor student progress during completion of activity Name: ___________________ ACTIVITY: MONTE CARLO METHOD FOR ESTIMATING THE AREA UNDER THE CURVE Lesson Objectives: Reinforce concept of domain and codomain Calculate area under a curve on a given interval Determine percent of error Introduce integration for calculus In geometry, we have used the ratio of areas to calculate the probability that a dart would hit a shaded region within a given target. For example, suppose you want to win a stuffed animal for your moms birthday and decide to take a chance on the dart game. The game consists of 20 balloons (all congruent of course!) on a rectangular board as in Figure 1. If you throw a dart and it hits a balloon you win a prize. The probability of randomly hitting a balloon is the ratio of the area of all the balloons to the area of the entire rectangular region:  EMBED Equation.DSMT4    To simulate this activity, we could throw darts at a board and count the number of hits to determine the probability and then multiply the area of the board by this probability to determine the area of the target region, in this case the balloons. We are going to use this process to estimate the area of regions whose exact area cannot be calculated without the use of calculus. For each problem you are given an equation of a curve and an x-interval that together begin to define our region (board). You need to determine the y-interval that captures the region and set an appropriate viewing window based on these x and y-intervals on the calculator. Once the region, is set create 30 random ordered pairs with this region and create a scatter plot to represent where the darts hit the board. Next, count the number of hits (points on or under the curve) and use the ratio  EMBED Equation.DSMT4  to estimate the area. This is known as the Monte Carlo method for estimating the area under a curve. The first two problems are examples that can easily be solved geometrically in order to compare estimated area to actual area and verify the method used. Example: Estimate the area under the function  EMBED Equation.DSMT4 and above the x-axis on the interval  EMBED Equation.DSMT4 . Solution: Step 1: Determine the appropriate y-interval. Since we want the area above x-axis, the minimum y-value is 0. Also, since the graph is decreasing (How do know the function is decreasing? ________________________________________________), the maximum y-value will be at the left of the region along the boundary where  EMBED Equation.DSMT4 . To find the value, we evaluate  EMBED Equation.DSMT4 when  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Thus, the maximum y-value for the region is 8. Step 2: Set the window on the calculator, enter the equation of the function, and graph.  Looking at the window, the x-values of the points representing the darts need to be between 0 and 4 and the y-values need to be between 0 and 8. Step 3: The 30 coordinate points in our region are generated randomly. To generate 30 x-values, enter the key sequence [4] [MATH] PRB [1] [(] [3] [0] [)] [STO] [L1]. This statement will generate 30 random numbers with values between 0 and 4 and store them in L1 as x-values. Step 4: Generate 30 random numbers between 0 and 8 for the y-coordinates and store them in  EMBED Equation.DSMT4 . The screen should look like Figure 5 with the exception that the numbers in your list will be different because the numbers are randomly created by the calculator. Step 5: Turn ON the STAT PLOT and identify L1 and L2. (Figure 6) Graph these points with your equations. (Figure 7)  Step 6: Count the number of hits generated by your random points. (A hit is any point on or below the curve). The ratio  EMBED Equation.DSMT4  * the area of the viewing rectangle will be the estimated area of the region. (Why is the denominator 30? ________________________) In the example above, there are 15 hits, therefore, the estimated area is:  EMBED Equation.DSMT4  = 16. (Where did the 4*8 come from? ________________________________ (Hint:  EMBED Equation.DSMT4 )) Step 7: Record your results in the chart below. Repeat the experiment two times by reentering your two random number generator expressions for the x-values and y-values and complete the table. Calculate the average of your three trials then find the average of your groups trials and enter in the appropriate boxes below. Trial 1Trial 2Trial 3Average Area for 3 Trials.Groups Average AreaHits  Attempts 303030Estimated AreaStep 8: For this example, the exact area under the curve can be calculated geometrically to test the results of your estimated area. Notice the shape of the target region is a triangle with  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Calculate the exact area. Exact area under the curve is: _______________ Debrief Questions: How does your average area estimate compare to actual? What is the percent of error between your estimated area and the actual area? ________________________________________________________________________ How does your average area estimate compare to the groups average? What is the percent of error between your groups estimated area and the actual area? ________________________________________________________________________ What accounts for the differences between your average and the groups average? ________________________________________________________________________ Additional Examples: Estimate the area under the function  EMBED Equation  and above the x-axis on the interval  EMBED Equation . XMIN __________ YMIN __________ XMAX __________ YMAX __________ Expression for random values of x _____________________________________ Expression for random values of y _____________________________________ Trial 1Trial 2Trial 3Average Area for 3 Trials.Groups Average AreaHits  Attempts 303030Estimated Area Notice the shape of this region is a trapezoid (it may help to rotate the calculator  EMBED Equation ). Calculate the area of the region geometrically and compare to your estimate. Exact area under the curve is: _______________ Debrief Questions: How does your average area estimate compare to actual? What is the percent of error between your estimated area and the actual area? ________________________________________________________________________ How does your average area estimate compare to the groups average? What is the percent of error between your groups estimated area and the actual area? ________________________________________________________________________ What accounts for the differences between your average and the groups average? ________________________________________________________________________ Estimate the area under the function  EMBED Equation  and above the x-axis on the interval  EMBED Equation . XMIN __________ YMIN __________ XMAX __________ YMAX __________ Expression for random values of x _____________________________________ Expression for random values of y _____________________________________ Trial 1Trial 2Trial 3Average Area for 3 Trials.Groups Average AreaHits  Attempts 303030Estimated Area The area of this region cannot be determined geometrically. When you have completed the worksheet check with your teacher for the actual area. Exact area under the curve is: _______________ Debrief Questions: How does your average area estimate compare to actual? What is the percent of error between your estimated area and the actual area? ________________________________________________________________________ How does your average area estimate compare to the groups average? What is the percent of error between your groups estimated area and the actual area? ________________________________________________________________________ What accounts for the differences between your average and the groups average? ________________________________________________________________________ Estimate the area under the function  EMBED Equation  and above the x-axis on the interval  EMBED Equation . XMIN __________ YMIN __________ XMAX __________ YMAX __________ Expression for random values of x _____________________________________ Expression for random values of y _____________________________________ Trial 1Trial 2Trial 3Average Area for 3 Trials.Groups Average AreaHits  Attempts 303030Estimated Area In this case, the area of this region cannot be determined geometrically. Check with your teacher for the actual area. Exact area under the curve is: _______________ Debrief Questions: How does your average area estimate compare to actual? What is the percent of error between your estimated area and the actual area? ________________________________________________________________________ How does your average area estimate compare to the groups average? What is the percent of error between your groups estimated area and the actual area? ________________________________________________________________________ What accounts for the differences between your average and the groups average? ________________________________________________________________________ Estimate the area under the function  EMBED Equation  and above the x-axis on the interval  EMBED Equation . XMIN __________ YMIN __________ XMAX __________ YMAX __________ Expression for random values of x _____________________________________ Expression for random values of y _____________________________________ Trial 1Trial 2Trial 3Average Area for 3 Trials.Groups Average AreaHits  Attempts 303030Estimated Area In this case, the area of this region cannot be determined geometrically. Check with your teacher for the actual area. Exact area under the curve is: _______________ Debrief Questions: How does your average area estimate compare to actual? What is the percent of error between your estimated area and the actual area? ________________________________________________________________________ How does your average area estimate compare to the groups average? What is the percent of error between your groups estimated area and the actual area? ________________________________________________________________________ What accounts for the differences between your average and the groups average? ________________________________________________________________________ Estimate the area under the function  EMBED Equation  and above the x-axis on the interval  EMBED Equation . XMIN __________ YMIN __________ XMAX __________ YMAX __________ Expression for random values of x _____________________________________ Expression for random values of y _____________________________________ Trial 1Trial 2Trial 3Average Area for 3 Trials.Groups Average AreaHits  Attempts 303030Estimated Area In this case, the area of this region cannot be determined geometrically. Check with your teacher for the actual area. Exact area under the curve is: _______________ Debrief Questions: How does your average area estimate compare to actual? What is the percent of error between your estimated area and the actual area? ________________________________________________________________________ How does your average area estimate compare to the groups average? What is the percent of error between your groups estimated area and the actual area? ________________________________________________________________________ What accounts for the differences between your average and the groups average? ________________________________________________________________________ Summary Questions: What was the main goal of this activity? ________________________________________________________________________ ________________________________________________________________________ How does this connect with topics from previous courses? ________________________________________________________________________ ________________________________________________________________________ Additional Examples and Extension: Estimate the area under the function  EMBED Equation  and above the x-axis on the interval  EMBED Equation . XMIN __________ YMIN __________ XMAX __________ YMAX __________ Expression for random values of x _____________________________________ Expression for random values of y _____________________________________ Trial 1Trial 2Trial 3Average Area for 3 Trials.Groups Average AreaHits  Attempts 303030Estimated Area Calculate the area of the region geometrically and compare to your estimate. Exact area under the curve is: _______________ Debrief Questions: How does your average area estimate compare to actual? What is the percent of error between your estimated area and the actual area? ________________________________________________________________________ How does your average area estimate compare to the groups average? What is the percent of error between your groups estimated area and the actual area? ________________________________________________________________________ What accounts for the differences between your average and the groups average? ________________________________________________________________________ Estimate the area under the function  EMBED Equation  and above the x-axis on the interval  EMBED Equation . XMIN __________ YMIN __________ XMAX __________ YMAX __________ Expression for random values of x _____________________________________ Expression for random values of y _____________________________________ Trial 1Trial 2Trial 3Average Area for 3 Trials.Groups Average AreaHits  Attempts 303030Estimated Area In this case, the area of this region cannot be determined geometrically. Check with your teacher for the actual area. Exact area under the curve is: _______________ Debrief Questions: How does your average area estimate compare to actual? What is the percent of error between your estimated area and the actual area? ________________________________________________________________________ How does your average area estimate compare to the groups average? What is the percent of error between your groups estimated area and the actual area? ________________________________________________________________________ What accounts for the differences between your average and the groups average? ________________________________________________________________________     Project AMP Dr. Antonio R. Quesada Director, Project AMP ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( s{0 : r   :  # %ZpU`3"#CKǾDZǡǙNJǡzuǙ h>* hCJjhEHUj;H hCJUVaJjhUjhCJUmHnHu h6] hCJh5CJ\h h56h>*CJOJQJ^JaJ h56CJOJQJ^JaJhCJOJQJ^JaJ*9rs/ 0 q r    :  #  & F^ & F^ ^`$m^`ma$gd/KK# h %9XYZpTUa3 $dN$a$^ & F & F^ & F ^ & F3\qr!"$#$%NP^ & F     12IJKLQ\ˮˏ}ndUje;H hCJUVaJj hEHUj;H hCJUVaJ h>*jh5EHU\#j;H h5CJUV\aJjh5EHU\#je;H h5CJUV\aJjh5U\ h5\h5>*\hjhUj;hEHUj;H hCJUVaJPQ[\3|Yab~  /01256MNOP^`3456騞rha h6]jhEHUj;H hCJUVaJ hH*hOJQJjhCJUmHnHujhEHUj;H hCJUVaJjhEHUj;H hCJUVaJj4hEHUjZ;H hCJUVaJhjhUjahEHU&YZ 8!9!P!Q!R!S!!!7";"b"n""""""#######rjc*jhCJUmHnHu hH*h, X!Y!Z!""""""""$If $$Ifa$ """""""B99999 $$Ifa$kd$$$Ifl4ֈX h,"``064 laf4"""""#3kd%$$Ifl4ֈX h,"  064 laf4 $$Ifa$$If### # # ## $$Ifa$##### #!#B99999 $$Ifa$kdW&$$Ifl4ֈX h,"``064 laf4!#"###$#=$>$m$9777kd'$$Ifl4ֈX h,"  064 laf4 $$Ifa$###$$$$$$n$$&&&''$'%'&'''M'N'_'`'a'b'd'?(((((((())))*)鼱sdj`*\ h>*jv*hEHUju(?(@( $$Ifa$ & F@(H(P(X(s(($If $$Ifa$(((((((B99999 $$Ifa$kd1$$Ifl4ֈX h,"``064 laf4((((((3kd1$$Ifl4ֈX h,"  064 laf4 $$Ifa$$If((((((( $$Ifa$(((((((B99999 $$Ifa$kd2$$Ifl4ֈX h,"``064 laf4(((((})~)9777kd73$$Ifl4ֈX h,"  064 laf4 $$Ifa$*)+),))),=,>,O,P,Q,R,x,y,,,,,,i--------..1:1;1L1M1N1O1u1v11ݡ݂uj<h5EHU\#jb*hjhUj3hEHU&~))))))J*K***0+1+z+{+++,,,,,,,-h-i-j-r- $$Ifa$h^h & Fr-z-----3kd9$$Ifl4ֈX h,"``064 laf4$If $$Ifa$---------$If $$Ifa$-------B99999 $$Ifa$kdv:$$Ifl4ֈX h,"  064 laf4-------9kd-;$$Ifl4ֈX h,"``064 laf4 $$Ifa$-------97kd;$$Ifl4ֈX h,"  064 laf4 $$Ifa$-|.}.....H/I//-0.0w0x00011111112e2f2g2o2 $$Ifa$h^h & F11111f222222223356 616263646Z6[6l6m6n6o6r6K77777777y88:;;;;γΔu#jSc*h h5\jh5U\j>h5EHU\#jb*\ h>*hjLh5EHU\#jocE>>>*?+?|??????@M@N@@@@@AAeAfAAAABB'BIBBBBBBBBC%C$If $$Ifa$h^h & F%C&C+C,C-C.C/CB99999 $$Ifa$kdU$$Ifl4ֈX h,"``064 laf4/C0C1C2C3C*h h5\<[C\C]C^C_CCC9777kdW$$Ifl4ֈX h,"  064 laf4 $$Ifa$CCCCCxDDD]E^EEEEECFDFEFFFFGOGGGGGG $$Ifa$h^h & FGGGGGG3kd\$$Ifl4ֈX h,"``064 laf4 $$Ifa$$IfGGGGGGGG$If $$Ifa$GGGGGHHB99999 $$Ifa$kdZ]$$Ifl4ֈX h,"  064 laf4HHHHHHH9kd^$$Ifl4ֈX h,"``064 laf4 $$Ifa$HHHHHHH97kd^$$Ifl4ֈX h,"  064 laf4 $$Ifa$HHHHHH`IaIIIFJGJJJJJ,K-K.K/K1K2K4K5K7K8K:K;KK$a$gdkV9KKKKKK !#$%&/0234=>@ABCLMNP$a$KKKKKKKKKKKKKKKKL !"01>?NOijjohUjsmhUjjhUjhhUjfhUj dhUj_hUUh jh+( ( ( ( ( (  Figure 2  Figure 3  Figure 4  Figure 5  Figure 6  Figure 7 Figure 1  Pfghik,1h/ =!"#$% ;Dd lJ  C A? "2\q~e0yD`!q\q~e0+ ?xڥS1o@~wNMvHuJPĂT 7H*$!*YBّO@ :# ${wQ zݻw޽`@{rp88c)c|6 v]JxfW_il0#cl)8+o5$hP`MoJQyAD)?:[q*q#rOR<UYrYgGaCONSWDd J  C A? "2;ݶNbG'`!ݶNbG'0 dxڥRn@} Av0@ CRĽnbKPԤh8bzzP |0+ٷovg_b&wET RBtn20ںfa f_p?oUD~o:Xchm w4ruVꐮhou'3Eq4^ǍQ <;4ȬZDcezFQ= Dd @J  C A? "25-[LPQe+C-lX`! -[LPQe+C-l@0= xڕRAKQmj F `%5Y%%nR#$ɀPzj<2QHL&h--Y^RЩf:&2vxVЀs[Cw%reI VG:~_|%, w [v lӄt O7 '.鿸I>?'Y6j~j70(p2{i X? T۠zn4vn}T1?_JIvS:)fac/NX2zvg6"'z;#,J:ѦWB)?)37D$$xiPz&8gC?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklnopqrstwz{|}~Root Entry FC˒y7Data mb{WordDocument.ObjectPool' ˒C˒_1211883753F ˒ ˒Ole CompObjiObjInfo  #$%(+,-03458;<=>ADEFGJMNORUVWX[^_`abehijmpqruxyz} FMathType 5.0 Equation MathType EFEquation.DSMT49qxPDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  P(hit bEquation Native _1211884994 F ˒ ˒Ole  CompObj ialloon) =  Area of balloonsArea of board FMathType 5.0 Equation MathType EFEquation.DSMT49qObjInfo Equation Native (_1211885157F˒˒Ole  |,TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   hitsattempts FMathType 5.0 Equation MathType EFEquation.DSMT49qCompObjiObjInfoEquation Native _1211885277 F˒˒|TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  y=="-2x++8 FMathType 5.0 Equation MathTyOle CompObjiObjInfo!Equation Native "pe EFEquation.DSMT49q|TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  0d"xd"4 FMathType 5.0 Equation MathTy_1211885589F˒˒Ole &CompObj'iObjInfo)pe EFEquation.DSMT49q|,TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  x==0 FMathType 5.0 Equation MathTyEquation Native *_1211885658"F˒˒Ole .CompObj /ipe EFEquation.DSMT49q|,TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  x==0 FMathType 5.0 Equation MathTyObjInfo!1Equation Native 2_1211885731$F˒˒Ole 6CompObj#%7iObjInfo&9Equation Native :_1211885759O)F˒˒pe EFEquation.DSMT49q|,TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  y=="-2(0)++8Ole ?CompObj(*@iObjInfo+BEquation Native C FMathType 5.0 Equation MathType EFEquation.DSMT49q|TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  y==0++8y==8_1211887598.F˒˒Ole HCompObj-/IiObjInfo0K FMathType 5.0 Equation MathType EFEquation.DSMT49q|4TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  L 2Equation Native L_1211889760,63F˒˒Ole PCompObj24Qi FMathType 5.0 Equation MathType EFEquation.DSMT49q|TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   hits3ObjInfo5SEquation Native T _12118909348F˒˒Ole Y0 FMathType 5.0 Equation MathType EFEquation.DSMT49qa| TDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_ECompObj79ZiObjInfo:\Equation Native ]}_12118912601E=F˒˒_A  Area under curve =  1530*(4*8) FMathType 5.0 Equation MathType EFEquation.DSMT49q|TDSMT5WinAllBasicCodePagesOle cCompObj<>diObjInfo?fEquation Native gTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  A=lw FMathType 5.0 Equation MathType EFEquation.DSMT49q|TDSMT5WinAllBasicCodePages_1211891811BF˒˒Ole kCompObjACliObjInfoDnEquation Native o_1211891829@JGF˒˒Ole sCompObjFHtiTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  b==4 FMathType 5.0 Equation MathType EFEquation.DSMT49q| TDSMT5WinAllBasicCodePagesObjInfoIvEquation Native w_1211915879L F˒˒Ole {Times New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  h==8 FMathType Equation Equation Equation9qL y=2x+3%2CompObjKM|\ObjInfoN~Ole10NativeP_1211915922;^Q F˒˒ IoftTG FMathType Equation Equation Equation9q@ 0x3%2Ole CompObjPR\ObjInfoSOle10NativeD_1211916455V F˒p˒Ole CompObjUW\ObjInfoX FMathType Equation Equation Equation9q@ 90 o%2 FMathType Equation Equation Equation9qOle10NativeD_1211916559T[ Fp˒p˒Ole CompObjZ\\ObjInfo]Ole10Natived_1211916604Ym` Fp˒p˒Ole ` y=2x 2 +1&YesK%2P&N FMathType Equation Equation Equation9qH 0x2%2CompObj_a\ObjInfobOle10NativeL_1211916974he Fp˒p˒T%CT%C FMathType Equation Equation Equation9qH y= x %2 @T%CT%COle CompObjdf\ObjInfogOle10NativeL_1211917024j Fp˒p˒Ole CompObjik\ObjInfol FMathType Equation Equation Equation9qH 0x4%2T%CT%C FMathType Equation Equation Equation9qOle10NativeL_1211917087cwo Fp˒p˒Ole CompObjnp\ObjInfoqOle10Natived_1211917139t Fp˒p˒Ole ` y=x 3 +2P&YesK%2P&N FMathType Equation Equation Equation9q` y=-x 2 +25esK%2CompObjsu\ObjInfovOle10Natived_1211917167ry Fp˒p˒P&N FMathType Equation Equation Equation9qH 0x5%2T%CIOle CompObjxz\ObjInfo{Ole10NativeL_1211917328~ Fp˒p˒Ole CompObj}\ObjInfo FMathType Equation Equation Equation9q@ y=xab FMathType Equation Equation Equation9qOle10NativeD_1211917352| Fp˒p˒Ole CompObj\ObjInfoOle10NativeL_1211917453 Fp˒p˒Ole H -5x5%2 @T%CT%C FMathType Equation Equation Equation9q` y=sin(x)P&YesK%2CompObj\ObjInfoOle10Natived_1211917480 Fp˒p˒P&N FMathType Equation Equation Equation9qd 0 o x120 oP&NJIJIOh+'0Ole CompObj\ObjInfoOle10Nativehm#RF,^#k%6jUw'A:XFolWІavգ*~vWBO@Jѯ~Uʰ"=W*l hTless:Ab4U޵*5Q} ϸ)gkWGn]L`tn斬B`op=;xʰˠ024];Xpփd3c1$X.5(^C"54)o+ۖ$_i37`ۚrXԜrhzUTF)jDd 0J  C A? "2ʣ5RAAx`!ʣ5RAAkHxڕRMKQ=L AڴPBM( dLfaJhFN||ą 7 RfWhzˤHZ}̝{}s=>2QHL&h--Y^RЩf:&2vxVЀs[Cw%reI VG:~_|%, w [v lӄt O7 '.鿸I>?'Y6j~j70(p2{i X? T۠zn4vn}T1?_JIvS:)fac/NX2zvg6"'z;#,J:ѦWB)?)37D$$xiPz&8gCUE8^Hq4΃n-N|2eAZa 8V+aLQmԚ7AKw[n[+Ga= Țf {jT z1eU7ܺt[[~ ɫr3rr#O;r3[_0pI5]"KZՌ(H.JAU^N <o~LGo`NGeƕF)QSH^76tHTkDd pJ   C A ? "2P#C?c,ύj),`!$#C?c,ύj)@xڥSAkA5qa7VBXCCmM{"6$' B[G<# "  of76 x{vr @`rl(C$BV&&u?j{\Y8MzYO)pΙr\#_zEw^t# +dKS'KllA0DYo=?kQ/Wa# ( ZׂQÞDAA/u\j+n#Ǿy6㛖K%5mUyɒT:wɯ^"?2ujkP*xC{eV%,I;=~#;s*$.6 HTǤT˘Dd ,hJ  C A? " 26C0<¤-`!6C0<¤-@|xڝROKQyj5IP K'W]%tܶZJO{>}Nsނyo5f{fs zG]̶m6qٸ/R#֦$͛d'wD5"]!(蝋yi ?cͱ7(OL(\ؾэ6z3Ӱ^oPB 5~4pqQ?wM\}} |,[*w#GՆRv4<3цodO5:4ZU!-cjafV P,iJcy𼤀Z  Zr@TRsu-) IR~t6($Iɷ 0-x)/Q6;g~nߌځDd lJ  C A? " 2)ɯ)E K%α`!ɯ)E K%αxtxڥRMo@}Nh8 !HpZ^7%("f|TSKH#??H9; B>ĕYٷ<;X' (bEvTzn屫z+ BW^qs!'dYܕhI;9*dT3m#eNw2T%:ӂD[աzAM>eP8hƵQh <=nuk7KH/Jl{?5t6~Vi%IlgT9ory+PT(f݆>nѫuބ1*+Vg2x9s1:GXQ)5m2p6ismw?!iHA:!KtEv8~R\iH1Csf[z+EqqWnːcn˗EL5dp3p R_-aC0q :Dd lJ  C A? " 2']*irx^`!p']*irA>xڥSkA3hS7zjS@ n^"xܮɠlR65q*"ģG9HГI*vf7y{;o08@ @-30gdwٵ@vvI~d\afd-g34X%׆m`?_VCXa$E{SO2αTyk#A'h4"i둉 "RL(k`N ~#X"F<$L7R7ћ [XmRpQJ}:KD` +@UbNDd XJ  C A? " 2#rWJ;Y"`!#rWJ;YHxڕRJ@fSI9UЃ-`l^*V<ƨA i+m%"x#(3>Hnuv*ԓ7o33K>DYQHt]h&\PGd'D涰3fj⫪0 hg!&BD+FsalJ b.iFcїɼLƏli b0+:*uO| X&%hFQK3+.vpC Jov6< 7<~GeLw 0M&_>(OF[+׫9`Ԡ)5|~pg :Abޭ׬g~;,-3v7!HwN[ ?0+ Ɩk^~25lN6]F+n.Hyf]'mĈ O3*Ź,CDRH \koj31vw2bl5rE0T pDd J  C A? "2܀ :crB]*`!܀ :crB]`:HxڕRJ#Q=u;#qhY>MDiqٶccyHlƁqW~'N1ֽD]̥ܺũs/a~ Ђ\: ^M 1ȋ+AO&2ٮ_%5&¬r^t}`EK>!єاa ~=v0Ro$vrV +7<V2&O;1MM߽Q|Uáw'YЩh9S5hFkB o_4ݽNEs@R*7FGv#><_\ Z^HI'ͺI/첕Lcӵs?;t7\Fk VU4-ւu@~#/x> JϨw"rBJ|_|S}ޞQP:_;)W>=Mş-/VDd ,6  3 A?2nk??"(aJq-`!Bk??"(a` xcdd`` @bD"LGc21Bq3Lazd31ig_#7T7$# !k~35;a##B2sSRsnJ}`R[ɰ .R  6hjpenR~,Fop.bݥ0BLg5 F "ͬ c4a1 v0o8+KRsa!3X@BDd 6  3 A?2cIbNe`b?i/`!7IbNe`bRxcdd``~ @bD"LGc21Bq3Laj`\d31ig?5*XRYd50L11d++&Aݢ ˁф؀\d H("i*ܤ&Y`3? iR2\&RF\.Ub:#ܬ:pV0b 8 YF&&\nfR@ε$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4Dd |,6  3 A?2P%׃lNHn,24`!$%׃lNHnn`0xcdd`` @bD"LGc21Bq3Lazd31ig_i #7T7$# Ak~35;a#bbM-VK-WM̃E(00 48pAl$S@(V%20At3Ma+#Hq>C$\K8Dl;bF&&\ 220= 1} Dd `h6  3 A?2Y$@7Ȉ0|\ 6`!TY$@7Ȉ0|8@X |"xcdd``ad``beV dX,49=F !#T7&&v> KA?H;ؾ0rCx|K2B* Rl7S?&0:R $37X/\!(?71]4' F\ #@j+VEJA"@4SfM Ma`6`m.EYA"@QL3TRpCЈ\D $bXbz4̹1 " `p321)W"Bff:}6NDd 6  3 A?2c߈M^3xAc)?8`!7߈M^3xAc)Rxcdd``~ @bD"LGc21Bq3Laj`<d31ig5*XRYd50L11d++&Aݢ ˁф؀\d H("i*ܤ&Y`3? iR2\&RF\.Ub:#ܬ:pV0b 8 YF&&\nf!A $$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4#Dd |J  C A? "2Oo̷.NU4a<`!YOo̷.NU4``\0'x]JAsfdtB}A$EJ#DpUJK#X[#,} kǙ3Fw`΅&@c,`i{HbcL-O;0M->Rhwy0;V4fcn\ 'ɰWfßT*SfM>wIMݱS洧|G*rڃ^L6o_ܼ`Ľ<']僛6qԇ*/)VkTaZj/F`RJ#! ;dUX׫OiO|ѡ*(>Dd J  C A? "2cg& Fr)??`!7g& Fr)Rxcdd``~ @bD"LGc21Bq3Laj`<d31ig5*XRYd50L11d++&Aݢ ˁф؀\d H("i*ܤ&Y`3? iR2\&RF\.Ub:#ܬ:pV0b 8 YF&&\nfA$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4Dd hJ  C A? "2{u5i^ϙWVWC`!Ou5i^ϙWV&`@ |xcdd``fd``beV dX,49=F !#T7&6> KA?H; l_r< %! H6\Q Av)ZZu.P# ". 6hjpenR~,Fop.@\L.Ub:#Ԭ/ M &dK#YA0pAC  ``CI)$502> LDd 6  3 A?2c߈M^3xAc)?E`!7߈M^3xAc)Rxcdd``~ @bD"LGc21Bq3Laj`<d31ig5*XRYd50L11d++&Aݢ ˁф؀\d H("i*ܤ&Y`3? iR2\&RF\.Ub:#ܬ:pV0b 8 YF&&\nf!A $$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4%Dd hJ   C A ? "2Qr9>icJ`![Qr9>i:@e |)xcdd``ed``beV dX,49=F !#T7&&> KA?H;ؿ0rCx|K2B* Rl7S?&0:R $37X/\!(?71]4' F\ #@j+VE2D*"h\1l$332\ZIĂ$ ]?-ݮ s;>FY_X@ 0YsA P# bP0H0y{Ĥ\Y\_PKDd pJ ! C A!? "2c"v;`Ћ^;&?L`!7"v;`Ћ^;&Rxcdd``~ @bD"LGc21Bq3Laj`l`3p? aM,,He`P@L bB2sSRsn}`hRlp.$b 4 djpenR~,F̴e)]p. Aw)#.*\ 1nV8+V1pAC bOC,#RpeqIj.73@$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4 Dd hJ " C A"? "2k`BWW0/FGQ`!?`BWW0/F @d| x]NAC5Pa /PP%pb^# `삆I27@YIB۫S˔ p[B$;+7Є8ʭ:B|/3`nJm!O=.1ogF[gޣ͋Oz!u FҿϳYz=|H]_ @XOYdjLn:{sl8L۫@MPݛ'vͳϡ-_k9Dd J # C A#? "2hQn)Q(&w0DS`!<Qn)Q(&w0` R xcdd`` @bD"LGc21Bq3Laj`L`3p? aM,,He`P@A35;a#؉L I9 L Eg0r`a)e k1H LDt! ~ Ay {>d?#H)H1+LtF3@Mg S N\086 `px121)WBLCv33X6AV$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4Dd @J $ C A$? "2pyq1T;.qC*LX`!Dyq1T;.qC* xcdd`` @bD"LGc21Bq3Lab`j`3pv*XRYd50dБ! ~ AyP>a0J``dX 6hjpenR~,Fop] ܥ s,F^YgiLef8e+8M4t!vv0o8+KRsap;3X@C+Dd ,J % C A%? "2*\GIEuviZ`!a*\GIEuv6e /xuJPƿsҚ&- ,)Z|vY#|萱/#8IzKՄK9|BpkakD+j{)S97BkUvVЅ2hsUn?Pu0  '})ܓ<^pGcI^^vu] akv7ׅ6 &M.$=LJʙ])oBeirY1Q2;#Z6*e9/ߎXff'e"³[N/<1oq?V-xO+$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4$$If!vh555555#v:V l406++54f4Dd 0   # A b~+Ov(| O&Kn~+Ov(| O&KPNG  IHDR`@jVYsBIT3 sRGBvIDATx^[ l߹@E͇ɏ^Vr|<k]S>o~U60P\ןPJ6v X.D[ k͔ȃQ|jYs O2Hv\hwfWCGDy>agq' 'ԗB 鴒~J~[1;@K R95+ĴFzdfP׺Va JW$ew;I a7rn?t;Ma= || >8xq&O0`{W |a#4~ts%,Kb<r 1JIENDB` Dd @ @@0   # A b/s,`;X(_e n]/s,`;X(_PNG  IHDRu[gAMA|QIDATHA1 J +] ZakIȲDƦ@>"T0$ CS׀0d0Rh5߇W#m`H##!bn R0Rx4޼ F0R(wBP49>j )y?D G}'T"-"0Ҁ QC) 6ޮ܇y9]85IENDB`Dd @ p0  # AbDG S-;0.* nG S-;0.*PNG  IHDRu[gAMA|QIDATHA0 E 9RAW7d[Vr4T7V]z)GW1_K'Q8U}o !v J ݉_c\wB"7kP:y`f ̙4?]'c" k.I5>'El<OIR`!8hy u;DۃclW|4O6X@v6 ̯d^o>-3D5MEp""NEAEͽ2B:Үq4?|x\L#(xF[d,8U E=nV49wu$)iw6r@ɀ;βEj .V;3;;@R 1"Gqe}3c ٚI5hǼڀף?o )EIENDB`Dd @ @@0  # Ab#v@wPh$dQnv@wPh$dQPNG  IHDRu[gAMA|QIDATHj0 J W^AbUiS`PY;v팥 mhQkU!-ۇ9꾷RB4tUـǰ% 0%@UpCJA) 7*Eq=Ǐ`b!^ Z4G`2E XZ[uieKK>k݀G vR @q4<>hgݗ==6 -zuºf ^8KUu«]rYRpF%.LF{ԚD7`P@h>`Ecrqvd\q ֮!U;O|7P^p/[xܿ9cov'` O2x-*ڢgBTZ)ʪ /==]IENDB`|Dd @ 0  # Abiwy.ECwY:`sniwy.ECwY:`PNG  IHDRu[gAMA|QrIDATHǕM fz*@^MYt&mK%`3o{b)'}d(!c,L6d!kd ![e0LQhQfl#gk{99Bg.˅ $GNHkTb༿E#v+}y 7[B A]. <=Σ\+q7˺`p < +Ay9ԑ),"S2 '<B.9cG%B˓>ַꬊ=9r^ZslȦ6F&[l2̺I}{I򴃜ڒ&"kmVm&Oy'_v\,hGl?hv_k8tIˠ4ݦ6El R5s4(^֗J[<8Q-B=kEh=v'pn-:Qn?tOSkJ^FQC7BwB}GV>RɰG/BE:ZmX{dbK0vM=2 '_Vփ"-J b[KٚI4}4)_c'Y 2IK Ồ %)h/9PC~Ge4#ӡ /1΀Vך| "T8φXt ׁJYXB=`%Ah*Ԡ4^m*,Xbc/cNƓXo`)N$KM1;.Leq&¥a3<_bX+%Й64 ?|7oG /H#Axnt'[ZX +Rf B&az)=2++{{#\4p$CIvs_5p@僠$VNZ+geՒAVJzȳ\sQ)_ЍRzA\4\Ziyޠ9p=]a^foQ )/Ȣ^}EN= Pr$:dGh_nKOSg=I5e俿}rHGu5V(V{S:*Z)s鴼G!<&zVPOFM؟c'Zєa4X|15V AA)SSq44Xxa*'R}oţ 2K aU$d8Z҆J{ն_b5SC<"y.l{-#zPex@_!8 WC*f̅7٨9FP,v^JZG< K%*)rqKO(hI!d3zxĦ#4J%q v[˭\"w]&n7t e:цMPbPҳ}]EoL4jcǽʼ/.l6,܉k_>H)Y%H]"Y|ﴥnԊHAW%93 a<J9˫+CQy}N+hI֪VAkj=Ui kMF-fא9WuFhul۫u'Xs hIY#]k:VtS喬&SuZ5q~;1TableSummaryInformation(DocumentSummaryInformation8CompObjq   @ L Xdlt| Draft copyAdministrator Normal.dotRhonda Renker2Microsoft Office Word@F#@l˒@l˒  O8՜.+,D՜.+,L hp  The University of Akronx!B  Draft copy TitleH 6> MTWinEqns   FMicrosoft Office Word Document MSWordDocWord.Document.89q      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGH@@@ NormalCJ_HaJmH sH tH @@@ Heading 1$$@&a$5\F@F Heading 2$@&^ 5CJ\DAD Default Paragraph FontVi@V  Table Normal :V 44 la (k(No List e@ HTML Preformatted7 2( Px 4 #\'*.25@9CJOJPJQJ^JaJTOT MTDisplayEquation  ! B*ph4@4 Header  !4 @"4 Footer  ! #$%&'()*+,C          #$%&'()*+,/     C !z!z!z!z!z!z!z!z! z! z! z! zT$$ &,23g8>C w # 9rs/0qr: #h%9XYZpTUa3\ ! " #$%NPQ[\3ab~XYZ    #$=>mn  ST9:de=>?@HPXs} ~ J!K!!!0"1"z"{"""#######$h$i$j$r$z$$$$$$$$$$$$$$$$$$$$$$$$$$$|%}%%%%%H&I&&-'.'w'x'''((((((()e)f)g)o)w)))))))))))))))))))))))))))b*c*****.+/+x+y+,,^,_,,,,q-r---.J.K.L.T.\.d..........................G/H/I/x/y///00^0_000D1E111111Y2Z2~222233343<3D3L3g3|3}33333333333333333333333/404_4`4s444D5E555*6+6|6666667M7N7777788e88899'9I99999999:%:&:+:,:-:.:/:2:3:<:=:@:C:F:G:H:I:S:X:Y:Z:]:^:_:::::::x;;;]<^<<<<<C=D=E====>O>>>>>>>>>>>>>>>>>>>>?????????????????`@a@@@FAGAAAAA,B-B.B;BBBBBCC!C#C$C%C&C0C2C3C4C>C@CACBCCCMCNCPCfChCiCkCCCCCCCCCCCCCCC!!!!!!!!!!H!H!H!H!!!!H!! !Ճ!!!!!!!!H!H!!!!Ճ!Ճ!!!H!H!!!!!!H!H!H!H!!!v:!v:!!!!!6 !!!!!!!!!o!!!!! !!!!!!!!!!!! !!!!!Z !!!!!!!!!!YYY!!!!!!!!!!!!!!!!!!!!!!a!!!!!!!YYY!!?!!!!!!!!!!!!!!!!!!!!)!!!!!!YYY!!!!!!!!!!!!!!!!!!!!![!!!!!!YYY!!!!!!!!!!!!!!!!!!!)!!!!!!YYY!!!!!!!!!!!!!!!!!!!!!!)!!!!!!YYY!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)!!!!!!YYY!!!!!!!!!!!!!!!!!!!!)!!!!!!YYY!!!!!!!!!!!!!!!!!!!!!QQQQQc ' c c c c c gzc c c c ' c c c c     \ \ gz\  C c c c  \  9rs/0qr: #h%9XYZpTUa3\q r   ! " $ #$%NPQ[\3Yab~XYZ    !"#$=>mn  ST9:de=>?@HPXs} ~ J!K!!!0"1"z"{"""#######$h$i$j$r$z$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$|%}%%%%%H&I&&-'.'w'x'''((((((()e)f)g)o)w)))))))))))))))))))))))))))))))b*c*****.+/+x+y+,,^,_,,,,q-r---.J.K.L.T.\.d..............................G/H/I/x/y///00^0_000D1E111111Y2Z2~222233343<3D3L3g3|3}333333333333333333333333333/404_4`4s444D5E555*6+6|6666667M7N7777788e8f88899'9I99999999:%:&:+:,:-:.:/:0:1:2:3:<:=:@:C:F:G:H:I:S:X:Y:Z:[:\:]:^:_:::::::x;;;]<^<<<<<C=D=E====>O>>>>>>>>>>>>>>>>>>>>>>???????????????????`@a@@@FAGAAAAA,B-B.B/B;BBBBBBCC C!C#C$C%C&C/C0C2C3C4C=C>C@CACBCCCLCMCNCPCfCgChCiCkCCCCCCCCCCCCCCCCCCCC000000000 0 0 0 0000 00 0 000 0 0 000 0 0000 0 000 0 0000000 0 0 0 000000000000000000000000000000000000000000000000000000000000 0 0 0 0 0 0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 000000000000000000000 000000000 0 0 0 0 0 0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 000000000000000000000 00000000 0 0 0 0 0 0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0000000000000000000 00000000 0 0 0 0 0 0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 000000000000000000 00000000 0 0 0 0 0 0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 000000000000000000000 00000000 0 0 0 0 0 0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 000000000000000000000000000000000 00000000 0 0 0 0 0 0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0000000000000000000 00000000 0 0 0 0 0 0 00 0 00 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0000000000000000000B0`0n0h000000000000000000000000000000000000000000000000000  ^^^a#*)1;ICK&+-.5=ER_i# 3P ""##!#m$@((((((~)r------o222222\7777773<|<<<<<0=A%C/C?@ABCDFGHIJKLMNOPQSTUVWXYZ[\]^`abcdefghk(     1IK/15MO358PR$&M_a ) + =#O#Q#x###:(L(N(u(((-1-3-Z-l-n-222B2T2V2888888j=|=~====C::::::::::::::::::::::::::::::8@ (    C F. `vv`TT`TTv H  #  H  #  H  #  H  #  H  #  H   #      C F. `R`TR`T    C F . `LL`TU`TUL  B S  ? " YCT@ @ 8($ Dt@ @ ,I T,$ th, T P|D% T\ | T , TS[/BBC8QV(1HJ68DM  o#x#)%{%l(u(Q-Z-92B288==-B.B/BBC:::::::::::::::::: 89  25P , _::-B-B.B/B;BBBBBBBC C C!CCC LE:ȻjYS&>Sh_<[BY'al _k*wLR2E6zF+}h^`OJQJo(hHh^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@ @ ^@ `OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@ @ ^@ `OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hHh^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.h^`OJQJo(hHh ^`hH.h pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH.h ^`OJQJo(h ^`OJQJo(oh pp^p`OJQJo(h @ @ ^@ `OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh PP^P`OJQJo(h^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.h^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.h^`OJQJo(hHh^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@ @ ^@ `OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hHh ^`hH.h^`OJQJo(hHh pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH. *wjYS+}E6z:>Sh__kBY'a                                                                                  kV9    !"#$?@HPXsi$j$r$z$$$$$$$$$$$$$$$$$$$$$$$$$$f)g)o)w))))))))))))))))))))))))))K.L.T.\.d.........................3343<3D3L3g3|3}333333333333333333333399999:%:&:,:-:/:0:1:2:3:=:@:C:F:G:H:I:X:Y:Z:[:\:]:^:>>>>>>>>>>>>>>>>>????????????C@-B-B-)  -B-B4CC`@``<@`@UnknownGz Times New Roman5Symbol3& z ArialQTI-83 SymbolsSymbol?5 z Courier NewI& ??Arial Unicode MS;Wingdings"qh҆҆ O8 !x O8 !x!>4dBB S 3QHX ?2 Draft copy Administrator Rhonda Renker0