.MCAD 306000000 Z  docDocument<mcObjectIqq d2_graph_format graphData axisFormatLLtrace2D      dim_formatCmasslengthtimecharge temperature luminosity substanceNumericalFormat@dii shpRectE(tmcDocumentObjectStateJ mcPageModel:???L?mcHeaderFooter9MATHCAD TUTORIALLESSON 6: Graphing EquationsPage |P9 ComputeEngine=BuiltInsB? SerialAnyvalH@@@H @A@HB@HMbP?C@H units_classA TextState0 TextStyle/@ Bookman Old Style0,0,128Normal/@ArialHeading 1/@ ArialHeading 2 /@ ArialHeading 3 /@ ArialParagraph /@ ArialList /@ ArialIndent /@Times New RomanTitle/@Times New RomanSubtitle font_style_list> font_style?  VariablesTimes New Roman?  ConstantsTimes New Roman? TextArial? Greek VariablesSymbol? User 1Arial? User 2 Courier New? User 3System? User 4Script? User 5Roman? User 6Modern? User 7Times New Roman? SymbolsSymbol? Current Selection FontArial? Undefined Font? HeaderArial? FooterArial? Rotated Math FontTimes New Roman/ TextRegion docRegion7shpBoxD Oh CharacterMapRangeMap,Lesson 6: Graphing Equations ChrPropMap( RangeElem- ChrPropData) RangeData.mBookman Old Style0,0,128 ParPropMap*- ParPropData+EmbedMap"-LinkMap -LinkData!@NormalBookman Old Style @D*8VXll@hObjectives: Be able to create plots in Mathcad for: 1. data 2. functions, and 3. regressed equations ( h-)kBookman Old Style0,0,128-)kBookman Old Style0,0,128-)kBookman Old Style0,0,128-)kBookman Old Style0,0,128 - !)iBookman Old Style0,0,128"-]#)kBookman Old Style0,0,128 "*h$-h%+"&- h'-h(!@NormalBookman Old Style )@Dk$|xX (**-++",- --.!@NormalBookman Old Style /@Df3h^^BGraphing Data To graph a set of data, put the data in a vector or define the points in terms of indices. Then click on the graph button on the tool bar to open up the graph palette. While there are several types of plots than can be created in MathCad, we shall deal only with 2-dimensional scatter-line plots. These plots are initiated by clicking on the x-y button (top-left button on the graph palette). With the cursor in a blank spot on the page, click on the x-y plot button. This creates the plot (rather small at first) that can be sized in the usual ways. Notice that there is a black square to the left of the plot square and one at the bottom. Click on these one at a time and type in the corresponding y and x variable names.(0-1)/eBookman Old Style0,0,1282-3)/eBookman Old Style0,0,128020*4-5+"6- 7-8!@NormalBookman Old Style 9eqRegion3@DSCd"`:tree1 p;1 :<1d;i=1;>1t=0?1=6@@3@D{S'@A1 p@B1 @A@C1@@B@D1d@Cx@E1@Ci@F1@B@G1@@F@H1d@Gi@I1@G2@J1@F1@K3@Dp9~@L1 p@M1 @L@N1d@My@O1p@M@P10@O@Q10A@P@R10A@Q@S10A@R@T10A@S@U10A@T@V10A@U@W1@@V@X1@V1032@Y1@U600@Z1@T300@[1@S120@\1@R50@]1@Q20@^1@P3@_@D{b33@Here we have defined 7 (x,y) pairs. I have defined x in terms of an index variable i and y as a vector, just so that you can see that you can do it either way (*@`-@a+"@b- @c-@d!@NormalBookman Old Style @e3@Dpm|p@f1 p@g1@f@h1@@g@i1@@h@j1@@i@k1B@j@l1d@k1.032@m1@k@n1d@m10@o1@m3@p1@j3@q1@i@r1@@q@s1@q@t1@hy@u1@g@v1@@u@w1@@v@x1f@w13@y1@w1@z1@v@{1@@z@|1@z@}1@ux@~ LL     @3@D@1 p@1@@@1@@@1@@1@@@1@@@1@@@1@@@1@@1@@1@@@1@@1@@1@@1@@@1@@@1@@@1@@1@@1@@@1@@1@@ LL     @@D]?XeDeD6Click on the x-y plot button and this is what you get.(6*6@-6@+"@- 6@-6@!@NormalBookman Old Style @@Dp33@OThis is what you get when you type y and x into their respective place holders.(O*O@-O@+"@- O@-O@!@NormalBookman Old Style @@D/BNow, since we are plotting data and not a curve or trendline, we really want each datum to be a distinct point, not a sequence of line segments. To change the look of the plot double click anywhere within the plot itself and a formatting window opens. Click on the traces tab at the top. With trace 1 highlighted as in the figure at the right, click on the drop down box below the symbol column and choose box. Click on the drop down box below Type and choose points. 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     !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopUINU RINUNZPU@+@+@DRAWPATTERNS pقGpقGSTREAMSLISTSummaryInformation(Richard L. Rowley CORELDRWRichard L. Rowley1@@@@ lG@@ lGCDR8/.U@3@D \ @1 p@1@@1@@@1@@@1@@@1B@@1d@1.032@1@@1d@10@1@3@1@3@1@@1@@@1@@1@y@1@@1@@@1@@@1f@13@1@1@1@@1@@@1@@1@x@ LL     @@D@p@000BNearly all of the format changes that you will want to make to your plot can be accessed via this format window by simply double clicking on the plot. For example, there is a Labels tab that allows you to put titles and labels on your plots. There is an x-y Axes tab which can be used to change the type of plot. For example, through the x-y Axes tab you can add grid lines to the plot simply by checking the gridline box on both the x and y axes. You can change your plot to a semilog plot or a log-log plot by clicking on the log scale box for the y axis or for both axes respectively. The plots below illustrate a plot with gridlines and a semilog plot. (*@-@+"@- @-@!@NormalBookman Old Style @3@D=%@1 p@1@@1@@@1@@@1@@@1B@@1d@1.032@1@@1d@10@1@3@1@3@1@@1@@@1@@1@y@1@@1@@@1@@@1f@13@1@1@1@@1@@@1@@1@x@ NN     @3@DHmH)@1 p@1@@1@@@1@@@1@@@1B@@1d@1.032@1@@1d@10@1@3@1@3@1@@1@@@1@@1@y@1@@1@@@1@@@1f@13@1@1@1@@1@@@1@@1@x@ LM     @@Di;+`YYBThe default scale for the axes may be changed in order to zoom in on specific data or to format the plot more to your own individual taste. This is done by clicking once on the graph. Notice that this causes the maximum and minimum values of the data to be shown below (for the x axis) or to the left (for the y axis) of the axis tick labels. Click on the y axis maximum value, press space bar so the editing box encloses the entire value, delete the value and type in a new value to be the maximum y value. For example, let's change the maximum to 400 so as to just view the smaller values. Your plot will now look like that shown below. In this manner, you can adjust the minimum and maximum values displayed by either axis.(*@-A+"A- A-A!@NormalBookman Old Style A3@DG6H.A1 pA1AA1@AA1@AA 1@AA 1tA 400A 1A 3A 1AA 1@A A1A A1AyA1AA1@AA1@AA1fA13A1A1A1AA1@AA1AA1AxA NN     A@DKjX0hbCbCA)Graphing Functions Functions are also easily graphed in MathCad. Simply define the function name and place the function in the y axis place holder and the independent variable in the x axis place holder as illustrated below. The equation given is for the vapor pressure of acetone in mm Hg.()A-A)AeBookman Old Style0,0,128A-A)AeBookman Old Style0,0,128AAA*)A-)A +"A!- )A"-)A#!@NormalBookman Old Style A$3@DQ8A%1 pA&1 A%A'1@A&A(1dA'PvpA)1pA'A*1A)TKA+1A&A,1@A+A-1tA,10A.1A,A/1tA.7.02447A01A.A11tA01161A21A0A31dA2TKA41A249A51A+torrA6@Dl=xt3t3@Notice that by dividing the function when put into the axis place holder, that I can make what is being displayed be in the units I want. (*A7-A8+"A9- A:-A;!@NormalBookman Old Style A<3@D, 2A=1 pA>1A=A?1@A>A@1@A?AA1@A@AB1BAAAC1dAB 1.788311AD1ABAE1dAD10AF1AD29AG1AAAH1dAG 6.631661AI1AGAJ1dAI10AK1AI23AL1A@AM1@ALAN1ALAO1A?AP1@AOAQ1dAPPvpAR1pAPAS1ARTKAT1AOatmAU1A>AV1@AUAW1@AVAX1fAW10AY1KAWAZ1AY10A[1AVA\1@A[A]1A[A^1AUTKA_. LL     A`@D; b H ?`RURUAIn the above plot, MathCad automatically plots a function from -10 to 10. To change this to the range we want, we either need to set up a range of TK values or go in and change the minimum and maximum values on the x axis as discussed above. By clicking on the graph, and then changing the minimum x value to 300 and the maximum x value to 500, our plot now looks like the one below. (*Aa-Ab+"Ac- Ad-Ae!@NormalBookman Old Style Af3@D(  ( @Ag1 pAh1AgAi1@AhAj1@AiAk1@AjAl1fAk 37.100251Am1Ak0.32973An1AjAo1@AnAp1AnAq1AiAr1@AqAs1dArPvpAt1pArAu1AtTKAv1AqatmAw1AhAx1@AwAy1@AxAz1tAy500A{1Ay300A|1AxA}1@A|A~1A|A1AwTKA. LL     A@D c  G`SDSDAGOne can plot more than one set of data on the same plot by separting each set with a comma when putting the x and y variables into the axes place holders. For example, on the plot below, we plot some experimental data as well as the function. See if you can start from scratch and make the graph look just like the one below.(G*GA-GA+"A- GA-GA!@NormalBookman Old Style A3@D5 Tr )X LA1 pA1 AA1dATexpA1pAA10AA10AAA10AAA1@AA1A450A1A400A1A350A3@D5 r X MA1 pA1 AA1dAPexpA1AA1p@AA10AA10AAA10AAA1@AA1A17.5A1A7.1A1A2.0A1AatmA3@D(  ( KA1 pA1AA1@AA1@AA1@AA1fA 37.100251A1A0.32973A1AA1@AA1AA1 AA1@AA1@AA1dAPvpA1pAA1ATKA1AatmA1AA1dAPexpA1AatmA1AA1@AA1@AA1tA500A1A300A1AA1@AA1AA1 AA1dATKA1ATexpA. NN     A@D c  Ph[C[C@Regression and Fitting Lines Built into MathCad are interpolation functions and smoothing functions. To fit a smooth curve through data we can use a spline function in conjunction with MathCad's interp function. This is illustrated below.(A-A)AeBookman Old Style0,0,128A-A)AeBookman Old Style0,0,128AAA*A-A+"A- A-A!@NormalBookman Old Style A3@D" Au P UA1 pA1 AA1dAxA1pAA10AA10AAA10AAA10AAA1@AA1A12A1A7A1A4A1A2A3@Dh" u vP VA1 pA1 AA1dAyA1pAA10AA10AAA10AAA10AAA1@AA1A30A1A12A1A9A1A1A3@DC #T P WA1 pA1 AA1dAvsA1AA1dAcsplineA1pAA1 AA1dAxA1AyA@D@C ov @P Y0/3/3@This just fits a smoothing function (a c-spline) through the (x,y) data. The vector vs will be used by the interp function to plot the data.(*A-A+"A- A-A!@NormalBookman Old Style A3@D w  \A1 pA1 AA1@AA1dAysmoothA1pAA1AxxA1AB1dAinterpB1pAB1 BB1 @BB1 @BB1dBvsB1BxB1ByB1BxxB @D h  ^UU@This will allow our graph to interpolate from the x, y data, using the c-spline smoothing function, a corresponding yy value for any given xx value.(*B -B +"B - B -B!@NormalBookman Old Style B3@D(t(`B1 pB1BB1@BB1@BB1@BB1fB73.02B1KBB1B16.98B1BB1@BB1BB1 BB1dByB1BB1dBysmoothB1pBB 1BxxB!1BB"1@B!B#1@B"B$1tB#15B%1B#0B&1B"B'1@B&B(1B&B)1 B!B*1dB)xB+1B)xxB, LL     B-@DjdDD@vThe result of using a c-spline fitting function with the interp function to plot a smooth function through a data set.(v*vB.-vB/+"B0- vB1-vB2!@NormalBookman Old Style B3@Dn f`^f^fAIf you prefer to actually regress an equation (a c-cpline is a numerical technique and you don't get an actual equation with coefficients) through the data, MathCad has a number of such functions built in. We won't go into them here, but you should check out the slope and intercept functions for linear regression as well as linfit and genfit for general purpose regression. These can be found from the on-line help. Here, we will put a straight line and a 3rd-order polynomial through the above data.(*B4-B5+"B6- B7-B8!@NormalBookman Old Style B93@D0,g(gB:1 pB;1 B:B<1@B;B=1dB<ylineB>1pB<B?1B>xxB@1B;BA1@B@BB1@BABC1dBBslopeBD1pBBBE1 BDBF1dBExBG1BEyBH1BAxxBI1B@BJ1dBI interceptBK1pBIBL1 BKBM1dBLxBN1BLyBO3@D0CTDPiBP1 pBQ1 BPBR1dBQvsBS1BQBT1dBSregressBU1pBSBV1 BUBW1 @BVBX1dBWxBY1BWyBZ1BV3B[@DCTPk  *Do regression using a 3rd order polynomial(***B\-*B]+"B^- *B_-*B`!@NormalBookman Old Style Ba3@D0ctoplBb1 pBc1 BbBd1@BcBe1dBdyquadBf1pBdBg1BfxxBh1BcBi1dBhinterpBj1pBhBk1 BjBl1 @BkBm1 @BlBn1dBmvsBo1BmxBp1BlyBq1BkxxBr@Dctpnx#Find the fit value for any given xx(#*#Bs-#Bt+"Bu- #Bv-#Bw!@NormalBookman Old Style Bx3@D((oBy1 pBz1ByB{1@BzB|1@B{B}1@B|B~1fB}73.02B1KB}B1B16.98B1B|B1@BB1BB1 B{B1 @BB1 @BB1dByB1BB1dBysmoothB1pBB1BxxB1BB1dBylineB1pBB1BxxB1BB1dByquadB1pBB1BxxB1BzB1@BB1@BB1tB15B1B0B1BB1@BB1BB1 BB1 @BB1 @BB1dBxB1BxxB1BxxB1BxxB/% LL     B@D e`qh]U]UA We have just scratched the surface of the capabilities of Mathcad with regard to plotting and regressing functions from data. But, you have been introduced to enough features that you can go on your own from here to learn more techniques as your applications warrant them. The fundamentals illustrated in this tutorial will provide the main plotting tools that you will need in the Chemical Engineering curriculum.(*B-B+"B- B-B!@NormalBookman Old Style