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TextState? TextStyle>@ Times New RomanSerial_ParPropDefaultW>Normal>@Arial@W Heading 1>@ Arial@W Heading 2 >@ Arial@W Heading 3 >@ Arial@W Paragraph >@ Arial@WList >@ Arial@WIndent >@Times New Roman@W Title>@Times New Roman@W Subtitle font_style_listO font_styleP  VariablesTimes New Roman@P  ConstantsTimes New Roman@P TextArial@P Greek VariablesSymbol@P User 1Arial@P User 2 Courier New@P User 3System@P User 4Script@P User 5Roman@P User 6Modern@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Romanv TextRegion* docRegionGshpBoxUD<<<;<; CharacterMap-RangeMap;KMathcad 8.0 Tutorial 1. GETTING STARTED 1.1 Introduction Mathcad combines the live document interface of a spreadsheet with the WYSIWYG (What You See Is What You Get) interface of a word processor. With Mathcad you can typeset equations on the screen exactly the way you see them in a book. But Mathcad equations do more that look good on the screen. You can use them to actually do math. Like a spreadsheet, as you make a change anywhere in the document, Mathcad goes straight to work, updating results and redrawing graphs. With Mathcad you can easily read data from a file and do mathematical chores ranging from evaluating integrals and derivatives to solving systems of differential equations. In fact, just about anything that you can think of doing with math you can do with Mathcad. 1.2 The Mathcad Tutorials It is recommended for first time users to spend some time studying the on-line Mathcad Tutorial. To access the tutorial, first start Mathcad. Under the Help Menu, select resource center. Click on the "Getting Started" option. Select the appropriate tutorial. 1.3 The Mathcad Window Like other Windows applications, Mathcad contains a menu bar at the top of the screen. To pull down a menu, you can click on it with the mouse. Below the menu bar is the tool bar. Many menu commands can be accessed more quickly by clicking a button on the tool bar. To learn what a button does, you can can place the mouse cursor over the button, and after a moment, the name of the command will appear. The font bar is immediately under the tool bar. This contains scrolling lists and buttons used to specify for characteristics in equations and text. On the right of the screen, in a small window labeled "Math" is the Math toolbar. Each button on this menu opens another toolbar palette. From left to right then top to bottom, the buttons activate the 1. Calculator Toolbar 2. Graph Toolbar 3. Vector and Matrix Toolbar 4. Evaluation Toolbar 5. Calculus Toolbar 6. Boolean Toolbar 7. Programming Toolbar 8. Greek Symbol Toolbar 9. Symbolic Keyword Toolbar Each toolbar contains commonly used symbols, functions, and objects (e.g. matrices or graphs) which may be inserted into the worksheet. To close a toolbar palette, click on the x in the upper right corner. If you close the Math toolbar palette, you can reopen it in the "View" menu, under "toolbars". 1.4 On-line Help Help can be accessed in several ways. In general, one can click on the Help menu and activate either Mathcad Help or the Resource Center. Mathcad help is similar to the online help for other programs - there is a table of contents with many topics and subtopics, alphabetical index of help topics, or a search feature. The Resource Center contains tutorials and quick sheets, as well as other resources. The quick sheets can be an especially useful reference - they are working Mathcad sheets which demonstrate how to perform many sample calculations. 2. SOME SIMPLE COMPUTATIONS  ChrPropMap7   RangeElem< ChrPropData8 RangeData=gArial<8<8<8<8oTimes New Roman ParPropMap9` ` ] < ParPropData:@W<:@W <]!:@W EmbedMap1"<LinkMap/ #< $LinkData0@NormalTimes New Roman %*@UI^XVVV-A2.1 A simple Calculation Mathcad can be used as a simple calculator. For multiplication use "*" and for division "/". After entering the full denominator (21) press [SPACE] and continue normally. Note that the symbol "=" is used to instruct Mathcad to perform the calculation. This "=" sign corresponds to the keyboard's equal sign and to the first button of the Evaluation and Boolean palette. 79&<':@W1(</)<*0@NormalTimes New Roman +eqRegionB@UzB ,tree@ p-@@,.@@@-/@@@.0@@t/15.31@@/2@@t1143@@1214@@.5@@t44.16@@46.77@@-8@@+@7Serial_DisplayNodeX9@@7:*@U2 ***-A2.2 Building Expressions The key to editing expressions in Mathcad is to enclose the correct part of the equation in the selection box. Several Mathcad symbols like division, power, and radical are "sticky" in Mathcad. This means that after you type on the these operators, everything you type next will be part of the denominator, exponent, or radical until you explicitly move the cursor by pressing the SPACEBAR. The following example illustrates how to build the expression:79;<<:@W1=</><?0@NormalTimes New Roman @@@B@UeQ@A@@ p@B@@@A@C@@@@B@D@@@@C@E@@@@D@F@@d@Ex@G@@@E3@H@@@D@I@@@@H@J@@t@I1@K@@@I@L@@d@Ka@M@@@Kb@N@@@H@O@@d@Nx@P@@@N@Q@@d@Pc@R@@@Pd@S@@{@C@T@@@S@U@@d@Tx@V@@@T1@W@@@B1@X*@UtJtt-AG (i) Type x^3. Note that there is a vertical bar next to 3. Any character that you type next will be a superscript next to 3. Press [SPACE]. Note that selection box now encloses the x3. You are back to the baseline. (ii) Type +1/a+b[SPACE][SPACE]*x^c+d (iii) Type [SPACE][SPACE]+\x+1[SPACE][SPACE]+17EG@Y<@Z8@X@[<@\8@X_@Y@]@@@1@</@<@0@NormalTimes New Roman @@B@U:-aP@@@ p@@@@@@@@@@@@@@@@@@@@@@d@x@@@@3@@@@@@@@@@@@t@1@@@@@@@d@a@@@@b@@@@@@@d@x@@@@@@@d@c@@@@d@@@{@@@@@@@@d@x@@@@1@@@@1@@B@U1@@@ p@@@@@@@@@@@@@@@@@@@@@@d@x@@@@3@@@@@@@@@@@@t@1@@@@@@@d@a@@@@b@@@@@@@d@x@@@@@@@d@c@@@@d@@@{@@@@@@@@d@x@@@@1@@@@1@@B@U" 9I %8 @@@ p@@@@@@@@@@@@@@@@@@@@@@d@x@@@@3@@@@@@@@@@@@t@1@@@@@@@d@a@@@@b@@@@@@@d@x@@@@@@@d@c@@@@d@@@{@@@@@@@@d@x@@@@1@@@@1@*@U DS !JDD-EdIf you make a mistake editing, you can always click on the "Edit" menu and select "Undo". Mathcad remembers the last several operations, so you can continue undoing several steps. For further help on editing, see Mathcad help under "Equations" and "Equation editing". 3. ENTERING TEXT AND TEXT MANAGEMENT Mathcad handles text as easily as it does equations, so you can make notes about the calculations that you are doing. TO begin typing text, click in an empty space and click on and press " (i.e., [SHIFT]-[APOSTROPHE]). Mathcad has a series of word-processing capabilities including the use of different fonts and font sizes, subscripts and superscripts, text alignment, spell checking, etc. If you want to use special fonts, italics, superscripts, etc., the best approach is to first just type normally and then select the text region that you want to format by clicking at its beginning or end and then dragging the mouse over it, then formatting it. Selecting first the formatting option (e.g. bold) and then typing the text can lead to some extra difficulties. 4. EQUATIONS AND COMPUTATIONS 4.1 Defining Variables and Function Use the definition symbol ":=" from the evaluation and boolean palette (or press ":"). Note the difference, "=' instructs Mathcad to calculate something numerically, ":=" to define something and keep it for later use. For example: 7d9{{d@<{@:@W@<@:@W>@@@1@</d@@B@U25aHAA?@@ pA@@@ A?AA@@dA@accAB@@A@AC@@tAB11.2AD@@ABAE@@dADmAF@@ADAG@@dAFsecAH@@AF2AI*@UaREpFJJJ-BWhenever you enter an expression involving units, Mathcad checks it for dimensional consistency. If you add or subtract values with incompatible units, Mathcad displays an appropriate error message. Mathcad automatically displays results in terms of fundamental units: m, kg, coul, and K. You can ask Mathcad redisplay a particular result in terms of any unit or combination of units you want. To do so: (1) Click on the result. An empty placeholder will appear to its right. (2) Click in the units placeholder and type the unit that you want Mathcad to use when it redisplays the result. Click on the UNITS button for help. (3) Press ENTER and Mathcad (in auto mode) converts the units. Try the following: 79yyTAJAJALAL1AN</AO<AP0@NormalTimes New Roman AQ@B@U@cvOpGAR@@ pAS@@ ARAT@@dASFAU@@ASAV@@dAUmassAW@@AUaccAX@B@UZJpOAY@@ pAZ@@AYA[@@dAZFA\@@AZA]@@+@A\@XA^@@A\A_@@@A^A`@@dA_kgAa@@A_mAb@@A^Ac@@dAbsecAd@@Ab2Ae@B@U,UAf@@ pAg@@AfAh@@dAgFAi@@AgAj@@+@Ai@XAk@@AiNAl@B@U3VAm@@ pAn@@AmAo@@dAnFAp@@AnAq@@+@Ap@XAr@@AplbfAs@B@U<WAt@@ pAu@@AtAv@@dAuFAw@@AuAx@@+@Aw@XAy@@AwdyneAz*@UJXJJJ-B6. ADVANCED CALCULATIONS 6.1 Iterative Calculations Mathcad can do repeated or iterative calculations as easily as individual calculations. Mathcad uses a special variable type called a range variable to perform iteration. Range variables take on a range of values, such as all integers from 0 to 10. Whenever a range variable appears in a Mathcad equations, Mathcad calculates the equation not just once, but once for each value of the range variable. For example assume that you want to calculate the position of an object traveling at the above acceleration for time values 0.1, 0.2, 0.3, ..., 1. Then you define time as a range variable.79A{<A|:@W1A}</A~<A0@NormalTimes New Roman A*@UB`bbb-ANote, the first number is the beginning of the range, the second is the next number in the range, and the third is the last number in the range. The first two numbers are separated by a comma and the last two by a pair of dots (choose m..n from the arithmetic palette, or press ";") and the calculation is automatically repeated 6 times as shown here (type "time =" to get the first column and "1/2[SPACE]*acc*time^2=" to get the second). 79A<A:@W1A</A<A0@NormalTimes New Roman A@B@U8YA@@ pA@@ AA@@dAtimeA@@AA@@ @AA@@@AA@@tA0A@@AsecA@@AA@@tA.3A@@AsecA@@AA@@tA1.2A@@AsecA@B@U4*#_A@@ pA@@AA@@dAtimeA@@AA@@@AA@@@AA@@AA@@<@AA CVSEqItem EqItem AAA CVSComboItemCVSItem< eCVSOleClientItem ࡱ> Root EntryaB.dK8`R(Contents(OlePres000 p di iTxArial4NyxM_!@_M_@Arial4NyxM_!@_M_@<T1c$A@@A@XA@@AA@B@U`2^A@@ pA@@AA@@@AA@@@AA@@@AA@@tA1A@@A2A@@AaccA@@AA@@dAtimeA@@A2A@@AA@@@AA@@@AA@@AA@@<@AA AAA< b!*eCVSOleClientItem ࡱ> Root EntryaB.dK8`R(Contents(OlePres000 Wp )di ixArial4NyxM_!@_M_@Arial4NyxM_!@_M_@<T1c$A@@A@XA@@AA*@UaJ&&-@f6.2 Defining a function The same calculation can be performed much easier by defining a function d(t)7f9fA<B?8B@ Times New RomanBBB<BC8B@ Times New RomanB@BD<BE8B@ SymbolBBBF<BG8B@ Times New RomanBDBH<BI8B@ SymbolBFBJ<BK8B@ Times New RomanBHBL<BM8B@ SymbolBJBN<BO8B@ Times New RomanBLBP<BQ8B@ SymbolBNBR<BS8B@ Times New RomanBPBT<BU8B@ SymbolBRBV<BW8B@ Times New RomanBTBX<BY8B@ SymbolBVBZ<B[8B@ Times New RomanBXB\<B]8B@ SymbolBZB^<B_8B@ Times New RomanB\B`<Ba8B@ SymbolB^Bb<Bc8B@ Times New RomanB`Bd<Be8B@ SymbolBbBf<Bg8B@ Times New RomanBdBh<Bi8B@ SymbolBfBj<Bk8B@ Times New RomanBhBl<Bm8B@ SymbolBjBn<Bo8B@ Times New RomanBlBp<Bq8B@ SymbolBnBr<Bs8B@ Times New RomanBpBt<Bu8B@ SymbolBrBv<Bw8B@ Times New RomanBtBx<By8B@ SymbolBvBz<B{8B@ Times New RomanBxB|<B}8B@ SymbolBzB~<B8B@ Times New RomanB|B<B8B@ SymbolB~B<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ Times New RomanBB<B8B@ SymbolBB<B8B@ SymbolBB<B8B@ Times New RomanBBB9bbbBBBB1B</B<B0@NormalTimes New Roman B*@UJ JJrJr-A6.5 Vectors and Matrices A single number in Mathcad is called a scalar. A column of numbers is a vector and a rectangular arrange of numbers is called a matrix. The general term for a vector or matrix is an array. One can create a vector or a matrix by using the Vector and Matrix Toolbar, or "Matrix" under the Insert menu. Mathcad creates an empty set of place holders and the user fills them manually by first clicking on them.79B<B:@W1B</B<B0@NormalTimes New Roman B@B@U B1pB@@ pB@@ BB@@dBAB@@pBB@@0BB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@0ABB@@@BB@@B3.33B@@KBB@@B1C@@B0C@@B0C@@B1C@@B4C@@B1C@@B2C@@KBC@@C2C@@B1C @@B0C @@B0C @@B3.1C @@B3C @@KBC@@C 2C@@B1C@B@UBpC@@ pC@@ CC@@dCxC@@pCC@@0CC@@0ACC@@0ACC@@0ACC@@@CC@@KCC@@C1.1C@@C3C@@C2C@@C1C*@UJ\JJJ-BoMatrix and vector operations are straightforward. For example to add two matrices A and B one has just to write A+B=, to multiply them A*B=, to invert the first one A^-1, etc. Important note: By default, Mathcad arrays begin at element zero. To change this select the Math menu and select options. Change the Origin option to one. Now Mathcad assumes that the arrays begin at one. This change has been done for the rest of this document. To use the (i,j) element of a matrix use two subscripts (achieved by typing "[" or by selecting subscript from the vector and matrix toolbar) separated by a comma. For example:7o9oC from the second symbol panel. For example:7IC1<C28C0C3<C48C0_C1C5<+C68C0C3C5C39IC7@@C=C?@@4@C>C@@@dC?ACA@@C?3CB@@C>CC@@+@CB@XCD@@CBCE*@UAJAA-AFor more information about the linear algebra capabilities of Mathcad consult its help feature for Vectors and Matrices. 6.5 Numerical operators (Sums, Products, Derivatives, Integrals) User of summations and products is straightforward by selection of the appropriate symbol from the Calculus toolbar and clicking on the placeholders (you can also use the [TAB] key to cycle through the placeholders). For example:79CF<CG:@W1CH</CI<CJ0@NormalTimes New Roman CK@B@U0rCL@@ pCM@@CLCN@@@@CMCO@@@CNCP@@tCO1CQ@@CO15CR@@CNCS@@dCRkCT@@CRCU@@dCTkCV@@CT1.3CW@@CMCX@@+@CW@XCY@@CWCZ@B@Uy]C[@@ pC\@@C[C]@@A@C\C^@@@C]C_@@tC^0C`@@C^12Ca@@C]Cb@@dCanCc@@pCaCd@@CcCe@@dCdnCf@@CdCg@@tCf1Ch@@Cf2Ci@@C\Cj@@+@Ci@XCk@@CiCl*@UFJF&F&-@You can use Mathcad's derivative operator for numerical derivative calculation. You need to define the function and the point at which you want to evaluate the derivative.79Cm<Cn:@W1Co</Cp<Cq0@NormalTimes New Roman Cr@B@U 5A?(Cs@@ pCt@@ CsCu@@@CtCv@@dCufCw@@pCuCx@@CwxCy@@CtCz@@@CyC{@@dCzexpC|@@pCzC}@@C|xC~@@CyC@@@C~C@@dCsinC@@pCC@@CC@@dCxC@@C2C@@C~C@@@CC@@tC1C@@CC@@dCxC@@C2C@@CC@@@CC@@dCxC@@CC@@dCcoshC@@pCC@@CxC@@CC@@dCexpC@@pCC@@CC@@K@CC@@CxC@@C3C*@UH!H0-@]Definition of function Definition of g(x)=f'(x) Calculation of f'(2) Range for plotting7]9KK]C<C:@WC<C:@W>CC<C:@WCC<C:@W>CC<C:@WCC<C:@W>CCC1C</]C<]C0@NormalTimes New Roman C@B@UQjz8hC@@ pC@@ CC@@@CC@@dCgC@@pCC@@CxC@@$CC@@dCxC@@CC@@dCfC@@pCC@@CxC@B@UT8C@@ pC@@CC@@@CC@@dCgC@@pCC@@C2C@@CC@@+@C@XC@@CC@B@U^&C@@ pC@@ CC@@dCxC@@CC@@ @CC@@tC0C@@C.1C@@C3C@B@U.C@@ pC@@CC@@@CC@@@CC@@@CC@@vC1.93C@@KCC@@C2.607C@@CC@@@CC@@CC@@ CC@@@CC@@dCfC@@pCC@@CxC@@CC@@dCgC@@pCC@@CxC@@CC@@@CC@@@CC@@vC3C@@C0C@@CC@@@CC@@CC@@CxC - )L)L&&&&&&&&& & & & & &&&C*@U!E0uurur-@FNote: Use a comma to separate the functions represented on the Y axis7F9FC@@dE=expE?@@pE=E@@@E?EA@@@E@EB@@dEA1EC@@EA2ED@@E@xEE@@E7EF@@dEEcosEG@@pEEEH@@EGxEI@@E6EJ@@@EIEK@@@EJEL@@dEK2EM@@EK5EN@@EJEO@@dENsinEP@@pENEQ@@EPxER@@EIES@@dERexpET@@pEREU@@ETEV@@@EUEW@@dEV1EX@@EV2EY@@EUxEZ*@UI$O$X$JGLGL-@Select the whole expression with the editing lines and click on the Symbolics menu, Evaluate, Symbolically. (b) Symbolic Differentiation. Use the differentiation symbol from the Calculus palette to write79E[<E\:@W1E]</E^<E_0@NormalTimes New Roman E`@B@U$$$Ea@@ pEb@@>EaEc@@tEb2Ed@@EbEe@@dEdxEf@@EdEg@@@EfEh@@dEgsinEi@@pEgEj@@EixEk@@EfEl@@@EkEm@@dElexpEn@@pElEo@@EnxEp@@EkxEq@B@U$D!%%Er@@ pEs@@ErEt@@@EsEu@@@EtEv@@@EuEw@@@EvEx@@dEw2Ey@@EwEz@@dEycosE{@@pEyE|@@E{xE}@@EvE~@@@E}E@@dE~expE@@pE~E@@ExE@@E}xE@@EuE@@@EE@@dE2E@@EE@@dEcosE@@pEE@@ExE@@EE@@@EE@@dEexpE@@pEE@@ExE@@EE@@dExE@@E2E@@EtE@@@EE@@dE2E@@EE@@dEsinE@@pEE@@ExE@@EE@@@EE@@dEexpE@@pEE@@ExE@@EE@@dExE@@E2E@@EsE@@@EE@@dE2E@@EE@@dEsinE@@pEE@@ExE@@EE@@@EE@@dEexpE@@pEE@@ExE@@EE@@dExE@@E3E*@U9%K%H%JCC-A\Select the whole expression in a box and click on Symbolics, Evaluate, Symbolically (or [SHIFT]-F9). Finding Roots Mathcad contains two built in functions to find the roots of functions. The first, root(f(x),x), uses a Newton-Rhapson method to find the root of the function, f(x), that is produced from a specific guess. (i) Define the function7 \E<E8EE< E8EkTimes New RomanEE&N8&E@@ pE@@ EE@@dExE@@E1E*@UI&\&X&Jxx-?(iii) Use the root function to find the root nearest the guess7?9?E@@F=xF?@@F)F@@@@F?FA@@@F@FB@@vFA9.5FC@@KFAFD@@FC10.5FE@@F@FF@@@FEFG@@FEFH@@F?xFI 1 )L)N&&&&&&&&& & & & & &&&FJ*@Ui(1(x(J1L1L-@The second function, polyroots(v), finds all of the roots to a function (i) Define a vector that contains the polynomial coefficients starting with the constant term and ending with the highest order polynomial coefficient. For the above example:79FK<FL:@W1FM</FN<FO0@NormalTimes New Roman FP@B@U (4)?)FQ@@ pFR@@ FQFS@@@FRFT@@dFSfFU@@pFSFV@@FUxFW@@FRFX@@@FWFY@@@FXFZ@@@FYF[@@@FZF\@@dF[xF]@@F[5F^@@FZF_@@tF^119F`@@F^Fa@@dF`xFb@@F`3Fc@@FYFd@@tFc174Fe@@FcFf@@dFexFg@@Fe2Fh@@FXFi@@tFh3052Fj@@FhxFk@@FW5880Fl@B@Up(=)~)Fm@@ pFn@@ FmFo@@dFnvFp@@pFnFq@@0FpFr@@0AFqFs@@0AFrFt@@0AFsFu@@0AFtFv@@0AFuFw@@@FvFx@@Fv1Fy@@Fu0Fz@@KFtF{@@Fz119F|@@Fs174F}@@Fr3052F~@@KFqF@@F~5880F*@Ua)8t)p)J88-4(ii) Use the polyroots function to find all 5 roots7494F<4F:@W1F</4F<4F0@NormalTimes New Roman F*@U)))J&&-@HNote the small amount of numerical error in finding the two roots at x=77H9HF