ࡱ> bh}Uq` sbjbjqPqP ::Jjdf   . R6R6R686n7t. ,T(7>:"?"?8?]FBHSI\SSSSSSS$TUhWS KF@]FKKS"?8?SRRRK"? 8?SRKSRR T R8?7 R6KRSS0,TRBXkN6BXRBX RI>I,RJ$=JIIISSR^III,TKKKK. . . $,R6. . . R6. . .   A Mathcad Primer This document will summarize many of the features of Mathcad 11 and how these features are used to solve physical chemistry problems. It is not a replacement for the Mathcad Reference Guide by Mathsoft. You need to enable Print Layout View in Word (View, Print Layout). You can CTRL-click any topic to go to any section or you can scroll down. Contents  TOC \o "1-3" \h \z  HYPERLINK \l "_Toc79981125" The Mathcad Window  PAGEREF _Toc79981125 \h 2  HYPERLINK \l "_Toc79981126" Entering Text  PAGEREF _Toc79981126 \h 3  HYPERLINK \l "_Toc79981127" Equations and Variables  PAGEREF _Toc79981127 \h 3  HYPERLINK \l "_Toc79981128" Important Warning about Typical Scientific Symbols  PAGEREF _Toc79981128 \h 6  HYPERLINK \l "_Toc79981129" Selecting and Editing  PAGEREF _Toc79981129 \h 7  HYPERLINK \l "_Toc79981130" Variable Names  PAGEREF _Toc79981130 \h 7  HYPERLINK \l "_Toc79981131" Units  PAGEREF _Toc79981131 \h 8  HYPERLINK \l "_Toc79981132" Defining Your Own Units  PAGEREF _Toc79981132 \h 9  HYPERLINK \l "_Toc79981133" Easy Graphs  PAGEREF _Toc79981133 \h 10  HYPERLINK \l "_Toc79981134" Range Variables and Graphs  PAGEREF _Toc79981134 \h 12  HYPERLINK \l "_Toc79981135" Zoom and Trace Functions in a Graph  PAGEREF _Toc79981135 \h 14  HYPERLINK \l "_Toc79981136" Data and Data Files: Excel Data  PAGEREF _Toc79981136 \h 15  HYPERLINK \l "_Toc79981137" Entering Data into Mathcad  PAGEREF _Toc79981137 \h 16  HYPERLINK \l "_Toc79981138" Copying Data from EXCEL  PAGEREF _Toc79981138 \h 17  HYPERLINK \l "_Toc79981139" Numerical Calculations  PAGEREF _Toc79981139 \h 17  HYPERLINK \l "_Toc79981140" Numerical Roots of Equations  PAGEREF _Toc79981140 \h 18  HYPERLINK \l "_Toc79981141" Symbolic Calculation  PAGEREF _Toc79981141 \h 19  HYPERLINK \l "_Toc79981142" Symbolic Integration  PAGEREF _Toc79981142 \h 20  HYPERLINK \l "_Toc79981143" The Assume Keyword  PAGEREF _Toc79981143 \h 21  HYPERLINK \l "_Toc79981144" Partial Derivatives  PAGEREF _Toc79981144 \h 22  HYPERLINK \l "_Toc79981145" Solving Equations  PAGEREF _Toc79981145 \h 22  The Mathcad Window When you start Mathcad, the screen should appear as shown below. A series of menu items are listed, and underneath the menu are toolbars. An ancillary toolbar called Math appears to the right. In this document, menu commands will be noted as Item 1, item 2, and so forth. As an example, the print preview command would be typed as: File, Print Preview. To execute the command you would point to File, then select Print Preview from the drop-down list.  If you do not see all the items in the graphic, go to View, Toolbars, and select the toolbars that you wish to see. Your first assignment is to check the various menu items to see what they contain. The page boundary shows you what will fit on a normal page when the document is printed. Material to the right of the boundary will print on the next page.  The toolbar under the menu is the Standard toolbar and contains icons for: New document, Open document, Save, Print, Print Preview, and so on. These icons have tooltips (little tags that inform you of their functions). If you forget what an icon does, hold the mouse pointer over it for a second and the tooltip will appear. Note the red crosshair in the document window. This is the insertion point in Mathcad and any typed information will appear at this point. Clicking the mouse in a new location moves the insertion point (try it!). Entering Text In Mathcad, text is different from mathematics. Text information and math information have their separate fonts, colors, and styles. The formatting toolbar (the one right above the actual document) has a style box that informs you of the type of information that is being typed or selected in the document. The default style is variables so when you type, Mathcad assumes you are typing mathematics. If you wish to type text, first type a quotation mark, . The quote opens a text region in Mathcad and you can type any text that you wish. Click anywhere outside the text region to stop. Any good Mathcad document should contain text to annotate and explain the calculations. You can, and should, change the appearance of text to distinguish it from math. You can do this by typing Format, Style. This brings up the format dialog box. To change the text font, choose Normal and click modify. Click the font box and you may change the font style, size, and color. A good choice is to make text font a different color than the black font used for variables. Equations and Variables Mathematical expressions are typed in the usual way, using the +, , /, and * keys for addition, subtraction, division, and multiplication. Exponents are entered with the ^ key. You must remember to include the multiplication symbol in all expressions. Mathcad will interpret xy as a new variable, and not as the product of x and y. Here we type 5*12= and you can see the result is 60. If we type 512= Mathcad interprets the second statement as the number 512 as you can see from the result.  EMBED Mathcad  Forgetting the multiplication symbol (*) is the most common source of headaches for Mathcad beginners. A convenient way to see multiplication is to go to Tools, Worksheet options, Display. If you make the multiplication symbol a dot, large dot, or (, it will be hard to miss. You must first define a variable before you use it in a calculation. In the example below, x is not defined at the beginning and is highlighted by Mathcad in red (an error):  EMBED Mathcad  To define a variable, use the := operator. You enter this by typing a colon. The := is an assignment operator in Mathcad; it states that the variable on the left is assigned the value or expression on the right. You cannot define an expression on the right side of this operator; 14 := sin(x) will produce an error. Some typical uses: Try the assignment operator by typing x:12.5 and observe the result. Common mathematical functions can be found by using the function icon, f(x) on the toolbar. Try defining a new variable to be the cosine of y and set y equal to 15.  EMBED Mathcad   To evaluate an expression, use the normal equals sign just as you would in normal mathematics.  EMBED Mathcad  Notice that the function is always on the left side of the assignment operator. An expression such as  EMBED Equation.DSMT4  makes no sense to Mathcad. Exercise: Define a variable to represent the distance as a function of time. The distance follows this expression: initial_position + 11.5*time + 1.2*time2. Set the initial position to 255 and calculate the position at time = 2, 12, 25. Exercise: Harmonic motion follows a sinusoidal pattern: position = amplitude*sin(time) + phase. Define a variable to represent the position as a function of time when the amplitude is 1.5 and the phase is zero. Calculate the position at time = 0, 12, 24, 44. What change to position occurs if the phase changes? Important Warning about Typical Scientific Symbols Mathcad has a built-in set of units that are used for cancellation and obtaining the units of an answer. This is a very powerful concept but sometimes these units can cause confusion when you try to make them variables. A common example is g. We all like to use g for grams, but Mathcad has already defined this symbol as the acceleration due to gravity:  This problem also occurs for T. We like to use T for temperature, but it represents Tesla, the built-in unit for magnetic flux density. You can use T in this manner: T = 325 K, where K is the symbol for kelvin temperature. Another problem exists for the letter e. We typically use e as the charge on an electron but in Mathcad, e represents the base of natural logarithms (e = 2.781828.). Mathcad also writes e as exp as in ex = exp(x). Selecting and Editing Mathcad expressions can be edited using the standard windows methods for cut, copy, and paste. Before editing part of an expression, the appropriate section must be selected. To select an area, click inside the expression; the red crosshair will become a blue selection cursor that looks like (, or (. The small foot on the cursor indicates which way the data will be selected. ( will select to the right, while ( will select to the left. Use the insert key to change the direction. The spacebar repeatedly expands the selection, and the ( and ( keys move it right and left respectively. Try it on an expression to see how these keys work The spacebar changes the selection by parts of an expression (pieces connected by +, , *, /), while the arrows move character by character. It is easier to observe than to explain. Build and expression such as  EMBED Equation.DSMT4  and try it. Once a part of an expression is selected, you can cut it, delete it, or copy it. You can also edit it by substituting different operators. Text is edited in the same manner. Any selected item can be pasted into another document such as a Word document or an Excel spreadsheet. All of the Mathcad work in this primer was pasted directly from Mathcad into Word. Variable Names Mathcad allows almost any name for a variable as long as the name is not a predefined function. Some example variables are:  EMBED Mathcad  Variable names can be any combination of alphanumeric characters. These names should begin with a letter and should not contain symbols except for the underscore( _ ), percent ( % ), or a prime( ` ). You cannot use unit symbols or built-in functions as names (to see a list of built-in functions, click the f(x) on the toolbar). Some examples of variables are: This_is_a_variable okay Mass_proton okay Variable_4 okay V okay 1st_value invalid; begins with a number. Sin(X) invalid; sin is a built-in function. A list of predefined functions can be found by clicking the f(x) icon on the toolbar. As simple example of variables, we calculate P from the ideal gas law. Define some values and evaluate the pressure.  EMBED Mathcad  The variables are defined in the first line; P is then defined (as nRT/V) and evaluated. If any variable is left undefined, the undefined quantities will be in red (an error). In this example, we delete the definition for V, therefore P cannot be calculated.  EMBED Mathcad  You can put a subscript as part of a variable name by using the period character on the keyboard. If we type v.initial (thats v period initial), we get  EMBED Equation.DSMT4  in the document. Subscripted variable names can be quite useful. Units By default, Mathcad will use the SI unit system for all calculations. You can change this in the Tools, Worksheet options, Display command. Go to the Units tab and choose a system of units or set it to none. You enter a unit for any quantity by multiplying a numerical value by the unit. If you want a volume of 15 liters, you type V:15*L. The previous gas law example, with units is:  EMBED Mathcad  Mathcad automatically cancels units and produces the result in Pascals (the default SI unit for pressure). If you prefer a different unit for the result, click the expression and a black box will appear after the Pa unit. Double-click this box and a set of possible units will be shown; you can choose the units you want. Here we change the answer to atmospheres.  EMBED Mathcad  An easy way to see the defined units in Mathcad is to click the measuring cup on the toolbar. Choose any type of quantity (length, mass, etc.) and Mathcad will display the possible units. Defining Your Own Units Mathcad does not contain all possible units for all possible quantities. For example, Mathcad does not have picometers in its units database. You can define units such as picometers in your Mathcad document.  EMBED Mathcad  Note that it is 1012 * m! Dont forget the multiplication operator. Mathcad also has a table of reference data that contains physical constants. These data can be copied and pasted into any Mathcad document. Click the Resource center icon on the toolbar (it looks like a book); click on quicksheets and reference tables; click reference tables and then physical constants. The Resource center is also an excellent source for help, and the quicksheets are quite useful to show you how to do certain calculations in Mathcad. When you use units in calculations, Mathcad checks the units and will not work with incompatible units. You will get an error message.  EMBED Mathcad  The bar is the preferred unit of pressure, but Mathcad does not have the bar in its units database. You must define it as we did for picometers earlier. By definition: 1bar(105Pa. Easy Graphs Mathcad has a method for developing an xy graph of any simple function. To illustrate the method, lets graph the sine of x. Start a new document and put the cursor where you want the graph. From the menu, choose Insert, Graph, XY Plot. You will see an empty graph with black boxes (placeholders). You can cycle through the placeholders by using the TAB key.  EMBED Mathcad  Type x in the placeholder on the x-axis and then type sin(x) in the box on the y-axis. The graph will change to:  EMBED Mathcad  Mathcad makes a guess at the limits, in this case: 10 to +10. Plot the function  EMBED Equation.DSMT4 . From the menu, choose Insert, Graph, XY Plot. You will see an empty graph. Type x in the x-axis box, and type the expression,  EMBED Equation.DSMT4 , in the y-axis box:  EMBED Mathcad  One more graph example. We can define a function and then graph it by placing the f(x) in the y-placeholder:  EMBED Mathcad  If the range of values on the x or y axis is not appropriate, you can change them by clicking the graph. Values in placeholders will show for the maximum and minimum x and y ranges. In the example, the maximum y is 22.636.  EMBED Mathcad  Click inside any value and edit it to your liking. Here the maximum y has been changed to 10 and the maximum x to 5.  EMBED Mathcad  If you double-click the graph, a dialog box will open that allows you to change any properties of the graph including whether it will be displayed as lines, points, or a combination of both; the color of the lines, labels for the axes, and a graph title. For the title to show, you must be sure to check the show title box (dont ask why). Range Variables and Graphs When we need to have numerous values for a variable in Mathcad, we use range variables. Range variables are also used to generate graphs of scientific data. A range variable is a variable that has different values based on an index. For example, we may have three different pressures: P1, P2, and P3 where P1 is 100 bar, P2 is 220 bar and P3 is 75 bar. In Mathcad, enter a range variable by typing the variable name, then type [ and then the index. The [ is a definition for a subscripted range variable. This is completely different than the subscripted variable we discussed earlier (obtain by using a period). As an example, lets produce a set of pressures that go from 1 to 12 atm in unit steps. First we define the range, i. Type i:1;12 (thats i colon 1 semicolon 12) Now define the pressures, type P[i := i You should see:  EMBED Mathcad  To see that this works, type P = to list the values of P. What if we did not want to increase the pressure in unit steps, but we want increase it by steps of 0.25 starting at 1? In this case, we need to define another variable. We cannot change the range of i to be 1, 1.25, 1.50because i is the index for P. We can have P1, P2, or P3, but we cant have P1.25. One way around this problem is shown below. We define i and an increment that depends on i.  EMBED Mathcad  The new range variable (incr) changes the increment values to 0.25 times the index. Suppose you want a range of x-data for a graph that starts at 12.5, increments at 0.01, and ends at 22. You enter this as: x colon 12.5 comma 12.51 semicolon 22.  EMBED Mathcad  Note that the second number (12.51) is the next value in the sequence and it is not the increment to be added to the initial value. We can also produce variables that are functional forms. For instance, we can define a range of pressures and then define a variable such as V(P) that depends on the pressure range. This method of using range variables is shown next. In this example, we use the pressure itself as the range variable. We define the compressibility in terms of both pressure and temperature, and that allows us the option of evaluating and plotting multiple functions simultaneously.  EMBED Mathcad  Note how Z(p,T). The pressure is defined in the range and we plot three different temperatures: 1379, 300, and 500 K. Multiple arguments on the y-axis are entered and separated with a comma. To enter these functions we would type: Z(p, 1379), Z(p, 300), Z(p, 500) into the y placeholder on the graph. Zoom and Trace Functions in a Graph Mathcad has built-in zoom and trace capabilities for graphs as shown in the next figure. On a Mathcad document, reproduce the work to produce the previous graph. Right-click the graph and choose either Trace or Zoom from the context menu.  Zoom: Move the Zoom dialog box to the side. Click and hold at the point in the graph that you wish to begin the Zoom; drag a rectangular box until it encompasses the region you wish. Finish by clicking Zoom or OK in the dialog box. Trace: Move the Trace dialog box to the side. Click anywhere in the graph; the points will show in the Trace box. If the Track Data Points box is checked, Mathcad will skip from one data point to the next. It will not provide data (interpolate) between the points. Data and Data Files: Excel Data Mathcad data are held in arrays and arrays have a range associated with them. Often we need to get scientific data into or out of Mathcad. Fortunately, Mathcad has built-in functions for this. Data can usually be saved as a text (ASCII) file from another program (such as EXCEL). In this example, the data were saved from a Word file (Saved a text file, not as a Word document!). To read the data file, we first define a variable for the data. Usually we have x and y data so this variable will have two columns of data; this is called a matrix. A typical matrix of data is:  EMBED Equation.DSMT4  To add our data to Mathcad, click the Insert Data File Input, and then choose the file format (usually Excel). Click Browse to find the data file. Then click finish. Put a variable name in the placeholder on the Mathcad screen and you're done. When done, you should have something like this:  The variable PZ is defined as the data set given by the url, C:\..\co2.txt. The variable PZ contains the pressure and compressibility data for the gas. To separate the pressure and compressibility data, we extract the columns of the data matrix. P is in the first column (Mathcad labels it as zero) and the compressibility is in the second column (Mathcad labels it as one). To extract each column use the Extract Column command from the matrix palette or use CTRL6. You need to type in the column number that you want to extract. We define P as the first column of our data matrix and z as the second column:  EMBED Mathcad  Now we have two variables of data and we can work with them just as we did for any other range of data. A graph of the data is:  EMBED Mathcad  Entering Data into Mathcad Often we need to enter data directly into Mathcad. Data are held in arrays and arrays have a range associated with them. First, define a range variable (j in this case). Then type P[j : Type in the first number followed by a comma; repeat for each number. The last line shows that P3 does equal 7.  EMBED Mathcad  Another method of entering data is to define a variable as a matrix. For example, we can define X as a matrix of five elements in this manner: Type X : to produce the assignment operator; then choose Insert, Matrix from the toolbar. In the dialog box, choose 5 rows and 1 column. You should have:  EMBED Mathcad  You can now type values into the matrix. Copying Data from EXCEL The easiest method to provide Mathcad with EXCEL data is to define a variable such as New_data (you can use any name you choose): Open EXCEL and highlight the data range in EXCEL. Copy the data. In Mathcad, place the cursor on the placeholder for New_data and choose Edit, Paste from the menu (or right-click and chose Paste). The selected data will now be in Mathcad. You can set up x and y variables in Mathcad and then paste the x data from EXCEL and then select and paste the y data from EXCEL. Numerical Calculations Mathcad can solve many problems numerically using common algorithms. As you have already seen, numerical computations are trivial:  EMBED Mathcad  Integrals are evaluated using a modified Romberg algorithm, a method that is superior to the trapezoidal rule:  EMBED Mathcad  Numerical derivatives are also easily obtained:  EMBED Mathcad  Numerical Roots of Equations Roots of equations can be obtained from the root function: root(f(x),x) The root function obtains roots by successive approximation and it needs an initial guess for the root. This guess can be supplied after graphing the function and observing where the function crosses the x-axis. As an example, we will find the roots of:  EMBED Equation.DSMT4 .  EMBED Mathcad  Now we can find the exact roots:  EMBED Mathcad  Warning: Always graph the function or try many initial estimates to be sure you have the correct root. Exercise: Find the third root of the function. Mathcad can also find the roots by symbolic calculation. In this example, we use the same function,  EMBED Equation.DSMT4 , but we do not write it as a definition (that is, we leave out the f(x) part). Select any x in the expression, and then select Symbolics, Variable, Solve  EMBED Mathcad  Note the decimal point after the 5 in the expression. Without a decimal number, Mathcad will solve analytically without substituting numbers into the expression. For numerical problems, this can be cumbersome. Exercise: Remove the decimal point after the 5 and solve to see the result. Also, note that Mathcad provides all three roots by this method. Symbolic Calculation A powerful feature of Mathcad is the ability to do symbolic algebra and calculus. The commands are located in the Symbolics menu on the toolbar and also on the Symbolics palette (the graduation hat icon). Symbolic calculations can either be static or live. A live symbolic calculation is automatically updated when a change is made to the document; a static calculation is not updated. Static symbolic work is accomplished by using the commands on the symbolics menu; live symbolic calculation is accomplished using the symbolics equal sign, (. As an example, lets find the derivative of sin(5x) using the static method. Type sin(5x) into a worksheet (dont forget the * for multiplication), select the x, and then click Symbolics, Variable, Differentiate. The result, 5cos(5x) will appear below your expression. Another option can be set to improve readability. Click Symbolics, Evaluation Style, and check the box that says Show Comments. Repeat the sin(5x) example and note the difference. If we use live symbolics to perform the same derivative, we would first enter the derivative symbols from the calculus palette (the one with the integral). Fill in the placeholders with sin(5x) and x. Type CTRL . (Control period) and you should see the live symbolics arrow (this is equivalent to an equals sign). Click anywhere and the result, 5cos(5x) should appear. The difference in live symbolics is that any change is immediately reflected in the result. Change the 5 to 17 and observe.  EMBED Mathcad  Symbolic calculations from the menu do not update in this manner as you can show for yourself. Symbolic Integration Symbolic integration works in the same manner as differentiation. We can evaluate both definite and indefinite integrals. Use the integral symbol from the calculus palette.  EMBED Mathcad  Mathcad can also evaluate double and triple integrals:  EMBED Mathcad  and  EMBED Mathcad  Notice that in the second example, the function was defined and then substituted into the integral. The Assume Keyword (Very Important) Sometimes, we know something about the problem that Mathcad does not know. For instance, the integral  EMBED Equation.DSMT4 is not defined for a < 0. If we put this expression into Mathcad, we get this result:  EMBED Mathcad  This is Mathcads way of telling you it cannot evaluate the expression; the result is the same as the original. We must inform Mathcad that a > 0, and we use the assume keyword for this. Write the integral and then press CTRL SHIFT. (Control shift, period, pressed simultaneously) to produce the symbolics equal sign and a placeholder. You can also use the (( from the symbolics palette if you prefer. In the placeholder, type assume, a>0, then click outside the integral to see the result.  EMBED Mathcad  Mathcad sometimes has problems with the assume command and multiple integrals.  EMBED Mathcad  The integral can be evaluated if it is written as a product of three integrals:  Partial Derivatives Mathcad easily evaluates partial derivatives. In fact, Mathcad always assumes it is taking the partial derivative.  EMBED Mathcad  Partial derivatives:  EMBED Mathcad  The partial derivative symbol is obtained by Math, Options, Display, Derivative. Solving Equations Mathcad can symbolically solve a single equation for an unknown or it can solve a set of equations for multiple unknowns. These calculations are done in solve blocks, a construct that is defined in Mathcad by the words given and find. A solve block always begins with the word given. This is followed by the equation or equations to be solved. The result is evaluated in the find command. Solve block use only the symbolics equal sign typed as CTRL =. It appears as a bold equal in the document. A typical example is:  EMBED Mathcad   Mathcad can solve multiple equations in multiple unknowns. Remember to use the symbolics equal sign here (CTRL=).  EMBED Mathcad  You can also solve for limits in integrals. The following example shows how to find the radius, d, where the probability of finding the electron is 90% (or 0.9).  EMBED Mathcad  Simple Method for Finding Eigenvalues Often we need to obtain an eigenvalue for a function. A convenient method for solving for eigenvalues is shown here. An eigenvalue is a constant in the equation,  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is an operator. A simple method for finding an eigenvalue is to rearrange the equation:  EMBED Equation.DSMT4 . Example: is the function  EMBED Equation.DSMT4  and eigenfunction of the derivative operator? If so, what is the eigenvalue?    No, we do not get a constant; we get a new function,     Mathcad Handbook  DATE \@ "M/d/yyyy" 9/16/2004 Page  PAGE 24 of  NUMPAGES 24 Menu Items Toolbars Math toolbar Insertion point Page boundary  EMBED Mathcad  Style box # I l x y     ) ˨˝}c˝2jhV7hZ}>*B*UmHnHphu j{hZ}UmHnHujhV7UmHnHuhV7mHnHu2jhV7hZ}>*B*UmHnHphuhV7mHnHuh ~hV70JmHnHu$jh ~hV70JUmHnHujhUhQ.hHvh## m n w x / P = K am !  ! 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Ole ObjInfo Contents.MCAD 309000000 \  docDocument MmcObject[  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV3*~mcDocumentObjectState\=x" mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U_#  CharacterMap-RangeMap;#no result because x is not defined. ChrPropMap7# ParPropMap9# RangeElem<# ParPropData: RangeData=@WEmbedMap1<LinkMap/# <#!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial  _1115706897@S|! Ole ObjInfo Contentsj.MCAD 309000000 \  docDocument MmcObject[  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV+mcDocumentObjectState\a`* mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/ <!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial @@@B@U0_=@A@@ p@B@@ @A@C@@d@Bx@D@@@B12.5@E@B@U0 {O@F@@ p@G@@ @F@H@@d@GSine@I@@@G@J@@d@Isin@K@@p@I@L@@@Kx@M@B@U0#6N0@N@@ p@O@@@N@P@@d@OSine@Q@@@O@R@@+@@Q@X@S@@@Q _1115706965d@S|! Ole ObjInfoContents .MCAD 309000000 \  docDocument MmcObject[II  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV3mcDocumentObjectState\;F mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/ <!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial @@@B@U8VD@A@@ p@B@@ @A@C@@@@B@D@@d@Cy@E@@p@C@F@@@Ex@G@@@B@H@@@@G@I@@t@H27@J@@@H@K@@d@Jx@L@@@J2@M@@{@G@N@@@Mx@O*@U;^^-define a function79@P<@Q:@W1@R</@S<@T0@NormalArial @U@B@U8gEC@V@@ p@W@@ @V@X@@d@Wx@Y@@@W12.5@Z*@U6-set x79@[<@\:@W1@]</@^<@_0@NormalArial @`@B@U8 UB@a@@ p@b@@@a@c@@@@b@d@@d@cy@e@@p@c@f@@@ex@g@@@b@h@@+@@g@X@i@@@g@j*@U 5::- evaluate y7 9 @k< @l:@W1@m</ @n< @o0@NormalArial @p@B@U8+d>E8A@q@@ p@r@@ @q@s@@d@rx@t@@@r195@u*@U+;8488- set new x7 9 @v< @w:@W1@x</ @y< @z0@NormalArial @{@B@U8KfU`@@|@@ p@}@@@|@~@@@@}@@@d@~y@@@p@~@@@@x@@@@}@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{T8?00-@xa faster and easier approach; the function is already defined as y(x) so we can place any value of x into the expression7x9x@@ ArialSerial_ParPropDefaultWNormal>@Arial@W Heading 1>@ Arial@W Heading 2>@ Arial@W Heading 3 >@ Arial@W Paragraph >@ Arial@WList >@ Arial@WIndent >@Times New Roman@W Title >@Times New Roman@W Subtitle font_style_listO font_styleP  VariablesTimes New Roman@P  ConstantsTimes New Roman@P TextArial@P Greek VariablesSymbol@P User 1Arial@P User 2 Courier New@P User 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/ <!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial @@@B@U8VD@A@@ p@B@@ @A@C@@@@B@D@@d@Cy@E@@p@C@F@@@Ex@G@@@B@H@@@@G@I@@t@H27@J@@@H@K@@d@Jx@L@@@J2@M@@{@G@N@@@Mx@O*@U;^^-define a function79@P<@Q:@W1@R</@S<@T0@NormalArial @U@B@U8gEC@V@@ p@W@@ @V@X@@d@Wx@Y@@@W12.5@Z*@U6-set x79@[<@\:@W1@]</@^<@_0@NormalArial @`@B@U8 UB@a@@ p@b@@@a@c@@@@b@d@@d@cy@e@@p@c@f@@@ex@g@@@b@h@@+@@g@X@i@@@g@j*@U 5::- evaluate y7 9 @k< @l:@W1@m</ @n< @o0@NormalArial @p@B@U8+d>E8A@q@@ p@r@@ @q@s@@d@rx@t@@@r195@u*@U+;8488- set new x7 9 @v< @w:@W1@x</ @y< @z0@NormalArial @{@B@U8KfU`@@|@@ p@}@@@|@~@@@@}@@@d@~y@@@p@~@@@@x@@@@}@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc ObjInfo#%Contents$_1115722266(@S|! Ole     ! #$8&'()*+,-./01234567L:;<=>?@ABCDEFGHIJKaNOPQRSTUVWXYZ[\]^_`defghijklmnopqrstuvwxyz{|}~.MCAD 309000000 \  docDocument MmcObject[ZZ  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV;qmcDocumentObjectState\PS; mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/ <!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial @@@B@U8VD@A@@ p@B@@ @A@C@@@@B@D@@d@Cy@E@@p@C@F@@@Ex@G@@@B@H@@@@G@I@@t@H27@J@@@H@K@@d@Jx@L@@@J2@M@@{@G@N@@@Mx@O*@U;^^-define a function79@P<@Q:@W1@R</@S<@T0@NormalArial @U@B@U8gEC@V@@ p@W@@ @V@X@@d@Wx@Y@@@W12.5@Z*@U6-set x79@[<@\:@W1@]</@^<@_0@NormalArial @`@B@U8 UB@a@@ p@b@@@a@c@@@@b@d@@d@cy@e@@p@c@f@@@ex@g@@@b@h@@+@@g@X@i@@@g@j*@U 5::- evaluate y7 9 @k< @l:@W1@m</ @n< @o0@NormalArial @p@B@U8+d>E8A@q@@ p@r@@ @q@s@@d@rx@t@@@r195@u*@U+;8488- set new x7 9 @v< @w:@W1@x</ @y< @z0@NormalArial @{@B@U8KfU`@@|@@ p@}@@@|@~@@@@}@@@d@~y@@@p@~@@@@x@@@@}@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@#i6P0X@@@ p@@@ @@@@d@V@@@@12@@B@U#60Z@@@ p@@@ @@@@d@R@@@@0.0821@@B@U#/60W@@@ p@@@ @@@@d@T@@@@225@@B@U`#6o0Y@@@ p@@@ @@@@d@n@@@@2.5@@B@U@JqO`U@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@USf`V@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@ ObjInfo')Contentsk$_1072856539L,FOle .MCAD 309000000 \  docDocument MmcObject[``  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV@qmcDocumentObjectState\K[? mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/ <!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial @@@B@U8VD@A@@ p@B@@ @A@C@@@@B@D@@d@Cy@E@@p@C@F@@@Ex@G@@@B@H@@@@G@I@@t@H27@J@@@H@K@@d@Jx@L@@@J2@M@@{@G@N@@@Mx@O*@U;^^-define a function79@P<@Q:@W1@R</@S<@T0@NormalArial @U@B@U8gEC@V@@ p@W@@ @V@X@@d@Wx@Y@@@W12.5@Z*@U6-set x79@[<@\:@W1@]</@^<@_0@NormalArial @`@B@U8 UB@a@@ p@b@@@a@c@@@@b@d@@d@cy@e@@p@c@f@@@ex@g@@@b@h@@+@@g@X@i@@@g@j*@U 5::- evaluate y7 9 @k< @l:@W1@m</ @n< @o0@NormalArial @p@B@U8+d>E8A@q@@ p@r@@ @q@s@@d@rx@t@@@r195@u*@U+;8488- set new x7 9 @v< @w:@W1@x</ @y< @z0@NormalArial @{@B@U8KfU`@@|@@ p@}@@@|@~@@@@}@@@d@~y@@@p@~@@@@x@@@@}@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U#60`@@@ p@@@ @@@@d@R@@@@0.08205@@B@U#/60_@@@ p@@@ @@@@d@T@@@@225@@B@U`#6o0^@@@ p@@@ @@@@d@n@@@@2.5@@B@U@JqO`U@@@ p@@@@@@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@USf`V@@@ p@@@@@@@@@@@@d@P@@@@@@@+@@@X@@@@ CompObj+- iObjInfo."Equation Native #_10728568431@S|!  FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  v initialOle (ObjInfo02)Contents%%_1072856905/l5@S|! .MCAD 309000000 \  docDocument MmcObject[dd  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV;mcDocumentObjectState\w; mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/ <!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial @@@B@U8VD@A@@ p@B@@ @A@C@@@@B@D@@d@Cy@E@@p@C@F@@@Ex@G@@@B@H@@@@G@I@@t@H27@J@@@H@K@@d@Jx@L@@@J2@M@@{@G@N@@@Mx@O*@U;^^-define a function79@P<@Q:@W1@R</@S<@T0@NormalArial @U@B@U8gEC@V@@ p@W@@ @V@X@@d@Wx@Y@@@W12.5@Z*@U6-set x79@[<@\:@W1@]</@^<@_0@NormalArial @`@B@U8 UB@a@@ p@b@@@a@c@@@@b@d@@d@cy@e@@p@c@f@@@ex@g@@@b@h@@+@@g@X@i@@@g@j*@U 5::- evaluate y7 9 @k< @l:@W1@m</ @n< @o0@NormalArial @p@B@U8+d>E8A@q@@ p@r@@ @q@s@@d@rx@t@@@r195@u*@U+;8488- set new x7 9 @v< @w:@W1@x</ @y< @z0@NormalArial @{@B@U8KfU`@@|@@ p@}@@@|@~@@@@}@@@d@~y@@@p@~@@@@x@@@@}@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uc~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@ Ole *ObjInfo46+Contents9%_1072597823C9@S|! .MCAD 309000000 \  docDocument MmcObject[dd  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV;mcDocumentObjectState\7:Y mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/ <!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial @@@B@U8VD@A@@ p@B@@ @A@C@@@@B@D@@d@Cy@E@@p@C@F@@@Ex@G@@@B@H@@@@G@I@@t@H27@J@@@H@K@@d@Jx@L@@@J2@M@@{@G@N@@@Mx@O*@U;^^-define a function79@P<@Q:@W1@R</@S<@T0@NormalArial @U@B@U8gEC@V@@ p@W@@ @V@X@@d@Wx@Y@@@W12.5@Z*@U6-set x79@[<@\:@W1@]</@^<@_0@NormalArial @`@B@U8 UB@a@@ p@b@@@a@c@@@@b@d@@d@cy@e@@p@c@f@@@ex@g@@@b@h@@+@@g@X@i@@@g@j*@U 5::- evaluate y7 9 @k< @l:@W1@m</ @n< @o0@NormalArial @p@B@U8+d>E8A@q@@ p@r@@ @q@s@@d@rx@t@@@r195@u*@U+;8488- set new x7 9 @v< @w:@W1@x</ @y< @z0@NormalArial @{@B@U8KfU`@@|@@ p@}@@@|@~@@@@}@@@d@~y@@@p@~@@@@x@@@@}@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm Ole ,ObjInfo8:-ContentsMe'_1072682090=@S|! .MCAD 309000000 \  docDocument MmcObject[gg  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV+mcDocumentObjectState\@+ mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/ <!LinkData0@NormalArial "@B@U8;^NEH #@@ p$@@ #%@@d$x&@@$14'*@U;KH---define x79(<):@W1*</+<,0@NormalArial -@B@U8c~hx .@@ p/@@.0@@@/1@@@02@@d1x3@@124@@05@@t456@@4x7@@/8@@+@7@X9@@7:*@Uk{x__-result of function79;<<:@W1=</><?0@NormalArial @@@B@U8VD@A@@ p@B@@ @A@C@@@@B@D@@d@Cy@E@@p@C@F@@@Ex@G@@@B@H@@@@G@I@@t@H27@J@@@H@K@@d@Jx@L@@@J2@M@@{@G@N@@@Mx@O*@U;^^-define a function79@P<@Q:@W1@R</@S<@T0@NormalArial @U@B@U8gEC@V@@ p@W@@ @V@X@@d@Wx@Y@@@W12.5@Z*@U6-set x79@[<@\:@W1@]</@^<@_0@NormalArial @`@B@U8 UB@a@@ p@b@@@a@c@@@@b@d@@d@cy@e@@p@c@f@@@ex@g@@@b@h@@+@@g@X@i@@@g@j*@U 5::- evaluate y7 9 @k< @l:@W1@m</ @n< @o0@NormalArial @p@B@U8+d>E8A@q@@ p@r@@ @q@s@@d@rx@t@@@r195@u*@U+;8488- set new x7 9 @v< @w:@W1@x</ @y< @z0@NormalArial @{@B@U8KfU`@@|@@ p@}@@@|@~@@@@}@@@d@~y@@@p@~@@@@x@@@@}@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m Ole .ObjInfo<>/ContentscE_1072598498A@S|! g.MCAD 309000000 \  docDocument MmcObject[ d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV0EmcDocumentObjectState\I0E mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman8eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. 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TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman  TextRegion* docRegionGshpBoxU@Sc@`  {{ CharacterMap-RangeMap;Roots of the equation ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1<LinkMap/<LinkData0@NormalArial eqRegionB@UH{d tree@ p@@ @@@@@df@@p@@x@@@@@@@@@@@ @@dx!@@3"@@#@@t"31$@@"%@@d$x&@@$2'@@(@@t'700)@@'x*@@5+*@UHHPP-A+Graph the function to estimate roots. 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ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1 <LinkMap/!<"LinkData0@NormalArial #@B@U8;^NEH $@@ p%@@ $&@@d%x'@@%14(*@U;KH---define x79)<*:@W1+</,<-0@NormalArial .@B@U8c~hx /@@ p0@@/1@@@02@@@13@@d2x4@@225@@16@@t557@@5x8@@09@@+@8@X:@@8;*@Uk{x__-result of function79<<=:@W1></?<@@0@NormalArial @A@B@U8VD@B@@ p@C@@ @B@D@@@@C@E@@d@Dy@F@@p@D@G@@@Fx@H@@@C@I@@@@H@J@@t@I27@K@@@I@L@@d@Kx@M@@@K2@N@@{@H@O@@@Nx@P*@U;^^-define a function79@Q<@R:@W1@S</@T<@U0@NormalArial @V@B@U8gEC@W@@ p@X@@ @W@Y@@d@Xx@Z@@@X12.5@[*@U6-set x79@\<@]:@W1@^</@_<@`0@NormalArial @a@B@U8 UB@b@@ p@c@@@b@d@@@@c@e@@d@dy@f@@p@d@g@@@fx@h@@@c@i@@+@@h@X@j@@@h@k*@U 5::- evaluate y7 9 @l< @m:@W1@n</ @o< @p0@NormalArial @q@B@U8+d>E8A@r@@ p@s@@ @r@t@@d@sx@u@@@s195@v*@U+;8488- set new x7 9 @w< @x:@W1@y</ @z< @{0@NormalArial @|@B@U8KfU`@@}@@ p@~@@@}@@@@@~@@@d@y@@@p@@@@@x@@@@~@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m@@B@U ;?N-Hj@@@ p@@@ @@@@d@x@@@@x@@B@U@ql@@@ p@@@?@@@@$@@@@@d@x@@@@@@@d@sin@@@p@@@@@@@@t@17@@@@x@@@@ ObjInfo~Contentsm_1072679246@S|! BBOle       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdejklmnopqrstuvwxyz{|}~.MCAD 309000000 \  docDocument MmcObject[  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectVmcDocumentObjectState\% mcPageModelK????mcHeaderFooterI@I CHeaderFooterJ@J@J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultW?Normalfont_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 Roman^ TextRegion* docRegionGshpBoxU-(99 CharacterMap-RangeMap;Symbolic Calculus Using Mathcad ChrPropMap7 RangeElem< ChrPropData8 RangeData=~ ParPropMap9 <  ParPropData:@W,EmbedMap1 <LinkMap/ < LinkData0@NormalArial *@U S0 `pp-AIn this document we will illustrate some of the symbolic calculations that Mathcad can perform. Before we do so, you need to be aware that there are two methods of performing symbolic calculations: we can use the symbolic menu commands or we can use symbolic operators. In general, the symbolic operators are preferred because the calculations are "live", that is, they will automatically update if you change a parameter. Let's see the difference. Compute the derivative of sin(5x).79<:@W,1</<0@NormalArial *@U -Symbolics Menu Method7<8v 9<:@W,1</<0@NormalArial *@U  ?   -@\Choose Symbolics, Evaluation style and check the boxes that say horizontal and show comments7\9\<\:@W,1</\ <\!0@NormalArial "*@U #5S 0(00-@Place the cursor or selection box on or near the x and choose symbolics, variable, differentiate. What do you observe?. Try some other mathematical expression and obtain the derivative. Will you be able to do this any time you need to?79#<$:@W,1%</&<'0@NormalArial (eqRegionB@U0kc~?x)tree@ p*@@)+@@d*sin,@@p*-@@,.@@t-5/@@-x0*@U(8((  -@hTo integrate, place the cursor or selection box on or near the x, choose symbolics, variable, integrate.7h9h1@@t=5?@@=x@@*@UxxExx-by integration, yields79@A<@B:@W,1@C</@D<@E0@NormalArial @F@B@U;F@G@@ p@H@@@G@I@@d@Hsin@J@@p@H@K@@@J@L@@t@K5@M@@@Kx@N*@U( <;((00-@Now in the space below, repeat these calculations but change the 5 to 33 and note any change when you recalculate the worksheet. Try integrating other mathematical functions for practice.79@O<@P:@W,1@Q</@R<@S0@NormalArial @T*@U  0-Symbolic Operator Method7@U<@V8@Tv 9@W<@X:@W,1@Y</@Z<@[0@NormalArial @\*@U $    -@From the calculus palette, choose the derivative operator and fill in the placeholders. Type CTRL period (the period key) to insert the symbolic equals sign (an arrow).79@]<@^:@W,1@_</@`<@a0@NormalArial @b*@U.3 RhN N -@JRepeat the instructions to see how the equation to the left was developed.7J9J@c@A A"@@tA!2A#@@A!A$@@dA#xA%@@A# _n_u_l_l_A&@@>A A'@@tA&2A(@@A&A)@@dA(yA*@@pA(A+@@A*A,@@dA+ _n_u_l_l_A-@@>A+A.@@tA-2A/@@A-A0@@dA/zA1@@A/ _n_u_l_l_A2*@U`;K`H^^-define a function79A3<A4:@W,1A5</A6<A70@NormalArial A8@B@U;N,HA9@@ pA:@@ A9A;@@@A:A<@@dA;fA=@@pA;A>@@ A=A?@@ @A>A@@@dA?xAA@@A?yAB@@A>zAC@@A:AD@@@ACAE@@@ADAF@@dAEsinAG@@pAEAH@@AGxAI@@ADAJ@@dAIsinAK@@pAIAL@@AKAM@@tAL2AN@@ALyAO@@ACAP@@dAOsinAQ@@pAOAR@@AQAS@@tAR3AT@@ARzAU@B@U@STAV@@ pAW@@?AVAX@@p@AWAY@@AXAZ@@>@AYA[@@tAZ2A\@@AZA]@@dA\xA^@@A\A_@@dA^fA`@@pA^Aa@@ A`Ab@@ @AaAc@@dAbxAd@@AbyAe@@AazAf@@>AYAg@@tAf2Ah@@AfAi@@dAhyAj@@pAhAk@@AjAl@@@AkAm@@dAlfAn@@pAlAo@@ AnAp@@ @AoAq@@dApxAr@@ApyAs@@AozAt@@>AkAu@@tAt2Av@@AtAw@@dAvzAx@@AvAy@@dAxfAz@@pAxA{@@ AzA|@@ @A{A}@@dA|xA~@@A|yA@@A{zA@@AWA*@U  0-2. Symbolic integrals7A<A8Av 9A<A:@W,1A</A<A0@NormalArial A*@U ,8-@OIntegral calculus is also relatively easy in both indefinite and definite forms7O9OA84A@@ pA@@ AA@@@AA@@dA\yA@@pAA@@ AA@@ @AA@@dArA@@A\qA@@A\fA@@AA@@@AA@@dAeA@@AA@@K@AA@@A7A@@ArA@@AA@@dAcosA@@pAA@@A\qA*@U@Ss@`L  -@You must know what normalization means. Define 'normalization factor'. How is the normalization factor used to produce a normalized function?79A<A:@W,1A</A<A0@NormalArial A@B@U@G5BA@@ pA@@?AA@@%@AA@@@AA@@tA0A@@AA@@tA2A@@A\pA@@AA@@dA\fA@@%AA@@@AA@@tA0A@@A\pA@@AA@@dA\qA@@%AA@@@AA@@tA0A@@A\A@@AA@@dArB@@AB@@@BB@@@BB@@@BB@@dB\yB@@pBB@@ BB@@ @BB@@dBrB @@B\qB @@B\fB @@B2B @@BB @@dB rB@@B 2B@@BB@@dBsinB@@pBB@@B\qB@@AB*@UWQ<-?Recall that   is the volume element in spherical coordinates7 ?B< B8B128,0,0B<B8B?BB<B8B128,0,0BB<B8B?BB<0B8B128,0,0BBB9?B*@U00C-So p/1029 = 1/N2, thus7B?<B@8B>BA<BB8B>SymbolB?BC< BD8B>BABE<BF8B>_BCBG<BH8B>BEBGBA9BI<BJ:@W,1BK</BL<BM0@NormalArial BN*@U0 U# 0 E %%-Check79BO<BP:@W,1BQ</BR<BS0@NormalArial BT@B@U O 8 GBU@@ pBV@@?BUBW@@@BVBX@@@BWBY@@dBXNBZ@@BX2B[@@%BWB\@@@B[B]@@tB\0B^@@B\B_@@tB^2B`@@B^\pBa@@B[Bb@@dBa\fBc@@%BaBd@@@BcBe@@tBd0Bf@@Bd\pBg@@BcBh@@dBg\qBi@@%BgBj@@@BiBk@@tBj0Bl@@Bj\Bm@@BiBn@@dBmrBo@@BmBp@@@BoBq@@@BpBr@@@BqBs@@dBr\yBt@@pBrBu@@ BtBv@@ @BuBw@@dBvrBx@@Bv\qBy@@Bu\fBz@@Bq2B{@@BpB|@@dB{rB}@@B{2B~@@BoB@@dB~sinB@@pB~B@@B\qB@@BVB*@U0c B 0p N  -@Repeat the normalization using the same function but this time it contains sin(q) instead of cos(q). What is the correct normalized function? Prove it as we did here for the cos function.7bOB@@C8C?@@dC>\fC@@@C>1CA@@%C7CB@@@CACC@@tCB0CD@@CB\pCE@@CACF@@dCE\qCG@@CECH@@dCGsinCI@@pCGCJ@@CI\qCK@@pC6CL@@%CKCM@@@CLCN@@tCM0CO@@CM\CP@@CLCQ@@dCPrCR@@CPCS@@@CRCT@@@CSCU@@dCTRCV@@pCTCW@@ CVCX@@dCWZCY@@CWrCZ@@CS2C[@@CRC\@@dC[rC]@@C[2C^@@C5C_ BitmapRegion PictureRegionF@U H  H [C` mc_dibitmap 5x (5P Ca*@U   - 3.Solving equations symbolically7 Cb< Cc8Cav 9 Cd< Ce:@W,1Cf</ Cg< Ch0@NormalArial Ci*@U0  0   -@bWhat is the maximum in our radial function shown above? How do you find the maximum in a function?7b9bCjD K R Zagmr!v(y/z7y>xDuJpOkSdV]WVWOUHRAN;H6 "--&&l3 "--F$!B<877:>E{MuWqaplpwsw~}rg\RJB--&&Mi~P "--%Ok{M--&&&n "--%k--&&ihn "--%kek--&&xn "--%k "-- $  z--&&rn "- -%kR- - $UpK--&--- !gz---=- 2 ==r@Times New Roman- 2 N=1- 2 _= '--- !g---H4-  2 3Hr- 2 3H2- 2  3H '-C*@U{0-@H________________________________________________________________________7H9HC@@@D=D?@@@D>D@@@dD?cDA@@D?5DB@@D>DC@@tDB96DD@@DB\pDE@@D=DF@@dDErDG@@DE2DH@@D@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman"eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1 <LinkMap/!<"LinkData0@NormalArial #@B@U8;^NEH $@@ p%@@ $&@@d%x'@@%14(*@U;KH---define x79)<*:@W1+</,<-0@NormalArial .@B@U8c~hx /@@ p0@@/1@@@02@@@13@@d2x4@@225@@16@@t557@@5x8@@09@@+@8@X:@@8;*@Uk{x__-result of function79<<=:@W1></?<@@0@NormalArial @A@B@U8VD@B@@ p@C@@ @B@D@@@@C@E@@d@Dy@F@@p@D@G@@@Fx@H@@@C@I@@@@H@J@@t@I27@K@@@I@L@@d@Kx@M@@@K2@N@@{@H@O@@@Nx@P*@U;^^-define a function79@Q<@R:@W1@S</@T<@U0@NormalArial @V@B@U8gEC@W@@ p@X@@ @W@Y@@d@Xx@Z@@@X12.5@[*@U6-set x79@\<@]:@W1@^</@_<@`0@NormalArial @a@B@U8 UB@b@@ p@c@@@b@d@@@@c@e@@d@dy@f@@p@d@g@@@fx@h@@@c@i@@+@@h@X@j@@@h@k*@U 5::- evaluate y7 9 @l< @m:@W1@n</ @o< @p0@NormalArial @q@B@U8+d>E8A@r@@ p@s@@ @r@t@@d@sx@u@@@s195@v*@U+;8488- set new x7 9 @w< @x:@W1@y</ @z< @{0@NormalArial @|@B@U8KfU`@@}@@ p@~@@@}@@@@@~@@@d@y@@@p@@@@@x@@@@~@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m@@B@U ;?N-Hj@@@ p@@@ @@@@d@x@@@@x@@B@U@ql@@@ p@@@?@@@@$@@@@@d@x@@@@@@@d@sin@@@p@@@@@@@@t@17@@@@x@@@@A@B@U(nA@@ pA@@?AA@@C@AA@@dAyA@@CAA@@dAxA@@AA@@dAxA @@AA @@dA yA @@A 2A @@A ObjInfoContents82_1072679902FBBOle .MCAD 309000000 \  docDocument MmcObject[]] d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV;^;mcDocumentObjectState\:] mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman%eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1 <LinkMap/!<"LinkData0@NormalArial #@B@U8;^NEH $@@ p%@@ $&@@d%x'@@%14(*@U;KH---define x79)<*:@W1+</,<-0@NormalArial .@B@U8c~hx /@@ p0@@/1@@@02@@@13@@d2x4@@225@@16@@t557@@5x8@@09@@+@8@X:@@8;*@Uk{x__-result of function79<<=:@W1></?<@@0@NormalArial @A@B@U8VD@B@@ p@C@@ @B@D@@@@C@E@@d@Dy@F@@p@D@G@@@Fx@H@@@C@I@@@@H@J@@t@I27@K@@@I@L@@d@Kx@M@@@K2@N@@{@H@O@@@Nx@P*@U;^^-define a function79@Q<@R:@W1@S</@T<@U0@NormalArial @V@B@U8gEC@W@@ p@X@@ @W@Y@@d@Xx@Z@@@X12.5@[*@U6-set x79@\<@]:@W1@^</@_<@`0@NormalArial @a@B@U8 UB@b@@ p@c@@@b@d@@@@c@e@@d@dy@f@@p@d@g@@@fx@h@@@c@i@@+@@h@X@j@@@h@k*@U 5::- evaluate y7 9 @l< @m:@W1@n</ @o< @p0@NormalArial @q@B@U8+d>E8A@r@@ p@s@@ @r@t@@d@sx@u@@@s195@v*@U+;8488- set new x7 9 @w< @x:@W1@y</ @z< @{0@NormalArial @|@B@U8KfU`@@}@@ p@~@@@}@@@@@~@@@d@y@@@p@@@@@x@@@@~@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m@@B@U ;?N-Hj@@@ p@@@ @@@@d@x@@@@x@@B@U@ql@@@ p@@@?@@@@$@@@@@d@x@@@@@@@d@sin@@@p@@@@@@@@t@17@@@@x@@@@A@B@U(nA@@ pA@@?AA@@C@AA@@dAyA@@CAA@@dAxA@@AA@@dAxA @@AA @@dA yA @@A 2A @@AA @B@U@c~zx]A@@ pA@@ AA@@@AA@@dA\yA@@pAA@@ AA@@ @AA@@dArA@@A\qA@@A\fA@@AA@@@AA@@dAeA@@AA@@K@AA@@A7A@@ArA@@AA @@dAcosA!@@pAA"@@A!\qA#@B@UHm7rA$@@ pA%@@?A$A&@@%@A%A'@@@A&A(@@tA'0A)@@A'A*@@tA)2A+@@A)\pA,@@A&A-@@dA,\fA.@@%A,A/@@@A.A0@@tA/0A1@@A/\pA2@@A.A3@@dA2\qA4@@%A2A5@@@A4A6@@tA50A7@@A5\A8@@A4A9@@dA8rA:@@A8A;@@@A:A<@@@A;A=@@@A@@dA=\yA?@@pA=A@@@ A?AA@@ @A@AB@@dAArAC@@AA\qAD@@A@\fAE@@A<2AF@@A;AG@@dAFrAH@@AF2AI@@A:AJ@@dAIsinAK@@pAIAL@@AK\qAM@@A%AN*@U;s<<-=Recall that  is the volume element in spherical coordinates7 =AO< AP ChrPropData8AN128,0,0AQ<AR8AN?AOAS<AT8AN128,0,0AQAU<AV8AN?ASAW<.AX8AN128,0,0AUAWAO9=AY<=AZ:@W1A[<A\ EmbedData2 4EmbedObj3A] EmbedObjPtr4A^@B@UYA_@@ pA`@@A_Aa@@@A`Ab@@dAarAc@@Aa2Ad@@A`Ae@@dAdsinAf@@pAdAg@@Af\qAh<Ai2)3Aj4Ak@B@U;$ZAl@@ pAm@@Al drd\qd\fA[AhA[/=An<=Ao0@NormalArial  CompObjiObjInfoEquation Native F_1072679897@S|! BBTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  qe "-ax 2 dx "-"" +"Ole ObjInfoContents 5_1072680044@S|! BB.MCAD 309000000 \  docDocument MmcObject[aa d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV@"_mcDocumentObjectState\=@" mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman'eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1 <LinkMap/!<"LinkData0@NormalArial #@B@U8;^NEH $@@ p%@@ $&@@d%x'@@%14(*@U;KH---define x79)<*:@W1+</,<-0@NormalArial .@B@U8c~hx /@@ p0@@/1@@@02@@@13@@d2x4@@225@@16@@t557@@5x8@@09@@+@8@X:@@8;*@Uk{x__-result of function79<<=:@W1></?<@@0@NormalArial @A@B@U8VD@B@@ p@C@@ @B@D@@@@C@E@@d@Dy@F@@p@D@G@@@Fx@H@@@C@I@@@@H@J@@t@I27@K@@@I@L@@d@Kx@M@@@K2@N@@{@H@O@@@Nx@P*@U;^^-define a function79@Q<@R:@W1@S</@T<@U0@NormalArial @V@B@U8gEC@W@@ p@X@@ @W@Y@@d@Xx@Z@@@X12.5@[*@U6-set x79@\<@]:@W1@^</@_<@`0@NormalArial @a@B@U8 UB@b@@ p@c@@@b@d@@@@c@e@@d@dy@f@@p@d@g@@@fx@h@@@c@i@@+@@h@X@j@@@h@k*@U 5::- evaluate y7 9 @l< @m:@W1@n</ @o< @p0@NormalArial @q@B@U8+d>E8A@r@@ p@s@@ @r@t@@d@sx@u@@@s195@v*@U+;8488- set new x7 9 @w< @x:@W1@y</ @z< @{0@NormalArial @|@B@U8KfU`@@}@@ p@~@@@}@@@@@~@@@d@y@@@p@@@@@x@@@@~@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m@@B@U ;?N-Hj@@@ p@@@ @@@@d@x@@@@x@@B@U@ql@@@ p@@@?@@@@$@@@@@d@x@@@@@@@d@sin@@@p@@@@@@@@t@17@@@@x@@@@A@B@U(nA@@ pA@@?AA@@C@AA@@dAyA@@CAA@@dAxA@@AA@@dAxA @@AA @@dA yA @@A 2A @@AA @B@U@c~zx]A@@ pA@@ AA@@@AA@@dA\yA@@pAA@@ AA@@ @AA@@dArA@@A\qA@@A\fA@@AA@@@AA@@dAeA@@AA@@K@AA@@A7A@@ArA@@AA @@dAcosA!@@pAA"@@A!\qA#@B@UHm7rA$@@ pA%@@?A$A&@@%@A%A'@@@A&A(@@tA'0A)@@A'A*@@tA)2A+@@A)\pA,@@A&A-@@dA,\fA.@@%A,A/@@@A.A0@@tA/0A1@@A/\pA2@@A.A3@@dA2\qA4@@%A2A5@@@A4A6@@tA50A7@@A5\A8@@A4A9@@dA8rA:@@A8A;@@@A:A<@@@A;A=@@@A@@dA=\yA?@@pA=A@@@ A?AA@@ @A@AB@@dAArAC@@AA\qAD@@A@\fAE@@A<2AF@@A;AG@@dAFrAH@@AF2AI@@A:AJ@@dAIsinAK@@pAIAL@@AK\qAM@@A%AN*@U;s<<-=Recall that  is the volume element in spherical coordinates7 =AO< AP ChrPropData8AN128,0,0AQ<AR8AN?AOAS<AT8AN128,0,0AQAU<AV8AN?ASAW<.AX8AN128,0,0AUAWAO9=AY<=AZ:@W1A[<A\ EmbedData2 4EmbedObj3A] EmbedObjPtr4A^@B@UYA_@@ pA`@@A_Aa@@@A`Ab@@dAarAc@@Aa2Ad@@A`Ae@@dAdsinAf@@pAdAg@@Af\qAh<Ai2)3Aj4Ak@B@U;$ZAl@@ pAm@@Al drd\qd\fA[AhA[/=An<=Ao0@NormalArial Ap@B@U@"_H^Aq@@ pAr@@?AqAs@@%@ArAt@@@AsAu@@K@AtAv@@Au\Aw@@At\Ax@@AsAy@@dAxxAz@@AxA{@@dAzeA|@@AzA}@@K@A|A~@@A}aA@@A|A@@dAxA@@A2A@@ArA@B@U@uaA@@ pA@@?AA@@@AA@@%@AA@@@AA@@K@AA@@A\A@@A\A@@AA@@dAxA@@AA@@dAeA@@AA@@K@AA@@AaA@@AA@@dAxA@@A2A@@ AA@@dAassumeA@@(AA@@dAaA@@A0A@@A Ole ObjInfoContents 5_1072680439@S|! BB.MCAD 309000000 \  docDocument MmcObject[aa d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV@umcDocumentObjectState\`@u mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman'eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1 <LinkMap/!<"LinkData0@NormalArial #@B@U8;^NEH $@@ p%@@ $&@@d%x'@@%14(*@U;KH---define x79)<*:@W1+</,<-0@NormalArial .@B@U8c~hx /@@ p0@@/1@@@02@@@13@@d2x4@@225@@16@@t557@@5x8@@09@@+@8@X:@@8;*@Uk{x__-result of function79<<=:@W1></?<@@0@NormalArial @A@B@U8VD@B@@ p@C@@ @B@D@@@@C@E@@d@Dy@F@@p@D@G@@@Fx@H@@@C@I@@@@H@J@@t@I27@K@@@I@L@@d@Kx@M@@@K2@N@@{@H@O@@@Nx@P*@U;^^-define a function79@Q<@R:@W1@S</@T<@U0@NormalArial @V@B@U8gEC@W@@ p@X@@ @W@Y@@d@Xx@Z@@@X12.5@[*@U6-set x79@\<@]:@W1@^</@_<@`0@NormalArial @a@B@U8 UB@b@@ p@c@@@b@d@@@@c@e@@d@dy@f@@p@d@g@@@fx@h@@@c@i@@+@@h@X@j@@@h@k*@U 5::- evaluate y7 9 @l< @m:@W1@n</ @o< @p0@NormalArial @q@B@U8+d>E8A@r@@ p@s@@ @r@t@@d@sx@u@@@s195@v*@U+;8488- set new x7 9 @w< @x:@W1@y</ @z< @{0@NormalArial @|@B@U8KfU`@@}@@ p@~@@@}@@@@@~@@@d@y@@@p@@@@@x@@@@~@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m@@B@U ;?N-Hj@@@ p@@@ @@@@d@x@@@@x@@B@U@ql@@@ p@@@?@@@@$@@@@@d@x@@@@@@@d@sin@@@p@@@@@@@@t@17@@@@x@@@@A@B@U(nA@@ pA@@?AA@@C@AA@@dAyA@@CAA@@dAxA@@AA@@dAxA @@AA @@dA yA @@A 2A @@AA @B@U@c~zx]A@@ pA@@ AA@@@AA@@dA\yA@@pAA@@ AA@@ @AA@@dArA@@A\qA@@A\fA@@AA@@@AA@@dAeA@@AA@@K@AA@@A7A@@ArA@@AA @@dAcosA!@@pAA"@@A!\qA#@B@UHm7rA$@@ pA%@@?A$A&@@%@A%A'@@@A&A(@@tA'0A)@@A'A*@@tA)2A+@@A)\pA,@@A&A-@@dA,\fA.@@%A,A/@@@A.A0@@tA/0A1@@A/\pA2@@A.A3@@dA2\qA4@@%A2A5@@@A4A6@@tA50A7@@A5\A8@@A4A9@@dA8rA:@@A8A;@@@A:A<@@@A;A=@@@A@@dA=\yA?@@pA=A@@@ A?AA@@ @A@AB@@dAArAC@@AA\qAD@@A@\fAE@@A<2AF@@A;AG@@dAFrAH@@AF2AI@@A:AJ@@dAIsinAK@@pAIAL@@AK\qAM@@A%AN*@U;s<<-=Recall that  is the volume element in spherical coordinates7 =AO< AP ChrPropData8AN128,0,0AQ<AR8AN?AOAS<AT8AN128,0,0AQAU<AV8AN?ASAW<.AX8AN128,0,0AUAWAO9=AY<=AZ:@W1A[<A\ EmbedData2 4EmbedObj3A] EmbedObjPtr4A^@B@UYA_@@ pA`@@A_Aa@@@A`Ab@@dAarAc@@Aa2Ad@@A`Ae@@dAdsinAf@@pAdAg@@Af\qAh<Ai2)3Aj4Ak@B@U;$ZAl@@ pAm@@Al drd\qd\fA[AhA[/=An<=Ao0@NormalArial Ap@B@U@"_H^Aq@@ pAr@@?AqAs@@%@ArAt@@@AsAu@@K@AtAv@@Au\Aw@@At\Ax@@AsAy@@dAxxAz@@AxA{@@dAzeA|@@AzA}@@K@A|A~@@A}aA@@A|A@@dAxA@@A2A@@ArA@B@U@uaA@@ pA@@?AA@@@AA@@%@AA@@@AA@@K@AA@@A\A@@A\A@@AA@@dAxA@@AA@@dAeA@@AA@@K@AA@@AaA@@AA@@dAxA@@A2A@@ AA@@dAassumeA@@(AA@@dAaA@@A0A@@A Ole ObjInfoContents9_1115724263@S|! BB.MCAD 309000000 \  docDocument MmcObject[ee d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectVCGmcDocumentObjectState\pC mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman)eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1 <LinkMap/!<"LinkData0@NormalArial #@B@U8;^NEH $@@ p%@@ $&@@d%x'@@%14(*@U;KH---define x79)<*:@W1+</,<-0@NormalArial .@B@U8c~hx /@@ p0@@/1@@@02@@@13@@d2x4@@225@@16@@t557@@5x8@@09@@+@8@X:@@8;*@Uk{x__-result of function79<<=:@W1></?<@@0@NormalArial @A@B@U8VD@B@@ p@C@@ @B@D@@@@C@E@@d@Dy@F@@p@D@G@@@Fx@H@@@C@I@@@@H@J@@t@I27@K@@@I@L@@d@Kx@M@@@K2@N@@{@H@O@@@Nx@P*@U;^^-define a function79@Q<@R:@W1@S</@T<@U0@NormalArial @V@B@U8gEC@W@@ p@X@@ @W@Y@@d@Xx@Z@@@X12.5@[*@U6-set x79@\<@]:@W1@^</@_<@`0@NormalArial @a@B@U8 UB@b@@ p@c@@@b@d@@@@c@e@@d@dy@f@@p@d@g@@@fx@h@@@c@i@@+@@h@X@j@@@h@k*@U 5::- evaluate y7 9 @l< @m:@W1@n</ @o< @p0@NormalArial @q@B@U8+d>E8A@r@@ p@s@@ @r@t@@d@sx@u@@@s195@v*@U+;8488- set new x7 9 @w< @x:@W1@y</ @z< @{0@NormalArial @|@B@U8KfU`@@}@@ p@~@@@}@@@@@~@@@d@y@@@p@@@@@x@@@@~@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m@@B@U ;?N-Hj@@@ p@@@ @@@@d@x@@@@x@@B@U@ql@@@ p@@@?@@@@$@@@@@d@x@@@@@@@d@sin@@@p@@@@@@@@t@17@@@@x@@@@A@B@U(nA@@ pA@@?AA@@C@AA@@dAyA@@CAA@@dAxA@@AA@@dAxA @@AA @@dA yA @@A 2A @@AA @B@U@c~zx]A@@ pA@@ AA@@@AA@@dA\yA@@pAA@@ AA@@ @AA@@dArA@@A\qA@@A\fA@@AA@@@AA@@dAeA@@AA@@K@AA@@A7A@@ArA@@AA @@dAcosA!@@pAA"@@A!\qA#@B@UHm7rA$@@ pA%@@?A$A&@@%@A%A'@@@A&A(@@tA'0A)@@A'A*@@tA)2A+@@A)\pA,@@A&A-@@dA,\fA.@@%A,A/@@@A.A0@@tA/0A1@@A/\pA2@@A.A3@@dA2\qA4@@%A2A5@@@A4A6@@tA50A7@@A5\A8@@A4A9@@dA8rA:@@A8A;@@@A:A<@@@A;A=@@@A@@dA=\yA?@@pA=A@@@ A?AA@@ @A@AB@@dAArAC@@AA\qAD@@A@\fAE@@A<2AF@@A;AG@@dAFrAH@@AF2AI@@A:AJ@@dAIsinAK@@pAIAL@@AK\qAM@@A%AN*@U;s<<-=Recall that  is the volume element in spherical coordinates7 =AO< AP ChrPropData8AN128,0,0AQ<AR8AN?AOAS<AT8AN128,0,0AQAU<AV8AN?ASAW<.AX8AN128,0,0AUAWAO9=AY<=AZ:@W1A[<A\ EmbedData2 4EmbedObj3A] EmbedObjPtr4A^@B@UYA_@@ pA`@@A_Aa@@@A`Ab@@dAarAc@@Aa2Ad@@A`Ae@@dAdsinAf@@pAdAg@@Af\qAh<Ai2)3Aj4Ak@B@U;$ZAl@@ pAm@@Al drd\qd\fA[AhAh/=An<=Ao0@NormalArial Ap@B@U@"_H^Aq@@ pAr@@?AqAs@@%@ArAt@@@AsAu@@K@AtAv@@Au\Aw@@At\Ax@@AsAy@@dAxxAz@@AxA{@@dAzeA|@@AzA}@@K@A|A~@@A}aA@@A|A@@dAxA@@A2A@@ArA@B@U@uaA@@ pA@@?AA@@@AA@@%@AA@@@AA@@K@AA@@A\A@@A\A@@AA@@dAxA@@AA@@dAeA@@AA@@K@AA@@AaA@@AA@@dAxA@@A2A@@ AA@@dAassumeA@@(AA@@dAaA@@A0A@@AA@B@UHucA@@ pA@@@AA@@ AA@@@AA@@dARA@@pAA@@ AA@@dAZA@@ArA@@AA@@p@AA@@AA@@tA2A@@AA@@@AA@@dAZA@@ArA@@Aa.oA@@AA@@dAeA@@AA@@@AA@@K@AA@@AZA@@ArA@@Aa.oA@B@UHG0eA@@ pA@@@AA@@?AA@@@AA@@%@AA@@@AA@@tA0A@@AA@@tA2A@@A\pA@@AA@@dA\fA@@%AA@@@AA@@tA0A@@A\pA@@AA@@dA\qA@@%AA@@@AA@@tA0A@@A\A@@AA@@dArA@@AA@@@AA@@@AA@@@AA@@dARA@@pAA@@ AA@@dAZA@@ArA@@A2A@@AA@@dArA@@A2A@@AA@@dAsinA@@pAA@@A\qA@@ AA@@ @AA@@dAassumeA@@(AA@@dAa.oA@@A0A@@(AA@@dAZA@@A0A@@A Ole ObjInfoContents:_1115724280@S|! BB      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~.MCAD 309000000 \  docDocument MmcObject[ff d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectVHmcDocumentObjectState\>)G mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman*eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1 <LinkMap/!<"LinkData0@NormalArial #@B@U8;^NEH $@@ p%@@ $&@@d%x'@@%14(*@U;KH---define x79)<*:@W1+</,<-0@NormalArial .@B@U8c~hx /@@ p0@@/1@@@02@@@13@@d2x4@@225@@16@@t557@@5x8@@09@@+@8@X:@@8;*@Uk{x__-result of function79<<=:@W1></?<@@0@NormalArial @A@B@U8VD@B@@ p@C@@ @B@D@@@@C@E@@d@Dy@F@@p@D@G@@@Fx@H@@@C@I@@@@H@J@@t@I27@K@@@I@L@@d@Kx@M@@@K2@N@@{@H@O@@@Nx@P*@U;^^-define a function79@Q<@R:@W1@S</@T<@U0@NormalArial @V@B@U8gEC@W@@ p@X@@ @W@Y@@d@Xx@Z@@@X12.5@[*@U6-set x79@\<@]:@W1@^</@_<@`0@NormalArial @a@B@U8 UB@b@@ p@c@@@b@d@@@@c@e@@d@dy@f@@p@d@g@@@fx@h@@@c@i@@+@@h@X@j@@@h@k*@U 5::- evaluate y7 9 @l< @m:@W1@n</ @o< @p0@NormalArial @q@B@U8+d>E8A@r@@ p@s@@ @r@t@@d@sx@u@@@s195@v*@U+;8488- set new x7 9 @w< @x:@W1@y</ @z< @{0@NormalArial @|@B@U8KfU`@@}@@ p@~@@@}@@@@@~@@@d@y@@@p@@@@@x@@@@~@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m@@B@U ;?N-Hj@@@ p@@@ @@@@d@x@@@@x@@B@U@ll@@@ p@@@?@@@@$@@@@@d@x@@@@@@@d@sin@@@p@@@@@@@@t@17@@@@x@@@@A@B@U(nA@@ pA@@?AA@@C@AA@@dAyA@@CAA@@dAxA@@AA@@dAxA @@AA @@dA yA @@A 2A @@AA @B@U@c~zx]A@@ pA@@ AA@@@AA@@dA\yA@@pAA@@ AA@@ @AA@@dArA@@A\qA@@A\fA@@AA@@@AA@@dAeA@@AA@@K@AA@@A7A@@ArA@@AA @@dAcosA!@@pAA"@@A!\qA#@B@UHm7rA$@@ pA%@@?A$A&@@%@A%A'@@@A&A(@@tA'0A)@@A'A*@@tA)2A+@@A)\pA,@@A&A-@@dA,\fA.@@%A,A/@@@A.A0@@tA/0A1@@A/\pA2@@A.A3@@dA2\qA4@@%A2A5@@@A4A6@@tA50A7@@A5\A8@@A4A9@@dA8rA:@@A8A;@@@A:A<@@@A;A=@@@A@@dA=\yA?@@pA=A@@@ A?AA@@ @A@AB@@dAArAC@@AA\qAD@@A@\fAE@@A<2AF@@A;AG@@dAFrAH@@AF2AI@@A:AJ@@dAIsinAK@@pAIAL@@AK\qAM@@A%AN*@U;s<<-=Recall that  is the volume element in spherical coordinates7 =AO< AP ChrPropData8AN128,0,0AQ<AR8AN?AOAS<AT8AN128,0,0AQAU<AV8AN?ASAW<.AX8AN128,0,0AUAWAO9=AY<=AZ:@W1A[<A\ EmbedData2 4EmbedObj3A] EmbedObjPtr4A^@B@U YA_@@ pA`@@A_Aa@@@A`Ab@@dAarAc@@Aa2Ad@@A`Ae@@dAdsinAf@@pAdAg@@Af\qAh<Ai2)3Aj4Ak@B@UZAl@@ pAm@@Al drd\qd\fA[AhAh/=An<=Ao0@NormalArial Ap@B@U@"_H^Aq@@ pAr@@?AqAs@@%@ArAt@@@AsAu@@K@AtAv@@Au\Aw@@At\Ax@@AsAy@@dAxxAz@@AxA{@@dAzeA|@@AzA}@@K@A|A~@@A}aA@@A|A@@dAxA@@A2A@@ArA@B@U@uaA@@ pA@@?AA@@@AA@@%@AA@@@AA@@K@AA@@A\A@@A\A@@AA@@dAxA@@AA@@dAeA@@AA@@K@AA@@AaA@@AA@@dAxA@@A2A@@ AA@@dAassumeA@@(AA@@dAaA@@A0A@@AA@B@UHucA@@ pA@@@AA@@ AA@@@AA@@dARA@@pAA@@ AA@@dAZA@@ArA@@AA@@p@AA@@AA@@tA2A@@AA@@@AA@@dAZA@@ArA@@Aa.oA@@AA@@dAeA@@AA@@@AA@@K@AA@@AZA@@ArA@@Aa.oA@B@UHG0eA@@ pA@@@AA@@?AA@@@AA@@%@AA@@@AA@@tA0A@@AA@@tA2A@@A\pA@@AA@@dA\fA@@%AA@@@AA@@tA0A@@A\pA@@AA@@dA\qA@@%AA@@@AA@@tA0A@@A\A@@AA@@dArA@@AA@@@AA@@@AA@@@AA@@dARA@@pAA@@ AA@@dAZA@@ArA@@A2A@@AA@@dArA@@A2A@@AA@@dAsinA@@pAA@@A\qA@@ AA@@ @AA@@dAassumeA@@(AA@@dAa.oA@@A0A@@(AA@@dAZA@@A0A@@AA@B@UHpfA@@ pA@@?AA@@$@AA@@dAxA@@AA@@dAxA@@AyA@@A Ole ObjInfoContents_1115724408@S|! B O.MCAD 309000000 \  docDocument MmcObject[{{  dim_formatTmasslengthtimecharge temperature luminosity substance)o`^3)/Z^3*`%-a.o`^3*sin(`%--m%q`)-2/Z^3*`NumericalFormatQdii shpRectV[mcDocumentObjectState\Z mcPageModelK????mcHeaderFooterI@I CHeaderFooterJ@J@J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultW?Normalfont_style_listO font_styleP  VariablesTimes New Roman@P  ConstantsTimes New Roman@P TextArial@P Greek VariablesSymbol@P User^1Arial@P User^2 Courier New@P User^3System@P User^4Script@P User^5Roman@P User^6Modern@P User^7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman^ TextRegion* docRegionGshpBoxU-(99 CharacterMap-RangeMap;Symbolic Calculus Using Mathcad ChrPropMap7 RangeElem< ChrPropData8 RangeData=~ ParPropMap9 <  ParPropData:@W,EmbedMap1 <LinkMap/ < LinkData0@NormalArial *@U S0 `pp-AIn this document we will illustrate some of the symbolic calculations that Mathcad can perform. Before we do so, you need to be aware that there are two methods of performing symbolic calculations: we can use the symbolic menu commands or we can use symbolic operators. In general, the symbolic operators are preferred because the calculations are "live", that is, they will automatically update if you change a parameter. Let's see the difference. Compute the derivative of sin(5x).79<:@W,1</<0@NormalArial *@U -Symbolics Menu Method7<8v 9<:@W,1</<0@NormalArial *@U  ?   -@\Choose Symbolics, Evaluation style and check the boxes that say horizontal and show comments7\9\<\:@W,1</\ <\!0@NormalArial "*@U #5S 0(00-@Place the cursor or selection box on or near the x and choose symbolics, variable, differentiate. What do you observe?. Try some other mathematical expression and obtain the derivative. Will you be able to do this any time you need to?79#<$:@W,1%</&<'0@NormalArial (eqRegionB@U0k]~?x)tree@ p*@@)+@@d*sin,@@p*-@@,.@@t-5/@@-x0*@U(8((  -@hTo integrate, place the cursor or selection box on or near the x, choose symbolics, variable, integrate.7h9h1@@t=5?@@=x@@*@UxxExx-by integration, yields79@A<@B:@W,1@C</@D<@E0@NormalArial @F@B@U5F@G@@ p@H@@@G@I@@d@Hsin@J@@p@H@K@@@J@L@@t@K5@M@@@Kx@N*@U( <;((00-@Now in the space below, repeat these calculations but change the 5 to 33 and note any change when you recalculate the worksheet. Try integrating other mathematical functions for practice.79@O<@P:@W,1@Q</@R<@S0@NormalArial @T*@U  0-Symbolic Operator Method7@U<@V8@Tv 9@W<@X:@W,1@Y</@Z<@[0@NormalArial @\*@U $    -@From the calculus palette, choose the derivative operator and fill in the placeholders. Type CTRL period (the period key) to insert the symbolic equals sign (an arrow).79@]<@^:@W,1@_</@`<@a0@NormalArial @b*@U.3 RhN N -@JRepeat the instructions to see how the equation to the left was developed.7J9J@c@A A"@@tA!2A#@@A!A$@@dA#xA%@@A# _n_u_l_l_A&@@>A A'@@tA&2A(@@A&A)@@dA(yA*@@pA(A+@@A*A,@@dA+ _n_u_l_l_A-@@>A+A.@@tA-2A/@@A-A0@@dA/zA1@@A/ _n_u_l_l_A2*@U`;K`H^^-define a function79A3<A4:@W,1A5</A6<A70@NormalArial A8@B@U;N,HA9@@ pA:@@ A9A;@@@A:A<@@dA;fA=@@pA;A>@@ A=A?@@ @A>A@@@dA?xAA@@A?yAB@@A>zAC@@A:AD@@@ACAE@@@ADAF@@dAEsinAG@@pAEAH@@AGxAI@@ADAJ@@dAIsinAK@@pAIAL@@AKAM@@tAL2AN@@ALyAO@@ACAP@@dAOsinAQ@@pAOAR@@AQAS@@tAR3AT@@ARzAU@B@U@~ygTAV@@ pAW@@?AVAX@@p@AWAY@@AXAZ@@>@AYA[@@tAZ2A\@@AZA]@@dA\xA^@@A\A_@@dA^fA`@@pA^Aa@@ A`Ab@@ @AaAc@@dAbxAd@@AbyAe@@AazAf@@>AYAg@@tAf2Ah@@AfAi@@dAhyAj@@pAhAk@@AjAl@@@AkAm@@dAlfAn@@pAlAo@@ AnAp@@ @AoAq@@dApxAr@@ApyAs@@AozAt@@>AkAu@@tAt2Av@@AtAw@@dAvzAx@@AvAy@@dAxfAz@@pAxA{@@ AzA|@@ @A{A}@@dA|xA~@@A|yA@@A{zA@@AWA*@U  0-2. Symbolic integrals7A<A8Av 9A<A:@W,1A</A<A0@NormalArial A*@U ,8-@OIntegral calculus is also relatively easy in both indefinite and definite forms7O9OA84A@@ pA@@ AA@@@AA@@dA\yA@@pAA@@ AA@@ @AA@@dArA@@A\qA@@A\fA@@AA@@@AA@@dAeA@@AA@@K@AA@@A7A@@ArA@@AA@@dAcosA@@pAA@@A\qA*@U@Ss@`L  -@You must know what normalization means. Define 'normalization factor'. How is the normalization factor used to produce a normalized function?79A<A:@W,1A</A<A0@NormalArial A@B@U@=+BA@@ pA@@?AA@@%@AA@@@AA@@tA0A@@AA@@tA2A@@A\pA@@AA@@dA\fA@@%AA@@@AA@@tA0A@@A\pA@@AA@@dA\qA@@%AA@@@AA@@tA0A@@A\A@@AA@@dArB@@AB@@@BB@@@BB@@@BB@@dB\yB@@pBB@@ BB@@ @BB@@dBrB @@B\qB @@B\fB @@B2B @@BB @@dB rB@@B 2B@@BB@@dBsinB@@pBB@@B\qB@@AB*@U3W<<-?Recall that   is the volume element in spherical coordinates7 ?B< B8B128,0,0B<B8B?BB<B8B128,0,0BB<B8B?BB<0B8B128,0,0BBB9?B*@U00C-So p/1029 = 1/N2, thus7B?<B@8B>BA<BB8B>SymbolB?BC< BD8B>BABE<BF8B>_BCBG<BH8B>BEBGBA9BI<BJ:@W,1BK</BL<BM0@NormalArial BN*@U0 U# 0 E %%-Check79BO<BP:@W,1BQ</BR<BS0@NormalArial BT@B@U O 8 GBU@@ pBV@@?BUBW@@@BVBX@@@BWBY@@dBXNBZ@@BX2B[@@%BWB\@@@B[B]@@tB\0B^@@B\B_@@tB^2B`@@B^\pBa@@B[Bb@@dBa\fBc@@%BaBd@@@BcBe@@tBd0Bf@@Bd\pBg@@BcBh@@dBg\qBi@@%BgBj@@@BiBk@@tBj0Bl@@Bj\Bm@@BiBn@@dBmrBo@@BmBp@@@BoBq@@@BpBr@@@BqBs@@dBr\yBt@@pBrBu@@ BtBv@@ @BuBw@@dBvrBx@@Bv\qBy@@Bu\fBz@@Bq2B{@@BpB|@@dB{rB}@@B{2B~@@BoB@@dB~sinB@@pB~B@@B\qB@@BVB*@U0c B 0p N  -@Repeat the normalization using the same function but this time it contains sin(q) instead of cos(q). What is the correct normalized function? Prove it as we did here for the cos function.7bOB@@C8C?@@dC>\fC@@@C>1CA@@%C7CB@@@CACC@@tCB0CD@@CB\pCE@@CACF@@dCE\qCG@@CECH@@dCGsinCI@@pCGCJ@@CI\qCK@@pC6CL@@%CKCM@@@CLCN@@tCM0CO@@CM\CP@@CLCQ@@dCPrCR@@CPCS@@@CRCT@@@CSCU@@dCTRCV@@pCTCW@@ CVCX@@dCWZCY@@CWrCZ@@CS2C[@@CRC\@@dC[rC]@@C[2C^@@C5C_ BitmapRegion PictureRegionF@U H  H [C` mc_dibitmap 5x (5P Ca*@U   - 3.Solving equations symbolically7 Cb< Cc8Cav 9 Cd< Ce:@W,1Cf</ Cg< Ch0@NormalArial Ci*@U0  0   -@bWhat is the maximum in our radial function shown above? How do you find the maximum in a function?7b9bCjD K R Zagmr!v(y/z7y>xDuJpOkSdV]WVWOUHRAN;H6 "--&&l3 "--F$!B<877:>E{MuWqaplpwsw~}rg\RJB--&&Mi~P "--%Ok{M--&&&n "--%k--&&ihn "--%kek--&&xn "--%k "-- $  z--&&rn "- -%kR- - $UpK--&--- !gz---=- 2 ==r@Times New Roman- 2 N=1- 2 _= '--- !g---H4-  2 3Hr- 2 3H2- 2  3H '-C*@U{0-@H________________________________________________________________________7H9HC@@@D=D?@@@D>D@@@dD?cDA@@D?5DB@@D>DC@@tDB96DD@@DB\pDE@@D=DF@@dDErDG@@DE2DH@@D?@ABCDEFGHIJKLMPQRSTWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~.MCAD 309000000 \  docDocument MmcObject[ d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectVpmcDocumentObjectState\XZ mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman3eqRegionB docRegionGshpBoxU8 &h tree@ p@@@@@@@@@@@@@dx@@2@@@@t5@@x@@@@+@Serial_DisplayNodeX@@ TextRegion*@U*#  CharacterMap-RangeMap;no result x is not defined. ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=@WEmbedMap1 <LinkMap/!<"LinkData0@NormalArial #@B@U8;^NEH $@@ p%@@ $&@@d%x'@@%14(*@U;KH---define x79)<*:@W1+</,<-0@NormalArial .@B@U8c~hx /@@ p0@@/1@@@02@@@13@@d2x4@@225@@16@@t557@@5x8@@09@@+@8@X:@@8;*@Uk{x__-result of function79<<=:@W1></?<@@0@NormalArial @A@B@U8VD@B@@ p@C@@ @B@D@@@@C@E@@d@Dy@F@@p@D@G@@@Fx@H@@@C@I@@@@H@J@@t@I27@K@@@I@L@@d@Kx@M@@@K2@N@@{@H@O@@@Nx@P*@U;^^-define a function79@Q<@R:@W1@S</@T<@U0@NormalArial @V@B@U8gEC@W@@ p@X@@ @W@Y@@d@Xx@Z@@@X12.5@[*@U6-set x79@\<@]:@W1@^</@_<@`0@NormalArial @a@B@U8 UB@b@@ p@c@@@b@d@@@@c@e@@d@dy@f@@p@d@g@@@fx@h@@@c@i@@+@@h@X@j@@@h@k*@U 5::- evaluate y7 9 @l< @m:@W1@n</ @o< @p0@NormalArial @q@B@U8+d>E8A@r@@ p@s@@ @r@t@@d@sx@u@@@s195@v*@U+;8488- set new x7 9 @w< @x:@W1@y</ @z< @{0@NormalArial @|@B@U8KfU`@@}@@ p@~@@@}@@@@@~@@@d@y@@@p@@@@@x@@@@~@@@+@@@X@@@@@*@USc`3::- evaluate y7 9 @< @:@W1@</ @< @0@NormalArial @*@U8{8?-a faster and easier approach79@<@:@W1@</@<@0@NormalArial @@B@U8e>@@@ p@@@@@@@@@@@@d@y@@@p@@@@@12.5@@@@@@@+@@@X@@@@@@B@U8A:P@@@ p@@@@x@@B@U`kcO@@@ p@@@@y@@B@UN@@@ p@@@@ velocity@@B@U;M@@@ p@@@@ temperature@@B@UhL@@@ p@@@@ initial_conc@@B@U@t.P(d@@@ p@@@ @@@@d@V@@@@@@@t@15@@@@L@@B@U 9(b@@@ p@@@ @@@@d@R@@@@@@@t@0.08205@@@@@@@@@@@@d@L@@@@atm@@@@@@@d@mol@@@@K@@B@UP.`(a@@@ p@@@ @@@@d@T@@@@@@@t@225@@@@K@@B@U.(c@@@ p@@@ @@@@d@n@@@@@@@t@2.5@@@@mol@@B@U@bOxU@@@ p@@@ @@@@d@P@@@@@@@@@@@@@@@@@d@n@@@@R@@@@T@@@@V@@B@Uk~xV@@@ p@@@@@@@d@P@@@@@@@+@@@X@@@@atm@*@U00f-Defining a unit in Mathcad79@<@:@W1@</@<@0@NormalArial @@B@U8Pg@@@ p@@@ @@@@d@pm@@@@@@@@@@@@t@10@@@K@@@@@12@@@@m@@B@U ;?N-Hj@@@ p@@@ @@@@d@x@@@@x@@B@U@ql@@@ p@@@?@@@@$@@@@@d@x@@@@@@@d@sin@@@p@@@@@@@@t@17@@@@x@@@@A@B@U(nA@@ pA@@?AA@@C@AA@@dAyA@@CAA@@dAxA@@AA@@dAxA @@AA @@dA yA @@A 2A @@AA @B@U@c~zx]A@@ pA@@ AA@@@AA@@dA\yA@@pAA@@ AA@@ @AA@@dArA@@A\qA@@A\fA@@AA@@@AA@@dAeA@@AA@@K@AA@@A7A@@ArA@@AA @@dAcosA!@@pAA"@@A!\qA#@B@UHm7rA$@@ pA%@@?A$A&@@%@A%A'@@@A&A(@@tA'0A)@@A'A*@@tA)2A+@@A)\pA,@@A&A-@@dA,\fA.@@%A,A/@@@A.A0@@tA/0A1@@A/\pA2@@A.A3@@dA2\qA4@@%A2A5@@@A4A6@@tA50A7@@A5\A8@@A4A9@@dA8rA:@@A8A;@@@A:A<@@@A;A=@@@A@@dA=\yA?@@pA=A@@@ A?AA@@ @A@AB@@dAArAC@@AA\qAD@@A@\fAE@@A<2AF@@A;AG@@dAFrAH@@AF2AI@@A:AJ@@dAIsinAK@@pAIAL@@AK\qAM@@A%AN*@UsQ<-=Recall that  is the volume element in spherical coordinates7 =AO< AP ChrPropData8AN128,0,0AQ<AR8AN?AOAS<AT8AN128,0,0AQAU<AV8AN?ASAW<.AX8AN128,0,0AUAWAO9=AY<=AZ:@W1A[<A\ EmbedData2 :EmbedObj3A] EmbedObjPtr4A^@B@UYA_@@ pA`@@A_Aa@@@A`Ab@@dAarAc@@Aa2Ad@@A`Ae@@dAdsinAf@@pAdAg@@Af\qAh<Ai2)3Aj4Ak@B@UZAl@@ pAm@@Al drd\qd\fA[AhAh/=An<=Ao0@NormalArial Ap@B@U@"_H^Aq@@ pAr@@?AqAs@@%@ArAt@@@AsAu@@K@AtAv@@Au\Aw@@At\Ax@@AsAy@@dAxxAz@@AxA{@@dAzeA|@@AzA}@@K@A|A~@@A}aA@@A|A@@dAxA@@A2A@@ArA@B@U@uaA@@ pA@@?AA@@@AA@@%@AA@@@AA@@K@AA@@A\A@@A\A@@AA@@dAxA@@AA@@dAeA@@AA@@K@AA@@AaA@@AA@@dAxA@@A2A@@ AA@@dAassumeA@@(AA@@dAaA@@A0A@@AA@B@UHucA@@ pA@@@AA@@ AA@@@AA@@dARA@@pAA@@ AA@@dAZA@@ArA@@AA@@p@AA@@AA@@tA2A@@AA@@@AA@@dAZA@@ArA@@Aa.oA@@AA@@dAeA@@AA@@@AA@@K@AA@@AZA@@ArA@@Aa.oA@B@UHG0eA@@ pA@@@AA@@?AA@@@AA@@%@AA@@@AA@@tA0A@@AA@@tA2A@@A\pA@@AA@@dA\fA@@%AA@@@AA@@tA0A@@A\pA@@AA@@dA\qA@@%AA@@@AA@@tA0A@@A\A@@AA@@dArA@@AA@@@AA@@@AA@@@AA@@dARA@@pAA@@ AA@@dAZA@@ArA@@A2A@@AA@@dArA@@A2A@@AA@@dAsinA@@pAA@@A\qA@@ AA@@ @AA@@dAassumeA@@(AA@@dAa.oA@@A0A@@(AA@@dAZA@@A0A@@AA@B@UHpfA@@ pA@@?AA@@$@AA@@dAxA@@AA@@dAxA@@AyA@@AA*@Uhhz-find the maximum in a function79A<A:@W1A</A<A0@NormalArial A@B@Up $tA@@ pA@@@AA@@ AA@@@AA@@dARB@@pAB@@ BB@@dBZB@@BrB@@AB@@p@BB@@BB@@tB2B@@BB @@@BB @@dB ZB @@B rB @@Ba.oB @@BB@@dB eB@@B B@@@BB@@K@BB@@BZB@@BrB@@Ba.oB*@U0 0{^^-define a function79B<B:@W1B</B<B0@NormalArial B@B@Up+>8sB@@ pB@@BGivenB*@U+c;8j{{-Begin the solve block79B<B :@W1B!</B"<B#0@NormalArial B$*@UKkX: -.this is a symbolic equals sign typed by CTRL =7.9.B%<.B&:@W1B'</.B(<.B)0@NormalArial B*@B@UpIr`B+@@ pB,@@,B+B-@@$@B,B.@@dB-rB/@@B-B0@@dB/RB1@@pB/B2@@ B1B3@@dB2ZB4@@B2rB5@@B,0B6@B@UpB7@@ pB8@@?B7B9@@@B8B:@@dB9FindB;@@pB9B<@@B;rB=@@B8B>*@UQZii-End of solve block79B?<B@:@W1BA</BB<BC0@NormalArial  Ole ObjInfoContents _1087798566@S|! .MCAD 309000000 \  docDocument MmcObject[  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV{V"mcDocumentObjectState\K{V mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New RomaneqRegionB docRegionGshpBoxU[nhtree@ p@@Given@B@U{@@ p@@,@@dR@@@@dr.2@@r.1 TextRegion*@U    CharacterMap-RangeMap;1definition of the center of mass of two particles ChrPropMap71 ParPropMap91 RangeElem<1 ParPropData: RangeData=@WEmbedMap1<LinkMap/1<1LinkData0@NormalArial @B@U@@ p @@,!@@@ "@@d!m.1#@@!r.1$@@ %@@d$m.2&@@$r.2'@B@U4"(@@ p)@@?(*@@@)+@@d*Find,@@p*-@@ ,.@@d-r.1/@@-r.20@@) Ole ObjInfoContentsj_1156832046F.MCAD 309000000 \  docDocument MmcObject[ d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectV@mcDocumentObjectState\(@ mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @J@{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs18 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs20 \par } @{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq15 Arial;}{\f3\fswiss Arial;}{\f4\fswiss\fprq15 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f4\fs18 \par } @J@JMbP?MbP? 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