Chapter 2: Introduction to Maple V

Transcription

1 Chapter 2: Introdution to Maple V 2-1 Working with Maple Worksheets Try It! (p. 15) Start a Maple session with an empty worksheet. The name of the worksheet should be Untitled (1). Use one of the standard methods for your platform to maximize the worksheet (that is, expand the Maple worksheet so that it ompletely fills the window). The result should be very similar to Figure 2-1. See Figure 2-1 on page 15 of the text. Try It! (p. 16) Ativate balloon help in your urrent session. Then, use it to display the desriptions of eah of the ions in the toolbar. Follow the diretions found in the text on page 16 immediately preeding this Try It! (p. 16). Try It! (p. 17) Use the toggle swithes at the top of the View menu to see how the interfae hanges when various menu bars are hidden from view. When finished, be sure that all bars are visible. See the disussion that immediately preedes this Try It! (p. 17) This exerise is intended to familiarize you with Maple s interfae and the terminology used to desribe the different omponents of the interfae. I reommend having all three ion bars visible at all times. Try It! (p. 20) Use the Copy and Paste method just desribed to omplete the verifiation that the seond solution satisfies the two equations. restart; To get to a state from whih it is possible to omplete the Try It!, exeute the following ommands (from pages ): f := x^3 + x; f := x 3 + x g := a*x^2; g := ax 2 df := diff( f, x ); df := 3 x dg := diff( g, x ); dg := 2 ax SOL := solve( { f = g, df = dg }, { x, a } ); SOL := { a= 2, x= 1 }, { x= -1, a = -2} subs( SOL[1], [ a, x, f = g, df = dg ] ); [ 21,, 2= 2, 4= 4] To verify that the seond solution in SOL also satisfies the two equations, opy the previous line to an input region, hange SOL[1] to SOL[2], and exeute the ommand. The new ommand and its result should look like: subs( SOL[2], [ a, x, f = g, df = dg ] ); [-2,-1,-2 = -2, 4= 4] Try It! (p. 21) Open the worksheet first.mws loated on the Toolkit homepage. Compare this Page 1

2 worksheet with the one you have just reated, myfirst.mws. See the worksheet first.mws that an be obtained from Addison-Wesley or from one of the authors (Meade). The relevant URLs are ftp://ftp.aw.om/seng/authors/meade/first.mws and Note to Publisher: Please onfirm that the AW URL is/will be appropriate. What do you need to make this possible? Try It! (p. 21) To explore the ramifiations of the fat that all worksheets within the same Maple session aess the same Maple kernel, reate a new worksheet (the default name should be alled Untitled (2). ) Enter and exeute the ommand f; in the new worksheet. The result should be the expression representing the funtion f that was entered in the worksheet now titled myfirst.mws. Note that even though there is nothing in Untitled (2) to indiate how f reeived a value, the result learly shows that Untitled (2) is sharing information with myfirst.mws, and any other ommand exeuted within this Maple session. The output from the ommand f; in Untitled (2) should be x 3 + x If you have not followed the text, you will likely see that there is no value assigned to the name f. In this ase you should repeat enough of the disussion in the text to give a value to f in another worksheet. Try It! (p. 23) Use the subs ommand to determine the exat loation of the seond point of intersetion for the derivatives. Are the answers you obtained in reasonable agreement symbolially and graphially? Why is this point not a solution to the tangeny problem? restart; The following assignments from the text are needed f := x^3 + x; f := x 3 + x g := a*x^2; g := ax 2 G2 := subs( a=2, g ); G2 := 2 x 2 df := diff( f, x ); df := 3 x dg2 := diff( G2, x ); dg2 := 4 x The intersetion(s) of the derivatives an be found using solve( df=dg2, x ); 1, 3 1 The intersetion at x = other intersetion ours at x = 1 is the point of tangeny disussed on page 22. The 1 (this an also be estimated from the plot and 3 the tools provided on the Maple interfae). To omplete this exerise, examine the values of the funtions and their derivatives at x = 1. This produes: 3 Page 2

3 subs( x=1/3, [ f, G2, df, dg2 ] ); ,,, While it is obvious that f and G2 both have slope 4 3 at x = 1, the funtion values 3 1 are not the same. Sine the two funtions do not interset at the point x = 3, this annot be a point of tangeny. Try It! (p. 24) The insertion of an exeution group after the ursor and onversion of a region between text and input are used so often that shortuts are provided on the tool bar. Use balloon help to loate the ions on the tool bar that orrespond to a) inserting a new exeution group after the ursor, b) inserting and formatting inert text, and ) inserting Maple ommands in a text region. These ions are a) the 11th ion from the left in the tool bar ( [ ) b) the 10th ion from the left in the tool bar ( T ) ) the 9th ion from the left in the tool bar ( \Sigma ) Note to Publisher: it would be nie to be able to inlude these ions as inserts in the IG. What do you need to make this possible? 2.2 Using Online Help Try It! (p. 25) The online help page that desribes most features of the Maple worksheet an be aessed using help(worksheet);. Load this help doument. Clik the highlighted string Help System Guide. This opens another help doument. Loate and read the information about searhing the table of ontents. To get started, exeute the following ommand help(worksheet); Note: alternate methods of aessing online help Two alternate methods of aessing the same help doument are to lik on this hyperlink or to exeute the ommand?worksheet Note that the help ommand requires a semi-olon, the? ommand does not. The only other ommand in Maple that does not require a semi-olon or olon is the quit ommand. Following the instrutions in the Try It!, you should see the following setion (whih was inserted into this worksheet via opy and paste) : How do I searh the table of ontents? Choose Contents from the Help menu. The ensuing help page is the top-level table of ontents of the help system. Clik on a hyperlink to see more details. Try It! (p. 27) Plae the ursor on the word plot in the plot ommand toward the bottom of the myseond.mws worksheet. Use ontext-based help to aess the help for the plot ommand. Read this information, and the examples, to find out how to modify the plot ommand to ause f and f to be displayed in blue and the graphs of g and g in red. To get an idea of how the worksheet myseond.mws should appear, obtain a opy of Page 3

4 the worksheet seond.mws from Addison-Wesley or from one of the authors (Meade). The relevant URLs are and To omplete this exerise, the following definitions are needed: f := x^3 + x; f := x 3 + x g := a * x^2; g := ax 2 df := diff( f, x ); df := 3 x G1 := subs( a=2, g ); G1 := 2 x 2 dg1 := subs( a=2, dg ); dg1 := 4 x To reate the plot desribed in the Try It!, it is neessary to speify the olor= option in the plot ommand. The final ommand and plot should look something like: plot( [ f, G1, df, dg1 ], x=-2..2, olor = [ blue, red, blue, red ], title= Graphial verifiation when a=2 ); Graphial verifiation when a= x What If? (p. 37) Suppose the material is heated. How would this hange the stress-strain urve? Would the toughness inrease or derease? Why? As the temperature inreases, the Young s modulus (E) dereases. As a result, assuming the strain is held onstant, the stress ( σ = E ε) also dereases. In terms of the stress-strain urve, the temperature inrease will lower the urve. Sine the lowering of the stress-strain urve redues the area under the urve, the toughness will also be redued. 2.3 Advaned Worksheet Features Page 4

5 Try It! (p. 38) Open the worksheet[howto] help doument. This worksheet ontains a long olletion of setions. Loate and expand the setions related to exeution groups and to setions. Consult these referenes for detailed information about exeution groups and setions. The help doument assoiated with the keyword worksheet[howto] an be aessed via any of the standard methods (hyperlink, help ommand, or? ommand). For example,?worksheet[howto] The information about exeution groups and setions has been opied here for your onveniene. Exeution Groups The exeution group is the fundamental omputation and doumentation element for the worksheet. Eah exeution group is enlosed in a large square braket at the left. The text you are reading now is embedded in an exeution group. It an ontain Maple input ommands, the output from a Maple omputation, explanatory text and graphis. Notie that an exeution group may ontain zero, one, or more Maple input ommands. If you plae the ursor in an input ommand and press [Enter], Maple V exeutes all the input ommands in the urrent exeution group. How do I delete an exeution group? See the setion on Deleting. How do I enlose several exeution groups in a setion? Highlight the exeution groups that you would like to enlose in a setion. Next, hoose Indent from the Format menu. How do I insert an exeution group? See the setion on Inserting. How do I insert text above an exeution group? Clik the exeution group braket (the large square braket to the left of the prompt(s)). Then hoose Paragraph from the Insert menu and Before from the ensuing submenu. How do I insert text below an exeution group? Clik the braket enlosing the exeution group to selet it. Then hoose Paragraph from the Insert menu and After from the ensuing submenu. How do I join exeution groups? See the setion on Joining. How do I remove an exeution group from a setion? Position your ursor in the exeution group that you would like to remove from a setion. Next, hoose Outdent from the Format menu. How do I show (hide) exeution group ranges? By default, Maple V displays the exeution group ranges. To hide them, selet Show Group Ranges from the View menu. Notie that if you return to the View menu, the hek mark beside Show Group Ranges has disappeared. How do I split an exeution group? See Splitting. Also see the setions on Deleting., Inserting, Joining, and Splitting. Setions A setion is enlosed in a large square braket with a box at the top. How do I ollapse a setion? With your mouse, lik on the box ([-]) to the left of the setion heading. How do I ollapse all the setions in a worksheet? Choose Collapse All Setions from the View menu. Page 5

6 How do I delete a setion? See Deleting. How do I enlose one or more exeution groups in a setion? Highlight the exeution groups that you would like to enlose in a setion. Next, hoose Indent from the Format menu. How do I expand a setion? Clik on the box ([+]) to the left of the setion heading. How do I expand all the setions in a worksheet? Choose Expand All Setions from the View menu. How do I insert a setion? See Inserting.. How do I join setions? See Joining. How do I show (hide) setion ranges? Choose Show Setion Ranges from the View menu. How do I split a setion? See Splitting. Problems (pp ) Problem 1 Repeat the graphial and symboli verifiation that a = 2 is another solution to the tangeny problem. Add these results, with appropriate doumentation and explanation to the worksheet mythird.mws. The worksheet third.mws presents one solution to this problem. A opy of the worksheet third.mws an be obtained from Addison-Wesley or from one of the authors (Meade). The relevant URLs are ftp://ftp.aw.om/seng/authors/meade/third.mws and Problem 2 Convert all textual mathematial expressions in the worksheet reated in Problem 1 to inline math expressions. Also, in a new setion at the end of the worksheet, reate hyperlinks to the help douments for eah of the Maple ommands used in this worksheet and to the help douments most relevant to the interfae features used in this worksheet. Call the resulting worksheet myfourth.mws. The worksheet fourth.mws, available from Addison-Wesley or from one of the authors (Meade), provides one possible solution to this problem. The relevant URLs are ftp://ftp.aw.om/seng/authors/meade/fourth.mws and Problem 3 Create a new worksheet, alled referene.mws, that ontains links to examples and help and other introdutory material. Use hyperlinks to reate links to ommonly aessed online help douments, inluding worksheet, worksheet[howto], worksheet[glossary], student, and help. As you progress through this module, you should update this worksheet with new links, summaries of main tehniques, examples, and any other information you find useful. A barebones version of the worksheet referene.mws an be downloaded from Addison-Wesley or from one of the authors (Meade). The relevant URLs are ftp://ftp.aw.om/seng/authors/meade/referenes.mws and Problem 4 Estimate the modulus of toughness of the omposite material disussed in Appliation 2 when the area under the third segment of the stress-strain urve is Page 6

7 approximated by two trapezoides with a ommon side of ε=5.3%. What is the orresponding error in the modulus of toughness, relative to the smallest estimate of the toughness? (What benefit is obtained by using the smallest estimate in this omparison?) In preparation to answer this question we repeat the ommands (with some minor simplifiations) from the Appliation setion of the text (pp ) restart; with(student): pt1 := [0,0]: pt2 := [0.028,330]: pt3 := [ 0.032, 330 ]: pt4 := [ 0.053, 440 ]: pt5 := [ 0.074, 360 ]: E := ( pt2[2]-pt1[2] ) / ( pt2[1] - pt1[1] ); E := tough[elast] := evalf( 1/2 * ( pt2[1] - pt1[1] ) * ( pt2[2] - pt1[2] ), 2 ); tough[onst] := evalf( ( pt3[1] - pt2[1] ) * pt2[2], 2 ); tough[min] := evalf( ( pt5[1] - pt3[1] ) * pt5[2], 2 ) ; tough[max] := evalf( ( pt5[1] - pt3[1] ) * pt4[2], 2 ) ; tough elast := 4.6 tough onst := 1.3 tough min := 15. tough max := 18. Ttough[min] := evalf( tough[elast] + tough[onst] + tough[min], 2 ); Ttough[max] := evalf( tough[elast] + tough[onst] + tough[max], 2 ); Ttough min := 21. Ttough max := 24. Ttough[err] := ( Ttough[max] - Ttough[min] ) / Ttough[min]; Ttough err := QUAD := sigma = a*epsilon^2 + b*epsilon + ; EQ1 := evalf( subs( epsilon=pt3[1], sigma=pt3[2], QUAD ), 2 ); EQ2 := evalf( subs( epsilon=pt4[1], sigma=pt4[2], QUAD ), 2 ); EQ3 := evalf( subs( epsilon=pt5[1], sigma=pt5[2], QUAD ), 2 ); QUAD := σ = aε 2 + bε+ EQ1 := 330. =.0010 a b+ EQ2 := 440. =.0028 a b+ EQ3 := 360. =.0055 a b+ SOLN2 := solve( { EQ1, EQ2, EQ3 }, { a, b, } ): SOLN2 := evalf( SOLN2, 2 ); stress := subs( SOLN2, rhs(quad) ); SOLN2 := { = -210., b = , a = } stress := ε ε 210. tough[quad4r] := rightsum( stress, epsilon= ): tough[quad4r] := evalf( tough[quad4r], 2 ); Ttough[quad4R] := evalf( tough[elast] + tough[onst] + tough[quad4r], 2 ); tough quad4r := 18. Ttough quad4r := 24. tough[quad4t] := evalf( trapezoid( stress, epsilon= ), 2 ); tough quad4t := 16. tough[quad4s] := evalf( simpson( stress, epsilon= ), 2 ); tough quad4s := 17. Ttough[quad4S] := evalf( tough[elast] + tough[onst] + tough[quad4s], 2 ); Page 7

8 Ttough quad4s := 23. error[min] := abs( Ttough[quad4S] - Ttough[min] ) / Ttough[quad4S]; error[max] := abs( Ttough[quad4S] - Ttough[max] ) / Ttough[quad4S]; error[quad4r] := abs( Ttough[quad4S] - Ttough[quad4R] ) / Ttough[quad4S]; error min := error max := error quad4r := And, now, the solution of the urrent problem is found by obtaining the trapezoidal approximation to the toughness between ε =.032 and ε =.074. This is the same as the quad4t approximation exept that the optional seond argument of the trapezoid ommand must be used to indiate that only 2 trapezoids are to be used. tough[quad2t] := trapezoid( stress, epsilon= , 2 ): tough[quad2t] := evalf( tough[quad2t], 2 ); tough quad2t := 16. The total modulus of toughness is Ttough[quad2T] := evalf( tough[elast] + tough[onst] + tough[quad2t], 2 ); Ttough quad2t := 22. The relative error, ompared to the lowest estimate, is error[quad2t] := abs( Ttough[quad2T] - Ttough[min] ) / Ttough[min]; error quad2t := The omputation of the error relative to the lowest estimate is not likely to underestimate the atual error. (This is a result of the fat that the denominator is smaller.) Problem 5 This hapter used rightbox and rightsum to approximate the area under the quadrati urve with the default number of retangles. How many retangles are needed to approximate the area to four digits of auray. (Hint: This question an be answered by trial-and-error; onsult the on-line help for the neessary modifiation to the syntax of rightbox and rightsum.) This solution requires the parts of the Appliation that diretly relate to the omputation of the quadrati setion of the stress-strain urve. restart; with(student): pt3 := [ 0.032, 330 ]: pt4 := [ 0.053, 440 ]: pt5 := [ 0.074, 360 ]: QUAD := sigma = a*epsilon^2 + b*epsilon + ; EQ1 := evalf( subs( epsilon=pt3[1], sigma=pt3[2], QUAD ), 2 ); EQ2 := evalf( subs( epsilon=pt4[1], sigma=pt4[2], QUAD ), 2 ); EQ3 := evalf( subs( epsilon=pt5[1], sigma=pt5[2], QUAD ), 2 ); QUAD := σ = aε 2 + bε+ EQ1 := 330. =.0010 a b+ EQ2 := 440. =.0028 a b+ EQ3 := 360. =.0055 a b+ SOLN2 := solve( { EQ1, EQ2, EQ3 }, { a, b, } ): SOLN2 := evalf( SOLN2, 2 ); stress := subs( SOLN2, rhs(quad) ); SOLN2 := { b= , a = , = -210.} stress := ε ε 210. tough[quad4r] := evalf( rightsum( stress, epsilon= ) ); tough quad4r := Page 8

9 The plan is to double the number of retangles until the first four digits of the approximations stabalize. evalf( rightsum( stress, epsilon= , 8 ) ); evalf( rightsum( stress, epsilon= , 16 ) ); evalf( rightsum( stress, epsilon= , 32 ) ); evalf( rightsum( stress, epsilon= , 64 ) ); evalf( rightsum( stress, epsilon= , 128 ) ); evalf( rightsum( stress, epsilon= , 256 ) ); evalf( rightsum( stress, epsilon= , 512 ) ); At this point it an now be said that the four-digit approximation to the area under the stress-strain urve for <= ε <= is This estimate is obtained with 128 retangles. Problem 6 Repeat the previous problem with trapezoid and simpson. What do these results tell you about the approximation errors assoiated with the use of retangles, trapezoids, and quadratis to approximate the area under the parabola? This solution requires the parts of the Appliation that diretly relate to the omputation of the quadrati setion of the stress-strain urve. restart; with(student): pt3 := [ 0.032, 330 ]: pt4 := [ 0.053, 440 ]: pt5 := [ 0.074, 360 ]: QUAD := sigma = a*epsilon^2 + b*epsilon + ; EQ1 := evalf( subs( epsilon=pt3[1], sigma=pt3[2], QUAD ), 2 ); EQ2 := evalf( subs( epsilon=pt4[1], sigma=pt4[2], QUAD ), 2 ); EQ3 := evalf( subs( epsilon=pt5[1], sigma=pt5[2], QUAD ), 2 ); QUAD := σ = aε 2 + bε+ EQ1 := 330. =.0010 a b+ EQ2 := 440. =.0028 a b+ EQ3 := 360. =.0055 a b+ SOLN2 := solve( { EQ1, EQ2, EQ3 }, { a, b, } ): SOLN2 := evalf( SOLN2, 2 ); stress := subs( SOLN2, rhs(quad) ); SOLN2 := { = -210., b = , a = } stress := ε ε 210. The doubling algorithm used in Problem 5 will be used here with trapezoid and simpson replaing rightsum. To begin the searh for the four-digit approximation with trapezoid, reall the approximation obtained in the Appliation tough[quad4t] := evalf( trapezoid( stress, epsilon= ) ); tough quad4t := evalf( trapezoid( stress, epsilon= , 8 ) ); evalf( trapezoid( stress, epsilon= , 16 ) ); Page 9

10 evalf( trapezoid( stress, epsilon= , 32 ) ); evalf( trapezoid( stress, epsilon= , 64 ) ); evalf( trapezoid( stress, epsilon= , 128 ) ); evalf( rightsum( stress, epsilon= , 256 ) ); evalf( rightsum( stress, epsilon= , 512 ) ); As in Problem 5, the four-digit approximation to the area under the stress-strain urve for <= ε <= is But, now this estimate is obtained with only 64 retangles. For Simpson s Rule we begin by realling the estimate when four subintervals (two parabolas) are used to estimate the area tough[quad4s] := evalf( simpson( stress, epsilon= ) ); tough quad4s := As the number of subintervals is doubled: evalf( simpson( stress, epsilon= , 8 ) ); evalf( simpson( stress, epsilon= , 16 ) ); These results do not hange as the number of subintervals is inreased (exept possibly for the last digit - whih is not signifiant). The reason for this is that the stress is a quadrati funtion and Simpson s Rule is exat for quadrati funtions. The four-digit approximation agrees with our previous results, but is obtained with muh less work (in fat, only 2 subintervals are needed). evalf( simpson( stress, epsilon= , 2 ) ); While this is only one example, the fat that the different quadrature rules have different onvergene rates. The onvergene rate for Simpson s Rule, whih uses quadrati approximations to the funtion, is faster than the Trapezoid Rule, whih uses linear funtions. And, the "Right-Hand" Rule, whih uses onstants to approximate the funtion, requires the most subintervals to obtain the same auray. Problem 7 The integral of a nonnegative funtion is the limit of the area of retangles. In the previous problems, you estimated the area under the quadrati urve to two digits of auray. Compute the exat area under the urve as an integral. First, use the online help to find the Maple ommand for omputing definite integrals. How does this value ompare to the approximation in this hapter and in Problem 6? Corretion The previous problems estimated the area under the quadrati urve to four (not two) digits of auray. This does not hange the urrent problem in any way. This solution requires the parts of the Appliation that diretly relate to the omputation of the quadrati setion of the stress-strain urve. restart; with(student): pt3 := [ 0.032, 330 ]: pt4 := [ 0.053, 440 ]: pt5 := [ 0.074, 360 ]: Page 10

11 QUAD := sigma = a*epsilon^2 + b*epsilon + ; EQ1 := evalf( subs( epsilon=pt3[1], sigma=pt3[2], QUAD ), 2 ); EQ2 := evalf( subs( epsilon=pt4[1], sigma=pt4[2], QUAD ), 2 ); EQ3 := evalf( subs( epsilon=pt5[1], sigma=pt5[2], QUAD ), 2 ); QUAD := σ = aε 2 + bε+ EQ1 := 330. =.0010 a b+ EQ2 := 440. =.0028 a b+ EQ3 := 360. =.0055 a b+ SOLN2 := solve( { EQ1, EQ2, EQ3 }, { a, b, } ): SOLN2 := evalf( SOLN2, 2 ); stress := subs( SOLN2, rhs(quad) ); SOLN2 := { = -210., b = , a = } stress := ε ε 210. This problem an be solved using either int or Int. The advantage of using Int is that it gives the user a hane to hek that the orret integral has been entered before the alulation is performed. tough[exat] := Int( stress, epsilon= ); := tough exat ε ε 210. dε tough[exat] := evalf( tough[exat] ); tough exat := Note that this result is exatly the same as the result obtained by Simpson s Rule in Problem 6. This is further verifiation that Simpson s Rule is exat for this problem. Problem 8 Investigate the usage of stats[fit] as a means of fitting the data to a quadrati funtion. Verify results in a worksheet. Then ollet more data, and ompute the revised fit and orresponding ontribution to the modulus of toughness. Investigate the benefits of olleting more data and using fit to obtain a better approximation to the Young s modulus. There are many ways to approah this problem. For information on the fit ommand, see the setion Least Squares Fit to Data (p. 105) in Chapter 4. Problem 9 This hapter presented tehniques for the insertion of new setions, exeution groups, text regions, and other ommon elements of a Maple worksheet. Elements an also be removed from a worksheet. The only menu seletion that relates to deletion is Delete Paragraph (under Edit). Use the on-line help to learn how to delete a setion and exeution group. See the worksheet,howto help doument. If you look at the Exeution Group or Setion setion you will be referred to the Deleting setion. These onnetions are made via bookmarks (whih are not otherwise disussed in this module). For more information about bookmarks, see the Bookmarks setion in the worksheet,howto help doument. Problem 10 Create doumented worksheets, inluding hyperlinks to relevant help douments for the solution to Example1-1 and for the five-step problem-solving proess used to analyze Appliation 1 in Chapter 1. s are similar to previously disussed sample worksheets. No detailed solution provided. Page 11

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