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# AnalyticGeometryandCalculusB MATH242

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MAPLEquot TUTORIAL FOR MATH 242 Rakes hf August 307 2006 Contents 1 COMMANDS FOR MATH 242 MATERIAL 2 11 Differentiation7 Integration7 Editing 2 12 Destroying or Starting a Worksheet 3 13 Manipulating Windows 3 14 Getting Help in Maple 3 15 Quitting Maple 4 16 Substituting values into expressions7 Numerical value of an expression 4 17 Time Saving Tricks 4 18 Interrupting Maple 6 19 Some Common Errors 6 110 Plotting Planar Curves I 7 111 Exact Solution Of Systems Of Equations 9 112 Approximations Of Solutions Of Equations 10 1121 Using the avoid option with fsolue 12 113 Commands Longer Than A Single Line 12 114 Procedures 13 Version 1000 IfPlease send suggestions and corrections to rakesh mathudeledu 115 Finite sequences and series Taylor Polynomials 14 116 Manipulating Expressions Extracting Parts of Expressions 14 117 Plotting Planar Curves II 16 118 Summary 17 119 Problems 18 2 PREPARING A MAPLE REPORT 19 1 COMMANDS FOR MATH 242 MATERIAL The commands given below have been tested only for the Worksheet Mode of Maple 11 Differentiation Integration Editing Start Maple To differentiate sinm2 type diff sinxquot2 x hit Return The e a T 1 keys may be used to navigate in the worksheet and the Backspace Delete keys may be used to correct typing errors Don t forget the semicolon every command ends with a semicolon To compute de nite and inde nite integrals use int sinx x 0 Pi and hit Return int axquot2 x and hit Return Note that Pi upper case P is the symbol used for 7139 in Maple The rst command computes the de nite integral and the second command computes the inde nite integral Notice in the second command azz could have been integrated with respect to z or a One convenient editing trick is the following Suppose we wish to differentiate msinm3 We have already differentiated sinz2 so using the e T a 1 keys on the right hand side of the keyboard move the cursor to that line Then using the Backspace Delete and arrow keys modify that line to read diff xsinxquot3 x and hit Return We now continue with other Maple commands move the cursor to the last line using the H T a 1 To compute higher order derivatives note the example below f sqrt1x 3expx ln1xquot2 difff X3 Here we rst de ne an expression f and then we compute its third derivative note the x3 Also note the use of the square root function the exponential function and the natural log DO NOT use equotx for the exponential function 12 Destroying or Starting a Worksheet To start a new worksheet because you want to work on a new problem left click the File menu and in the new menu which pops up left click the New line and choose Worksheet Mode You will get a new worksheet called Untitled 2 To destroy the old worksheet called Untitled 1 click the 8 button associated with Untitled 1 in the top left corner Reply No to the question about saving the old worksheet 13 Manipulating Windows Open a Maple worksheet if you don t have one To plot the graph of the function 2 over the range 71 g x g 1 type plot xquot2 x11 This will result in the plot of y 2 over 71 g x g 1 The plot may be manipulated click on the plot and a box appears around the plot move the mouse arrow to the bottom right corner of the plot box and the arrow will turn into another shape holding down the left mouse button drag that corner till you have resized the plot to your liking From the Plot menu at the top choosing the Scaling Constrained option will result in a plot in which the scale used in the z and the y direction are the same the default is unconstrained scaling where the scales chosen are those which ll the box with the plot The menu appearing on the main Maple window depends on whether the plot box is active or not Click on the plot command line and notice that the headings in the main Maple window change Click on the plot again and see the change 14 Getting Help in Maple Maple has an excellent help facility Suppose you wish to nd out the Maple command to simplify an expression Move to the the last Maple command line and type simplify and a separate window pops up giving information about the simplify command The most useful part of this help window are the examples at the bottom scroll down to see them Also useful is the line at the end which says See Also this mentions other commands which may be appropriate You can get information on any of the commands listed there by clicking on that word for example7 click on collect and analyze the examples at the bottom of the help window seems like a useful command When you are done with the Help windows you can either destroy it or iconify it for later use Destroy the Help window now Help is also available from the menu at the top of the main Maple window you can explore this further later 15 Quitting Maple To quit Maple7 from the File menu choose Exit answer No to prompts for saving this work later we will see how to save your work This will get you out of Maple 16 Substituting values into expressions Numerical value of an expression Start Maple if you do not have a Maple window To substitute a value for a variable in an expression use the command eval To obtain a numerical value of an expression use evalf We show their use by an example We compute the third order m derivative of 2 sinkz3 at z 171 37 accurate to 7 digits Try fxxx diff xquot2sinkxquot3 x3 computes 3rd order derivative myval eval fxxx x 1 k 3 evaluate fxxx at x1 k3 evalf myval 7 obtain numerical value of myval accurate to 7 digits eualf is the Maple command to obtain a numerical value for an expression and the answer is calculated accurate to 7 digits To obtain greater accuracy change 7 to a higher number say 9 try it Note the use of curly parenthesis and 7 in eval7 to specify all the variables that are being replaced 17 Time Saving Tricks Start Maple if you don t have a Maple window 0 Use space between characters judiciously7 not too much and not too little7 to make error detection easier The following three lines do exactly the same thing substitute values for variables in an expression but the rst is preferred over the other two In the rst line7 note how the command7 the expression and the variables are judiciously spaced so that the different units stand out eval sinxy x2 y3 Good evalsinxy x2 y3 correct but terrible eval sinx y x 2y 3 correct but room for improvement o If the output of a certain calculation is to be used later it is best to give it a name and choose a meaningful name For example myder diff xquot2 x int myder X o Occasionally Maple responds to a command with quotsyntax errorquot eg try diff xquot3 3xquot2 x x Maple responds with the error message Also note that it points to the probable source of the error Can you determine the error The error is that 3xquot2 should have been entered as 3xquot2 Now instead of retyping the line it is quicker to correct the old line by using the arrow keys then hit Return Remember modify only the red colored lines and never retype a line it is usually faster to correct the incorrect line 0 To use part of an old command one can use the copy and paste technique Type diff sqrt1x xexpx ln1x x Hit Return Now suppose we want to integrate the same function xl z mem ln1 z from 2 to 5 Instead of retyping this expression we copy it Highlight the expression to be copied in the red colored part then from Edit choose Copy not Cut Next using the arrow keys move the cursor to a new line and try int from Edit menu choose Paste x 2 5 You should obtain int sqrt1x xexpx ln1x x 25 Hit Return 0 ln Maple several commands may be executed together This is particularly convenient if the desired output is the result of several commands and one is not interested in the intermediate results For example we will compute the value of 3 i m3 sinkz t cost k dt dm 1 when x 2 k 4 accurate to 9 digits a diff sinkxxquot3 x while pressing Shift hit Return b int tcostk t1 3 while pressing Shift hit Return 6 eval ab x2 k4 while pressing Shift hit Return evalfc 9 Hit Return All the commands are executed together and the last result is the desired answer make sure you understand why the last result is what we wanted The problem was broken into several parts then combined to get the nal result 0 In the previous problem we were not really interested in the intermediate values a b c In Maple one can suppress the output of a command by replacing the semicolon by a colon In the preVious commands replace the semicolon by a colon for the rst three commands only these are the intermediate commands whose output does not interest us Now hit return and note the output Compare this with the earlier situation by putting back the semicolons Now is a good time to do the exercises below They are quite straightforward and will help you assimilate the commands and ideas we have discussed so far Exercise 11 Compute the fourth deriuate of cosm 1 lnm 7 1 wem21 with respect to m Also compute the integral of the original function with respect to m ouer the interual 24 do not retype the expression to be integrated Exercise 12 What is the ualue of e 2 u2 sinuu when u 73 39u 2 Obtain this ualue accurate to 12 digits What is the ualue of sin3 accurate to 9 digitsf2 All trigonometric angles in Maple are in radians and not degrees by default Exercise 13 Using Maple determine the sine of 20 degrees accurate to 10 digits Hint conuert degrees to radians and then apply the sine function Answer 03420201433 Exercise 14 Determine the ualue of the following expression whenp 3 q 1 accurate to 10 digits p 2 s d 3 s q U 81112914du dig pq Slnpqpq16 q Answer 78373549981 18 Interrupting Maple lf Maple gets stuck in a calculation to stop the calculation left click the STOP button eg type sum sinnquot2 n110quot7 Hit Return This attempts to nd the sum of sinn2 as n ranges from 1 to 107 This will take too long so to stop the calculation left click the red hand stop button at the top of the window 19 Some Common Errors 0 Do not type my when you really want z y or type cosz when you want cosz o A common mistake is the inappropriate use of the solve command The solve command is useful only for solving equations explained later it is uselessincorrect to use it for simplifying expressions use simplify or obtaining numerical values for expressions use evalf o A common error is to use a symbol as a variable even though the symbol was assigned a value earlier For example suppose at some stage you used z 4 and you forgot about it Some commands later you differentiate sinz expecting cosz but instead you get an error message because x has the value 4 One clears the value of z is shown below Try x 4 diff sinx x Error x x use quote key right of semicolon key this clears value of x diff sinx x now it works as you want it to 0 Since the commands in a Maple worksheet are often modi ed and then rerun it is a good idea to start every worksheet with the restart command Execute the following commands diff sinx X diff cosy y x 4 Now reexecute each of the commands by just hitting Enter You will notice a problem with the rst command That is so because Maple remembered that z 4 from the third command so in the rerun the rst command does not make sense So try the following move the cursor to the rst line then from Insert choose Execution Group choose Before Cursor and you should have a new execution group as your rst line In that line type restart and then reexecute the next set of commands and there should be no problem 110 Plotting Planar Curves I Planar curves may be given explicitly as in y z2 a parabola implicitly as in 2 y2 1 a circle or in parametric form as in z t sin t y 17 cos t 76 g t g 6 a cycloid Curves may also be given in polar form as in the three leaved rose r sin30 To draw the graph of the explicitly given curve y 2 9 sinz over the interval 73 3 use plot xquot2 9sinx x3 3 To draw the graph of two or more explicitly given curves y z 7 2 and y 1 7 2x over the interval 01 in the same window with y z 7 2 colored red and y 1 7 2x colored blue use plot xxquot2 12x x01 colorredblue To determine the point of intersection of the two curves in 0 lt z lt 1 left click the point of intersection to obtain its approximate location as 038023 this is displayed in the box in the left corner of the main window To draw curves given implicitly ie they are not given in the form y m but in the form gzy c use the implicitplot command However this command is in the plots package so to draw the ellipse 2 4342 4 use withplots Load the plots package only once in a worksheet implicitplot xquot2 4yquot2 4 x3 3 y3 3 Note that for implicitplot both the z and y ranges have to be given implicitplot plots the part of the curve which lies in the rectangle determined by the z and the y ranges Compare this with plotting using the plot command for example to plot y 2 where only the z range is required Examine what happens if you increase or decrease the ranges for z and y try several changes For implicitplot it is advisable to start with a large range and then decrease the range until the graph is satisfactory To draw more than one implicitly given curve for example the ellipse 2 4342 4 and the circle 2 y2 2 with ellipse blue and circle colored green use implicitplot xquot2 4yquot2 4 xquot2 yquot2 2 Shift Return x33 y33 colorblue green Again the approximate locations of the points of intersection may be read with the help of the mouse Some remarks are in order here 0 Curves which can be plotted using plot may also be plotted using implicitplot but not vice versa Wherever possible use plot instead of implicitplot because plot is faster and gives more accurate graphs 0 Notice the use of the curly brackets when plotting two or more curves all the equa tions functions being plotted must be enclosed in these square brackets These square brack ets are not necessary but may be used when plotting a single curve To superimpose two plots obtained from different commands use the display command in the plots package For example to display the graphs of y H y 2m and the circle 2 y2 1 in the same picture use withplots picl plot 2x xquot2 x22 colorred blue picQ implicitplot xquot2 yquot2 1 x22 y22 colorgreen display picl picZ The second and third lines de ne two graphs named pic and 10262 The last line displays the two pictures superimposed 111 Exact Solution Of Systems Of Equations To obtain solutions of one equation in one variable or to nd the solutions of a system of equations the solve command is useful To nd all solutions of the quadratic equation 2 7 3x 2 0 use solve xquot2 3 2 O x In the command solve the rst parameter is the equation to be solved and the second parameter is the variable to be solved for To solve the system of equations z y 1 2 7 y 1 for the variables m y use solve x y 1 Xquot2 y 1 LY 5 and we obtain the two sets of solutions Now consider the following example nd all solutions of 3x 7 2y 5 2 7 y3 2 As before we use mysol solve 3x 2y 5 xquot2 yquot3 2 xy where mysol is the name given to the list of solutions generated The solutions are x 1 y 71 and z Z y 3Z27 52 where Z is one of the roots of the quadratic equation 9Z2713Z77 0 Now the quadratic equation has two solutions so there are two choices for Z To obtain both these solutions coming from the Z we rst isolate the part containing Z Try mysol 1 first part of mysol mysol 2 second part of mysol So mysol 2 contains the terms dependent on Z To get all the solutions that is expanding the Rootof part apply the allvalues command to mysol 2 othersolns allvalues mysol 2 So we have the other two solutions of the system of equations giving a total of three solutions To obtain the numerical values of the other two solutions accurate to 5 digits we can apply evalf evalf othersolns 5 Often the solutions of a system of equations are to be substituted in another expression This can be done very ef ciently as shown below Suppose we wish to nd the value of lnm2 yz at the solution of above system which lies in the rst quadrant that is z gt 0 and y gt 0 Of the three solutions obtained above it is the z 29080 y 18621 solution which lies in the rst quadrant So one could use eval lnxquot2yquot2 x29080 y18621 However7 this is not quite correct and also ine icient Firstly7 z 290807 y 18621 is only an approximation to the solution7 the correct solution being y 1318 1 VIM187 z so we should be substituting this into lnm2 yz and not its numerical approximation The second more important issue is that one should not have to retype results which Maple has generated instead the result should be given a name and the name should be used In our case7 the solution of interest is othersolns 17 try othersolns 1 Note that the solution is already in the form x 7 y exactly the form needed for using the command eval7 so a more e icient way of getting the value of lnm2 yz at this solution would be myval eval lnxquot2yquot2 othersolns 1 evalf myval 10 numerical value Please try the exercise below before moving to the next section The ideas in this section are very useful Exercise 15 Find the numerical ualue of sinm2 m where m is the positiue root of the equation m3 1 3m2 2 Answer 09544912329 Exercise 16 Eualuate m3 em at the solution of y m 17 m2 y2 4 which lies in the rst quadrant Ans 5038878215 112 Approximations Of Solutions Of Equations To nd all solutions of the equation 7 cosz z 2 15 try solve 7cosx x xquot2 15 x Maple tries to nd the exact solutions and fails So we attempt to obtain the approximations to these solutions using the command fsolve First we must nd out how many solutions this equation has For large lzl 7cosm z 2 is very large and hence will not equal 15 Hence the equation will not have a solution x for which lml is large so the solutions of 7cosm z 2 15 are in a nite interval The solutions of 7cosm z 2 15 are solutions of 7cosm z 2 7 15 07 that is the values of z where the graph of 7 cosz z 2 715 cuts the z axis WHYY So we plot the graph of 7cosm z 2 715 over a reasonably large interval plot 7cosx x xquot2 15 x101O From the graph using the mouse we observe that the graph cuts the z axis at two points hence the equation has two solutions one between 75 and 74 and the other between 3 and 4 To nd these solutions we use the fsolve command So we can obtain the approximate value of the two solutions using fsolve 7cosx x xquot2 15 x x54 fsolve 7cosx x xquot2 15 x x34 The fsolve command needs the equations and the variables just as solve but fsolve also needs an interval which encloses a solution When using fsolve it is important to choose an interval which contains only the solution one wants the narrower the interval the better This is usually done by rst plotting the appropriate graphs so always draw the appropriate graphs before using f solve Finally to complete the solution of the problem we must show somehow using pencil and paper that 7cosm z 2 15 has no solutions outside 710 10 That takes some effort and we will skip that Exercise 17 Find all the solutions of m2 2m msinm 4 Check your answer by substituting these ualues ofz into the equation We now study how to nd solutions of systems of equations for which solve fails to work Consider the system of equations 2 y2 4 and sinm y cosz 1 If we try solve xquot2 yquot2 4 sinxy cosx 1 xy then Maple is unable to nd the exact solutions We plot the two curves 2 y2 4 and sinm y cosz 1 and the solutions of this system of equations are the points which lie on both the curves withplots unnecessary if already done once this session implicitplot xquot2 yquot2 4 sinxy cosx 1 x33 y33 Using the mouse we note that the system has two solutions in the rectangle 73 3 x 73 3 one near 71 16 and the other near 1 17 The system will not have any solutions outside the rectangle 73 3 x 73 3 WHY To nd these solutions more accurately use fsolve xquot2 yquot2 4 sinxy cosx 1 xy x2O y02 fsolve xquot2 yquot2 4 sinxy cosx 1 xy x02 y02 Again fsolve needed the system of equations the variables to be solved for and the range over which the search for the solution is to be conducted The ranges for z and y were chosen carefully using the graph they were chosen to include only one solution at a time So again plot the appropriate curues before using the fsolve command to help determine the ranges to be used in fsolve 1121 Using the avoid option with fsolve The graphical technique of obtaining the rectangle containing the solution we want fails in three and higher dimensions ie solving k equations in k unknowns with k 2 3 because the graphs are hard to visualize in three dimensions and non existent visually in higher dimensions however fsolve will still work soll fsolve xquot2 yquot2 zquot2 4 xyz0 xsinyz1 Shift Return xyz x22 y22 z22 gives one solution of the system of equations ls it the only solution How do we nd others if there are any Maple s solve command has an option where it nds solutions which avoid certain speci ed solutions Try 5012 fsolve xquot2 yquot2 zquot2 4 xyz0 xsinyz1 Shift Return xyz avoid 011 which gives a solution different from 011 Are there more solutions We try 5013 fsolve xquot2 yquot2 zquot2 4 xyz0 xsinyz1 Shift Return xyz avoid 50115012 to nd solutions which avoid soll 012 and it seems that there are none Use fsolvedetails to nd about other options for fsolve Exercise 18 Find all solutions of m2 y2 9 m3 y3 7 sinxy 7 Why are you sure that there are no more Answer There are two solutions 113 Commands Longer Than A Single Line As we saw in the previous example some commands need more space than a single line To split up a command over several lines do as below solve xyz2 x2y3z7 x2z10 holding down Shift press Return xyz 114 Procedures So far we have used expressions instead of functions For example7 if we want to work with the function 3 sinz then we have chosen to use p xquot2 sinx However7 the p de ned above is an expression and not a function If we want the value of that expression at some point then we use the eval command A better way may be to de ne a Maple function or procedure as f procx Shift Return xquot2 sinx Shift Return end Every Maple procedure has the form name of procedure proc variables separated by commas value of the function end Given next is an example of a procedure dependent on two variables g procxa Shift Return xquot3 aquot3 xsina Shift Return end Return Examine the following use of the procedures 1 and g carefully and make sure that the output of each command is what you expect it to be f2 exact value of f at x2 f20 numerical value of f2 fp Df fp is the derivative of f it is also a procedure Df3 exact value of derivative of f at x3 Df30 numerical value of derivative of f at x3 plotf 22 graph of f over the interval 22 g21 exact value of g at x2 a1 g20 10 numerical value of g at x2 a1 D1 g D2 g Procedures are very useful in implementing algorithms We will see the use of procedures when implementing Newton s method and Simpson s rule 13 115 Finite sequences and series Taylor Polynomials To nd the sum of the series accurate to seven digits use a Sum sinn13nquot2 n569 evalfa7 The rst statement assigns a the sum of the series the second obtains a 7 digit numerical value for a The upper case S in the Sum command tells Maple to return only the symbolic sum Repeat the above commands after replacing Sum by sum see what happens A command useful for generating sequences of numbers expressions etc arising from one formula is seq Observe the result of executing the following statements seq iquot2 i37 seq iquot2 i341020 5 seq sinkx k14 For example to plot the graphs of sin m sin2z sin5m over the interval 7713 7139 one could use funs seq sinkx k15 plot funs xPi Pi Of course it would be nicer if we knew which graph corresponded to which function This may be done as follows click on the plot and then from the Legend menu choose Show Legend and the graphs are labeled The Maple command to nd the Taylor expansion of a function is mtaylor To nd the Taylor expansion of lnz around z 1 of order 6 use mtaylor lnx x1 7 use 7 to get expansion up to order 6 116 Manipulating Expressions Extracting Parts of Expressions Maple has many commands to manipulate algebraic expressions A few of them are collect expand simplify combine use the to nd out more about them To extract parts of an expression two useful commands are op and nops Consider the expression m jag3 2m 7 y3z2 y2 7 pm If we want the expansion of this in powers of z and y then we would use restart f xpyquot3 2x y3xquot2 yquot2 Px g simplifyf if simplify had not worked we would try expand hl collectg x h2 collectg xy h3 collectg xy distributed Note the three different ways of collecting the terms in g or f h1 organizes the terms of g by powers of z alone ignoring the powers of the other variables h2 organizes the terms of 9 rst by powers of z and then the coefficients of each power of z are organized by powers of y where as in h37 g is organized in powers of z and y as a polynomial or more correctly a multinomial Each of these representations is useful depending on what we use we have in mind Expressions may be organized not just in powers of x but in powers of any expression Try k xsinx 3x sinx 2sinx xquot2 collect expandk sinx Now we reexamine the expression h3 obtained earlier We will extract various pieces of hS Suppose we want to extract an expression which consists of the rst three terms in ha and another expression which consists of the rest of the terms in h3 We will be using two commands nops a which counts the number of termsoperands in the expression a7 and op ka which returns the kth termoperand of the expression a So hS just to see the expression nopsh3 counts the number of terms in h3 not needed op1h3 first term in h3 not needed op2h3 second term in h3 not needed a op1h3 op2h3 op3h3 the first three terms of h3 b h3 a the rest of the terms of h3 To extract all the coefficients of the various powers of 71 in h3 use coeffsh3 xy It is a little trickier to get just the coefficient of a single term such as zzy We use Maple s coeff command which works only on expressions expanded in terms of one variable hS just to see the expression not necessary t1 collecth3 xy expand h3 in powers of x then coeff in powers of y t2 coefft1 xquot2 extract terms which have an xquot2 t3 coefft2 y extract term with y hence the xquot2 y term is extracted Exercise 19 Consider the expression azz bx siny csin y2 a sin2 y bx3 15 i Write the above as a polynomial in m and extract the coe icient of 2 Ans 39 b23a 1sin2y 2acsiny ii Write the above as a multinomial in the variables m and sing and extract the coe icient of zsinzy Ans 39 2bc 117 Plotting Planar Curves II We now learn the commands for plotting planar curves given in parametric form or in polar coor dinates To draw curves given in parametric form one still uses the plot command For example to plot the cycloid z t sin t y 17 cost use plot tsint 1cost t66 where the range t 766 was chosen somewhat arbitrarily What happens if you change the range for t try it Also the location of the range of t 6 6 on the command line is very important If the part t 6 6 is moved out of the then the graphs obtained are different Try plot tsint 1cost t66 You will obtain the graph of two curves y t sint and y 1 7 cost as t varies over 76 6 see the section 110 where this was discussed To draw the two or more parametric curves the cycloid z t sin t y 1 7 cos t and the circle z 2 cos t y 2 sin t use plot tsint 1cost t66 2cost 2sint t07 Shift Return colorredgreen The circle does not really look circular but that may be corrected by choosing the Constrained option for the plot The ranges t 766 and t 07 were chosen somewhat arbitrarily What is the effect of decreasing or increasing these ranges Try it and observe what happens and why To plot curves given in polar coordinates use the command polarplot in the plots package To draw the circle r cos0 with 0 g 9 7r and the cardiod r 1 cos0 with 0 g 9 27r in the same picture use withplots polarplot cost t t0Pi 1cost t t02Pi Here we have used the symbol t instead of 0 choose the Constrained option for the plot To superimpose two plots obtained from different commands use the display command in the plots package For example to display the graphs of y z2 and the cardiod r 1 2 cos0 in the same picture use withplots picl plot x2 x14 colorred picQ polarplot 12cost t t02Pi colorgreen display piclpi62 5 The second and third lines de ne two pictures pic and pic and the last line displays the two pictures superimposed There are some interesting plotting commands in the plottools package Use plottools to nd out more about it Exercise 110 Generate a plot showing the three leaued rose r sin3t9 and the circle r 1 coloring the rose red and the circle blue Do the same showing only the right hand side of the rose and the full circle Rewrite the two curves in parametric form m y and plot them using the plot command instead of the polarplot command 118 Summary We have used the following commands in the tutorial so far diff int 7 differentiate integrate evalf 7 getting a numerical approximation of a number eval 7 evaluating an expression when variables are assigned values solve allvalues fsolve 7 solving systems of equations plot implicitplot polarplot display 7 plotting curves in 2D simplify collect expand op nops 7 manipulating and extracting parts of expressions sum 7 summing the terms of a nite series seq 7 generates the terms of a sequence mtaylor 7 Taylor polynomial of a function with 7 loading a package We have seen three kinds of parentheses used in Maple each plays a different role 0 is used in expressions as in m 7 y m 7 y or surrounds the argument of a standard function as in sinm y or surrounds the arguments of a Maple function as in solve usual stuff here o is used in lists of expressions equations or variables where the order of the listing is unimportant as in solve x y 1 2 7y 0 xy 5 eval xquot2 2quot3 x2 24 5 o l is used in ordered lists as in 119 1 to 03 F U a plot tsint 1cost t6 6 when plotting the parametric curve z t sin t y 17 cos t Note the order of the elements in this list is important the rst element representing the z variable the second the y variable etc Another situation where order is important and where is used is in plot x xquot2 x1 1 color red blue Here the graphs of y z and y 2 colored red and y z2 are plotted and the order is important because y z is is colored blue to help use recognize which graph is which is also used to refer to one of the solutions of a system of equations obtained by using the solue command see subsection 111 g l is also used in vectors and matrices Problems Graphically determine the number of solutions of the equation 2 7 zsin3m 7 mlnz 2 1 in the interval 03 and the approximate values of these solutions Then nd the solution closest to z 2 accurate to 6 digits Ans 39 z 13 22 28 x 2199377 Find the following sum accurate to 8 digits 1 1 1 i i i 234 m Ans 41873775 Find all solutions of the system of equations 902 242 24 2x 2342 2coszy 3 which lie in the region 71 g x g 1 72 g y g 2 Your answer should be accurate to at least four digits Ans z 702895 y 09076 and z 043877y 026026 Find the largest and the smallest value of the function 2 7 zsin3z 7 mlnz 2 over the interval 0 3 at which points are these values attained Your answer should be accurate to four digits Ans mam 293533 at m 3 min 7081567 at m 06639 Let f Acosz Bsinz sz Qm R Find the value of A BP QR so that 121 df 7 3 dm2 dm Ans1a732b12p1q73r4 2f5sinz2x21 Find all solutions of the system of equations m2 y2 22 9 m y z 2 m 2y 32 5 Then compute the value of empzy z 22 at the solution where z is the largest Ans two solutions largest 2 solution is x 125958y 7151914z 225958 ualue is 65192 18 2 PREPARING A MAPLE REPORT When submitting a report of your Maple work you may wish to include comments and graphs between the commands Below we give an example of preparation of a report which presents the solution of the problem Find all solutions of the equation msinm 5 in the interual 0 10 0 Open a new worksheet by choosing New and then Worksheet Mode from the File menu 0 We rst provide a heading for our report From the Insert menu choose Section Next from the Insert menu choose Paragraph Before Cursor to insert a paragraph before the cursor Next move cursor to the top red arrow and delete that execution group by choosing Delete Element from the Edit menu Then change the font size from 12 to 18 click the B to use boldface and from the Format menu choose Paragraph and then Center to center your text Now we may type our heading Type First Report For Math 242243302 hit Shift Return Your name hit Shift Return The date hit Shift Return A report may contain the solution of one or more problems So in most cases each problem must be solved in a separate section of the report To start a section from Insert choose Section We give this section a heading type Finding all solutions of x sinx5 in 010 0 From Insert choose Execution Group After Cursor and we get the prompt gt Then from Insert choose Paragraph Before cursor to enter a comment about what you will do next Type without hitting Return We start by finding the approximate location of the solutions by plotting the graph of x sinx5 and then move the cursor to right of gt to be ready to type the Maple commands Now type restart Hit Shift Return plot xsinx5 x010 Notice the above is typed in red the color for Maple commands Hit Return The places where the plot crosses the x axis gives the approximate locations of the solutions of z sin z 5 in the interval 0 10 Reading off these points we nd that one solution is between 6 and 8 and the other is between 8 and 10 We enter this as a comment as follows Move the cursor to the new line generated which contains a gt From Insert choose Paragraph Before and type 19 We observe the solutions are between 6 and 8 and 8 and 10 So the solutions are obtained by Now move cursor to position after gt Now type the command to get the solutions soll sol2 fsolve xsinx5 x68 fsolve xsinx5 x810 Hit Shift Return Hit Return and we obtain the two solutions We state our conclusion The cursor should again be positioned after gt in the new line From the Insert menu choose Paragraph Before and then type the comment The equation x sinx 5 has two solutions in 010 Accurate to 8 digits they are 70688914 and 88222213 and our report is complete M342 Fall 2008 Practice Problems and Notes If you do not work on the problems regularly and simply try to study for a few hours before each test most likely this strategy will fail miserably Before you start on the practice problems go over the illustrated examples rst and check if you can do them rst Week One Please review the materials in Section 36 on row and column spaces of a rectangular matrix In particular go over the three examples Example 3 to Example 5 in that section Please pay particular attention to how a basis for the column space of A was selected in Example 4 For problem 13 this is a simple geometric review The answer is no Think of 1 as the vector i x2 as j How do you pick a vector 33 that is perpendicular toj but not to i Week Two Theorems 522 and 523 are important Example 4 is a concrete illustration of these four fundamental subspaces Please verify that the basis vectors found for NA are orthogonal to those in RAi Same question for the other two subspaces Another simple review problem is to nd the four fundamental subspaces for the row matrix A 123 Let me go over a simple solution to 2 on p233 with you Read the problem rst and then nd the solution on the Solution Page Please go over all the illustrated examples in this section rst before you attempt the practice problems listed below Practice Problems for 52 1a d 2 to 5911121315 53 Least Squares Problems This section will turn out to be useful in any future course that covers curve tting image processing or regression analysis in the area of statistics Please go through the derivation of the projection matrix P de ned at the top of p238 Try each of the three illustrated examples in this section and see if you obtained the same set of answers given well have a different formulation of the projection matrix if the vector space is set equipped with an inner product structure See 55 Then try 1 a c 2 3b 9 11 for practice partial hints 9 As explained at the end of Theorem 532 top of p238 the matrix P AATA 1AT gives 7the7 orthogonal projection of every vector b onto the subspace RA If the vector already lies in RA b is its own projection For part b use the Fundamental Subspaces Theorem 11 Write P2 as P P and simplify the product b P3P2PPPP c Use the hint given in the text 54 Inner Product Spaces 1 want you to be familiar with the inner product space de ned in Go over the illustrated examples 1 and 3 Work out 17 27 7a7 b7 and 8 for practice 8 1 039 39 lllllll ll lt1x gt folzdx 12 1 Hzllzf01zwdx 13 Therefore ll ll13 Week Three 55 Orthonormal Sets The main theme here is that once you have an orthonormal set set up in a subspace U7 then the projection can be computed handily See Corollary 559 ie there is no need to use that awful looking projection matrix de ned in the Least Squares section Of course7 the problem is how to come up with an orthonormal set The orthogonal matrices de ned on p258 are important items in linear algebra and in computer aid design Prove that the product of two orthogonal matrices is also orthogonal problem 16 Practice problems for 55 27 37 4 7 87 97 147 187 197 287 33 14 This is a famous matrix called Householder matrix Compute HT rst which is H7 done in the same way as I reviewed problem 177 p234 in class Now recall what the de nition of orthogonal matrix rst HTH I 7 2uuT I 7 2uuT Simplify this product carefully to get the identity matrix I Use the fact that uTu 1 since u is given as a unit vector Before you get frustrated with the simpli cation recall the difference between uTu and uuT that I pointed out in class 15 Done in class 16 Again7 this is a game on taking transposes If A and B are orthogonal matrices7 then ATA I7 BTB I SO ABTAB BTATAB BTB I Done 28 a Compute lt 17 2x 7 1 gt 012z 7 1dx 0 after integration b Normalize each each 1 and 2x 7 1 56 Gram Schmidt Process This is a pretty straight forward section The 7accounting7 method of labeling the entries in R of A QR requires some practice An important application is Theorem 563 showing how QR can be applied to solving for a 7normal7 equation Please go over Example 3 on p278 and 4 on p280 Then try 37 4 57 77 8 and 12 for practice p 296 Test B part Please go over 7 by QR method as done in class Fill in the blanks for 10 Week Four 61 is a review on eigenvectors that you should have seen in Math 341 Please go over Examples 4 5 and 8 The two formulas at the top of p309 on trace and determinant are important Work out as many parts as possible in 1 Add 3 4 6 9 10 19 25 is ultra important Before your work on it see if you can show that a square matrix of the form uuT is a rank one matrix Here u is a column vector 63 This is one of the 7mothers7 of all linear algebra Again this was covered in Math 341 and we7ll go over it once more Try 1 24 6 78a 13 To understand the usefulness of eigenvalues please read Application 2 in relation to Google search given on pp333 335 64 Hermitian Matrices Theorems 641 and 644 are fundamental in linear algebra Please skip Schur7s Theorem 643 and the materials on Normal matrices As pointed out in 53 diagonalization of a square matrix will reveal a great deal of structure hidden inside the matrix In particular if the matrix is Hermitian or real symmetric its eigenvectors form an orthonormal set The image Ax of any vector x can then be computed easily via inner product See the formula displayed in the top half of p351 Complex orthogonal unitary matrices appear in the study of FFT fast fourier transform Please verify the matrix F4 listed at the bottom of p268 has four column vectors that are orthogonal to one another How do you convert the matrix F4 into an unitary matrix Please do these two Please go over Example 4 on p350 and then try 2 5a c g 6 13 14 23 for practice 65 SVD l have posted a matlab galinm le to run a simple example of using SVD to do baby image compression 1 never got a chance to demonstrate it in class For practice try 1 2 5 DE 91 92 Fourier Series All these problems are pretty routine In higher mathematical terminology we are still computing the eigenfunctions of the second derivative operator 91 11 15 17 19 Review again the trick of di erentiating the answer in 17 to get the answer for Example 1 92 2 6 7 12 13 95 1 2 3 Please distinguish the case between 7iced7 at both end points to 7insulated at both end points The Fourier series aspect is pretty straight forward 97 Separation of Variables We only touch on a sample of solving for one of the basic mathematical physics equation Laplace equation As mentioned in the class7 this method of separation of variables is quite cumbersome to carry The method of nite difference is much easier to implement Try 17 37 5 for practice 71 Laplace transforms The major thrust here is to show its usefulness in solving for a second order differential equation with a more 7practical7 forcing function ft Back in M3417 a simple non homogeneous second order can be solved easily by the method of undetermined coef cients This method fails if the forcing function displays some switch off and on mentality Problems in 71 and 72 are pretty routine Please go over the illustrated examples in each section You may skip those examples with highly complicated partial fractions like Example 6 on p471 for example 71 37 77 97 237 277 31 use partial fractions here7 41 see how I did 40 72 1359101117181927323337 73 1351671113152729 74 to 76 74 1 3 5 7 8 13 15 36 to 38 75 13511131521253135 76 1 3 5 7 9 11 Note Please use the Laplace transform Maple le that l posed to inspect the behavior of the solution curve You may be able to get a further understanding of the simple RLC circuit theory 81 Power Series Review Please go over the rst four examples to review what you were supposed to have seen in Math 242 Basically to get the solutions of a second order differential equation by series method you need rst be able to handle the algebra coming out of simple differentiation as shown in Example 4 on p514 Example 2 on p512 is pretty straight forward too Do you know how to get the theoretical solution 1z 7 32 by separation of variables way back in M341 not the method as seen in 95 7 Please do it as a quick review As seen back in Chapter 3 for a second order DE there are two linearly independent solutions in the neighborhood of a regular point The best way to nd the rst one is to set the initial conditions yz0 0 y x0 1 while the second one is obtained from yz0 1 y x0 0 Since the Wronskian does not vanish the set form a fundamental ie linearly independent set Theorem 3 in 31 82 11 5 11 13 23 27 28 33 look at the Legendre functions on pp534 536 for this famous problem 83 Please go over Example 5 on p539 and 8 on p 543 85 Please solve for the Bessel function of the rst kind with p 1 The answer is in 4 on p562 with p 1 Also nd the rst four non zero terms of the Bessel function with p 32 Read Examples 1 and 2 Work on 5 and 23

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