ࡱ; xRoot Entry FAK :tCompObjbWordDocument*ObjectPool1n1n 4@  l  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijkmn pqrstuvyz{|}~oSummaryInformation( @ R@%@dEMicrosoft Word 6.031 FMicrosoft Word 6.0 Document MSWordDocWord.Document.6; _Oh+'0$ H l   D ht Parameters: * nMin C:\WINWORD\TEMPLATE\M-BOOK.DOT Module # ONEWayne State UniversityAuthorized Gateway Customer@ܥe= e"*-LKъъъъъۋ:ѓLR"????Rd#:D~DŸgTo5ъjot ??jjjъъ?jjjjъ?ъ?$\Aъъъъjjj Module # ZERO CALCULUS Symbolic & Numeric using MatLab By Professor Mohamad H. Hassoun Department of Electrical & Computer Engineering Wayne Satate University Detroit, MI 48202 October 22, 1996 Contents: I. Functions: Graphs, Extreme Points & Roots II. Differentiation III. Integration IV. Series clear % Just to make sure that no MatLab variables are present from a previous session. NOTE on executing commands (green colored text) in this document: place the cursor on the command/MatLab expression you wish to execute and then type Ctrl-Enter. Make sure that variables/array definitions commands are executed before executing any following commands which depend on such definitions. Definitions in bold blue font execute automatically when you open this document; i.e., you nedd not re-execute them over. You can get full explanation (help) of any MatLab command in this document by highlighting that command and clicking on the '?' in the bar menu above (or type Ctrl-F1). I. Functions: Graphs, Roots, Extreme Points, & More Graphs Consider the single variable function: f = 'sin(x)/x' % defining a symbolic function f = sin(x)/x x=1; eval(f) % evaluting f at x = 1 ans = 0.8415 A simple plot may now be generated by using the fplot command: fplot(f , [-20 20 -.4 1.2]) % The vector argument specifies the x & y axis ranges  One may want to add titles and axis labels as follows title('plot of f(x) = sin(x)/x') xlabel('x') ylabel('f(x)')  Alternatively, one may evaluate the function f(.) over a range of values of its argument and the plot the data using the plot command: z = linspace(-20, 20, 60); y = sin(z) ./ z; % here,component-wise pretty" form of the above answer using the pretty command: pretty(g) cos(x) sin(x) ------ - ------ x 2 x The second derivative of f(x) can now be evaluated as diff(g) ans = -sin(x)/x-2*cos(x)/x^2+2*sin(x)/x^3 pretty(ans) sin(x) cos(x) sin(x) - ------ - 2 ------ + 2 ------ x 2 3 x x The second derivative may also be evaluated directly from f(x) by diff(f,2) ans = -sin(x)/x-2*cos(x)/x^2+2*sin(x)/x^3 pretty(ans) sin(x) cos(x) sin(x) - ------ - 2 ------ + 2 ------ x 2 3 x x Here is how to differentiate a function h(b) = 3a -2b2 with respect to b: diff('3*a - 2*b^2', 'b') ans = -4*b In fact, the same answer will be generated even if we do not declare b to be the variable of differentiation in the diff expression; it is because diff defaults to the symbolic variable closest to 'x' as the variable of differentiation in the expression to be differentiated: diff('3*a - 2*b^2') ans = -4*b We conclude this section by generating a plot for the function h(x) = tanh(x) and its derivative over the range -4 < x < 4: h = 'tanh(x)'; hprime = diff(h); fplot(h,[-3,3,-1,1]) hold % this command holds current plot so that multiple % plots may be displayed. fplot(hprime,[-3,3,-1,1],'r') % r for red hold % this second execution of 'hold' releases the plot Current plot held Current plot released  Alternatively, we may display h and its derivative side by side using the subplot command subplot(1,2,1) % pick the left of two subplots fplot(h,[-3,3,-1,1]), title('tanh(x)') subplot(1,2,2) % pick the right of two subplots fplot(hprime,[-3,3,-1,1]), title('Derivative of tanh(x)')  III. Integration The command int(f) attempts to integrate the symbolic function f. It may be used for definate as well as indefinate integration. The following are some examples. h = '1/(x^2 + a^2)'; pretty(h) 1 ------- 2 2 x + a int(h) ans = 1/a*atan(x/a) pretty(ans) atan(x/a) --------- a We may also integrate w.r.t. 'a', as follows int(h,'a') ans = 1/x*atan(a/x) pretty(ans) atan(a/x) --------- x Other examples: pretty(int('exp(a*x)*sin(b*x)')) b exp(a x) cos(b x) a exp(a x) sin(b x) - ------------------- + ------------------- 2 2 2 2 a + b a + b Next, we integrate the standard normal distribution: N(0,1) = (2p)-1exp(-.5x2) t = int('1/sqrt(2*pi)*exp(-0.5*x^2)') t = .5000000000000000*erf(.7071067811865475*x) which may also be written as 0.5erf(2-1/2 x). erf is itself an integral with no closed form expression. It is a MatLab defined function. fplot(t,[-3,3,-.5,.5])  Next we integrate N(0,1) over the interval [0 inf]. This should give 0.5 as answer (this is standard knowledge from probability theory!). int('1/sqrt(2*pi)*exp(-0.5*x^2)',0,inf) ans = .6266570686577500*2^(1/2)/pi^(1/2) numeric(ans) ans = 0.5000 Here is a function with no antiderivative int('log(x)/exp(x^2)') ans = int(log(x)/exp(x^2),x) pretty(ans) / | log(x) | ------- dx | 2 / exp(x ) However, we can obtain a numerical result if the integration is defined over a numerical range (say from 1 to 10). The MatLab commands trapz or quad or quad8 may be used. quad and quad8 generally leads to more accurate integrations. x = 1:.01:10; y = log(x) ./ exp(x.^2); result = trapz(x,y) result = 0.0359 YOU ARE NOW READY TO MOVE ON TO OTHER EXCITING COMPUTATIONS WITH MATLAB !!! (Other Modules Available: Calculus, Linear Algebra, Complex Numbers, & Differential Equations) who % Please go to page 1 and read the small print!! 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Extreme Points: Numerical Search for Minima & Maxima The MatLab function fmin searches for the minima of a one-dimensional function: xmin = fmin(f,0,5) % search for the minimum over the range 0 < x < 5. xmin = 4.4934 x = xmin; eval(f) % evaluate the function at the minimum ans = -0.2172 Similarly, the minimum point inside 10 < x < 15 is fmin(f,10,15) ans = 10.9041 fmin may also be used to find the maxima of f(x), by noting that the local minima of -f(x) correspond to local maxima of f(x). Let us find the maximum of f(x) in the range 5 < x < 10: xmax = fmin('-sin(x)/x',5,10) xmax = 7.7252 The value of f(x) at this extreme point is x = xmax; eval(f) ans = 0.1284 Finding Roots: f(x) = 0 solve(f) % solves for roots of f(x) near x = 0. Symbolic solutions are attempted. ans = 3.141592653589793 Another example (multiple solutions) t = solve('tan(2*x) = sin(x)') t = [ 0] [acos(1/2+1/2*3^(1/2))] [acos(1/2-1/2*3^(1/2))] numeric(t) ans = 0 0 + 0.8314i 1.9455 II. 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"-#!- . 2 y0 ti "-pi-i "-#i-d . 2 yd1 t "-p- "-#- . 2 y2 t "-p-ܥe= e"-L=ÊÊÊÊÊy͋yyy:ÓLyR"1111Dd#+DoDXToџ5Ê\ot 11\\џ\ÊÊ1. 2 P-1L-H-- -] . 2 P0LB-HB-B- B-]? . 2 P?1L-H-- -] . 2 P2L-H-- -] . 2 P3-LA-LE-L-L-Q' . 2 D'-0.5.A-.E-.-.-3' . 2 &'-0.4A-E---' . 2 '-0.3A-E---' . 2 '-0.2A-E---' . 2 '-0.1A-E---6 . 2 60A-E---+ . 2 +0.1wA-wE-w-w-|+ . 2 o+0.2YA-YE-Y-Y-^+ . 2 Q+0.3:A-:E-:-:-?+ . 2 2+0.4A-E---!+ . 2 +0.5LA-L-A--LA-A-L--LA-LA----MA-~%=ALBLCLFKMKYKeJrHxG~FDB@>:73.+%" "{(q.g4^7Z=RANGGJDP>S<Y7_2f.l+r(x&~$"! --''-''Let us integrate the normal distribution suing the trapz command!<BfgmrjPSZ[w|+ - . 2  % ( ) + , q Ɑ uDS]b PUVbPUbPb uDt:]b]b uD"]bcPVbcb] U]bc]bc]bcPV]c]cPU]cUbcUbcUc?   # - \ _ p q t ~  ( + < ?   + , / Z [ m p @_bae+19<FI @BJMwzc]cbUbccPPVP]bcbcUbc uDg]bThk$=@KMcdx{SUGHpqxDd]^su24;>RT`c';>Ubc]b uD]buDȍVbb PUVbPb]chPcPS>JM ;<=?GHIKqt/HIJKL*+.=>NOS|~ !!/!4!8! ?   ! . / Z [ e o p v !!!!!! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !- ?@abf~;<HIK{  AB!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!-\\\\Ê1Ê1#׊\3ÊÊÊÊ\\\ Module # ZERO CALCULUS Symbolic & Numeric using MatLab By Professor Mohamad H. Hassoun Department of Electrical & Computer Engineering Wayne Satate University Detroit, MI 48202 October 22, 1996 Contents: I. Functions: Graphs, Extreme Points & Roots II. Differentiation III. Integration IV. Series clear % Just to make sure that no MatLab variables are present from a previous session. NOTE on executing commands (green colored text) in this document: place the cursor on the command/MatLab expression you wish to execute and then type Ctrl-Enter. Make sure that variables/array definitions commands are executed before executing any following commands which depend on such definitions. Definitions in bold blue font execute automatically when you open this document; i.e., you nedd not re-execute them over. You can get full explanation (help) of any MatLab command in this document by highlighting that command and clicking on the '?' in the bar menu above (or type Ctrl-F1). I. Functions: Graphs, Roots, Extreme Points, & More Graphs Consider the single variable function: f = 'sin(x)/x' % defining a symbolic function f = sin(x)/x x=1; eval(f) % evaluting f at x = 1 ans = 0.8415 A simple plot may now be generated by using the fplot command: fplot(f , [-20 20 -.4 1.2]) % The vector argument specifies the x & y axis ranges  One may want to add titles and axis labels as follows title('plot of f(x) = sin(x)/x') xlabel('x') ylabel('f(x)')  Alternatively, one may evaluate the function f(.) over a range of values of its argument and the plot the data using the plot command: z = linspace(-20, 20, 60); y = sin(z) ./ z; % here,component-wise division (./) must be used plot(z,y)  Here is the same plot but without connecting the 'blue' plot points plot(z,y,'b.')  Extreme Points: Numerical Search for Minima & Maxima The MatLab function fmin searches for the minima of a one-dimensional function: xmin = fmin(f,0,5) % search for the minimum over the range 0 < x < 5. xmin = 4.4934 x = xmin; eval(f) % evaluate the function at the minimum ans = -0.2172 Similarly, the minimum point inside 10 < x < 15 is fmin(f,10,15) ans = 10.9041 fmin may also be used to find the maxima of f(x), by noting that the local minima of -f(x) correspond to local maxima of f(x). Let us find the maximum of f(x) in the range 5 < x < 10: xmax = fmin('-sin(x)/x',5,10) xmax = 7.7252 The value of f(x) at this extreme point is x = xmax; eval(f) ans = 0.1284 Finding Roots: f(x) = 0 solve(f) % solves for roots of f(x) near x = 0. Symbolic solutions are attempted. ans = 3.141592653589793 Another example (multiple solutions) t = solve('tan(2*x) = sin(x)') t = [ 0] [acos(1/2+1/2*3^(1/2))] [acos(1/2-1/2*3^(1/2))] numeric(t) ans = 0 0 + 0.8314i 1.9455 II. Differentiation MatLab allows for the differentiation of symbolic functions using the diff command. Let us try differentiating the function f(x) = sin(x)/x which was defined earlier. g = diff(f) g = cos(x)/x-sin(x)/x^2 We can generate a "BLMSyz1fjk1i#$?@FMNcdz{!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-(=GHZptuvwx/e]^u04=>DTbc!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!-ce&'-=>LMO|   JJKstx/HIL-!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(!!!!!!!!!!!!!!!!--.=>DQRS}~ K w !!!!!!!!!!!'"7"8"S""""""""!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-""rsbct CGHYZh!!!!!!!!!!!! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-K @ Normal ]a c @ Heading 1]c&@& Heading 2xU]c"A@"Default Paragraph FontOInput U]b cOOutput]bcO!Error U]bcO1AutoInit U]b cOACalc \]bc (we will use 500 to approximate inf) x = 0:.1:5001/sqrt(2*pi)*-0.5* result = 0.5000 0x = 1:.10result = 0.0356 IVSeries The MatLab command symsum returns the sum of its symbolic function f(n) from a lower limit to a higher limit on n. Here is an example: f = '(2*n-1)^2'; symsum(f,1,'n') % summing f(n) from 1 to n ans = 11/3*n+8/3-4*(n+1)^2+4/3*(n+1)^3 factor(ans) ans = 1/3*n*(2*n-1)*(2*n+1) pretty(ans) 1/3 n (2 n - 1) (2 n + 1) Another example: f = 'x^k/k!'; symsum(f ,'k',0, inf) % summation over k. ans = exp(x) This can be shown to be the correct answer by using MatLab's Taylor series expansion comEFGHXYZfghpqrstuvwxy|~ +,-3cdegv$%+,-./017QRThhu]cPPVPc^-3efgwx017ST +ADHsuE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!@!!- f = sin(x)/x < 4: Current plot held Current plot released  t = .5000000000000000*erf(.7071067811865475*x)  zL""6D   5 OP^C D U $!(!)!:!;!I!x!y!!!!!!!!!!!!a"""""!!!!!!!!!ABCDrstuEEEEEEFPFeFfFFF=GGGGGGGGHHHHHHHHIIIIISITIUIUcUbc]b uD5]b uD]buDuPPPchcSmand (taylor):  taylor('exp(x)') ans = 1+1*x+1/2*x^2+1/6*x^3+1/24*x^4+1/120*x^5+O(x^6) Other examples: Geometric series: taylor('1/(1-x)') ans = 1+1*x+1*x^2+1*x^3+1*x^4+1*x^5+O(x^6) symsum('x^n', 'n',0,inf) ans = -1/(x-1) Power series expansion of cos(x) taylor('cos(x)') ans = 1-1/2*x^2+1/24*x^4+O(x^6) The mth term in this answer takes the general form -1mx2m/(2m)!. Let us sum the series from 0 to infinity: symsum('((-1)^m)*x^(2*m)/((2*m)!)', 'm',0,inf) ans = cos(x)  "-#- . 2 y3 -tH "-tL-t "-t-{  . 2 h -0.5  H "-L- "--8 . 2 80 H "-L- "--&' . 2 '0.5  tH "-t-H "--tH "-H-t "--tH "-tH- "---uH "-~%=HtItKsNsUscsqqpomkigd`\WROIE>91," !/6=DK~RsYi\ec\gWnPqLxE{B=840-*(&%$"! --''-''f = sin(x)/xSymbolic Integration Numeric IntegrationIf symbolic integration is not possible, one may still Here is an example involving the above integration.As another example, l The Laplace Integral (Laplace Transform)One very important application of integration is in finding the Laplace Transformation of a function. This transformation is very helpful in analysing linear systems; it is also useful in solving linear ordinary differential equations (refer to Module Three on Differential Equations for details). The Laplace integral of a function f(x) takes the form int('f(t)*exp(-s*t)', 't', 0, inf); pretty inf / | | f(t) exp(- s t) dt | / 0 The following are Laplace transforms for some basic functions: Example 1: f(t) = c, where c is a constant int('c*exp(-s*t)','t',0,inf) ans = limit(-1/s*exp(-s*t)*c+1/s*c,t = inf,left) pretty exp(- s t) c limit - ------------ + c/s t -> inf- s Here, s is a complex variable. Evaluating the above limit as t approaches infinity eliminates the first term in the limit. Thus the answer is c/s. Matlab has a build-in command for finding the Laplace integral directly. The command is laplace(f) where f is a symbolic function of the variable t (not x). If f is a constant (i.e., t does not appear in the expression for f) then the command laplace(f,'s','t') must be used. Here is the Laplace transform for the function f(t) = c pretty(laplace('c','s','t')) c/s Example 2: f(t) = ce-at, pretty(laplace('c*exp(-a*t)')) c ----- s + a Example 3: f(t) = cos(w*t), pretty(laplace('cos(w*t)')) s ------- 2 2 s + w f = sin(x)/xf = sin(x)/x PAGE  PAGE 2 f = sin(x)/xfplot(hprime,[-3,3,-1,1],'bb for blueCurrent plot held Current plot released , 'b--' 15INPUT_64 OUTPUT_64INPUT_65 OUTPUT_65INPUT_66 OUTPUT_66INPUT_67 OUTPUT_67INPUT_82 OUTPUT_82INPUT_73 OUTPUT_73INPUT_75 OUTPUT_75INPUT_70 OUTPUT_70INPUT_71 OUTPUT_71INPUT_85 OUTPUT_85INPUT_86 OUTPUT_86INPUT_87 OUTPUT_87INPUT_88 OUTPUT_88INPUT_89 UIcIfIhIIIIIIIIIIIIIIIIIIIIIyJzJ{J|JJJJJJ KKVK\KrKKKL L^L`LgL{L|L}L~LLLLLLLLLLLLLLLLLLLLLLLLrMsMtMuM}M~MMMMMMMMMMMM`NbNPbchc]cbccPPbbcZbNcNdNeNiNuNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNOX}~#%/;<=Ub uDj]buDX PUVbPP uDPPu2EEfFGGHHH!HBHbHHHHHIIITIUIIIIIIIIIIIJKJxJ{J|J_L`LLLLLLLLLL!! !h!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-LLLLLLMFMqMuMMMMMMM N5N`NdNeNmNyNNNNNNNNNNNNN}3<!!!!!!!!!!!!!!!!!!!!!!!!!!h`%&:M."R@Y M T/G_wgO7'?WooW?'7Ogw_G/45DJ*''''FOh LOC- 0"Helvetica------------------''45-''- "-0----''tH "-t "-H "--tH "-H-t "--H "-H-t "-t-tH "-t-tH "-H-tH "-tH--tH "-pH-H "-#H-< . 2 y<-3 t "-p- "-#- . 2 y-2 t "-p- "-#- . 2 y-1 t! 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