Maple and Math CoursesHere are examples of some typical problems from some 200 and 300 level Math courses at UBC, and how Maple might be used to help solve them.
<Text-field style="Heading 1" layout="Heading 1">Math 200</Text-field>Let D be the region inside the polar curve 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 and above the x axis. Evaluate the following integral: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restart; with(Student[MultivariateCalculus]):plot(1-cos(theta),theta=0..Pi,coords=polar,scaling=constrained);f:=changecoords(y/sqrt(x^2+y^2),[x,y],polar,[r,theta]);MultiInt(f, r = 0 .. 1-cos(theta), theta = 0 .. Pi,
coordinates = polar);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 215
</Text-field>JSFHSolve the following system of differential equations with initial conditions.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restart;
sys:= D(x1)(t)=3*x1(t)-2*x2(t), D(x2)(t)=4*x1(t)-x2(t);ics:= x1(0)=1,x2(0)=1;dsolve({sys,ics});LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiKUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EldHJ1ZUYnLyUqc2VwYXJhdG9yR1EmZmFsc2VGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZERi8=
<Text-field style="Heading 1" layout="Heading 1">Math 217</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Find the maximum and minimum values of 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 subject to the constraints 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 and 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.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=restart;
with(Student[MultivariateCalculus]):
LagrangeMultipliers(4*x-2*y,[2*z-x-y-4,x^2+z^2-1],[x,y,z],output=detailed);
<Text-field style="Heading 1" layout="Heading 1">Math 220</Text-field>JSFHJSFHDetermine whether the following series converges or diverges. 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; Maple doesn't really do proofs, which are the main focus of Math 220. Also there isn't a command to decide whether a series converges or diverges. But you can do the usual tests for convergence, e.g. ratio test.a:= n -> n/(3*n)^n;limit(a(n+1)/a(n),n=infinity);So: it converges.JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 221</Text-field>JSFHConsider the following linear system: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,
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,
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,
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For which values of a and b, if any, does the system have: (i) No solution? (ii) Exactly one solution?(iii) Exactly two solutions? (iv) More than two solutions?restart;
sys := {x + 3*y - 2*z + 2*w = 1,
y + z - 2*w = 2,
x + 2*y - 2*z + a*w = 0,
2*x + 8*y - z + w = b};with(LinearAlgebra):A,B:= GenerateMatrix(sys,[x,y,z,w]);solve(Determinant(A)=0,a);eval(sys,a=5);solve(%,{x,y,z,w,b});So: (i) 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; (ii) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjVGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GV0Y5; (iii) never; (iv) 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
<Text-field style="Heading 1" layout="Heading 1">Math 223</Text-field>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 an orthonormal basis of eigenvectors. restart;
with(LinearAlgebra):
A:= Matrix(<<0,2,1>|<2,3,2>|<1,2,0>>,shape=symmetric);E,V:=Eigenvectors(A);Q:= [Column(V,1..3)];Normalize~(Q,Euclidean);
<Text-field style="Heading 1" layout="Heading 1">Math 226</Text-field>Prove that the line given by the parametric equations
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, 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, 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, isparallel to the plane 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.restart;
2*(1+4*t) + 5*(2-t) + (-3*t);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 227</Text-field>A bug is flying through space so that its coordinates at time t are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidEYnRjZGOUZARkBGQEYrRkBGK0ZA = (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, 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiQtSSVtc3VwR0YkNiUtRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GPVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkNGK0ZDRitGQw==).Find the bug\342\200\231s velocity and acceleration for all LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo=, and the curvature of its trajectory at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjBGJ0ZARkBGK0ZARitGQA==.restart;
X := <t^2 + t, t^2 - t, t^3>;V:= map(diff, X, t);A:= map(diff, X, t$2);K:= eval(VectorCalculus[Curvature](X,t),t=0);simplify(%);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 300</Text-field>Find all complex solutions to the equation
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.Express each solution in the form 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, where x and y are real numbers.restart;
solve(cos(z)=2*I*sin(z),z,AllSolutions);convert(%,ln);
<Text-field style="Heading 1" layout="Heading 1">Math 301</Text-field>Evaluate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNigtSSNtbkdGJDYkUSIwRidGMi8lJ2l0YWxpY0dGPC8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxwbGFjZWhvbGRlckdGPC9GM1EnaXRhbGljRictRiM2Jy1GLzYtUSgmaW5maW47RidGMi9GNkZXL0Y5RlcvRjtGVy9GPkZXL0ZARlcvRkJGVy9GREZXRkVGSEZQRlUvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGWi8lMXN1cGVyc2NyaXB0c2hpZnRHRk8vJS9zdWJzY3JpcHRzaGlmdEdGTy1JJm1mcmFjR0YkNigtSSNtaUdGJDYlUSJ4RidGUEZaLUYjNiktRk02JFEiOEYnRjItRi82LVEiK0YnRjJGW29GXG9GXW9GXm9GX29GYG9GYW8vRkZRLDAuMjIyMjIyMmVtRicvRklGaXAtSSVtc3VwR0YkNiVGXHAtRiM2Jy1GTTYkUSIzRidGMkZQRlVGYm9GWkZlb0ZQRlVGYm9GWi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaHEvJSliZXZlbGxlZEdGVy1GLzYtUSJ+RidGMkZbb0Zcb0Zdb0Zeb0Zfb0Zgb0Zhb0ZFRkgtRi82LVEwJkRpZmZlcmVudGlhbEQ7RidGMkY1RjgvRjtGN0Y9L0ZARjdGQUZDRkVGSEZccEZVRjI=restart;
int(x/(8+x^3),x=0..infinity);
<Text-field style="Heading 1" layout="Heading 1">Math 302</Text-field>Let X and Y be independent standard normal random variables, and let 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.
Find the cumulative distribution function of Z, and be sure to specify your answer for all values of z.restart;
with(Statistics):X:= RandomVariable(Normal(0,1)): Y:= RandomVariable(Normal(0,1)):
Z:= X^2 + Y^2;CDF(Z,z);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 303</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=A Markov chain has state space 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 and transition matrixLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkobWFjdGlvbkdGJDYkLUkobWZlbmNlZEdGJDYoLUYjNiZGKy1JJ210YWJsZUdGJDY6LUkkbXRyR0YkNistSSRtdGRHRiQ2KC1GIzYlLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lLHBsYWNlaG9sZGVyR1EldHJ1ZUYnRkgvJSlyb3dhbGlnbkdGLi8lLGNvbHVtbmFsaWduR0YuLyUrZ3JvdXBhbGlnbkdGLi8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZWRj8tRkA2KC1GIzYlLUkmbWZyYWNHRiQ2KC1GRTYkUSIzRidGSC1GIzYlLUZFNiRRIjRGJ0ZILyUnaXRhbGljR0ZNL0ZJUSdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHRlYvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGam8vJSliZXZlbGxlZEdRJmZhbHNlRidGS0ZIRk5GUEZSRlRGV0Y/LUZANigtRiM2JS1GaG42KC1GRTYkRlZGSEZdb0Zmb0Zob0ZbcEZdcEZLRkhGTkZQRlJGVEZXRj9GTkZQRlItRj02K0Y/LUZANigtRiM2JS1GaG42KC1GRTYkUSIyRidGSC1GIzYlLUZFNiRRIjVGJ0ZIRmJvRmRvRmZvRmhvRltwRl1wRktGSEZORlBGUkZURldGP0Y/LUZANigtRiM2JS1GaG42KEZmcEZjcUZmb0Zob0ZbcEZdcEZLRkhGTkZQRlJGVEZXRmpwRk5GUEZSLUY9NitGPy1GQDYoLUYjNiUtRmhuNihGYHEtRiM2JUZqbkZib0Zkb0Zmb0Zob0ZbcEZdcEZLRkhGTkZQRlJGVEZXRj8tRkA2KC1GIzYlLUZobjYoRmZwRmZyRmZvRmhvRltwRl1wRktGSEZORlBGUkZURldGP0Y/Rk5GUEZSLUY9NistRkA2KC1GIzYlLUZobjYoRmBxLUYjNiUtRkU2JFEiN0YnRkhGYm9GZG9GZm9GaG9GW3BGXXBGS0ZIRk5GUEZSRlRGV0Y/LUZANigtRiM2JS1GaG42KEZlcUZmc0Zmb0Zob0ZbcEZdcEZLRkhGTkZQRlJGVEZXRj9GP0Y/Rk5GUEZSLUY9NitGP0Y/Rj9GPy1GQDYoLUYjNiVGZnBGS0ZIRk5GUEZSRlRGV0Y/Rk5GUEZSLUY9NitGP0ZjdEY/Rj9GP0Y/Rk5GUEZSLyUmYWxpZ25HUSVheGlzRicvRk9RKWJhc2VsaW5lRicvRlFGam8vRlNRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0ZNLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGZXUvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGYHYvJSZmcmFtZUdGYHYvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZfcC8lLWVxdWFsY29sdW1uc0dGX3AvJS1kaXNwbGF5c3R5bGVHRl9wLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRl12RitGSEZIL0krbXNlbWFudGljc0dGJFEnTWF0cml4RicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGY3cvJSthY3Rpb250eXBlR1EucnRhYmxlYWRkcmVzc0YnRitGSEYrRkg=(a) Find the communicating classes.
(b) What is the probability, starting in 3, that the Markov chain never visits state 4?.restart;
with(GraphTheory):
P := Matrix([[0,0,3/4,0,1/4,0],
[0,2/5,0,0,1/5,2/5],
[0,2/3,0,1/3,0,0],
[2/7,0,5/7,0,0,0],
[0,0,0,0,1,0],
[0,1,0,0,0,0]]);
G:= Digraph(P);
DrawGraph(G);
Classes:= StronglyConnectedComponents(G);eqs:= {seq(u[j] = add(P[j,k]*u[k],k=[1,3,4])+add(P[j,k],k=[5,2,6]),j=[1,3]),
u[4]=0};solve(eqs);
<Text-field style="Heading 1" layout="Heading 1">Math 307</Text-field>Let V denote the row space of the matrix
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Find bases of V and 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;
with(LinearAlgebra):
A:= <<1,2,1>|<1,0,5>|<0,1,-2>>;RowSpace(A);NullSpace(A);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 308</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Consider the triangle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkQUJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= whose vertices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY1LUYsNiVRIkNGJ0YvRjJGOQ== have coordinates 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRIjBGJ0Y+Rj5GPi1GMTYjUSFGJ0Y+, respectively.Find the coordinates of the centroid LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5of the orthocentre H.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=restart; with(geometry):
assume(t <> s):
triangle(ABC,[point(A,0,1),point(B,t,0),point(C,s,0)]);
coordinates(centroid(G,ABC));coordinates(orthocenter(H,ABC));
<Text-field style="Heading 1" layout="Heading 1">Math 312</Text-field>Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be a four-digit integer 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. Show that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is divisible by 7 if and only if 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 is divisible by 7.restart;
n:= add(a[j]*10^j, j=0..3);n mod 7, a[0] + 3*a[1] + 2*a[2] - a[3] mod 7;JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 313</Text-field>JSFHFind positive integers LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= such that 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW5HRiQ2JFEiNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GLDYkUSMxM0YnRi9GMi1GLDYkUSMxN0YnRi9GMi1GLDYkUSMyOUYnRi9GLw==restart;
isolve(x^2 + y^2 = 5*13*17*29);remove(hastype,[%],negative);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 316</Text-field>Consider the differential equation 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. Show that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjFGJ0ZARkBGK0ZARitGQA== is a regular singular point, and find the indicial equation at this point.restart;
de:= (x^3 - 1)*diff(y(x),x$2) + x*diff(y(x),x) + 2*y(x)=0;with(DEtools):
singularities(de);indicialeq(de,r,1,y(x));JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 317</Text-field>Evaluate 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 where S is that part of the sphere JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiK0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGWC1GLDYlLUYvNiVRInlGJ0YyRjVGOEZARkMtRiw2JS1GLzYlUSJ6RidGMkY1RjhGQC1GRDYtUSI9RidGPkZHRkpGTEZORlBGUkZUL0ZXUSwwLjI3Nzc3NzhlbUYnL0ZaRmNvRjpGPg== above the plane LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=, n is the upward unit normal, and
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restart;
with(VectorCalculus):
SetCoordinates(cartesian[x,y,z]):
S:= Surface(<x,y,sqrt(2-x^2-y^2)>,[x,y]=Circle(<0,0>,sqrt(2))):
F:= VectorField(<y^2,x^3,exp(x)+exp(y)+z>):
Flux(Curl(F),S);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Math 318</Text-field>A closet contains 10 different pairs of shoes (each pair consists of a left shoe and a right shoe, so there are 20 shoes in total). 6 shoes are chosen at random. Find the probability that exactly one complete pair is chosen.restart;
with(combinat):
shoes:= {seq(right[i],i=1..10),seq(left[i],i=1..10)};SampleSpace:= choose(shoes,6):SampleSpace[1..3];countpairs:= w -> add(piecewise({left[j],right[j]} subset w, 1, 0),j=1..10);nops(select(t -> (countpairs(t)=1),SampleSpace))/nops(SampleSpace);binomial(6,2)* 1/19 * 18/18 * 16/17 * 14/16 * 12/15;JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 321</Text-field>Evaluate 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.restart;
J:= int(t^5*exp(-I*n*t),t=-Pi..Pi) assuming n::integer;J2:=simplify(abs(J)^2) assuming n::posint;The case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+Rj5GK0Y+ must be done separately.int(t^5,t=-Pi..Pi);2*sum(J2,n=1..infinity);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 322</Text-field>JSFHFactor the following elements of the given rings LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= into irreducible elements of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=(a) 15, an element of 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[factorEQ](15,-2);(b) 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, an element of 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(5*x^4 - 20*x^3 + 30);(c) 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, an element of 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Factor(x^3+1) mod 7;
<Text-field style="Heading 1" layout="Heading 1">Math 340</Text-field>Solve the linear programming problem
maximize LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIitGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlgtRiw2JUYuLUYjNiUtRjs2JFEiMkYnRj5GMkY1RkBGQy1GLDYlRi4tRiM2JS1GOzYkUSIzRidGPkYyRjVGQC1GLzYjUSFGJ0Zjb0Y+ subject to 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 ,
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 ,
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 ,
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restart;
with(simplex):
maximize(x[1]+x[2]+x[3],
{x[1]+2*x[2]+3*x[3]<=6,
2*x[1]-2*x[2]+x[3]>=5,
x[1]-x[2]+x[3]=4},NONNEGATIVE);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 342</Text-field>Show that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ is generated by the cycles (1,2) and (2,3). restart;
with(group):permgroup(3,{[[1,2]],[[2,3]]});grouporder(%);JSFH
<Text-field style="Heading 1" layout="Heading 1">Math 345</Text-field>Find a conserved quantity for the two-dimensional system
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,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;
sys := {D(x)(t)=y(t), D(y)(t)=x(t)^2-4};with(DEtools):casesplit(sys);xde:= op([1,2],%);mu:= intfactor(xde);firint(xde*mu);solve(subs(diff(x(t),t)=y(t),%),_C1);JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Math 361</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Schnakenberg (1979) considered the following simplified model of glycolysis:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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
where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk=. As the parameter a varies, the steady state of the system changes its behaviour.Determine the sequence of steady state classifications for increasing values of a (e.g. stable node-to-saddle-to-unstable node ). Does the system undergo a Hopf bifurcation for any value(s) of a? If so, atwhat value(s) does the Hopf bifurcation(s) occur?restart;
F:= [x^2*y - x, a - x^2*y];S:=solve(F,{x,y});M:= eval(VectorCalculus[Jacobian](F,[x,y]),S);with(LinearAlgebra):solve({discrim(CharacteristicPolynomial(M,t),t)=0,a>0});solve({Trace(M)=0,a>0});Eigenvalues(eval(M,a=0.1));Eigenvalues(eval(M,a=sqrt(2)-1));Eigenvalues(eval(M,a=0.5));Eigenvalues(eval(M,a=1));Eigenvalues(eval(M,a=1.1));Eigenvalues(eval(M,a=sqrt(2)+1));Eigenvalues(eval(M,a=2.5));Conclusion: unstable node to unstable spiral at 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, to stable spiral at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=, to stable node at 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 bifurcation occurs at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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TTdSMApJNlJUQUJMRV9TQVZFLzEzOTY4MDgyMFgqJSlhbnl0aGluZ0c2IjYiW2dsKCMlISEhIiMiIzNGRDdGRkZGRkZGRkZGRkUzRkQ1MkE3RkE5RDJGOEU5M0ZEN0ZGRkZGRkZGRkZGRUJGRDUyQTdGQTlEMkY4RTlGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTY4MDk5NlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiMiI14jIiIiXiMhIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTY4MTE3MlgqJSlhbnl0aGluZ0c2IjYiW2dsKCMlISEhIiMiI0JGQkFFMTQ3QUUxNDdBRTMzRkYxODUwNkM3OTdCQ0ZFQkZCQUUxNDdBRTE0N0FFM0JGRjE4NTA2Qzc5N0JDRkVGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTY4MTA2OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiMiIywmISIiIiIiKiQiIiMjRilGK0YoRidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTY5ODM2NFgqJSlhbnl0aGluZ0c2IjYiW2dsKCMlISEhIiMiI0JGRkQzMTk5QjMzMDAwODAwMDAwMDAwMDAwMDAwMDAwQzAwQjY3MzMyNjY3RkZDMDAwMDAwMDAwMDAwMDAwMDBGJQ==