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UNF - MHF - 2312
Chapter 73822(b) fs (x) =112.30 (a) fc (x) =1(b) fs (x) =sin 3 2 sin 2cos x d02 cos 2 cos 3 1sin x d01 cos 2cos x d202 sin 2sin x d2012.29 (a) fc (x) =13.2 f (x) =i21 2inxenn =013.4 (c) q (t) = CV 1 2(1 e1/2 )13.6 f (t) =(
UNF - MHF - 2312
Chapter 739cos 2 12 cos 2 1, f (x) =sin x d,i08 cos sin2 (/2)cos x d, /813.16 f (x) =0213.15 g () =13.190|f (x)|2 dx =0|gc ()|2 d =2 sin2 a13.20 g () =, a3 /3 20|gs ()|2 d13.23 2 /8/4
UNF - MHF - 2312
Chapter 81.41.51.61.7x = k 1 gt + k 2 g (ekt 1)x = A 2 sin t + v0 t + x0(a) 15 months (b) t = 30(1 21/3 ) = 6.19 monthsx = (c/F )[(m2 c2 + F 2 t2 )1/2 mc]2.12.32.52.72.92.112.132.152.172.19y = mx, m = 3/22.22.4ln y = A(csc x cot x),
UNF - MHF - 2312
Chapter 84113.13 x = 2 ey + Cey3.14 x = y 2/3 + Cy 1/313.15 S = 2 107 (1 + 3t/104) + (1 + 3t/104 )1/3 , where S = numberof pounds of salt, and t is in hours.3.16 I = AeRt/L + V0 (R2 + 2 L2 )1 (R cos t + L sin t)3.17 I = Aet/(RC ) V0 C (sin t RC c
UNF - MHF - 2312
Chapter 842y = (Ax + B ) sin x + (Cx + D) cos x + (Ex + F )ex + (Gx + H )ex = 0 cos t, = g /lT = 2 R/g 85 min.= = 1/ LCoverdamped: R2 C > 4L; critically damped: R2 C = 4L;underdamped: R2 C < 4L.2t4 22t16y+y = 0, y = e8t/15 A sin+ B cos5.40
UNF - MHF - 2312
Chapter 87.107.117.127.137.167.177.187.197.207.217.227.237.257.288.88.108.128.178.228.258.2743x = 1 + t2x = (1 3t)1/3xt = 1 2 (1 u4 )1/2 duut = ( 2 )1 (cos )1/2 d(a) y = Ax + Bx3(b) y = Ax2 + Bx2(c) y = (A + B ln x)/x3(d) y
UNF - MHF - 2312
Chapter 810.310.510.710.810.910.1110.1310.154412 t sinh t10.4b(b a)tebt + a[ebt eat ]10.6(b a)2ata cosh bt b sinh bt aea2 b 2eatebtect++(b a)(c a) (a b)(c b) (a c)(b c)(2t2 2t + 1 e2t )/410.10 (1 cos at 1 at sin at)/a421cos bt
UNF - MHF - 2312
Chapter 8x 2 sin x,x < /412.13 y = x 2 cos x, x > /4212.15 y = x sinh x cosh x ln cosh x12.16 y = x ln x x x(ln x)2 /22112.17 y = 4 sin x12.18 y = x2 /2 + x4 /61linear 1st ordery = 3 x2 + Cx2(ln y ) (ln x)2 = Cseparabley = A + Bex sin(x
UNF - MHF - 2312
Chapter 92.12.32.52.62.72.9(y b)2 = 4a2 (x a2 )2.2 x2 + (y b)2 = a2ax = sinh(ay + b)2.4 ax = cosh(ay + b)y = aex + bex or y = A cosh(x + B ), etc.x + a = 4 (y 1/2 2b)(b + y 1/2 )1/23ex cos(y + b) = C2.8 K 2 x2 (y b)2 = K 42x = ay + b2.10
UNF - MHF - 2312
Chapter 95.55.65.85.95.105.115.125.135.145.155.165.175.185.195.205.215.225.235.245.2511L = 2 mx2 2 kx2mx + kx = 01L = 2 m(r2 2 + r2 sin2 2 ) mgr cos a sin cos 2 g sin = 0a(d/dt)(sin2 ) = 012L = 2 m(2r + r2 2 ) mgr2 + g =
UNF - MHF - 2312
Chapter 96.16.4catenarycatenary8.2I=8.3I=8.4I=486.26.5x2 y21+yy dy2x2+1circlecircle6.36.6circular cylindercircledx, x2 (2y + y 3 ) = K (1 + y 2 )3/222, x y 2 = C 2 (1 + x )3r2 + r4 2 dr ,drd= Kr r4 K 2y = aebx(x a)2 + (y
UNF - MHF - 2312
Chapter 104.44.54.64.7 1 0221 0 Principal moments:I=( 1, , + 1); principal151500axes along the vectors: (1, 1, 0), (0, 0, 1), (1, 1, 0).4004 2 Principal moments: (2, 4, 6); principal axes along theI = 00 24vectors: (0, 1, 1), (1, 0
UNF - MHF - 2312
Chapter 108.58.68.78.88.98.118.128.138.1450V = er cos e sin e r sin hu = hv = (u2 + v 2 )1/2 , hz = 1ds = (u2 + v 2 )1/2 (eu du + ev dv ) + ez dzdV = (u2 + v 2 ) du dv dzau = iu + jv = (u2 + v 2 )1/2 euav = iv + ju = (u2 + v 2 )1/2 evaz =
UNF - MHF - 2312
Chapter 109.351See 8.21 UU1 U+ e+ err r sin 111 VV = 2r2 Vr +(sin V ) +r rr sin r sin U1U12U12r2+2sin +2 2U= 2r rrr sin r sin 21VV =er(sin V ) r sin 1Vr1 Vr sin (rV ) e +(rV ) +er sin rr r9.6 See 8.11
UNF - MHF - 2312
Chapter 109.189.199.209.2152 r 1 e , r 1 e r , 32e , er cos e sin , 3r 1 , r 3 , 02r1 , 6, 2r4 , k2 eikr cos 11.4 Vector11.5 ds2 = du2 + h2 dv 2 , hu = 1, hv = u(2v v 2 )1/2 ,vdA = u(2v v 2 )1/2 du dv, ds = eu du + hv ev dv ,eu = i(1 v ) + j
UNF - MHF - 2312
Chapter 113.23.53.83.113.143/232/35(5/3)1(4/3)3.33.63.93.123.159/1072(5/4)(2/3)/3(2/3)/417.12 B (5/2, 1/2) = 3/1617.33 B (1/3, 1/2)7.5 B (3, 3) = 1/3017.72 B (1/4, 1/2)7.10 4 B (1/3, 4/3)37.12 (8/3)B (5/3, 1/3) = 32 2 3/27
UNF - MHF - 2312
Chapter 115412.15 2 E 23 E12.14 12E 35 15.86=12.16 2 2 E 1/ 2 3.820=a12.23 T = 8 5g K 1/ 5 ; for small vibrations, T 2 2a=3g13.713.913.1113.1313.1513.1713.1913.2113.2213.24(4) = 3!3/22.4222E135 F arc sin 4 , 4/5 = 0.1834 sn u d
UNF - MHF - 2312
Chapter 121.11.31.51.71.9yyyyy2.1= a1 xex= a1 x= a0 cosh x + a1 sinh x= Ax + Bx3= a0 (1 x2 ) + a1 x3= a0 e x= a0 cos 2x + a1 sin 2x= (A + Bx)ex= a0 (1 + x) + a2 x22= (A + Bx)ex1.21.41.61.81.10yyyyySee Problem 5.32.4Q0 =
UNF - MHF - 2312
Chapter 125610.5 sin (35 cos3 15 cos )/210.4 sin 10.6 15 sin2 cos 11.111.311.511.611.711.811.911.10y = b0 cos x/x211.2 y = Ax3 + Bx332y = Ax + Bx11.4 y = Ax2 + Bx31/21/2y = A cos(2x ) + B sin(2x )y = Aex + Bx2/3 [1 3x/5 + (3x)2 /(5 8
UNF - MHF - 2312
Chapter 1257= Ax + B x sinh1 x x2 + 1= A(1 + x) + Bxe1/x22= A(1 x ) + B (1 + x )exx= Ax Be= A(x 1) + B [(x 1) ln x 4]= A x + B [ x ln x + x]xx= A 1x + B [ 1x ln x + 1+x ]2= A(x2 + 2x) + B [(x2 + 2x) ln x + 1 + 5x x3 /6 + x4 /72 + ]= Ax2 +
UNF - MHF - 2312
Chapter 132.1T=2.2T=2.32.42.7T=nx(1)n+1 ny/10esinn101200 41odd nn22even n22n=2+4knx 1 ny/20sinen20neny sin nx1x 1 3y/30120 y/303x1 5y/305xesinesin+esin2309302530n4sinh n(1 y ) sin nxT= 2 (n2 1) sinh
UNF - MHF - 2312
Chapter 13592.14 For f (x) = x 5: T = 4021odd nnx ny/101cose2n10For f (x) = x: Add 5 to the answer just given.2.15 For f (x) = 100, T = 100 10y/3401For f (x) = x, T = (30 y) 263.23.33.43.53.63.7u=4001odd nu = 100 u=40 1od
UNF - MHF - 2312
Chapter 134.64.74.8y=y=y=604hl2 v9hl3 v4l2 v4.91odd n1nnwnxnvt1sinsinsinsin2n2lllnnxnvt1sinsinsinn33ll1sin(n/2)xvt2x2vtnxnvtsinsin+sinsinsinsin3ll16lln(n2 4)lln=31. n = 1, = v/(2l)2. n
UNF - MHF - 2312
Chapter 136116, n odd, the six solutions on (0, ) aren (4 n2 )1. T = bn eny sin nxbnsinh n(H y ) sin nx2. T =sinh nH23. u =bn e(n) t sin nxh 2 n24. =bn sin nx eiEn t/h , En =2m5. y = bn sin nx cos nvtbn6. y =sin nx sin nvtnv12(1)n+1
UNF - MHF - 2312
Chapter 135.14 u =6250 ln r200+ln 2o dd nln r5.15 u = 50 1 ln 2r n r nsin nn(2n 2n )200o dd nn (2n1 2 n )r2nr2nsin n6.2The rst six frequencies are 10 , 11 = 1.59310 , 12 = 2.1351020 = 2.29510 , 13 = 2.65210 , 21 = 2.91710.6.4
UNF - MHF - 2312
Chapter 136322227.21 n (x) = e (x +y +z )/2 Hnx (x)Hny (y )Hnz (z ), = m/ ,h1113En = (nx + 2 + ny + 2 + nz + 2 ) = (n + 2 ) .hh(n + 2)(n + 1), n = 0 to .Degree of degeneracy of En is C (n + 2, n) =2M e4+12r7.22 (r, , ) = R(r)Ylm (, )
UNF - MHF - 2312
Chapter 1310.12 u =40064o dd n1rna2nsin 2n10.14 Same as 9.1210.15 u =1rn 104006nsin 6n =2(10r)6 sin 6200arc tan1012 r12o dd n10.16 v 5/(2 )10.17 mn , n = 0; the lowest frequencies are:11 = 1.5910 , 12 = 2.1410 , 13 = 2.6510, 21 = 2.
UNF - MHF - 2312
Chapter 141.11.31.51.71.81.91.101.111.121.131.151.171.181.191.201.21u = x3 3xy2 , v = 3x2 y y 31.2 u = x, v = yu = x, v = y1.4 u = (x2 + y 2 )1/2 , v = 0u = x, v = 01.6 u = ex cos y , v = ex sin yu = cos y cosh x, v = sin y sinh xu
UNF - MHF - 2312
Chapter 143.13.53.93.123.173.203.236612+i11(a) 5 (1 + 2i)3(a) 0(a) 072i3.2 (2 + i)/33.6 1, 13.10 (2i 1)e2i(b) 1 (8i + 13)3(b) i(b) 17i/43.24 17i/963.3 03.7 (1 i)/83.11 2i3.43.8i/2i/23.18 i 63/3.22 i 3/1083.16 03.19 16
UNF - MHF - 2312
Chapter 14676.286.306.326.346.146.186.226.266.276.316.3511R(3i) = 16 + 24 i6.29R(0) = 1/6!6.31R(2i) = 3ie2 /326.33R(0) = 7, R(1/2) = 76.35 i/46.15 06.1606.19 06.2006.23 sinh 2 6.2433 2 i(1 + cosh 3 + i 3 sinh 3 )316.29
UNF - MHF - 2312
Chapter 146810.4 T = 200 1 arc tan(y/x)10.5 V = 200 1 arc tan(y/x)2210.6 T = 100y/(x + y ); isothermals y/(x2 + y 2 ) = const.;ow lines x/(x2 + y 2 ) = const.10.7 Streamlines xy = const.; = (x2 y 2 )V0 , = 2xyV0 , V = (2ix 2jy )V010.9 Streamlines
UNF - MHF - 2312
Chapter 151.11.31.51.71.91/10, 1/91/3, 5/91/4, 3/4, 1/3, 1/29/26, 1/2, 1/133/10, 1/31.21.41.61.81.103/8, 1/8, 1/41/2, 1/52, 2/13, 7/1327/52, 16/52, 15/529/100, 1/10, 3/100, 1/103/82.122.142.152.17(a) 3/4(b) 1/5(c) 2/3(d) 3/4(e
UNF - MHF - 2312
Chapter 15704.94.124.174.2125/102, 25/77, 49/101, 12/254.11 0.097, 0.37, 0.67; 1354.14 n!/nnMB: 16, FD: 6, BE: 104.18 MB: 125, FD: 10, BE: 35C (n + 2, n)4.22 0.1354.23 0.305.1 = 0, = 35.2 = 7, = 35/65.3 = 2, = 25.4 = 1, = 21/25.5 = 1, =
UNF - MHF - 2312
Chapter 159.3Number of particles:012345Number of intervals: 406 812 812 541 271 1089.4 P0 = 0.018, P1 = 0.073, P4 = 0.1959.5 P0 = 0.37, P1 = 0.37, P2 = 0.18, P3 = 0.069.6 Exactly 5: 64 days. Fewer than 5: 161 days. Exactly 10: 7 days. Moretha
Michigan State University - MUS - 212
Josh SawyerMUS 212: Music History since 1750Essay #1Classical EnlightenmentMusical eras are defined by the change of musical styles, composers, and compositionalinterpretation of music. Frequently, these changes are brought about by a change in music
Michigan State University - MUS - 212
ROMANTICSYMPHONIESSymphony(alongwithopera)consideredtheprimarygenreforromanticexpression(lyricism,color,emotion,harmonicexploration,virtuosity)by19thcenturycomposers.ManyusedBeethovens9symphoniesasthestartingpointfortheirworks.GiventhebreadthofLvBssym
Michigan State University - TE - 250
Pedagogy of the OppressedPaulo FreireA review and evalution of the relevance of this work to contemporary education andyouth workIntroductionThis seminal work was published in 1968 in Portuguese. The author, Paulo Freire, was aneducationalist workin
Michigan State University - TE - 250
Josh SawyerDr. TattoTE 250 Sec. 16Thought paper based on 1/9/12Teaching IdentityIn this modern age of education, students are bombarded by a variety of external factors.A student is shaped by an ever increasing amount of positive and negative influe
Michigan State University - TE - 250
Josh SawyerTE 250- Section 161/18/12Cultural IdentificationWhen I entered college (and the realm of education) I had never given my personalculture any real thought. I have lived a pretty average, middle class life in suburban Detroit.However, key c
York University - ECON - 2400
Name: _ Date: _1. Which statement below best illustrates the art, rather than the science ofmacroeconomics?A) Macroeconomic data provide the motivation for new macroeconomic theory.B) Macroeconomic relationships can be expressed using symbols and equa
York University - ECON - 2400
Economics, Faculty of Liberal Arts & Professional Studies, York UniversityFall 2011 Course OutlineCourse #: AP/ECON 2400 3.0 BFCOURSE TITLE: INTERMEDIATE MACROECONOMIC THEORY IFall 20111. Course Instructor Contact:Name: Dr Shadab QaiserOffice: 1078
York University - MATH - 2131
MATH 2131 3.00 MS2Assignment 3Total marks = 50Question 1: Let (X, Y ) have the joint pdffX,Y (x, y ) =ey , 0 < x < y < ,0,otherwise.(a) (4 marks). Find E (XY ).(b) (6 marks). Find Cov(X, Y ).(c) (4 marks). Find X,Y .Question 2: Let X1 , . . . ,
York University - MATH - 2131
MATH 2131 3.00 MS2Assignment 4Total marks = 40Question 1: Let r.v. X have pdff (x) =2/x3 , x > 1,0,otherwise.(a) (3 marks). Compute the rst two moments of X .(b) (1 mark). Except for the rst moment, what can you say about all theother moments of
York University - MATH - 2131
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York University - MATH - 2131
York University - MATH - 2131
York University - MATH - 2131
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York University - MATH - 2131
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York University - MATH - 2131
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