Maths 301 key_sem_1_2009_2010

FINAL EXAMINATION Semester I , 2009/2010 MATH-301 (Calculus III) Question 1. (a)Evaluate the iterated integral. 011002x+y+z dy dz dx = 0110x+zy+ y2202 dz dx = 01102x+z+2 dz dx = 01102x+2+2z dz dx = 012x+2z+ 2z2210 dx = 01-2x+2+ 12 dx = 01-2x-3 dx = -2x22- 3x01 = -1-3= -4 (b)Sketch the region bounded by the graphs of the following equations and find its area by using double integrals. ,y = 0, x = 1,x = 2. y = 0x =2x=1y=1x2Area= 1201x2dy dx = 12y01x2 dx = 12x-2 dx = -1x11= --12- 1 = 12 (c)A solid is bounded by the cone and the plane z = 3. The density at P(x, y, z) is directly proportional to the square of the distance from the origin to P. Find the mass of the solid using integration in cylindrical coordinates. z density= δ=kx2+y2+z2 =kr2+z2mass=m= QδdV = 02π03r3kr2+z2r dz dr dθ Z = 3 Z = r y x =k02π03r3r3+rz2dzdrdθ =k02π03r3z+ rz33r3 drdθ =k02π033r3+9r- r4+ r43 drdθ =k02π033r3+9r-43r4 drdθ =k02π3r44+ 9r22-43r5503 dθ =k7292002πdθ = 72920 k θ02π = 72910 πk (d)Use integration in spherical coordinates to find the centroid of a hemispherical solid Q of radius a = 8. z Mxy= Qz dv , z= ρ cosϕ =02π0π208z ρ2sinϕ dρ dϕ dθρ=a=8 Centroid= 0, 0,z , z= Mxyv y x = 02π0π208 ρ3sinϕcosϕ dρ dϕ dθ =02π0π2ρ4408sinϕcosϕ dϕ dθ = 1024 02π0π2sinϕcosϕ dϕ dθ = 10242 02πsin2 ϕ0π2 dθ= 10242 02πdθ = 10242 θ02π = 1024π z= MxyV = 1024π23 π83 = 1024π1024π3 = 3 Question 2. (a)If , prove that f(x, y) is a harmonic function. fx, y =lnx2+ y2 fx= 2xx2+ y2 , fxx= 2x2+ y2-2x 2xx2+ y22 = -2x2+2y2x2+ y22 A fy= 2yx2+ y2 , fyy= 2x2+ y2-2y 2yx2+ y22 = 2x2-2y2x2+ y22 (B) From (A) and (B) fxx+ fyy=0 -> f is harmonic. (b)Find an equation for the normal line to the graph of the equation: - 4 z2 = 2 x2 - 3 y2 - 10 at the point (-1, -2, 3). Fx,y, z=0 -> 2x2-3y2+4z2-10=0 , -1, -2, 3 x0 y0 z0 ∇F= Fx i+ Fy j+ Fz k = 4x i- 6y j+ 8z k ∇F-1, -2, 3= -4 i+ 12 j+ 24 k -> a= -4, 12, 24 a1 a2 a3 Equation of the normal line x=x0+ a1t , y= y0+ a2t , z= z0+ a3t x= -1-4t , y= -2+12t , z=3+24t (c)The surface of a lake is represented by a region in the xy-plane such that the depth (in feet) under the point (x, y) is . At the point (10, 9), in what direction does the depth remain the same?. fx,y=400-3x2-2y2 , ∇f= fxi+ fyj ∇f= -6xi- 4yj , ∇f10 , 9= -60i-36j The depth remains the same in a direction orthogonal to ∇f, i.e. in the direction of 36i-60j or c36, -60 where c is any real number ≠0. (d)Find the extrema and saddle points, if any, of fx, y= -2x3+ 6xy-2y3 fx=0 -> -6x2+ 6y=0 , fy= 0 ->6x-6y2=0 y= x2 A x= y2 B Substitute from B into A ->y= y22 -> y4-y=0 , yy3-1=0 yy-1y2+y+1=0 -> y=0 or y=1 For y=0 ->x=y2=0 -> 0, 0 For y=1 ->x=y2=1 -> 1, 1 are the critical points. Dx, y= fxx∙ fyy- fxy2= -12x-12y- 62 Dx, y= 144xy-36 At 0, 0 ->D=0-36 <0 -> D is negative So P0, 0, f0,0 is a saddle point. At 1, 1 ->D=14411-36 >0 -> 1, 1 is an extrema. fxx1, 1= -12x1, 1 = -12 <0 ->f is maximum at 1, 1 fmaximum= -213+ 611-213=2 Question 3. ,[object Object], Total Distance=x+2x2+ 2x4+ 2x8+ ….. =x+x+x2+x4+… Total Distance =x+x 1+ 12+14+18+ …..x/2x x/4 x/8 x/16Geometric series with a=1 and r=12 <1 S= a1- r Total Distance=x+xa1-r; Total Distance=x+x11-12=x+2x=3x 15=3x ->x=153=5m is the initial height. (b)Determine if each of the following series converges or diverges or if the test is inconclusive. (i) an= n!10n -> an+1= n+1!10n+1= n+1n!10n 10 L= limn->∞an+1an= limn->∞n+1n!10n 10 ∙10nn! = limn->∞ n+110= ∞ So the series diverges. (ii) an= n2n , L= limn->∞ an1n L=limn->∞ n2n1n= limn->∞ n1n2nn = lim⁡n->∞ n1n2= 12 <1 since lim⁡n->∞ n1n=1, So the series converges. Or limn->∞an+1an= limn->∞n+12n+1∙2nn= limn->∞n+12n 2∙2nn= limn->∞12∙n+1n = 12limn->∞1+1n= 12 <1, so the series converges. (c)Use to approximate to two decimal places. sinx=x- x33!+ x55!- x77!+ ….. A Substituting sin x from A in 01sinxx dx = 01x- x33!+ x55!- x77!+ ….. x12dx =01x12- x523!+ x925!- x1327!+ ….. dx = x3232- x72726+ x112112120- x1521527!+ ……01 I II III = 23- 242+ 0.0015…. Since 0.0015 < 0.005, we neglect term III and the remaining terms of the series except I and II. So 01sinxx dx= 23- 242= 1321 ≅0.619 ≅0.62 to two decimal places. Question 4. y(a)If a rock is thrown from a point 6 ft above the ground with an angle of elevation of 30º and an initial speed of 20 ft/sec. If there is a 8-ft high fence 10 ft in front of you, will the rock sail over the fence?. θ=30°h0=20cos30° =20 32 =103 8’x0=0 v0 =20sin30°=2012=10 y0=6' x= x0+ h0t A xt=10'xy= y0+ v0t- 12 gt2 B g=32 ft/s2 Substituting known values in A and B C xt =0+103 t and D yt=6+10t-16t2 At fence xt=10' so C becomes 10 =103 t->t=13 secs. From D y13fence= 6+1013- 16132=6.44' <8' So the rock will not sail over the fence. (b) Find the points on the curve C at which the tangent line is horizontal. ,;t in R. dydx= dydtdxdt= 133t2-1122t-1= t2-1t-1 The tangent line is horizontal at dydx=0 or dydt=0 i.e, at t2- 1=0 -> t2=1 ->t= ±1 At t = 1 -> x, y= -12, -23 At t = -1 -> x, y= 32, 23 Points at which the tangentline is horizontal. (c)Use integration in polar coordinates to compute the area of the region that is inside the graph of the polar equation r = 3 + 3 cos θ. θ=πθ=0r = 3 + 3 cos θ Area= 12 αβr2 dθ By symmetry, A=2∙12 0π3+3cosθ2 dθ= 0π9+18cosθ+9cos2θ dθ =0π 9+18cosθ+921+cos2θdθ =9θ+18sinθ+ 92θ+94sin2θ0π = 9π+0+ 92π+0- 0=9π+ 9π2= 27π2 ≅42.41 Question 5. (a)A constant force of magnitude 5 N has the same direction as the positive x-axis. If the distance is measured in meters, find the work done if the point of application moves from the point (1, -1, 1) to the point ( 3, 1, -2). Force=5 1, 0, 0 = 5, 0, 0 d= 3, 1, -2 – 1, -1, 1 = 2, 2, -3 Work done= F∙d= 5, 0, 0 ∙ 2, 2, -3 =10 N∙m (b)Find an equation of the plane through P, Q and R. P( 2, 0, 0), Q ( 0, 2, 0) , R ( 0, 0, 2). PQ= OQ-OP= 0, 2, 0 – 2, 0, 0 = -2, 2, 0 PR= OR-OP= 0, 0, 2 – 2, 0, 0 = -2, 0, 2 PQ × QR= ijk-220-202= i 2002- j -20-22+ k -22-20 PQ × PR= i 4-0- j -4 + k 4 =4i+ 4j +4k Equation of the plane through P, Q, and R a1x-x0+ a2y-y0+ a3z-z0 =0 A Taking x0, y0, z0= 2, 0,0 , a1, a2, a3 = 4, 4, 4 So Equation A becomes 4x-2+ 4y-0+ 4z-0=0 or 4x+4y+4z=8 (c)Sketch the graph of the following equations in an xyz-coordinate system: (i) z 1 y x -zzxy(ii)

Maths 301 key_sem_1_2009_2010

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