a. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines.

Transcription

1 Mah A Final Eam Problems for onsideraion. Show all work for credi. Be sure o show wha you know. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A. Hin: The volume of he parallelepiped deermined by vecors abc,, is he riple produc, a ( b c.. onsider he lines r ( = +,+ 4,5+ 6 and r ( = 4 5,+,+ 7. a. Show ha hese lines inersec by finding he poin of inersecion. b. Find an equaion for he plane conaining hese lines. = =. Give an approimae formula for he small change Δz ha resuls from small ln. Le z y ( e y changes Δ and Δ y in he values of and y. Use his o approimae he value of The figure a righ is he conour plo of a funcion of wo variables, f ( y,, for and y ranging from o. The scale is uni = 5 cm; spacing beween conour levels is. a. Use he conour plo o deermine wheher f and f y are >, =, or < a (,.5 and (.,.6. b. The funcion ploed on he figure is f ( y, = y + y 4 +. alculae he acual values of he parial derivaives a (,.5 and (.,.6. c. In he diagram, here are hree places where he angen plane is horizonal. Find he eac coordinaes of each of hese and characerize each as a ma, a min or neiher. / 5. onsider he curve r( = cos,sin, a. Find he lengh of he curve from b. Find he curvaure as a funcion of. c. Find he angenial and normal componens of he acceleraion. 6. onsider (,, g y z = + yz a. Find a uni vecor in he direcion from (,, in which g decreases mos rapidly. b. Find a parameerizaion of he line from ha poin in he direcion of mos rapid decrease. 7. Le f ( y, = y + + yand g(, y y minimum and maimum values, if hey eis, of f ( y, subjec o he consrain =. Use he mehod of Lagrange mulipliers o find he g, y = wih >. In he case ha hey do eis, idenify all of he poins (, y a which hese values are aained. 8. Find he area of he par of he saddle z = y ha lies inside he cylinder + y = 4.

2 9. Le f ( y, y y = +, and be y =, beween (, and (,, direced upwards. a. alculae F = f. b. alculae he inegral F dr hree differen ways: i. direcly; ii. by using pah-independence o replace by a simpler pah. iii. by using he Fundamenal Theorem for line inegrals.. Verify Green s heorem in he normal form, i.e. F nds ˆ = div (, sides and showing hey are equal if and (,. F =, y. Verify Sokes heorem for he paraboloid z = 6 y for z and he vecor field F = y,4 z, 6. You may find i convenien o use polar coordinaes o evaluae he surface inegral of he curl. F y da, by calculaing boh and is he square wih opposie verices a (,. Use he divergence heorem o calculae dv where V is he region bounded by he cone z y How abou F =,, = + and he plane z =. To do his, you will need a simple field whose divergence is `.? Hin: You can parameerize he cone by r, θ = cos θ, sin θ,.. The posiion of a paricle moving in he plane a is given by r( =, + + a. Find he velociy of he paricle a ime. b. Find he speed of he paricle a ime. c. Wha is he paricle s highes speed and when does ha occur?. 4. Find equaions of he normal plane and he osculaing plane for he curve r( =,, a (,, θ θ R = cos d, sin d a. Show ha he arc lengh along he curve from R o R( θ is θ. b. Find he curvaure a θ. 5. onsider he curve ( θ

3 yz 6. Find he direcion in which f ( yz,, = e increases mos rapidly a he poin Wha is he maimum rae of increase? 7. Find he minimum value of f ( y, / e ( y f fyy fy > o verify his.,,. = + and use he second derivaive es: 8. escribe and skech a graph for he solid whose volume is given by he following inegral: ρ sinφdρdφdθ and evaluae he inegral. 5 /6 9. Using cylindrical coordinaes, se up an inegral for he volume under he paraboloid z above he disc r cosθ.. Find he work done by he force field F( y z r( = sin,cos, for.,, = y, z, F, y, z =, y, z. Verify Soke s heorem for he vecor field paraboloid = above he y-plane. z y F, y, z zan y, z ln, z. Le = ( + paraboloid = r and in moving a paricle along he heli where S is he par of he. Find he flu of F across he par of he z = + y ha lies below he plane z = and is oriened upwards.

4 Mah A Final EamSoluions Fall. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A. Hin: The volume of he parallelepiped deermined by vecors abc,, is he riple produc, a ( b c. SOLN: Take a = AB=,,, b = A =,, and c = A=,, so ha a ( b c =,, 8,, 5 = = 6. onsider he lines r ( = +,+ 4,5+ 6 and r ( = 4 5,+,+ 7. a. Show ha hese lines inersec by finding he poin of inersecion. SOLN: r( = r =,7, b. Find an equaion for he plane conaining hese lines. SOLN: Take one vecor along r ; r r( = 5,,7,7, =,4,6 =,, and anoher along r : r r = 7,,,7, = 4,,, whence a normal o he plane is given by 4,,,, = 5,,5 = 5,,, so ha he equaion of he plane is y+ z = = =. Give an approimae formula for he small change Δz ha resuls from small changes Δ and Δ y in he values of and y. Use his o approimae he value of.99.. yln z y yln y SOLN: z = e, so e y z = = and = yln y ( ln e = ln. Thus, near a poin y y y (, y, wih Δ = and Δ y = y y we have Δz y Δ + ln Δ y. Thus.99. = +Δz 8 + (. + 8ln (. = ln To be sure, his esimae is slighly above he TI86 approimaion ln. Le z y ( e y 4. The figure a righ is he conour plo of a funcion of wo variables, f ( y,, for and y ranging from o. The scale is uni = 5 cm; spacing beween conour levels is. a. Use he conour plo o deermine wheher f and f y are >, =, or < a (,.5 and (.,.6.,.5 f,.5 < whereas SOLN: f < and y f (.,.6 < and f y (.,.6 = b. The funcion is f y, = y + y 4 +. alculae he acual values of he parial derivaives a (,.5 and (.,.6. SOLN: f ( y, = y + y 8+ and f y ( y, y f (,.5 = =.5, f y (,.5 = + = f (.,.6 = =., = + whence f.,.6 = = c. In he diagram, here are hree places where he angen plane is horizonal. Find he eac coordinaes of each of hese and characerize each as a ma, a min or neiher. y

5 SOLN: Firs find criical poins where f (, y = and y y+ = = or = y. Now = means ha means ha criical poins: f, y =. For he laer, f, y = y = and = y also 6 ± 56 8 ± 4 f y, y = 5y 6y+ = y = =. Thus here are hree 5 8± 4 8± 4, and, 5. I is eviden from he level curves plo ha , 5 is a local ma and, 5 is a saddle. The poin (, on he y- ais is no so obvious so we look a he discriminan: f ( y, = 6+ y 8, f yy ( y, = and f fy f y ( y, = y+ so ha a (,, = = < his is also a saddle. I fy fyy hough i d be. / 5. onsider he curve r( = cos,sin, a. Find he lengh of he curve from / 6 6 = + + = + = + = 7 7 SOLN: r '( d 4sin 4cos 9d 4 9d ( 4 9

6 b. Find he curvaure as a funcion of. dtˆ dtˆ / d r' r'' κ ( = = = ds ds / d r' SOLN: = = ( ( ( sin(,cos(, 4cos(, 4sin(, + / ( 4+ 9 / ( 4+ 9 ( 4+ 9 / / sin,cos, / / / / cos sin, sin cos,8 = = / / c. Find he angenial and normal componens of he acceleraion. r' ( r'' ( SOLN: an = =, / / r' 4+ 9 a T ( ( ( ( 4+ 9 ( sin(,cos(, 4cos(, 4sin(, r' r'' = = / r = ( + / ' 4 9 / 6. onsider (,, / / / g y z = + yz a. Find a uni vecor in he direcion from (,, in which g decreases mos rapidly. 4,, SOLN: g(,, =, z, y = 4,, uˆ =. (,, b. Find a parameerizaion of he line from ha poin in he direcion of mos rapid decrease. SOLN: r( =,, + 4,, = 4,,+ 7. Le f ( y, = y + + yand g(, y y minimum and maimum values, if hey eis, of f ( y, subjec o he consrain =. Use he mehod of Lagrange mulipliers o find he g, y = wih >. In he case ha hey do eis, idenify all of he poins (, y a which hese values are aained. SOLN: A he opimal poin we require ha he normals are parallel: f ( y, = λ g( y, y +,y+ = λ y, Also, he consrain mus be me. Thus we have hree equaions in hree unknowns: y + = λ y y+ = λ y = From he las equaion we subsiue y = ino he second equaion and ge λ = 4 whence y = λ/4 λ λ 6 4 and subsiuing ino he firs equaion yields + = λ =± =± so ha y =± 4 4

7 and, correspondingly, =±. f, = is a ma and f, = is a min. 8. Find he area of he par of he saddle z = y ha lies inside he cylinder + y = 4. SOLN: ( y 4 S ds = + z + z da = + + y da so ha A = / / 7 ds = r + 4r drdθ = r ( + 4r dr = ( 4r 7 + = Le f ( y, y y = +, and be y =, beween (, and (,, direced upwards. a. alculae F = f. SOLN: F = f = y, + y b. alculae he inegral F dr hree differen ways: i. direcly; F dr =,, d 7 d 4 + = + = + = ii. by using pah-independence o replace by a simpler pah. SOLN: The simpler pah would be r( =, as goes from o. F dr,, = d d 4 + = + = + = iii. by using he Fundamenal Theorem for line inegrals. f, f, = = 4 SOLN:. Verify Green s heorem in he normal form, i.e. F nds ˆ = div (, sides and showing hey are equal if F =, y F y da, by calculaing boh and is he square wih opposie verices a (, and (,. Paramerize he four edges like so:. r( =, ; r( =, ; r( =, ; r4( =, whence r ' ( =, ; r ' ( =, ; r ' ( =, ; r 4' ( =, and he normal componens for hese are n =, ; n =,; n =,; n =, ( ( ( ( 4 SOLN: F nˆ ds = d + d ( d d + + = + = = ddy = div F, y da

8 . Verify Sokes heorem for he paraboloid z = 6 y for z and he vecor field F = y,4 z, 6. You may find i convenien o use polar coordinaes o evaluae he surface inegral of he curl. SOLN: If z =, hen we have r = 4, which can be parameerized by r( = 4cos,4sin, so ha F dr = sin,, 4cos 4sin,4cos, d S sin 48 sin d = = = On he oher side, F ds = r θ r θ rdrdθ = r θ r θ rdrdθ 4 4 4, 6, cos, sin, sin 8 cos = 4r sinθ r cosθ r dθ = 56sinθ cosθ 4dθ 5 = 56 cosθ sinθ 4θ = 48. Use he divergence heorem o calculae dv where V is he region bounded by he cone z y How abou F =,,? Hin: You can parameerize he cone by r(, θ = cos θ, sin θ,. SOLN: dv = is he volume of he cone. To apply Guass heorem, compue F ds =,, ds =,, ds +,, ds = + and he plane z =. To do his, you will need a simple field whose divergence is `. where S is he fla op of he cone and S S S S S is he curved surface of he cone. Now since he normal o fla par of he surface is perpendicular o he field lines, he firs inegral is zero. For he second, we have he normal o he surface, r rθ = cos θ,sin θ, sin θ, cos θ, = cos θ, sin θ, which poins upwards when we wan somehing poining ouwards. So we negae i and inegrae,, ds = cos θ,, cos θ, sin θ, ddθ cos θdθ d = = S. The posiion of a paricle moving in he plane a is given by r( =, + + a. Find he velociy of he paricle a ime. + + SOLN: r' ( =, =, b. Find he speed of he paricle a ime. SOLN: r' ( + = = + ( + / / / / / /.

9 c. Wha is he paricle s highes speed and when does ha occur? The speed is maimum of when =. 4. Find equaions of he normal plane and he osculaing plane for he curve r( =,, 4 SOLN: r' ( =,, and r'' ( =,,6 so Tˆ =,, / and a (,, Tˆ ' = 4 /, 4 /, so a (,, he angen vecor is parallel 4 / o T ˆ = k,, and he normal vecor is parallel o N ˆ = k,8,9 so ha he binormal is Bˆ = kk,,,8,9 = 6kk, 7, 5 whence he normal plane is + y + z = 6 and he osculaing plane is 8y 9z = 6. θ θ R θ = cos d, sin d as a funcion of he 5. Epress he curvaure of he pah direced disance s measured from (,. ( θ θ u u θ s = R ' u du = cos + sin du = du = θ so ha he curvaure is dtˆ dtˆ d θ θ θ θ κ = = = cos d, sin d cos, sin ds dθ dθ = = yz 6. Find he direcion in which f ( yz,, = e increases mos rapidly a he poin Wha is he maimum rae of increase? SOLN: The gradien vecor is f = e, ze, ye yz yz yz 7. Find he minimum value of f ( y, / e ( y f fyy fy > o verify his. SOLN: / / e e which a (,, is,,.,,. = + and use he second derivaive es: / / f = + y + e = + y + = + y + = fy = ye = y = so he e e f fyy fy = + y + 4 y e = y + 4, which is f, = > which means f (, = is a local minimum. e e criical poin is (-,, and since e >, we noe ha

10 8. escribe and skech a graph for he solid whose volume is given by he following inegral: 5 /6 ρ sinφdρdφdθ and evaluae he inegral. SOLN: This is he inersecion of a cone whose op is a he origin and which opens downwards wih an angle of / wih a spherical shell. Two views of his volume are shown below: 9. Using cylindrical coordinaes, se up an inegral for he volume under he paraboloid z = r and above he disc r cosθ. SOLN: / cosθ / rdrdθ. Find he work done by he force field F( y z,, = y, z, in moving a paricle along he heli r( = sin,cos, for. / / SOLN: F dr = cos,,sin cos, sin, d = cos + ( sin d = joules? 4

11 . Verify Soke s heorem for he vecor field F(, y, z =, y, z paraboloid z = y above he y-plane. SOLN: Sokes heorem saes ha F dr = F ds where S is he par of he cos + sin F dr = cos,sin, sin, cos, d = sin cos + sin cosd = = iˆ ˆj kˆ This is consisen wih F = = y z y z F, y, z zan y, z ln, z. Le = ( +. Find he flu of F across he par of he paraboloid z = + y ha lies below he plane z = and is oriened upwards. SOLN: The divergence heorem is helpful here. The ouwards surface inegral is y r F ds = FdV = dzdyd = rdzdrdθ = r r dr S E = = 4