UNIVERSITY COLLEGE DUBLIN. An Coláiste Ollscoile, Baile Átha Cliath MOCK SEMESTER 1 EXAMINATION 2008/2009 MST30070

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1 UNIVERSITY COLLEGE DUBLIN An Coláise Ollscoile, Baile Áha Cliah MOCK SEMESTER EXAMINATION 8/9 MST37 Inroducion o Differenial Geomery Professor P.J. Rippon Professor S. Dineen Dr. J. B. Quigley Time Allowed: hours Insrucions for Candidaes Aemp shor quesions from par one, for 4% of available marks. Aemp 4 quesions from par wo, for 6% of available marks. Noes for Invigilaors Non Programmable calculaors are permied. Mahemaical formulae are on page 4. Page of 5

2 par one: aemp of 4 quesions for 4% from par one. image, graph, conour,curves, circle, AS, cycloid Helix. Skech he conour f of fx,y x y.. Skech he conour f of fx,y x y. 3. Skech he graph of fx,y x y. 4. Compue he speed of c i /j. 5. Give a formula for he velociy vecor of c xi yj. 6. Compue he acceleraion vecor of c i /j. arc-lengh 7. For c as in 5 above wha quaniy is a ẋ τ ẏ τ dτ? 8. Wrie down he oal arc lengh L of one urn of he helix. curvaure 9. Give a formula for he curvaure c as in 5 above.. Wrie down he curvaure κ of he sraigh line y x.. Wrie down he curvaure a cos sini sin cosj of he Archimedean spiral.. Wrie down he curvaure of he circle c 3 cos7i 4 3 sin7j involue and evolue 3. Give formula for he involue of c as in Give formula for he evolue of c as in Skech he conical Helix c cosi sinj k. 6. Skech he cardioid. 7. Skech involue of he circle. 8. Skech he evolue of he cycloid. classificaion of saionary poins 9. Give he second order approximaion o fx, y expx y near i j.. Is i j a saionary poin of gx,y xy y x?. Is a local maximum poin a degenerae saionary poin?. Is a saddle poin a degenerae saionary poin? 3. Find a funcion fx,y wih a degenerae saionary poin a i j. 4. Find he Hessian of kx,y x 4xy 6y a i j. Page of 5

3 par wo: aemp 4 of 5 quesions for 6% from par wo.. Le f be he funcion fx,y xy. i Skech he conour f. ii Skech he conour f. iii Skech he conour f. iv Show, in a single skech, he 9 conours f d for d { ±, ±4, ±9, ±6}. v Draw he graph of f as a surface in R 3. Also draw he box wih 6 faces x ±,y ±,z ±. Clearly show ha he graph cus each of he four verical faces in a line. Find he equaions of all four such lines. vi [ Here f d R denoes he conour or level se on which f has value d R ]. Le Γ R be he curve paramerized by c xi yj zk i coshj, i Roughly skech Γ R he caenary. ii Compue he velociy vecor ċ. iii Compue he acceleraion vecor c. iv Compue he speed scalar ċ. v Compue he oal arc lengh L of Γ. vi Compue s he arc lengh measured along Γ from c o he general poin c. 3. Le c xi yj paramerize a curve Γ R. i Give a formula for he curvaure κ a c Γ in erms of ẋ,ẏ,ẍ and ÿ. ii Paramerize he cardioid curve. iii Roughly skech he cardioid. iv Compue, showing deails, κ for he cardioid. 4. The cardioid curve is paramerized by c?i?j, π. i Compue he velociy vecor ċ. and he acceleraion vecor c. ii Prove ha he speed ċ?. iii Compue he uni normal vecor n. iv Prove ha he curvaure κ?. Page 3 of 5

4 v Prove ha he evolue curve is paramerized by e? i? j. vi Skech he cardioid and is evolue. 5. Le f be he funcion fx,y x 3 3x 3 3xy 6y y 3 3y. i Compue he firs order parial derivaives f x and f y. ii Compue he second order parial derivaives f xx,f xy,f yx and f yy. iii Prove ha f has wo saionary poins and find boh. iv Compue boh saionary values. v Prove ha boh saionary poins are non-degenerae. vi Classify each saionary poin as local maximum, local minimum or saddle. ooo Page 4 of 5

5 MATHEMATICAL FORMULAE Noe: some formulae do no for hold for all variable values. i Trigonomeric Formulae cosx y cos xcos y sin x sin y { cos x sin x anx y an x an y/ an xan y cosx cos x sin x sinx sin x cos x anx an x/ an x an x /, an/ cos x /, an/ sin x /, an/ cos x cos x/ sin x cos x/ cos xcos y [cosx y cosx y]/ sin xcos y [sinx y sinx y]/ sin xsin y [cosx y cosx y]/ cos x cos y cos[x y/]cos[x y/] cos x cos y sin[x y/]sin[x y/] sin x sin y sin[x y/] cos[x y/] sin x sin y cos[x y/]sin[x y/] sin cos cosec co arcsec x arccos/x arccosec x arcsin/x cosh ix cos x sinh ix isin x ii Differeniaion iii Inegraion fx f x fx f x x n nx n lnx /x cos x sin x sin x cos x an x sec x sec x sec xan x cosec x cosec xco x co x cosec x e x e x a x a x log a arcsin x / x arcan x / x arcco x / x arccos x / x arcsec x /x x arccosec x /x x iv Mensuraion fx fx dx fx fx dx x n n x n /n /x log x cos x sin x sin x cos x an x logsec x sec x logsec x an x cosec x loganx/ co x log sin x e x e x a x a x /log a / x log x x / x arcsin x /x arcan x / x [log x log x]/ / x x arcsec x / x log x x Area of riangle ABC/ab sin C ss as bs c where s /a b c Cosine rule for riangle ABC: a b c bc cos A Disc, radius r: Areaπr : Circumferenceπr Cylinder, radius r,heigh h: Volumeπr h: Surface area πrh Cone, radius r, heigh h:volume/3πr h: Surface area πr r h Sphere, radius r: Volume4/3πr 3 : Surface area4πr Page 5 of 5

6 ANSWERS MOCK-EXAM ms37 Inroducion o Differenial Geomery Auumn 8 J.B.Quigley 7-3 K3 K K -8 K K Figure : iconical helix, ii cardioid par one: aemp of 4 quesions for 4% from par one. image, graph, conour,curves ec. i Skech he conour f of fx, y x y. { The conour is he se x } x y in R. This is he se { x x y } which is he uni circle cenered on he origin, skech omied. ii Skech he conour f of fx, y x y. The conour is he subse of R consising of poins which saisfy he equaion x y x y ;i.e. he single se { } conaining one poin, he origin, skech omied. iii Skech he graph of fx, y x y. Consider he graph of z x x in R which is a sraigh line hrough he origin. The graph of in R 3 consiss of he laer roaed abou he z-axis and is a cone surface, see figure3 iv Compue he speed of c i /j. ċ ẋ ẏ v Give a formula for he velociy vecor of c xi yj. Page 6 of 5

7 Figure : i cycloid and evolue, ii Archimedian spiral, involue of circle K K K5 K Figure 3: i cone surface, graph of z x y, ii conours of z xy Page 7 of 5

8 y x.5. Figure 4: i graph z xy, cus enclosing box, ii graph from Q K K K K K Figure 5: caenary cosh as average expx expx/: and sinh Page 8 of 5

9 K3 K K K3 K K K K K K Figure 6: cardioid and evolue a lile cardioid c ẋi ẏj ẋ ẏ vi Compue he acceleraion vecor of c i /j. c ẍi ÿj i j j arc-lengh vii For c as in 5 above wha quaniy is ẋ τ ẏ τdτ? a This quaniy is he arc lengh measured along he curve Γ R paramerized by c from he poin ca Γ o he poin c Γ viii Wrie down he oal arc lengh L of one urn of he helix. Wih usual paramerizaion c a cosi a sinjbk he oal arc lengh is L π a b curvaure ix Give a formula for he curvaure of c as in 5 above. κ deċ, c <ċ,ċ> 3/ ẋÿẏẍ ẋ ẏ 3/ x Wrie down he curvaure κ of he sraigh line y x. The curvaure of any line is κ /. This is because a line is a degenerae circle of radius. Alernaively use he formula of he previous wer wih c i j. xi Wrie down he curvaure a cos sini sin cosj This is he paramerizaion of he AS curve and was covered in he lecures. From memory κ /. xii Wrie down he curvaure of he circle c 3cos7i 4 3sin7j We can wrie c 3cos7i 4 3sin7j. This shows ha raversal is negaive i.e. C.W. Since he radius is 3 he curvaure is κ /3. involue and evolue xiii Give formula for he involue of c as in 5. i c s x s ẋ ẋ y ẏ ẏ where s a ẋ ẏ d. xiv Give formula for he evolue of c as in 5. e c n κ x y ẋ ẏ ẏ ẋÿ ẏÿ ẋ xv Skech he conical Helix c cosi sinj k. See figure. Page 9 of 5

10 xvi Skech he cardioid. See figure6. xvii Skech involue of he circle. See figure. xviii Skech he evolue of he cycloid. See figure. classificaion of saionary poins xix Give he second order approximaion o fx, y expx y near i j. f x f y f xx f xy f yy f expx y. A x y Thus hese are all equal o. Thus f, Df,, D f,. fx, y f, Df, x y x! x, y D f, y, x y x, y x y x y x xy y xx Is i j a saionary poin of gx,y xy y x? i j. Firs order pds are g x y a and g y x a. Thus i j is a saionary poin. xxi Is a local maximum poin a degenerae saionary poin? No. de D f >, xxii Is a saddle poin a degenerae saionary poin? No.de D f <, xxiii Find a funcion fx, y wih a degenerae saionary poin a i j. fx, y x 3 y 3, check ha Df,, and de D f,. xxiv Find he Hessian of kx, y x 4xy 6y a i j. D h hxx h xy h yx h yy 4 4 x which is so for all in paricular for y x i. y quesion Le f be he funcion fx, y xy. i Skech he conour f. ii Skech he conour f. iii Skech he conour f. iv Show, in a single skech, he 9 conours f d for d { ±, ±4, ±9, ±6}. v Draw he graph of f as a surface in R 3. Also draw he box wih 6 faces x ±, y ±, z ±. Clearly show ha he graph cus each of he four verical faces in a line. Find he equaions of all four such lines. vi [ Here f d R denoes he conour or level se on which f has value d R ]. wer The conour f R is he se {x xy } {x y } {x x } and is made up of wo sraigh lines, namely he x and y axes. See figure3.. wer The conour f R is he se {x xy } and is an hyperbola consising of wo unconeced pars, one in he second quadran and one in he fourh quadran. The lines f are asympoic o his hyperbola. See figure3.3 wer This conour is again an hyperbola bu now lying in he firs and hird quadrans. See figure3 Page of 5

11 .4 wer See figure3.5 wer See figure4. The graph is a saddle. I is he usual saddle z x y roaed by π/4. Noe ha xy X Y afer a change of variable x X Y, y X Y. This explains similariy of he graphs, boh are saddles. To see where he graph cus he verical side plane x suxsiue x ino z xy o obain x and z y. There are four verical faces x ±, y ±. Each is cu in a line x, z y x, z y y, z x y, z x Thes lines can be seen in figure4 quesion Le Γ R be he curve paramerized by c xiyjzk icoshj, i Roughly skech Γ R he caenary. ii Compue he velociy vecor ċ. iii Compue he acceleraion vecor c. iv Compue he speed scalar ċ. v Compue he oal arc lengh L of Γ. vi Compue s he arc lengh measured along Γ from c o he general poin c.. wer By definiion cosh expx expx. In figure5 we see he cosh curve as he average of growh exp x and decayexpx.. wer Firs we summarize some well known formulae. coshx expx exp x sinhx expx exp x cosh sinh We have sinh cosh cosh sinh c x y Thus he velociy is c x y cosh sinh.3 wer Also he acceleraion is c x y.4 wer Also he speed is.5 wer The oal arc-lengh is.6 wer c cosh x y sinh cosh cosh L The general arc-lengh is sinh c d cosh d sinh sinh sinh [ e ] e 3 quesion L c τ dτ coshτdτ sinhτ sinh sinh sinh [ e e ] Le c xi yj paramerize a curve Γ R. i Give a formula for he curvaure κ a c Γ in erms of ẋ,ẏ,ẍ and ÿ. ii Paramerize he cardioid curve. iii Roughly skech he cardioid. iv Compue, showing deails, κ for he cardioid. Page of 5

12 3. wer 3. wer κ deċ, c < ċ,ċ > 3/ ẋ ẏ de ẍ ÿ ẋ ẏ 3/ ẋÿ ẏẍ ẋ, ẏ 3/ The explanaion for he following parameerizaion of he cardioid will be supplied soon, when ime permis. c : [, π] Γ R x y 3.3 wer See figure6 3.4 wer cos cos sin sin The following curvaure compuaion is quie srenuous; if you can do i you will be well prepared for he much less demanding real exam. To compue κ we firs need ċ and c. x cos cos c y sin sin ẋ sin sin ċ Thus c κ ẏ ẍ ÿ cos cos cos 4cos sin 4sin sin sinsin sin 4 cos coscos cos sin sin 3/ 8 cos cos sin 3sin sin sin 4 cos 3cos cos cos sin cos 3/ 8 cos cos sin sin sin cos sin cos 3cos cos sin sin sin cos sin cos cos cos sinsin sin cos 3 cos cos 4 3 cos 8 cos 3/ 3/ 3/ ẋÿ ẏẍ ẋ, ẏ 3/ sin sinsin 4sin cos coscos 4cos sin sin 3/ cos cos sin / 84sin / 3/ 4 sin / 64 sin 3 / 3 8sin/ Page of 5

13 4 quesion This again is a srenuous quesion, he real exam quesion will be easier. The cardioid curve is paramerized by c?i?j, π. i Compue he velociy vecor ċ. and he acceleraion vecor c. ii Prove ha he speed ċ?. iii Compue he uni normal vecor n. iv Prove ha he curvaure κ?. v Prove ha he evolue curve is paramerized by e? i? j. vi Skech he cardioid and is evolue. sin cos cos cos sinsin sin cos cos cos / 4sin / / / / 4. wer c x y cos cos sin sin ẋ sin sin ċ ẏ cos cos ẍ cos 4cos c ÿ sin 4sin 4. wer The speed is ċ ẋ ẏ 4sin/ 4.3 wer The uni normal vecor is 4.4 wer n ẏ ẋ ẏ ẋ cos cos 4sin/ sin sin The curvaure was compued in wer o q3 above 3 κ 8sin/ 4.5 wer sin sin cos cos sin sin cos cos / / e c n κ See he lecure noes for his formula; for he cardioid e c n κ sin cos cos cos sin sin sin cos / c κ n cos cos sin sin Page 3 of 5

14 [ ] 3/8 sin/ cos cos sin sin [ ] 8sin/ 3 4sin/ 4sin/ cos cos sin sin 3 summary cos cos sin sin cos cos sin sin cos cos sin sin 6cos 3cos 4cos 4cos 3 6sin 3 sin 4sin 4 sin cos cos 3 sin sin cos cos c sin sin e cos cos 3 sin sin I is eviden ha he evolue is some sor of cardioid reduced in size by /3. See a maple skech, see figure6 In fac we can rearrange e e cos π cos π 3 sin π sin π This proves ha he evolue is a copy of he original cardoid bu a wih ime delay of π b scaled in size by /3 d refleced in he origin o accoun for he exra minus sign v Prove ha boh saionary poins are nondegenerae. vi Classify each saionary poin as local maximum, local minimum or saddle. 5. wer f x 3 3x 3 3xy 6y y 3 3y f x 3x 6x 3y f y 3y 6y 3x 6 5. wer 5.3 wer f xx 6x 6 f xy 3 f yx 3 Saionary ps occur when f x and f y f yy 6y 6 3x 6x 3y and 3y 6y 3x 6 x x y and y y x NOTE The soluion of hese wo equaions is very difficul, in fac more difficul han inended. The real exam will be easier The firs equaion ells us ha y x x. Subsiue his ino he second o obain x x x x x 4x 4x 3 x 4 4x x x x 4 4x 3 6x 5x x 4 4x 3 6x 5x x x 3 3x 3x x x x x The expression x x has only complex roos. Thus x or and y x x. Then x and y or x and y 5 quesion Le f be he funcion fx, y x 3 3x 3 3xy 6y y 3 3y. i Compue he firs order f x and f y. parial derivaives ii Compue he second order parial derivaives f xx, f xy, f yx and f yy. iii Prove ha f has wo saionary poins and find boh. iv Compue boh saionary values. summary There are wo saionary poins. x x and y y 5.4 wer f, Page 4 of 5

15 f, poin x 8 3 Also ded f, 93 7 ; and so y is also a nondegenerae saionary poin. summary The saionary values are f,, f, 5.5 wer ded fx, y 6x 6 3 de 3 6y 6 6x 66y 6 9 3x 3y 9 9[x y ] 9[x y ] 9[4x y ] 9[4xy 4y 4x 5] Thus ded f, ; x and so is a nondegenerae saionary y 5.6 wer Since ded f, 9 he nondegenerae saionary poin is a saddle poin. x y Since ded f, 7 > his is a loc.max. or loc.min. Bu rd f f xx f yy 6x 6y < x when. The nondegenerae saionary y x poin is a local maximum poin. y For good measure here is a maple plo of he graph of f, see figure4. Page 5 of 5