BSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2

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1 BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root tst with a n n + 2 n2 x n, w hav lim n a n n n lim + n x x, n n n sinc log, and hnc n n log n. hrfor, r. n n n For x, w hav to study th sris whos gnric trm is + n 2 n log+ n n n n2 n n n n2 n 2n 2 +o n 2 n 2 n, and thrfor it is divrgnt. Finally, for x, w hav th altrnating sris hr holds d dt log+ t t2 + n n n + n 2 n. n t t [ t + 2 log + 2 t t + t [ tt + 2 ] t + o t2 log+ 2 t 2 t t ] t2 log+ t t and sinc clarly 2 < for t > as this is quivalnt to t 2 /2 + t >, w s tt+ t 2 that th squnc + n n n2 n is dcrasing for n sufficintly larg, and it is infinitsimal from its asymptotic bhavior computd abov. hrfor, by Libniz s rul, th givn sris convrgs for x.

2 2. Find all th stationary points of th following scalar fild, dfind on R 2, fx, y x 2 2xy y + y and classify thm into rlativ minima, maxima and saddl points. 2 omput th gradint of th following function of x, y: W hav gx, y x xy. f x, y 2x 2y, x f y x, y 2x + y2. At any stationary point x, y, on has fx, y,. his happns xactly whn 2x 2y and 2x + y 2. his is quivalnt to y x and y 2 2y. By solving ths quations, th stationary points ar x, y,,,. o classify ths points, w comput th Hssian matrix 2 2 Hx, y 2 6y. 2 x y x x y 2 y 2 2 At th point x, y,, w hav Hx, y and its dtrminant is 8 and 2 6 th trac is 8, thrfor, it has positiv ignvalus and, is a rlativ minumum. 2 2 At th point x, y,, w hav Hx, y and its dtrminant is 2 2 8, thrfor, it has both positiv and ngativ ignvalus and, is a suddl. 2 By chain rul and Libniz rul, g x xy + x y xy, g y x2 y xy. :

3 . Lt fx, y x, y. omput th lin intgral f dα, whr is th sgmnt of th parabola y x 2 from, to,. By dfinition of lin intgral, w nd to tak a paramtrization of th parabola. On such paramtrization is αt t, t 2, t [, ]. With this αt, w hav α t, 2t and fαt t, t2. Now th lin intgral can b computd: f dα t, t2, 2tdt t + 2t t2 dt [ t + t2 ] + + 2

4 4. omput th intgral dxdy x log + x 2 + y 2, whr {x, y R 2 : x 2 + y 2 4, x }. Goint to polar coordinats, th rgion corrsponds to { r, θ : r 2, π 2 θ π }. 2 With th Jacobian Jr, θ r, th intgral bcoms dxdy x log + x 2 + y 2 r cos θ log + r rdrdθ 2 π 2 π 2 2 r cos θ log + r rdθdr 2 r 2 log + r dr [r 2 whr in th last stp w intgratd by parts. Not that ] 2 2 log + r r dr, + r r + r + r2 + r + + r + r2 r + r, and hnc Althogthr, 2 r + r dr r2 + r 9 2 log + r + onst., dxdy x log + x 2 + y 2 [r ] 2 [ + r log + r log log 2 6 log 4 log log + r 2 log + r2 ] log 2

5 5. Lt Fx, y, z zy2, yx2, xz2 b a vctor fild on R, b th boundary of th squar with vrtics,,,,,,,,,,,. omput th lin intgral F dα, whr α is a paramtrization of going countrclockwis. Lt S b a surfac which has as th boundary, and ru, v Xu, v, Y u, v, Zu, v b its paramtrization such that th invrs-imag of α is also going countrclockwis in th uv-plan. Stoks thorm says, if w lt n N r whr N, thn it holds that N u v F dα curl F n ds. S For th abov, w can tak S {x, y, z : x, y, z }, and a paramtrization Xu, v u, Y u, v v, Zu, v. h corrsponding rgion in th uv-plan is {u, v : u, v }. It follows that r r,,,,, u v and hnc r,,. his last vctor is alrady a unit vctor. u v Aftr som straightforward computations actually, w only nd th z-componnt, w obtain, curl F y 2 zy2 z 2 xz2, 2xy yx2 2yz zy2 hrfor, on th uv-plan, namly x u, y v, z, curl Fru, v nu, v,, 2uv vu2,, 2uv vu2. Altogthr, through Stoks thorm, th givn lin intgral bcoms F dα curl F n ds S 2uv vu2 dudv 2uv vu2 dudv [ vu2 ] dv v dv [ v v] 2.