MTH 206 Final W Solution, Spring 2010

LEBANESE AMERICAN UNIVERSITY Department of Computer Science and Mathematics Calculus IV inal Exam Spring 2010 (June 9, 2010) Name: t ID: QuestionNumber —‘| Grade Vt, 4% ; 2.6% [ 3. 6% 14.5% | 5.5% | te3% 7.6% 8.8% 4 | 9. 6% pm ' 11.5% 142. 6% | L 13.6% 14. 6% i 7 L _— 15. 6% | 116. 4% 17.10% Tota 10! i 1. Find the direction in which the function f(z, y¥) = zy? — yz? is increasing or, decreasing most ost rapidly a “y) TS aa Seat ~~ : &) Uf - Dla : ‘ ; " * ond Meet \ — dx. 4 pice Bie det ta i 99 | hit | 4 2, Approximate the value of 9(4.01)2(—7: 99)Pusing linearization. : 93 £ (%5)* A x® -o @ C¢,—-*%) £UG, -8)2 BEF : bre fy- 78 ; IL meee | ; ode x f(t, -8)t ( Cx-4) «f (4+ 8) . sagy ate (el) 102 (ee) 3. Find the ebeciies maximum and minimum values of the function F(z,y) 8 we sel» sf {Wo nes 3 gla> Ge) + 4. Find parametric equations for the normal line to the surface z = 7x? — 5y? @ the point (2,1, 23) - 23, —loy —K. /.) Op = I¢zi logy —E lea s,25) i fos i Q+e2e . ~ lg —lo€ Yet 5. Identity the type of surface represented by + 2 1 hegpertolas Sane CM 9* day 0 (Sat) ‘ Wot pasting Wsorgh ae - ALL Kindo J =m. 6. A rectangle with sides parallel to the e axes is inscribed in the region bounded by the axes and the e line ) 2+ 2y = 2. Find the maximum area of this rectangle. , By 7, Find the volume of the region bounded by the coordinate planes, the parabolic cylinder # = 49 — 2, ¢.) re y aga? ole dy da. o O 8. Evaluate the integral / | (y — 4x)(6x + y)dedy where R is the parallelogram bounded by the lines —6z +7, using a transformation. y=4e+8,y=40+10,y= 62+ 4y= ~ uryv $7. Use we f-ER 25 TS Vi baey > 64t ¢v , y) (eo ~Yo Vio C/o fio 6. rivd.t RBLuUClO- 2 /o lo * Dur < C5 wy [ol medy = eS, fo s 9. Calculate the flow in the field F = (x —y)@ — (z? + y”) j along the path C fro = the upper half of the circle r = 3. Gar ig ieee re a (30) sad) x | {.: a49(®- Men > 9°(e)-? 9 =0,2=1y=0,2=0 ; 14. Find the curvature of the space curve: FO=AT 4G m7 sant - a N= |ayfe [Ly xaj ve ig eet & Wat . - As ge 7 x ol - ‘ Thee t0u> * 17. cob the function f(z) = —I<at<\ “oe (a) Find its Fourier series O71ne < “ old = . Qa, +O Te27 by BL 2. ( # sin (WH) dy | | r aa Np sin Coad de = al6 wna | T Th a = Me (ont) = : Tn Ge yr odet Founnin Seve-~ - b sm Te + be Sm Te + L. Sw 5 Teg nes = Basu Te 4 DL sm>Tey 32 Sw Biv 4+: 4 T ei us