{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

211-2005&amp;2006-2-F10-June2006

# 211-2005&amp;2006-2-F10-June2006 - Kuwait University...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Kuwait University Math. 211 June lst, 2006 Dept. of Math. & Comp. Sc. Final Examination Duration: 2 hours Calculators and mobile phones are not allowed. Answer all questions. Each question is worth 4 points. 1. a. (2 Points) Test the convergence or divergence of the series on tarflrr2 . ”=] n2 + 1 b. (2 Points) Test the absolute convergence, conditional convergence or divergence of the series :12 cos 2n ”:1 (1.3)" ' 2. Letx = eucosv, y = e“sinv. Show that 2 2 a x = ﬂ and Q = a_x 61:2 6V 6v2 6v ' 3. Find the linear approximation of the function ﬁx, y) = ,1x2 + y2 at (3,4) and use it to approximate 113. 02, 3. 99). 4. Find the points on the surface x2 + 4y2 + z2 = 13 at which the tangent plane is parallel to the plane 2x+16y— 22 : 9. 4 2 5. Reverse the order of integration to evaluate the integral I I e‘yldya‘x. 0 J27 6. Find the volume of the solid bounded by the graphs of the equations .ic2 +y2 = 1,2 = 114—):2 —y2 andz : 0. 3 M? (m 7. Use cylindrical coordinates to evaluate J- I I MW. —3 _ 9%; o 8. Let ?(x,y,z) = a”? + cos y? + sinzzz. a. (2 Points) Find curl F3. b. (2 Points) Find div 2". 9. Let "ﬁ(x,y,z) = 3x2y2? + (1 + 2x33»)? 3. (1 Point) Show that the vector field ?(x,y,z) is conservative. (lU—b —-> b. (3 Points) Evaluate the line integral I ) F - dr. (1,4 10. Use Green‘s theorem to evaluate fxzydx + xdy, where C is the triangle with vertices (0,0), (1,0) and (1,2). C CO CO Kuwait University Math. 211 June lst, 2006 Dept. of Math. 35 Comp. Sc. Final Examination Solution 1 (a) The series 2— tan 1R2 diverges by LCT' .1m , . a.” _ 7r _ _ tan—1 n2 m 1 "1323(E) —§>0, an———-n2+1 andbn—E n2 cos 271 (-—l 3)” The series 2“” n=1( (b) S 1:32”) n2 nconverges by the Ratio Test ; TL lirn 30(a"+1) = lim (n+1)2 lim (1.3) = i < 1 n—roo an n—mo n So the given series is absolutely convergent. 2. E —e“c 7} 522—1- ﬂeucosv @ —e” 31} => @ —§§~ Bu _ OS ’ Bu? # ’ 81; ._ C0 Bu? _ BU 6y __u 623; _ u. as: u. 623; _83: 3—,; —-e cost), W ——e sun), 61!) ,5 81117.) I) w —8’U 3. _. 2 2 _ x _ :1: HIM!) ~vr +9!) fx(\$ayl _Wi 13(33an —W 3 4 f(3:4) =5: f\$(394) 2% fy(3:4l :3 The linearization of f at (3, 4) is I MSW) =f(3,4)+fx(314)(\$—3l+fy(3a4)(ye4) = 3(3w+4y) and the linear approximation of f at (3, 4) is 2 .12 L (3.02, 3.99): g (3 02) +: (3 (99): 5 1(9.06 + 15.96) = %— x 5.024 Thus, (3.02)2 + (3.99)? e. 5.024. 4. Vf (\$4,!) =< 23:,8y,22 >=> 25': 2 2k, 83; :16k, 22 = —2k => y = :c = —z. Substitute in the equation of the surface we have 3:2 = i1. Thus the points are (1, 2, —1) and um MN 4 2 2 y2 2 2 5. f/e‘yadydx = [fewygdxdy = jy2e’y3dy : *% [€910 2 g (1 - 6’8) 0 {/5 u 0 o .9 27: 1 = fdﬂ/rvtl — rﬁdr O 0 = % W— 2) w 7. 3 «9%? x/25—z2—y2 211' 3 25mr / f f de : [d9] / zdzdr —3 _ 9_:c 0 0 0 u 3 W/(25—T2JTdr= “3% O 8. (a) _’ _) _, 1 j k cur] Y; : 2 2 ﬁ- : “mew-I? =< O, U, ~scexy > 6517 8y 62 e39 cosy sinzz (b) 8 a (9 . —» a; . . . d1v F = E (e 3') + 3—1; (cosy) + 5; (Sin2 2) : ye” — 31111; + 251nzcogz 9 (a) - 65—11.: = (3ny — (2;: => F (31:, y, z) is conservative From ﬂ = 3322312 and a 2 1+ 2x3y we have f (3:,y) = 2233,12 + y + C. 63: dy (3,1) Thus, / "ﬁ’ - at"? = [333312 + yﬁ‘jﬁ = 28 — 20 : 8 (1:4) ' OO 99 10. OO 99 ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

211-2005&amp;2006-2-F10-June2006 - Kuwait University...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online