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101-2006&amp;2007-1-F10-Jan2007

# 101-2006&amp;2007-1-F10-Jan2007 - Kuwait University...

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Unformatted text preview: Kuwait University Math 101 Date: January 13, 2007 Dept. of Math. &: Comp. Sci. Final Exam Duration: Two Hours Calculators, mobile phones, pagers and all other mobile communication equipment are not allowed Answer the following questions. Each question weighs 4 points. tan (2:5) + 3:3 sin (E) 1. Evaluate the following limit, if it exists: ling] I r— .r 2. Let 6., - T 6 , if I): < 1 ir — 1| [email protected])= \$3 _ 1 'f > I ,1 .2: . ﬂ — 1 (a) Show that f is discontinuous at z 2 1. (b) Classify this discontinuity as removable, jump or inﬁnite. 3. Let r2 , if .r S 1 re}: (27.233 ,if.r> 1. Find the local maxima and the local minima of f. 4. Let f (:5) = g (8 — 2:2) 2%. Find the r-coordinate of the point at which the tangent line to the graph of f is horizontal and the x-coordinate of the point at which the tangent line to the graph of f is vertical. 5. Evaluate: 7r :t—i—l r a ——--————— dz. b /sin5:t cosa: air. () \3/32:2+63:+5 () '0 6. The graph of y : f(.t) intersects the line y = r at .r 2 0 and .‘L‘ = 1. Find f(:i:), if f”(.t) : 1+ 2:5 — 3:52. (drift: i t 7. Find an equation of the tangent line to the curve 3; : 7 + f m tit at .1: = l. 3 8. Find the‘average value, fan, of ffr) = 1 + V4 — :52 on [—2, 2]. 9. Find thearea of the region bounded by the curves x = y2 and 1: 2 —2y2 + 3. 10. The region bounded by the curves y = 3:2 and y : 4 is revolved about: (a) the line? 2 —1, (b) the-line 3: z 5. Set up an integral that can be used to find the volume of the resulting solid in each C386. 99 Kuwait University Math 101 Date: January 13, 2007 Dept. of Math. & Comp. Sci. Final Exam Answers Key 1. For :5 # 0 —— 1 5 sin: 5 1 and —x2 S xjsing S 3:2.Since.lin101:2 : 0 = ling—3:2), 1—. 14* then from the Squeeze Theorem MUEJIZ sin — = 0. ten (22:) 4- 1351116) 1 ta 2 1:351:11 t 2.1: lim——\$—:lim n( \$)+lim I 2211171 an( I)+lim.1czsi]1—: zqu 33 :m0 m 14.0 I 2—40 2]; :1: r-HO 2 x 1+ 0 2, (a) f is undeﬁned a.t :I: = 1 (also limf(\$ ) does not exist), so it is discontinuous. (bi .111.“ Fleiw ﬁiiwgﬂirw):ﬁfe” jump discontinuity at a: = 1. 3. For a: >1,f'(:c)= —3(2 — 2:)2, and for I < 1, f’ (z) z 25:. The critical numbers of f 113-. l—I-I lml—-——I - i ii) = 0 in n imni minimum oi f. l—I_ --| and f(1)— 1 is a local maximum of f. Another solution: from the graph of f, f(0 )— - 0 1s a local minimum off and f(1)= 1 is a local maximum of f. ‘7} 31 xvii). . 2 _ I2 Ln") l 41%): -Waf'in)=o=>_=mf gal, W“ 353 For vertical tangent: f’ is undeﬁned : :1: — 0 (f is continuous at m - 0). 5. (21) Put u=3\$2+61+6 : (111.:(63:+6)d3: d 1 1 1+1éi1wu§+ngwxg+6x+M 1/3z2+6x+5 W 4 (b) Putu=sinsc => U(0)6:0.1!.(%)21&du:COSI dz % um +0. 1 7'1 11 [sins 1: coszdcc=ju5 din: [if] i o D U 6. The points of intersection are: (0,0) and (1,1) .13., f (0) : 0 and f (1) : =/f”(z) d_z=/(1+23:—3\$2) d\$=I+I2713+CL .3 4 {L I . + —3_ - ~4— +01 3: -I- Cg. Since: OO 99 9. The points of intersection are: (1,1) 8c (171).}! is an 31,- region. 1 1 Area: f {(-2y2+3) — (1,2) de= / (—3y2+3) may 1 =2/(—3y2+3)dy: [jysmyj [email protected] U 10. (a) REVOLUTION ABOUT THE LINE y: —1: 2 Washer's Method: Volme : 7r] [(5)2 _ [x2 +1)? dz —2 (b) REVOLUTION ABOUT THE LINE :6 = 5: 2 (II) Cylindrical Shell’s Method: Volme : 2w] (5 g z) (4 — :52) dz. —2 a "\a OO 99 ...
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101-2006&amp;2007-1-F10-Jan2007 - Kuwait University...

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