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final-1998

# final-1998 - MATH 100 MATH 1m Final...

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MATH 100

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Unformatted text preview: MATH 100 MATH 1m Final Examinaﬁon {31311111911199} Student Name: Student ID: 1. (a) Let G be the upper hali'ei'the eirele .1:2 +1312 = 2113, y 2 ﬂ eriented munterelmlrwise. Calculate the line integral [{em sing; — 21;]111: 4— (ea: easy — 2} rig. {3" {11) Find the value if 11 sueh that the integral f etmn+yﬂ1 {3111: +1 yrﬁﬂ is independent of path. Fer this 11 ﬁndthe pntential ﬁmeﬁnn nfthe veeter ﬁeld {eimuyﬁm Egan—1311]?) s. {11) Find the EuIﬂIﬂ-l area nfthe pm'tirm ofthe eylindermﬂ+rg =4that lies inside the eylindermﬂ+12=i :1 11's— «his—mi— {h} Evaluate the integral LI— u_:f_ a- 2:113:11 +ya+e 112113113... 1 3. (a) Findthe shortest distance ﬁnmthe [mint {2,ﬂ,ﬂ]| tn the surface: = —, wheres: :1 ﬂ 3?} andyieﬂ. {11] Find the ahsnlute maﬁmumtﬂ' the funeﬁnn f{m,y} =4my—m’1 —2y2 on the reginn R={(m,1}|—25m52, 453,152}. 4. Locate all relative extrema and saddle points off{m,y} = (a: + y} e‘lmﬂﬂ'ﬂw. 5. {a} Let f{m,y,z] = (.1: +yj2 + {y -|—:=*:}|2 + [z + 3:12. Find the maximal rate ofdecrease of f at PM?! —1,2] and the direction in which this rate of decrease oocurs. {11] Find all points onthe ellipsoid 23:2 +31;2+J122 =9 at which the tangent plane is parallel totheplanem—2y+ﬂz=5. ii. (a) Let ardenorte the hﬂ‘lllﬂdﬂl'j' ofthe solid l'mundedhythe paraboloidsz =32 +3? and 2 213—32—32. Findthe suriaceintegral ffF-ndﬁ', wherermi—yj-I—zlr 0' and the surface a is oriented outward. (h) Calculate the surface integral ffF - ndS, where or is the upper half hemisphere ﬂ" 324—492 +1»:2 =412'2 ﬂoriented upwardandF =m’1yz’1i—2r1ygz‘j—mgk. ...
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