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Unformatted text preview: MA 174 EXAM II 04/01/2011 gm. Name: ___.___ PUID: ____._______ Section: SHOW ALL YOUR WORK. NO CALCULATORS, BOOKS, OR PAPERS ARE
ALLOWED. Points awarded
1. (10 points)
2. (10 points)
3. (10 points)
4. (10 points)
5. (15 points)
6. (10 points)
7. (10 points) 8. (10 points) 9. (15 points) Total Points: wmmnwxwxww mm“;WWa“m“w.mm“mmWmmmmmwmwmwnwmwmmmwwmwmmmwmwyaw g
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,3 1. (10 points) Find an equation for the tangent plane of the surface
way + cos (my) —I— 'y — 2:2 = M2 at point (0, Z712, l). 3“”) $11: I:(X.j.‘?>) 3 9063+ 00M} “1“ 9 ‘"’EZ"/h 1 Ct) /,,
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W B m: “* ‘ _
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D)”~ 2X+yv222 /n2/'L
. . 8w . .
2. (10 pomts) Fmd (Eh, at pomt (x,y,z) = (0,1,7r) 1f
w=m2+y2+22 and ysinz+zsinx=0.
N. as
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(ax )j , 2x+1i «DX
3‘? ” d 4): (but. th‘9‘.$): \‘jgw‘ﬁ'f 25L1><>
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ﬂCJOi‘fJVix 2X (0,],7v) QD/(JjLShD Wmmv«mm»«amﬁm'saﬁwxiwWWWWWWWWWMYWﬁWﬁW mw‘:wmmm\mwmm . 3. (10 points) Use Taylor’s formula to ﬁnd a quadratic approximation of ﬂag, y) = 62“” at jaz(0’0)'7z«xg>z few w 4+ m + ~~;—<><vwwfw+mg
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2X#j 4. (10 points) If the derivative of f (cc, y) at a point P in the direction of i + j is 3\/§ and in
the direction of i — j is 2\/§, what is the gradient of f (:6, y) at the point P? .J a.) at ﬁ/P : 0x6+£J ‘4
\J 'V
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M a/:) Q (Ma/f” MW M )ldil 5. (15 points) Find all critical points of function
f($,y) = 3:4 +y4 —— 4xy+1 and identify each as a local maximum, local minimum or saddle point. we
~~:> X 62 >< : O «‘5 wxﬂmxiocﬁwm : lxc VX3~¢Y :0 ’5 y‘xs} l7< Wk“ ‘D “(My ~— scaa 7mm:
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r 6. (10 points) Set up an iterated triple integral that gives the volume of the tetrahedron
in the ﬁrst ootant bounded by the coordinate planes and the plane passing through
(1,0,0), (0,2,0) and (O, O, 3). Do NOT evaluate the integral. (3.343: 7a FUR/ML ?W5W (raw/ramyﬁ Imam Vi)
éx+éy+22~26 l/vuz. g'L’W/J} (owe/WV c/l%0lyoix>
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} 24mg ML 0 m1 711 g“ Wij/vjcme V3 éYﬂycé y
03/ 2304 7/1} /‘ y >/J )nvrl’) W o [M Z'ZX /{ for the integral foz fag” f (m,y)dydm and
write an equivalent integral with the order of integration reversed. SAX. w W % {7L1 Fajita; v5 Um X—Swwth, , chum w @7wa
[It 9/ fwd/3 ’83 7541 JAN: 3:; LX) : Xlnml’B W D g J;
Ye/J‘M‘h W 0 £2 2/ 8. (10 points) Evaluate 0’” OWermdy.
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”X: ymg 3*: rim/[‘19 akéﬂg 3 )’ MIMI/”(9 ZJ
J‘f”lf{(’"”)"”em‘dw =3 f: f
52' '7»), l
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