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Unformatted text preview: 18. The function f(.1:,y) = 32(2+y)+y2+4 on the disk D = {32+(yi1)2 S 9/4} has critical
points at (0,0). (2,—2) and (—2, —2). The following graph is of z = f(:1:,y) restricted
to the boundary of D, parameterized by a: = guest, :1; = —1+%sint, 0 51:5 211'. Find
the absolute minimum value of f[:1:,y) or: D. (A) 0 (B) 15 ":3: .
(C) 215 ‘° ::;;:;::::3::::::::::;::';::'"'""'"“""“ 1
(D) 3 u 1 (E) 4 I II..—_:':::':::::::::.:_:: . (F) 4,25 11: "'‘.'".."::".:27:31:: 5 .1 .__._,_:.'::::.'
(G) 7.2 a ' "j ""::"I"::":: '
(H) 73 {7,5 —— ——  ——+———::.——'. n:  ... _:...“....
(I) 8 n 7.2.; ..1“::::_.  1”.“ l — —7—.: .1. ....... .......
(.1) 10.25 '3: xiii. ‘2‘}; hyjnﬂziy]+yzv4rullﬂulldioxzt[yvﬂazﬂ4 19. Refer to problem 18. Find the absolute maximum value of f (3,3,1) on D. (A) n
(B) 1.5
(C) 2.15
(D) 3 (E) 4
(F) 4.25
(G) 7.2
(H) 7.3
(1) s {.11 10.25 2_' Find the minimal value of ﬂat. y) = 2m2 + 31:;2 subject to the constraint 9:2 + y2 = 4
(A 6 :2);
. (H)11
(D)12 (1)13
E 14 ' (1)15 a.
() atx‘m‘ﬂ’ﬂl 234:“:
pm gemm A? two — Wu H m“ e42“ W": “M" °‘ 0
w: would W 42... abswcﬂ‘Jrﬂ 3. For the function f(a:, y) = a.“ + 3mg 1 33:9;"2, fhas: 2
. O2 ‘73 “ 5:34 3‘l = 36"”)
(A) saddle pomts at (1,1) and (I. — 1) '1 31M")
(B)Iocal max'ima men) and(— 1, ~ I) 0’ ’33 " 3“ * 3"
(C) local minima at (0,0) and (4,1) n.._ 3 ¢ X
(D) a local maximum at (0,1) and a local minimum at (3,  3) =3 x. "' a saddle point at (0,0) and a local minimum at (1,1) a. l 1 _. (a
a saddle point a: (0,1) and a local minimum at (0,0) 0" x
(H) a saddle point at (4,3) and a local minimum at (0,0)
(I) a local maximum at (1,1) and a local minimum at (3,1)
, (I) no critical points 9“ 9.1.1 ”‘1”: ' @a saddle point at ( — 1,3) and a local maximum at (0,0) =5 2.1.9.“ )ﬁ ’0 2 ‘1 2. Suppose the temperature in degrees Celsius at a point (as, g) on a metal plate is
given by the formula T(z, y) = lone(31W). Which of the following unit vectors gives the direction in which T decreases most rapidly
at the point (1,4)? (A)<—1,~1>/¢§ {(F)<3;4>25I (B) <1,4>!x/ﬁ (G) <2,3>/¢ﬁ
(C)<—3,~I>/\/E (H)<—4,—3>/5
(D)<1,0> ' (I) <1,1>/\/§
(E) <0.1> (J) <5.2>N'§9
Dutch.“ 1'. “(VT)“.UX ,: + ‘00 (Mi) ___ +¢'+>{~§
LQTT‘U‘ \ lite (9.8)} 4) Find the Linimum value of the function f (x,y,z ) = 4x + 4y + 42 that
lies on the intersection of the sphere :2 + 3:2 + 22 = 3 and the plane z+2y+z:4. F\
3 A)6
3):? .. 9??
3% VP :(waw , [7942:932»
Ifﬁg VA : < 1/ 2/ ‘7
12 _. Fran (/7 $19) Wire P?‘0
H} ,5: r) L/_ a) x +,u 11:14 1; «~— 3??) 459/" Fm 07+ a) M m (:3: J
I)? 3)L/:a>"2 ‘1'” Fro» itbﬂbrz) Q's=9" [ii/ii From 8’
X"+(z~x)“+x" = 3* "’9’ ”:3 0”
BXL"qX'F' £0 Jo
(EXPIYXH )io X;/ art/2v (Ix/ﬂ)
am! +(4//;)=/L Mitt/Mt" aimme w of the function le.y)= Ky Q the unit
:circle x2+y2=1 (cg. not the inside). ( ) X13: /
: 1 K : I
{(593) aura anJ'f/‘GHT‘ 3 lb 4) Find the glinimum value of the ﬁmction f (x,y,z ) = 4x + 4y + 42 that
lies on the intersection of the sphere :2 + y2 + 22 = 3 and the plane .z+2y+z:4. F\
B):
32% V‘F ~‘‘(‘4/9/‘/> / [73:<2x,ab,a?> VA : <1,2,t?
12 r) L/: svx ‘7” From a; um) um»: 9750 H 25: . ﬂ
I))§14 ”p.503 +50“ Pram aumeH/ag
"’3'? ”enema +,or an Mme) Q‘s=‘f—‘9X 3 9‘26
From 8'
y‘: (Man X’" = axis/x +‘/= 3 0”
3X"—‘/X+ I =0 30
(3% ~! TX" VD X::/ are}?! (bl/f)
am/ ‘F(’///U:/L MIA/Mr" ...
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 Summer '17
 Janey
 Critical Point, pH, Fran, .——

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