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Unformatted text preview: M
r Calculus II (Math 114) Fall 2006 The Makeup Final Exam There are nine questions on this examination. Some have multiple parts. It is important
to Show your work and justify each statement. You will receive partied credit for substantial
progress towards the answers. You will lose partial credit for answers that are not justiﬁed. Please write Iegibly. No ceiculators, books, or notes may be used except for one two~sided
8.5x11 sheet of notes. Name: Instructor: E3 Galvin E! Komendarczyk El Leidy TA: Problem 2. (lﬂpts) (a) The general ﬁrst—order linear differentiai equation is @ + Pom) m om). do
Under what conditions on (2(3) and/ox P(:z;) is this equation separabie?
(b) Suppose that yp(:1:) is a solution to the diﬁerential equation of the ﬁrst part, and that yam)
is a solution to the compiementary equation ‘ dy a; + P(x)y(:c) m 0.
Show that yp(:c) + Cyc(x) is a solution to the differential equation of the ﬁrst part, for any
constant C. ' ' (c) Solve the initial—value probiem' Solution 3 3"“!
“‘5 GuiFarah" C.
Lento _.... .%.3
= Nae, " ' 3 la“ Problem 3. (10pts) (a) Solve the homogeneous equation 3;” + 2y’+ 2;; m 0, M0) m 1, y’m) L 1.
(1:) Find the géperal solution to yrr+2yr+2y=$2 Solution 3.6% fin59 Hawk(3%“ M?“ "ii “4.26 4. = ZenLia.” Zb+2mxz+2bm+ Z:
a 7, 291x24. [LindPay +éa+ZJo+Zc>
a“+za‘+25: x" *9 Za= I} LlakllorO, hawk=0 ‘ 1'2." x+ .32 4. erg all Cnx +C1MX3 Problem 4. (IDpts) (a) Find the area of the parallelogram which has the points (~1, *2),
(0,2), (4,4) and (3,0) as its vertices. (1)) Let P be a parallelogram in the plane with all four
sides lengths equal. Use vectors to Show that; the two diagonals of P are perpendicular. Solution 10 .. ; Problem 5. (lOpts) Consider the circular helix Fm : (3 cos 1:, 3 sin ‘6, 4t) 3 ow Nth <~5mt,3ua.£,‘l§
(apt ) \Wltﬂﬁi‘imltﬂmlﬁllo = 5 sue) a 8 saw. = 5t (ZPJEB) ‘5) $19395 5 I
' “(H3% <5m‘g'j SAM
(2ft) (L) Tier 45,05» 55.3%.
55‘ > gem): 5m=5f =51?“ =='> 12”?” ll Problem 6. (lOpts) A ball is thrown into the air from the origin in myz—space (the my~piene
represents the ground and the positive y«a,xis points north). The initial velocity vector of the
baii is 270 m (G, 80, 80). The spin of the ball causes an eastward acceleration of 2 ft/s2 in addition to gravitational
acceleration. Thus the aoceieration vector produced by the combination of gravity and spin is a m (2,0, ~32). First ﬁnd the velocity vector 170i) of the ball and its position vector ﬁt). Then determine where
and with what speed the ball hits the ground. Solution "0,: <0! to, m
a=<2,0.3?> '
Wﬁ), Sam 413:. 42&,0,52+.$+<o,20,365 A 4A: w
M (3 o are: 421: week» 
P '—:(.+_"+‘80&$+ 40.6.65 VGA = $16 =5? 40:} = = 44:) got” Am 2 < a, gotfltﬁzﬁdﬁ
(3196) “Found: a; §3Lt\=0. 13 Problem 7. (lﬂpts) Let f(a:,y) m my be deﬁned on the domain Q m {(2:,y) : m2 + y2 g 4}. (3) Sketch Q. Is it closed, Open or neither ?
(b) Determine critical points in the interior of Q, and classify them as: local minima, maximal or saddle points. (c) Appiy the Lagrange multipliers method to ﬁnd critical points on the boundary of Q: {(x, y) : 51:3 + 312 = 4}.
(d) Determine absolute minima and maxima of f on 5'2, or Show that they do not exist. I m'm‘tm‘i "Z; I h " ‘ :a ” 141)
ﬂﬂosaluateme—faam W, Problem 8. (1013138) Evaiuate 1 w m
f f / x/E dz dy due.
a 0 a Solution ‘8‘ at": a”? n “iv 5’{ “9%?! E ﬂw‘w‘u an! Ahﬁﬁyo’iiyo} ; 1:; ~ mo) (FLFe: Mk fattec‘l”
Safe? bound? 3 F35 cc? (iftzaé ' 9L ‘Fﬁf chosﬁ 39": «CW mwdah‘iwr H 16 Problem 9. (Iﬂpts) Set 22 my) :{ E33; 7: (0,0); M (a) that the partial deriviatives g; and 9y both exist at (0, 0), What are their values at
i F (b) Show that Hmmypmm g(m,y) does not exist. Solution Kg. 01. 9‘) Sacha»: 1E3; atk,6\=&? w 0 17 ...
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This note was uploaded on 04/04/2012 for the course MATH 114 taught by Professor Temkin during the Fall '07 term at UPenn.
 Fall '07
 Temkin
 Math, Calculus

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