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Unformatted text preview: MATH 234 (A. Assadi) — Fall 2007 — MIDTERM II
Thursday, November 8, 2007 NAME (print):
Instructions: 1. Write your name on each page. 2. Circle the name of your TA: CHILDERS TONEJC 3. Closed book. Closed notes. Calculators allowed. No laptops.
4. 2 index cards of size 3” X 5” with formulas allowed. 5. Answer your questions on the exam‘paper. There are some extra pages in the
back if you need them. 6. Show all your work. Partial credit is given only if your work is clear.
7. Enclose your ﬁnal answers clearly in a rectangle. 8. Time allowed: 70 minutes for your work, plus 5 minutes for checking your
answers. ' 1—1
—
2o —
—
—
—  Total 100+5 1 + cos 20 . 1 — COS 29 1 sin"_1 aa: cosa — 1
cos2 0 = —2 , sm2 6 = —2 / Sin” ax day = ————————————~_na x + n / sin"~2 an: d2:
n sec2¢9=1+tan20, csc2621+cot20 ’ n COSn—l amsinaw n_1 _2
sin(A:i:B)=sinAcosBicosAsinB /COS awdm: no, + n fcos axd'ml
cos(A :l: B) =: cosAcosB 2;: smAsmB /Sin ax COS by; dx : _cos(a + (2)27 * cos(a —— b):r
' tanAitanB 2(a+b) 2(a—b)
tan(A:1:B)= — . .
letanAtanB . . s1n(a—b)zv sm(a+b)x
smaxsmbxda: : —————— ~— ————————— 
2(a—b) 2(a+b)
/_ﬂ2_~=3_tan_1 ax—b fwsamsbmdm=m+w
mx/aw—b b 2(0’5) 201+”)
2
/ dz: = _2_ln Va$+b~\/5 /sinaavcosamda:= —CO:aa$
xx/ax+b x/E x/am+b+\/5 1
dsc 1 m footamdx= —lnlsinax
/ = —tan_1— ‘1
a2+x2 a a 1
d3; 1 m+a /tana:cdx= Elnlsecaxl
— —l ,
/a2—:I:2 2anx—a dm 1t ax
/ d1” _11 33 /1+cosa:L‘—;an—§
— n
x(a:c+b) b ax+b / dx 1 am
— =———cot———
dac __ ‘x + 1 tan_1 m 1 — cos as: a 2
2 22 ” 2 2 2 "—E —
(a 2a ((1 +23) 2a a ftan2amdx=étanax—m
:1:
———~—————— =1n(a:+\/a2+:c2)
Va2+$2 footzaxdxz—icotaxsc
'1/ 2 2 ,/ 2 2
a—Hd$=‘/a2+$2_a1nw 1
‘ :1: ac /seca:vda;: Elnlsecamt—tanaxl  —lln W fescaxdx— ~llncsca$+cotax
xx/a2+:132_ a m . a
1
/ dzr __Va2+x2 /sec2a:1:d:z:=: atanacc
: x2x/a2+:c2 _ (12:5 1 1
/ d2: . _1 112 /CS()2 amdx: —; cotazz;
———=sm ——
. W a
2 /1naa:dx=a:1nax~m 2 2 a 1
eaxda: = —e‘”‘
dac _ 11n a+Va2—:1r:2 _/ a
——————2 — 2 — '—‘— an:
x, /a x a w / eaa: COS M d]: = (126+ b2 (a cos b2: + bsin bx) d3: _Sin_1<m—a,) M
\/ 2am — 3:2 a / 6‘” sin but d3: = (126+ b2 ((1 sin bx — b cos bar) as a
’ 2
/\/a2+m2d:r=gx/a2+x2+%ln(a:+ (12—11102)
2
/\/$2—a2da:=g32Vx2~a —%~lnlx+\/x2—a2l NAME (print): ' Problem 1. The plane :5 + y + z = 2 cuts the cylinder $2 + y2 = 4 in an ellipse (see ﬁgure). Find the points on the ellipse that lie Closest to and farthest from the origin.
' z NAME (print): Problem 2. A thin plate covers the triangular region bounded by the m—axis and the
lines a: = 2 and y = 2:3 in the ﬁrst quadrant. The plate’s density at the point (133/)
is 5(x, y) = m + 2y + 1. Find the plate’s mass and its center of mass. NAME (print): Problem 3. A conical hole is drilled in a ball of radius a so that the hole has diameter
a at the top. The picture shows the cross section. Compute the volume of the solid
that remains. NAME (print): Problem 4. Let R be the region in the ﬁrst quadrant bounded by the coordinate
axes and the curves 3/ = 4 — a: and ac = 2. (a) Sketch the region R and set up the integral of the function f (9:, y) = my over
the region R With day on the outside. (b) Reverse the order of integration in the above integral. (0) Compute the integral. NAME (print): . Problem 5. Let A >‘ 0. Compute the following integral over the solid bounded by
the paraboloid z = A2 — m2 — y2 from above and the Lug—plane from below: A «715—72 A2 —a:2—y2
/ / / 3/132 + 3/2 dz dy das.
—A —\/A§—m2 0 HINT. Switching to appropriate coordinates will make things easier. NAME (print): Extra credit problem. Pompeii is a buried and ruined Roman city near modern Naples
in the Italian region of Campania. It was destroyed, and completely buried, during a
catastrophic eruption of the volcano Mount Vesuvius spanning two days on 24 August 79.
The volcano buried Pompeii under many meters of ash and pumice, and it was lost for
nearly 1700 years before its accidental rediscovery in 1748. Since then, its excavation has '
provided an extraordinarily detailed insight into the life of a city at the height of the Roman
Empire. A new archeological expedition is planning an excavation of a tunnel that leads to the
mayor’s tomb. They know that the cross section of the tunnel is a parabola, so that the
tunnel is in the shape of a parabolic cylinder. Before proceeding with the excavation they want to determine the exact location of the tunnel to reduce the amount of digging. (a) What is the minimal number of holes they need to dig to determine the exact
location and dimensions of the tunnel? Assume that they know a spot under
which there is deﬁnitely some section of the tunnel, however they do not know
in which direction the tunnel runs. (b) Assume they have drilled a hole and hit the tunnel at 35 feet below surface.
They drill a second hole 5 feet to the east of the ﬁrst hole and they ﬁnd the
tunnel at 40 feet below surface. In which direction should they drill the third
hole to obtain as much information about the location of the tunnel as possible? ...
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This note was uploaded on 09/15/2009 for the course MATH 234 taught by Professor Dickey during the Fall '08 term at University of Wisconsin.
 Fall '08
 DICKEY
 Multivariable Calculus

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