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Unformatted text preview: SAMPLE
“KER Math 16C Sec 2 (Malkin) Name:
Mid—term exam 1 Student ID:
Wed April 30th 2008 Signature: DO NOT TURN OVER THIS PAGE
UNTIL INSTRUCTED TO DO SO! Write your name, student ID, and signature NOW! NO NOTES, CALCULATORS, OR BOOKS ARE ALLOWED.
N O ASSISTANCE FROM CLASSMATES IS ALLOWED. Read directions to each problem carefully. Show all work for
full credit. In most cases, a correCt answer with no supporting
work will NOT receive full credit. Be organized and neat, and
use notation appropiately. You will be graded on the proper use
of derivative and integral notation. Put units on answers where
appropriate. Please write legiblyll SAMPLE 1. (a) (3 points) Show that the equation x2 + my 2 C (C is a constant) is a solution
of the differential equation xzy” * 2(m + y) = 0. "214 13‘ (3:0 (<3 9L ‘ a — 213 ’2 4 IBM \3‘ 44%)?) (:3 (laugh; (13‘ :O (:3 adj“: 31513 .
SM 1H3“? ~'b(~ 3‘ : "21+ [Bulltj : ’Lbu 3) . Tim/s, 303% ‘waﬁ ’4 Navy Hwy} :3 w rectchd _ (b) (6 points) Use any method to ﬁnd the general solution of the differential equation
yzy’ — stem = 0. You must write y as a function of as. (jig 9 men 3 O (a Dink : 161 t) :ijax‘ t
I I Vile mt (in
La W150 will“ 10M lﬁl dV:€10l>(\ss vcem (00%;)
(KO/ll with c )C'ey~[fjdb( : )L€?£€n + C [(5acm/ran4)/
W Volv My Vdvt (c) (6 points) Use any method to ﬁnd the general solution of the differential equation
my’ + 2y = x2 + You must write y as a function of 3:. l A \ 1‘ A I _ x _ i : 14 I
xj+1fj ‘xl 3" (“A ll xvi
fpétmx /% d1 Huff? l L
LIAM:6 =5 :Q t~€nx =1: l .
7/300, : “(xijﬂﬂl “(lelx : ﬁf/thi‘hk‘dx :x’ljxhl (ht. .: 1’1( [96" 41* C) (CO (1 '
,L 50 \j: 3L"L [ H0414 O 2 SAMPLE 2. (a) (3 points) Find an equation for the sphere that has a diameter with endpoints
(0,0,0) and (1,2,2). I + +1 ’ j,
(ViiWO (9%“ (S Qil)q'2"“ f llifi).
{ 3 f WM a We \is \M m—oimi\<>):mj$ $57416 quanta 0; w Iﬂere u (X’%)1.i(:)~\)l+ (“91(31' (b) (5 points) Consider the function f (93,31) = ln(xy). Find the domain and range
of f. Sketch the domain. /
3‘ {h// '
I Domin: filth? 3L3>03 L : {QWBOOWO 3 \) 31403 40; H
ts} >L
(Range :1 [—00 I»)
(c) (5 points) Consider the surface 2 = 332+y2— 1. Graph separately the intersection
of the surface with the planes 1: = 0, y = 0, and z = 0 (i.e. graph the zytrace,
the xztrace, and the anytrace). Don’t forget to label the axes.
5% ~ “WK (k 12, hum XﬁWWZ ~
 ‘ x.
We“ 1:0. %=:) \ Loewe“ 23:0. 1:)L v \ (3(qu oﬂlwtq
% $6 _ SAMPLE 3. (a) (2 points) Find the slope of the surface 2 = f(:v,y) = V532 —— 2y2 at the point
(1, O, 1) in the x—direction and in the ydirection. \
’ .943 (‘ r X (x i _ l
j X xm’l%1 3 (78. £131 \ lg) ’ 5113531 trrxklv‘i
l 50, H0996 in 14thwale {s /m(l.0): Shiv =Eg Md.   d ' '  _ 4(0) . , y/o‘pe m ?« W3 ‘0‘" '5 /7 (“0) ' rm.» ‘ (b) (5 points Find all the critical points of f (x, y) = 2x21; 2:1;2 — y2 + 3. (Hint: there are three critical points). /x(’)(lt7)c they ~ [(‘DL :0 (:3 [6x (V’I);o 2) x20 52$ 33‘ /j(’(y): (211‘7—3 :0 (>3 21):?C . ‘71,Ql Jill/h :OL‘JO 9 Mama l:(“/ 50 X:l of 1:4. Wm, Hie critical WWW me (0.9, (l‘l)‘a“dl'\tl) (c) (5 points) The function f (x,y) = $3 — 33: + 2y2 — 8y has two critical points:
(1,2) and (—1, —2). Classify each of the critical points of f. Clwliﬂ“ ii (an
rel, mm 5a (idle . SAMPLE 4. (10 points) During a chemical reaction, substance A is converted into substance B.
Let y be the amount of substance A after t hours. The rate at which A changes to
B is proportional to square of the amount of A, i.e. 99 = [cg/2. Initially, there is 100 dt
grams of A present, and after one hour there is only 10 grams remaining. (a) (8 points) Find the particular solution to the differential equation (i.e. solve the
differential equation and determine all the constants) (i L _ vi” 4
hick JRDL ‘2) Sid)" : 2) jjlhlc (: j=(vktcj), \ _L
, 00 : Jalel~C (:) lOO =‘C
anbcﬁtjzmo‘ SO \ :3 C: p756
\
50) :j : +hl‘6O . l 'L \ J—
— =lO Sai \O =~anl+ too (2 13:41»: loo
[JEWAJCAHO I :33. _\L
\90 ’ i
6‘) h a ' \oo.
‘ l__._ \00 ﬁ‘j:%ot+nao @3501???" (b) (2 points) After how many hours is there only 1 gram of A remaining? lj Bil. W\ l = glii\ L15 (:3 (‘3) “b: . lo) ﬁtter: is l rel/naming 6% END OF EXAM SAMPLE
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