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Unformatted text preview: Math 17B Kouba
Exam 2 KEY Your Name : _________________________________________________________ __ Your EXAM ID Number __________ __ 1. PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO. 2. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY
WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS
A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM
ANOTHER STUDENT’S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR
CODE TO HAVE ANOTHER STUDENT TAKE YOUR EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE
EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK
YOU FOR YOUR COOPERATION. 3. No notes, books, or classmates may be used as resources for this exam. YOU MAY
USE A CALCULATOR ON THIS EXAM. 4. Read directions to each problem carefully. Show all work for full credit. In most
cases, a correct answer with no supporting work will receive LITTLE or NO credit. What
you write down and how you write it are the most important means of your getting a good
score on this exam. Neatness and organization are also important. 5. Make sure that you have 8 pages, including the cover page.
6. You will be graded on proper use of integral and derivative notation.
7. You will be graded on proper use of limit notation. 8. You have until 10:50 am. to ﬁnish the exam. 1.2 1.5 1.1 0.6
0.9 0 0 0
0 0.8 0 O
O 0 0.7 0 1.) (3 pts. each) Consider the Leslie matrix L = a.) How many age classes are in this population ? q b.) What percentage of 2—year old females survive to the end of the following breeding
season ? 70 ‘70 c.) What percenatge of 3year old females survive to the end of the following breeding season ? O ‘376 d.) What is an average number of female offspring for a 0year old female ? l o a
e.) What is an average number of female offspring for a 1year old female ? l v 5/
10
20 .
f.) If N(0) = 40 , determine N(1) .
30 f
L; L5“ 1.( 0.9 ‘0 xaL3o+q4+IP loq
.. &0 Ct at
( v o 1 :
N C l 0.1 08 o q o W W
o o. o 0 ,2 5» a
O 0 0 ' ,7 o 3 o X
. . $+y—z=4
2.) (8 pts.) Use matrix reductlon to solve the system { 2x + z 2 3
it~l4 ll‘lqKN1‘:i A) f
a o l 3 O ’ l 3 "é o i {2 3«
x Y 5— 3
l 0 4L /1 _L ~ 2 Q
C) ‘ (5L /1. Y ~' ‘3 % x 5/
Ti. ' /sL 48 d
3.) (8 pts.) Solve the following differential equation : E 11262” ——" dar:
g—L 0W :SCQX‘Q")? —‘3
\{a —_L _ 1 2X4
.Y ' (1/ 4.) (10 pts.) Determine the equilibria for the given autonomous DE. and determine their stability using either the sign chart method or the derivative method: id];— = 3N —N2 :: j
M .
/ ,_, N, v V v v i,  ,
n. o + o  ' t  v
M jaw/0% jag/3 pawn N:O [0:3 / 5‘60): 3>o M N30 W) 1‘
Mﬁim v// 3163/:3_é:3<o Aowz3M. / 1 0
5.) (8 pts.) Determine the angle 0 between the vectors < 0 ) and (3) .
i —2
W V  W) i —— y 7.) (14 pts.) Find eigenvalues and the corresponding eigenvectors for the matrix : Map/Ma’u—z ~<v{ : AKVA~é : CA~3/(A+2):o
_—, Az3/ ,\:—az (A—A1)X:©t
Alz3 (if; ‘OQFK—s Xl—2X&:o
MM K13‘é iii—A X,=&~é4_a
km: [1* :+[,"3 44, M WHY) 8.) (8 pts.) Determine P2 (36), the seconddegree Taylor Polynomial centered at ct = 0, for f(x)=\/m. “1/ 4:0“) )
~F(O() 3 2%: LX442 A M am I ’———'" A—U’ l
( l {L
ammo) «nil 434:9); all 1 i
0' o( 'T“/ L; ll l ‘1/
“Q ( “/
“a‘VFC): 3112 A. 1Z— M
a: v 3 9.) (8 pts.) (Mixture Problem) Let S be the amount (pounds) of sugar in a tank at time
t (minutes). A solution containing 2 pounds of sugar per gallon begins ﬂowing into the
tank at the rate of 5 gallons per minute and the wellstirred mixture ﬂows out of the tank at the rate of 3 gallons per minute. Initially, the tank holds 100 gallons with 20 pounds of d
sugar. SET UP BUT DO NOT SOLVE a differential equation for the rate d—f . ﬁr Jinx moM/at
Mn 42 <4 2 (I
(\ 79 E
v
E l (A
m E 10.) (8 pts.) The following data are plotted on a semi—10g graph on the following page.
dN Use the graph to solve for N and determine the growth rate R in autonomous form.
t N
0 7
1.5 19.8 W
3 56 Z
4 112 l _ f +
g, 0 x»
5.5 316.8 [OJ N W1 3 C
7 896 0.4"er —»
l035é23m+1037 ——», 3m:[o<75'é~/077__,
3lm:10\7(%)~3 Wt: milojii/j—«r
“At1‘55;— H [OjA/t‘ﬁ'l032+l057b a“ {V
lay/V ‘6.l03o'L—l/]oj7 [03% {0‘77
0 o ’ lo “5 IIII II I" II II III.
nllu nulnlllnllumﬁﬂﬂﬁﬂﬂh':
Illllllll IIIIII
I I All IIIIII
Ell 0 l 2 3 4 5 6 7 8 9 IO The following EXTRA CREDIT problem is OPTIONAL. It is worth 10 points. 1.) The points (0, 0, 0), (1, 2, —1), and (3, —1, 1) form a triangle in threedimensional space.
Prove that this triangle is a. right triangle. ...
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This note was uploaded on 08/15/2008 for the course MATH 17B taught by Professor Kouba during the Fall '07 term at UC Davis.
 Fall '07
 Kouba
 Calculus, Logic

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