San Jose State University
Math 70, Fall 2009
Midterm 2 solutions
Name:
Granwyth Hulatberi
Score
1
25
2
25
3
25
4
25
Total 100
Write legibly. Explain your work.
1. (25 points) Consider the following linear programming problem:
Maximize P = 5x + 5y
subject
Solutions to Odd-Numbered Exercises 407
2“", k=—1.—2,.
(c) In 2 < '2' < 3, Z akz", ah = 4 1
-00 3k+1_53+1! k=0,1,2,.
In 5 < |z| < oo, 25, a, = 2n-1—4(3)"—1+ 5H, n = 1,2, 3,
l
N
25. 00 is removable for f iff 0 is removable for g(z) = f(1/z) iff g is b
Solutions to Odd-Numbered Exercises 401
A = {t e [0, 1], u(z(t) = c}.
Since u is equal to c on some disc centered at 20, the interval [0, 6) lies in A for some 6 > 0;
6 is the radius of the disc. Since u is continuous at the point 2(5), we also have 6 e A
Appendix 2 A Table of Conformal Mappings 391
Mapping w-plane
1+z
w:i(1-z)
i —— w ,
z = _ IS the
1 + w
inverse mapping.
_22+2iz+1
w=—1 .
22—212+1 392 Appendix 2 A Table of Conformal Mappings
Mapping w_plane
inverse mapping.
0 and a ar
APPENDIX 2
A Table of Conformal Mappings
z = x + iy, w = u + iv. Lowercase letters are mapped to uppercase letters, cross-hatched regions onto
shaded regions.
z-plane Mapping w-plane
\
:\
389 390 Appendix 2 A Table of Conformal Mappings
Mappi
APPENDIX 3
A Table of Laplace Transforms
u(x); u(x) = 0 ifx < 0
. sin bx
4.
cos bx b2 + $2
. b
5. s1nh bx sz b2
5
6. cosh bx s2 b2
b
7. " si b
e n x (s — a)2 + b2
ax s - a
8. e cos bx (s _ “)2 + b2
2sb
9. ' b
x
Appendix 1 Locating the Zeros of a Polynomial 387
To ﬁt the theorem to our context, we take p1 = p and p2 = p’, the derivative
of p. Simple calculus shows that condition (3) holds; that is, that the sign of pp’
changes from minus to plus as x passes throu
1. If you were Palmer at the end of the case, how would you respond?
Working on two vastly different projects was beginning to have a negative effect on his personal
life, since the additional hours required to keep up with his workload was creating tensi
Math 70 Essentials
I. Mathematics of nance. Main topics:
Simple interest: A = P (1 + rt)
Compound interest: A = P (1 + i)n , i = r/m
)
Future value of an annuity: F V = P M T (1+ii
Present value of an annuity: P V = P M T
n 1
. Sinking funds.
1(1+i)n
San Jose State University
Math 70, Fall 2009
Sample Final Exam
Name:
Score
1
2
3
4
5
6
Total
1. Consider the following system of linear equations:
x1 x2
=4
3x1 + kx2 = 7.
(a) For which k does the system have a unique solution?
(b) For which k is the syst
San Jose State University
Math 70, Fall 2009
Quiz 3 solution
Name: Granwyth Hulatberi
An experiment consists of tossing three fair (not weighted) coins, except one of the three coins
has a head on both sides. Compute the probability of obtaining 2 heads.
San Jose State University
Math 70, Fall 2009
Quiz 2 Solution
Name: XYZ
Solve using the augmented matrix method:
3x1 2x2 = 3
6x1 + 4x2 = 6.
Solution: The augmented matrix of the system is
3 2 | 3
.
6 4 | 6
Multiplying the rst row by 2 and adding it to the
29.
31.
33.
35.
Plz) M_N_ ao/Z~+"'+alv *aN
Q(z)z _bo/z” + + bM bM $0
_P(Z)—aQ(z)
R(z) —a —~—Q(Z)—.
Now deg(P — (IQ) = max(N, M) except if M = N and ab” = a”. In this case,
deg(P — aQ) < N = deg Q = max(N, N).
Use Cauchy’s Formula: Let y be a contour sur