ucsc
econ/ams 11b Review Questions 8
fall 2008
Final Review
1. The annual output for a discount hotel chain is given by Q = 30K 2/5 L1/2 R1/4 , where K , L and R are the capital, labor and real estate inputs, all measured in $1, 000, 000s, and Q is measur
UCSC
MATH 21
FALL 2009
Final Review Questions
Note: The nal exam is comprehensive, but these questions only cover chapters 5 and 6. To prepare for the nal exam, you need to review all three sets of review questions (as well as the homework, class notes, e
UCSC
AMS 10
FALL 2009
Final Review Questions
Note: The nal exam is comprehensive, but these questions only cover chapters 5 and 6. To prepare for the nal exam, you need to review all three sets of review questions (as well as the homework, class notes, et
UCSC
MATH 21
FALL 2009
Review For Midterm 2
(1) Consider the set of vectors 2 0 1 1 , 0 , 1 . B= 1 1 1 a. Show that B is a basis of R3 .
2
b. Find
3 1 425 3
2
, (the coordinate vector of
B
3 1 425 3
with respect to the basis B).
c. Find the change of basi
UCSC
AMS 10
FALL 2009
Review For Midterm 2
(1) Consider the set of vectors 2 0 1 1 , 0 , 1 . B= 1 1 1 a. Show that B is a basis of R3 .
2
b. Find
3 1 425 3
2
, (the coordinate vector of
B
3 1 425 3
with respect to the basis B).
c. Find the change of basis
UCSC
MATH 21
FALL 2009
Review For Midterm 1
(1) Polynomial interpolation. The polynomial f (x) = cn xn + cn1 xn1 + + c1 x + c0 interpolates the points (x1 , y1 ), (x2 , y2 ), . . . , (xk , yk ) if f (xj ) = yj for j = 1, 2, . . . , k . In other words, f (
MATH 21
NOVEMBER 2, 2009
EXAM 1.2 Solutions
Instructions
Please turn OFF all cell phones, iPods, etc.
There are 5 questions worth a total of 42 points. 100%=40 points. Read all the questions before you begin. Answer the ones you nd easiest rst. Please wr
ams 10/10A
supplementary notes
ucsc
A Primer on Complex Numbers
c 2009, Yonatan Katznelson
1.
Imaginary and complex numbers.
One of the fundamental properties of the real numbers is that the square of a real number is always nonnegative. I.e., if x is a r
ucsc Review Questions 2
econ/ams 11b
Solutions
1. Compute the following integrals a. Substitute u = x2 + 1, du = 2x dx, then 3x dx = (3/2)du and 3 9 3x dx 3 u2/3 = + C = (x2 + 1)2/3 + C . u1/3 du = 3 2+1 2 2 2/3 4 x 3 2 2 b. Substitute u = x + 3x 1, du =
ucsc Review Questions 2.5
econ/ams 11b
Table of integrals and separable dierential equations
1. Compute the following integrals a. 5x 7 2x dx = 5 = x 7 2x dx
10(6x 14)(7 2x)3/2 (3x + 7)(7 2x)3/2 +C = +C . 60 3 2(3bu 2a)(a + bu)3/2 u a + bu du = + C , with
Solutions to HW #10
1. Suppose that X is a random variable with mean = 121 and variance 2 = 15. Use Tchebychevs inequality to estimate the probabilities a. P (|X 121| > 25) < 15 = 0.024. 252 15 0.9112. 132
b. P (108 X 134) > 1
2. Suppose that X is the nu
econ/ams 11b
winter 2010
Midterm 2 Review Solutions
1. Find the critical point(s) of the functions below subject to the given constraint. a. h(u, v ) = 5u + 8v ; 2u2 + 3v 2 = 30. The Lagrangian in this case is F (u, v, ) = 5u + 8v (2u2 + 3v 2 30), and the
econ/ams 11b
winter 2010
Midterm 2 Review
1. FInd the critical point(s) of the functions below subject to the given constraint. a. h(u, v ) = 5u + 8v ; b. k (x, y, z ) = xy 2 z 3 ; 2u2 + 3v 2 = 30. 5x + 8y + 12z = 300.
2. A households utility function is
econ/ams 11b
winter 2010
Midterm 1 Review Solutions
1. Compute the indicated partial derivatives. a. f (x, y, z ) = 2x3 yz 2 3xy 3 z + 5x2 y 2 7yz 5 + 11x 1; fx = 6x2 yz 2 3y 3 z + 10xy 2 + 11. fy = 2x3 z 2 9xy 2 z + 10x2 y 7z 5 . fzx = fxz = 12x2 yz 3y 3
econ/ams 11b
winter 2010
Midterm 1 Review
1. Compute the indicated partial derivatives. a. f (x, y, z ) = 2x3 yz 2 3xy 3 z + 5x2 y 2 7yz 5 + 11x 1; nd fx , fy and fzx . b. q (u, v ) = q q u2 v 3uv 3 ; nd and . 2u + 3v u v c. w = r2 ln(3r + s2 ); nd wr , w
AMS/ECON 11B
SPRING 2008
Homework assignment #8
1. Compute the degree 2 Taylor polynomial for the function f (x, y ) = x3 y + 3x2 y 2 2xy 2 + 3x 5y, centered at the point (x0 , y0 ) = (1, 1). 2. Compute the degree 2 Taylor polynomial for the function Q(K,
ucsc
econ/ams 11b Review Questions 8
winter 2008
Solutions
1. a. Q(20, 15, 5) 575.866 b. First compute the marginal products of labor and real estate at the given point: QL = 15K 2/5 L1/2 R1/4 QR = 7.5K
2/5 1/2
= =
QL (20, 15, 5) 19.195, QR (20, 15, 5) 28
UCSC
MATH 21
FALL 2007
Review Questions 3
1. Consider the matrix 1 2 A = 1 2 3 1 3 2 1 2 1 4 3 0 1 161 2 15 0 3 11 3 . 2 9 4 1 13 1
a. Find the reduced-row-echelon form of A. Suggestion: keep track of the row operations that you use. b. Find a basis for t