Exam 2 Sample SOLUTIONS 1. True or False, and explain: (a) There exists a function f with continuous second partial derivatives such that fx (x, y) = x + y 2 fy = x y 2
SOLUTION: False. If the function has continuous second partial derivatives, then Clair
Hua Luo
Week 10 Questions Intersection of Science and Painting
Due Date: Tuesday, March 26
1. Describe the difference in chemical bonding that characterizes organic and inorganic
compounds.
The primary difference between organic compounds and inorganic co
Lenarz
Math 105
Probability Extra Credit
Solution
Name:
Calculate the probability of drawing the following poker hands. Assume a 5-card stud
poker game, without replacement, from a standard 52 card deck. There are no wild cards.
Ace can be used as both a
Probabilities of Poker Hands with
Variations
Jeff Duda
Acknowledgements:
Brian Alspach and Yiu Poon for providing a means to check my numbers
Poker is one of the many games involving the use of a 52-card deck of playing
cards. The 52 cards are categorized
Name:
Solution
QUIZ 2 FEBRUARY 22, 2007 (1) Suppose that cfw_an is a sequence converging to a number a, and a > 0. Show that there exists an N N such that an > 0 for n N . Proof. Since cfw_an converges to a, for any > 0, there exists an N N with |an -a|
Analysis Concepts
Math 425A
October 26, 2006
1
Supremum and inmum
Completeness property of the real numbers:
Let S be a non-empty set of real numbers that is bounded above. Then S
has a supremum (least upper bound).
Let S be a non-empty set of real number
HOMEWORK 10
SOLUTIONS
1. Page 202
(2) Compute the third Taylor polynomial for each of the following functions at the indicated point.
x
1
(a) f (x) =
dt, x0 = 0
2
0 1+t
f (0) = 0
d
dx x=0
1
=
1 + 02
=1
x
f (0) =
0
1
dt
1 + t2
1
d
dx x=0 1 + x2
2(0)
=
(1 +
HOMEWORK 9
SOLUTIONS
1. Page 159
(3) Suppose that f : [a, b] R is continuous, and
there is a point x0 [a, b] with f (x0 ) = 0.
b
af
= 0. Show that
Proof. The intuitive idea is as follows: if f (x) is ever positive, then it
must be negative somewhere for t
HOMEWORK 8
SOLUTIONS
1. Page 149
(2) Let P1 and P2 be partitions of [a, b]. If P1 is a renement of P2 , show
gap(P1 ) gap(P2 ). Is the converse necessarily true?
Proof. The gap of a partition is the maximum of the lengths of the
subintervals. Each subinte
HOMEWORK 7
SOLUTIONS
1. Page 108
(1) For each of the following, determine if it is true or false and justify.
(a) If f : R R is strictly increasing, then f (x) > 0 for all x.
False. The function f (x) = x3 is a counterexample.
(b) If the dierentiable func
HOMEWORK 6
SOLUTIONS
1. Page 93
(3) For m1 , m2 numbers with m2 = m2 , dene
m1 x + 4
m2 x + 4
f (x) =
if
if
x0
x 0.
Prove that f : R R is continuous but not dierentiable at x = 0.
Proof. It is clear that f is continuous on (, 0) and (0, ), because
f (x) i
HOMEWORK 5
SOLUTIONS
1. Page 73
(2) Dene f (x) = x for all x 0. Verify the criterion for
continuity at x = 4 and at x = 100.
Proof. Given > 0, we wish to nd a > 0 such that if |x 4| <
then | x 2| < . First, we show for any x 0, x0 > 0,
|x x0 |
| x x0 |
HOMEWORK 4
SOLUTIONS
1. Page 61
(1) For each of the following statements, determine whether it is true or
false, and justify your answer.
(d) Every continuous function f : (0, 1) R has a bounded image.
1
False. f (x) = is a counterexample.
x
(e) If the im
HOMEWORK 3
SOLUTIONS
1. Page 37
(1) For each of the following statements, determine whether it is true or
false, and justify your answer.
(a) Every bounded sequence converges.
False. The sequence cfw_(1)n is a counterexample.
(b) A convergent sequence of
HOMEWORK 2
SOLUTIONS
1. Page 32
(1) For each of the following statements, determine whether it is true or
false, and justify your answer.
(a) If the sequence cfw_a2 converges, then the sequence cfw_an conn
verges.
False. The sequence cfw_(1)n is a coun