Homework #1, Math 311:02, Fall 2008 Sample Solutions 0.1#6 TASK Suppose that A; B; C are sets. (a) Prove that If A B, then C B C
A:
(b) Either prove the converse or provide a counterexample. PROOF (a) Assume that A B: [Note: Our book uses for the weak inc
Information for Maple assignment 3
Each student will get individual data for the assignment.
Where to find .
Pictures of degree two polynomials in two variables
General information about conic sections
Pictures of degree two polynomials in three variables
Information for Maple assignment 1
Each student will get individual data for this assignment.
Here is some help for the first Maple assignment. The assignment requests several pictures. Pictures are
very important. Not many people can get much information
Information for Maple assignment 2
Each student will get individual data for the assignment.
Below is helpful background information for the second Maple assignment. I repeat what I wrote in the
last set of instructions:
You are allowed, and indeed encour
Information for Maple assignment 3
Each student will get individual data for the assignment.
Where to find .
Pictures of degree two polynomials in two variables
General information about conic sections
Pictures of degree two polynomials in three variables
Information for Maple assignment 3
Each student will get individual data for the assignment.
Where to find .
Pictures of degree two polynomials in two variables
General information about conic sections
Pictures of degree two polynomials in three variables
Information for Maple assignment 3
Each student will get individual data for the assignment.
Where to find .
Pictures of degree two polynomials in two variables
General information about conic sections
Pictures of degree two polynomials in three variables
Information for Maple assignment 3
Each student will get individual data for the assignment.
Where to find .
Pictures of degree two polynomials in two variables
General information about conic sections
Pictures of degree two polynomials in three variables
Information for Maple assignment 3
Each student will get individual data for the assignment.
Where to find .
Pictures of degree two polynomials in two variables
General information about conic sections
Pictures of degree two polynomials in three variables
Information for Maple assignment 5
Each student will get individual data for the assignment.
Lagrange multipliers using Maple
Background
Here is the problem:
Theoretical results imply that x+3yz has a maximum and a minimum on the sphere
x2+y2+z2=1. Use La
Information for Maple assignment 4
Each student will get individual data for the assignment.
Sketching regions in the plane
and computing double integrals
Sketching regions in space
and computing triple integrals
Two dimensions
Integration in more than on
Homework # 4 , Math 311:02, Fall 2008 Sample Solutions TASK 1.3 #25 See Workshop #4 TASK 1.3 #27 Suppose that (an )1 and (bn )1 are sequences in R such that lim (an ) = A 6= 0 and lim (an bn ) exists. Show that (bn )1 converges. PROOF: First let introduce
Homework #2, Math 311:02, Fall 2008 Sample Solutions 0.3 #24 TASK: Dene a function f from N into N by f (1) = 1 f (2) = 2 whenever n 4; f (n) = f (n Show that for all n in N; f (n) when when when when when n n n n n = = = = = 1; 2; 3; 4; 5; 2n EXPLORATION
xw q i q a 'R q &g cW4U d q g ` vVGhG VC IR QI E 9 QI dR q !'HhSgr'!y P 1 q o4&!'H&"EW!eyHcB1Fw'vh&!'8u g ` R V CU I R QI b V RU CE G h CV A G h R Q G I R Q E 9 QI d R R U h!q @o W'@Hc'!pq e!C I G G QI CU d 9 C q V G A A 9 d R xEyBHWVF4'BFS1DBB!1eEb '@&H
v WH V U S Q V U if di SY W Vf vrXW6gAt)Y y6$C6p T RV hzp6!#P$cc l if di SY qib f H WHVHXWj Ak$i hlYHqfpF1HX6V PaVRP$i AV Acfw_!16$)p6!$6V !%#ap)pA$q$cfw_q d U W f q H V H Q Wq U S h H I H d i i f d i H I U Y S i H Y S d b H h i f d i H i S V S Y ciSCd!H
1
1.1
Math 311:02
Introduction
Fall 2008
The Exponential Series
An "innite series" is a formal sum of innitely many terms. A series can be expressed as
1 X
tk tk tk
= t0 + t1 + t2 + : + tk + : = t1 + t2 + t3 + : + tk + : = tK + tK+1 + tK+2 + : + tk + :
k=
4.3. LIMIT OF A SEQUENCE: THEOREMS
4.3
115
Limit of a Sequence: Theorems
These theorems fall in two categories. The rst category deals with ways to
combine sequences. Like numbers, sequences can be added, multiplied, divided,
. Theorems from this category
SUGGESTED SOLUTIONS FOR PROBLEM SET 11
FALL 2010, MATH 311:01
6.1.1. Let (an ) be a sequence of real numbers. Prove that
n=1
(an an+1 )
n=1
converges i (an ) converges. If
n=1
Proof. () Suppose that
n=1 (an
an+1 ) converges, what is the sum?
an an+1 conv
SUGGESTED SOLUTIONS FOR PROBLEM SET 10
FALL 2010, MATH 311:01
5.5.19. Suppose f R(x) on [a, b] and
1
f
is bounded on [a, b]. Prove that
1
f
R(x) on [a, b].
1
Pre-proof Remark. If f is continuous, then the proof is very simple. Since f is bounded on [a, b
Dirichlet Function Functional Limits Continuous Functions Workshop #6 Compact Sets Uniform Continuity The Inter
Math 311: Advanced Calculus
Wolmer V. Vasconcelos
Set 4
Spring 2008
Dirichlet Function Functional Limits Continuous Functions Workshop
Uniform Convergence and Differentiability
Series of Functions
Power Series
Taylor Series
Workshop #10
Math 311: Advanced Calculus
Wolmer V. Vasconcelos
Set 5
Spring 2007
Wolmer Vasconcelos Advanced Calculus
Set 5
Uniform Convergence and Diff
Advanced Calculus I 311
Review Exercises 1. Prove the following assertions about a, b R. You must give reasons for every step! -(a + b) = -a + -b -(a/b) = (-a)/b = a/(-b) if b = 0 a a = 1 if and only if a = -1 or a = 1 If 0 < a < 1 then 0 < an < am
Homework # 5 , Math 311:02, Fall 2008 Sample Solutions 2.1 #2 TASK Set D = ( 2; 0) and dene f on D by f (x) = 2x2 + 3x x+2 2
Show that f has a limit at 2, nd the value of the limit, and prove the result. EXPLORATION. We need to validate two statements 1.
Homework # 6 , Math 311:02, Fall 2008 Sample Solutions 2.1 #16 Task Dene f by Dom(f ) = (0; 1) and f (x) = Prove that f has a limit at 0 and nd that limit. Result limx!0 f (x) = 1=6: x3 + 6x2 + x x2 6x
Proof For x in the domain, we know x 6= 0 and x 6= 6;
Homework Sets # 9 , Math 311:02, Fall 2008 Sample Solutions 4.1 #1 Task Consider the function dened by f (x) = x2 and an arbitrary point (x0 ; y0 ) on the graph of f subject only to the condition x0 6= 0: Without using the notion of derivative nd the equa
Homework Sets # 10 , Math 311:02, Fall 2008 Sample Solutions p 4.1 #16 Task Dene f on [0; 2] by f (x) = 2x x2 : (a) Verify the hypotheses of Rolle theorem and s (b) nd a c with 0 < c < 2 and f 0 (c) = 0: p Proof (a): Set r(s) = s for non-negative s: Set P
Homework Sets # 7 and #8 , Math 311:02, Fall 2008 Sample Solutions 3.1 #2 Task Dene function f on [ 4; 0] by f (x) =
2x2 18 x+3
12
if if
x 6= x=
3 3
Show that f is continuous at 3: Proof Note that 3 is an accumulation point of Dom(f ) and 3 2 Dom(f ): To