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School: Michigan State University
School: Michigan State University
School: Michigan State University
Math 235 section 202 HW #1 Solution 1 dy - y = te2t , y(0) = 2 2 dt 1. (3 pts) Find the solution of the initial value problem: y - 2y = 2te2t , (t) = e e-2t y - 2e-2t y = 2t, -2dt = e-2t , (e-2t y) = 2t, e-2t y = t2 + c, c = 2, therefore y = (t2
School: Michigan State University
School: Michigan State University
Course: CALCULUS II
Math 133 Dr. Kurtz Exam 1 Name_ Section No._ TA_ Instructions: Please show all of your work. Credit will not be given for answers with no supporting work. dy 1. (21 pts) Compute . dx 2 e- x y= 2 (a) x 2 2 x 2 e - x (-2 x) - e- x (2 x) = Quotient r
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 2 Notes: Plane Conics 2.1 Plane conics A conic is a plane projective curve of degree 2. Such a curve has the form C=k : ax2 + by2 + cz 2 + dxy + exz + f yz with a; b; c; d; e; f 2 k. Assuming the characteristic of k is not 2, we can make d = e = f
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 7 Notes: Field Norms 7.1 o f g . Field norms and traces Let L=K be a _nite _eld extension of degree n = [L : K]. Then L is an ndimensional K-vector space, and each 2 L determines a linear operator T : L ! L T h i s corresponding to multiplication
School: Michigan State University
Course: Introduction To Arithmetic Geometry
r i n g Lecture 4 Notes: Inverse Limits 4.1 Inverse limits Z De_nition 4.1. An inverse system is a sequence of objects (e.g. sets/groups/rings) (An ) together with a sequence of morphisms p h a s (e.g. functions/homomorphisms) (fn ) _ ! An+1 A2 ! An ! _ !
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 3 Notes: Quadratic Reciprocity 3.1 Quadratic reciprocity Recall that for each odd prime p the Legendre symbol ( p ) is de_ned as 8if a is a _ nonzero _ quadratic residue < 1 modulo p; if p a is zero modulo p; = otherwise: : 0 1 The Legendre symbol
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 8 Notes: Completions of Q We already know that R is the completion of Q with respect to its archimedean absolute value j j1 . Now we consider the completion of Q with respect to any of its nonarchimedean absolute values j jp . Theorem 8.1. The com
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 5 Notes: P_ Adic Numbers As a fraction _eld, the elements of Qp are by de_nition all pairs (a; b) 2 Zp , typically written as a=b, modulo the equivalence relation a=b _ c=d whenever ad = bc. But we can represent elements of Qp more explicitly by e
School: Michigan State University
Michael Wenstrup Mrs. Ann Dunayczan Character Analysis The Crucible 19 October 2009 Wenstrup 1 A Plethora of Lies To all stories comes an antagonist, in The Crucible by Arthur Miller, Abigail Williams is definitely the antagonist behind almost all the pro
School: Michigan State University
Wenstrup 1 Michael Wenstrup Mrs. Nott English 11 May 18, 2010 At The University of _, we are committed to building an academically superb and widely diverse educational community. What would you as an individual bring to our campus community? I could say
School: Michigan State University
Universities are committed to building an academically superb and widely diverse educational community. What would I as an individual bring to the campus community? I could say I am an inner city kid, that I lost my dad at a very young age to repeated dru
School: Michigan State University
Universities are committed to building an academically superb and widely diverse educational community. What would I as an individual bring to the campus community? I could say I am an inner city kid, that I lost my dad at a very young age to repeated dru
School: Michigan State University
While I live in a predominantly a white, Christian, middle-class community so typical of those found in Southwest Michigan, one of my best friends is Ranjeet Ghorpade who we effectively just call Jeet. Jeet is a first generation American of Hindu parents
School: Michigan State University
DescribetheuniquequalitiesthatattractyoutothespecificundergraduateCollegeorSchool(includingpreferredadmission anddualdegreeprograms)towhichyouareapplyingattheUniversityofMichigan.Howwouldthatcurriculumsupportyour interests?(500wordsmaximum) Mycareerambi
School: Michigan State University
Course: Transitions
Lectures 1 and 2: Chapter 1. Sets Math 299 Denition: A set is a welldened collection of distinct objects called the elements (or members). Sets are conventionally denoted with capital letters. 1. We enclose a set with braces (curly brackets) cfw_. 2. The
School: Michigan State University
Course: Transitions
Lectures 8 and 9: Chapter 3 Math 299 0. Ch3.1 A trivial proof and a vacuous proof (Reading assignment) 1. Ch3.2 Direct proofs 2. Ch3.3 Proof by contrapositive 3. Ch3.4 Proof by cases 4. Ch3.5 Proof evaluations (Reading assignment) Ch 3.2: Direct proofs A
School: Michigan State University
Course: Transitions
Lectures 6 and 7: Sec 2.9 and Sec 2.10 Math 299 2.9: Some Fundamental Properties of Logical Equivalence Theorem: For statements P , Q, and R, the following properties hold. 1. Commutative Laws (1) P Q Q P (2) P Q Q P 2. Associate Laws (1) P (Q R) (P Q) R
School: Michigan State University
Course: Transitions
Math 299 Lecture 5: Sections 2.6 and 2.7. Biconditional, Converse, Tautologies and Contradictions Converse The statement B A is the converse of the statement A B . Give an example for each of the following cases: The statement is true and its converse is
School: Michigan State University
Course: Transitions
Math 299 Lecture 3: Chapter 2. Logic Sec 2.1: Statements Mathematics is the business of proving mathematical statements to be true or false . Logic lays the foundation for rigorous mathematical proofs. Denition: A statement is a sentence that is either tr
School: Michigan State University
Course: Transitions
Math 299Lecture 4: Chapter 2. Logic. Sections 2.4 and 2.5: Implications If ., then . Statements Denition: Statements of the form If statement A is true, then statement B is true. are called implications. Mathematically this is denoted by A B . AB If A th
School: Michigan State University
School: Michigan State University
Course: CALCULUS II
Math 133 Dr. Kurtz Exam 1 Name_ Section No._ TA_ Instructions: Please show all of your work. Credit will not be given for answers with no supporting work. dy 1. (21 pts) Compute . dx 2 e- x y= 2 (a) x 2 2 x 2 e - x (-2 x) - e- x (2 x) = Quotient r
School: Michigan State University
School: Michigan State University
Name: ID Number: TA: Section Time: Math 20D. Exam 2 May 19, 2008 No calculators or any other devices are allowed on this exam. Read each question carefully. If any question is not clear, ask for clarication. Write your solutions clearly and legibly; no cr
School: Michigan State University
Name: ID Number: TA: Section Time: Math 20D. Exam 1 April 23, 2008 No calculators or any other devices are allowed on this exam. Read each question carefully. If any question is not clear, ask for clarication. Write your solutions clearly and legibly; no
School: Michigan State University
School: Michigan State University
Math 235 section 202 HW #1 Solution 1 dy - y = te2t , y(0) = 2 2 dt 1. (3 pts) Find the solution of the initial value problem: y - 2y = 2te2t , (t) = e e-2t y - 2e-2t y = 2t, -2dt = e-2t , (e-2t y) = 2t, e-2t y = t2 + c, c = 2, therefore y = (t2
School: Michigan State University
MAT 4200, eksamen 12.12.13, solutions Oppgave 1 La p1 , p2 , p3 vre forskjellige primtall, og la n = p1 p2 p3 . 2 3 1a. Hva er de maksimale idealene i Z/nZ? Hva er nilradikalet? Gi et eksempel p et a primrt ideal i Z/nZ som ikke er et primideal. (2pt) Sol
School: Michigan State University
Homework 1 (due: 10-7-11). All rings are commutative with identity! (1) [4pts] Let R be a nite ring. Show that R = NZD(R). (2) [8pts] Suppose that R is a subring of a ring S and that R is a direct summand of S as an R-module (i.e. S = R N as an R-module).
School: Michigan State University
Solutions Ark 1 From the book: Number 8, 9, 10 and 12 on page 11. Number 8 : Show that the set of prime ideals of A has a minimal element with respect to inclusion. Solution: By Zorns lemma it sucies to show that any descending chain of prime ideals conta
School: Michigan State University
Solutions Ark2 From the book: Number 10, 11 and 12 on page 32. Number 10 : Let A be a ring, a an ideal contained in the Jacobson radical of A; let M be an A-module and N a nitely generated A-module, and let u : M N be homomorphism. If the induced homomorp
School: Michigan State University
PNHSPhysics IBPhysicsDensityComparisonofDensityCubes MichaelWenstrup Date: 9/29/09 Codes: 1.2.11 Hours: 1 Evaluating: DPP & CE Discussion: Your task is to calculate the density of three different density cubes. One cube must be the wood block. You may use
School: Michigan State University
IBPhysicsHorizontalProjectileMotionNameMichaelWenstrup Codes: 9.1.1 Date: Materials: steel ball ramp Meter stick Carbon paper Hours: 1.5 Evaluating: DCP plumb line unlined white paper Procedure: Set up the apparatus as shown in the diagram. Use the appara
School: Michigan State University
PLANNINGLAB:HELICOPTERMOTION IBCRITERIA:TOPIC2MECHANICSHOURS:3 MICHAELWENSTRUP EVALUATING:DESIGN,DCP,CEDATE: _9/14/10_ ABSTRACT RESEARCHQUESTION Asthelength(independentvariable)ofthehelicoptersrotorsincrease,howistheamountoftime(dependent)the helicopterr
School: Michigan State University
PLANNINGLAB:HELICOPTERMOTION IBCRITERIA:TOPIC2MECHANICSHOURS:3 MICHAELWENSTRUP EVALUATING:DESIGN,DCP,CEDATE: _9/14/10_ ABSTRACT Theaimofthislabwastoinvestigateonefactorthataffectsthemotionofthepaperhelicopter.SoIdesignedthelabto measurehowthechangeinroto
School: Michigan State University
School: Michigan State University
1.8: 1. g(x)=x^2+2x+3 g(2+h)-g(2)= 2. f(x)=2x^2 and g(x)=x+3 f(g(x)= f(x)+k, move graph up k f(x+k), move graph left k cf(x), stretch graph vertically (c>1) cf(x), shrink graph vertically (0<c<1) -f(x), reflect graph 1,9 y=kx^p, find k and p 1.y=3/(x
School: Michigan State University
Course: Transition To Formal Mathematics
SYLLABUS TRANSITIONS: MATH 299, SECTION 6 SPRING 2014 Instructors Name: David Duncan Instructors Email: duncan42[at]math.msu.edu Instructors Oce: Wells Hall C-315 Oce Hours: TBD TAs Name: Alex A. Chandler Lecture Times and Location: M, W 5-6:20, Wells Hal
School: Michigan State University
Course: Intro To Algebra
Math 110. Finite Mathematics and Elements of College Algebra Fall 2007 Syllabus Students who do not satisfy the prerequisites for Math 110 will be dropped from this course by the Registrar's office. Course supervisor: Irina Kadyrova, Ph.D. Office Loc
School: Michigan State University
Course: Calc I
1 MTH 132 Calculus I SYLLABUS Spring 2008 Section: 18 TIME: MWF 3:00-3:50pm Room: C110 Wells Hall Instructor: Tsung-Lin Lee (Jules) Email: leetsung@msu.edu Office: D-316 Wells Hall Office Hours: M W F 13:50pm 14:50pm and by appointment Homepage
School: Michigan State University
School: Michigan State University
School: Michigan State University
Math 235 section 202 HW #1 Solution 1 dy - y = te2t , y(0) = 2 2 dt 1. (3 pts) Find the solution of the initial value problem: y - 2y = 2te2t , (t) = e e-2t y - 2e-2t y = 2t, -2dt = e-2t , (e-2t y) = 2t, e-2t y = t2 + c, c = 2, therefore y = (t2
School: Michigan State University
School: Michigan State University
Course: CALCULUS II
Math 133 Dr. Kurtz Exam 1 Name_ Section No._ TA_ Instructions: Please show all of your work. Credit will not be given for answers with no supporting work. dy 1. (21 pts) Compute . dx 2 e- x y= 2 (a) x 2 2 x 2 e - x (-2 x) - e- x (2 x) = Quotient r
School: Michigan State University
School: Michigan State University
1.8: 1. g(x)=x^2+2x+3 g(2+h)-g(2)= 2. f(x)=2x^2 and g(x)=x+3 f(g(x)= f(x)+k, move graph up k f(x+k), move graph left k cf(x), stretch graph vertically (c>1) cf(x), shrink graph vertically (0<c<1) -f(x), reflect graph 1,9 y=kx^p, find k and p 1.y=3/(x
School: Michigan State University
1 LIFE ASSURANCE MATHEMATICS W.F.Scott c 1999 W.F.Scott Department of Mathematical Sciences Kings College University of Aberdeen Aberdeen AB24 3UE U.K. 2 Preface This book consists largely of material written for Parts A2 and D1 of the U.K. actuarial exam
School: Michigan State University
$ % &'( & !" $ # ) *! , + - . 0 1 / 2 / 0 1 * 2 / % % % + 3 / % 24 5 ' 3 6 15 $ 0 20 / 24 6 / 5 1 /% 6 15 / % 5 28 /% 1 1 7 4 / / $ 1 % / 0 1 9 / 4 / 1 1 :1 / 1 % 1 1 1 1 5 % / % / 1 < 5 * % :4 3 :; 0 % + % 0 : =; 6 : ; =' ; : ' 1 5 % !"#!$ ! % * + % & >
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 2 Notes: Plane Conics 2.1 Plane conics A conic is a plane projective curve of degree 2. Such a curve has the form C=k : ax2 + by2 + cz 2 + dxy + exz + f yz with a; b; c; d; e; f 2 k. Assuming the characteristic of k is not 2, we can make d = e = f
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 7 Notes: Field Norms 7.1 o f g . Field norms and traces Let L=K be a _nite _eld extension of degree n = [L : K]. Then L is an ndimensional K-vector space, and each 2 L determines a linear operator T : L ! L T h i s corresponding to multiplication
School: Michigan State University
Course: Introduction To Arithmetic Geometry
r i n g Lecture 4 Notes: Inverse Limits 4.1 Inverse limits Z De_nition 4.1. An inverse system is a sequence of objects (e.g. sets/groups/rings) (An ) together with a sequence of morphisms p h a s (e.g. functions/homomorphisms) (fn ) _ ! An+1 A2 ! An ! _ !
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 3 Notes: Quadratic Reciprocity 3.1 Quadratic reciprocity Recall that for each odd prime p the Legendre symbol ( p ) is de_ned as 8if a is a _ nonzero _ quadratic residue < 1 modulo p; if p a is zero modulo p; = otherwise: : 0 1 The Legendre symbol
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 8 Notes: Completions of Q We already know that R is the completion of Q with respect to its archimedean absolute value j j1 . Now we consider the completion of Q with respect to any of its nonarchimedean absolute values j jp . Theorem 8.1. The com
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 5 Notes: P_ Adic Numbers As a fraction _eld, the elements of Qp are by de_nition all pairs (a; b) 2 Zp , typically written as a=b, modulo the equivalence relation a=b _ c=d whenever ad = bc. But we can represent elements of Qp more explicitly by e
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 11 Notes: Quadratic forms over Qp The Hasse-Minkowski theorem reduces the problem of determining whether a quadratic form f over Q represents 0 to the problem of determining whether f represents zero over Qp for all p _ 1. At _rst glance this migh
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 12 Notes: Field extensions Before beginning our introduction to algebraic geometry we recall some standard facts about _eld extensions. Most of these should be familiar to you and can be found in any standard introductory algebra text, such as We
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 10 Notes: The Hilbert symbol De_nition 10.1. For a; b 2 Qp the Hilbert symbol (a; b)p is de_ned by ax2 + by 2 = 1 has a solution in Qp ; ( (a; b)p = 1 otherwise: It is clear from the de_nition that the Hilbert symbol is symmetric, and that it only
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 9 Notes: Quadratic forms We assume throughout k is a _eld of characteristic dierent from 2. De_nition 9.1. The four equivalent de_nitions below all de_ne a quadratic form on k. 1. A homogeneous quadratic polynomial f 2 k[x1 ; : : : ; xn ]. 2. Asso
School: Michigan State University
Course: Calc I
Version 1 FINAL EXAMINATION, M A T 2010 April 28, 2011 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possib
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, M A T 2010 April 29, 2010 Write your solutions in a blue book. To receive full credit you must show ull work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but us
School: Michigan State University
Course: Calc I
FINAL EXANIIIUATION, M A T 2010 April 26, 2012 Write your solutions ini a blue book. To receive full credit you must show all work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, M A T 2010 April 30, 2009 Write your solutions in a blue book. To receive full credit you must show all work. You are tl allowed t o use an u p p r o ~ ~ egraphing calculator unless otherwise indicated. Simplify your answer when possibl
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, MAT 2010 December 15, 2011 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed to use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but us
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, M A T 2010 December 16, 2010 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, MAT 2010 December 17, 2009 Write your solutions in a blue book. To receive full credit you must show a l l work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, M A T 2010 December 18, 2006 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed t o use an approved graphing calculator unless otherwise indicated. There are 15 problems worth a total of
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, MAT 2010 April 24, 2008 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed to use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but use t
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, MAT 2010 December 15, 2008 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed to use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but us
School: Michigan State University
School: Michigan State University
0 a @) Cc;t a = % ,iLexdx 2 = ~e - L . Y ,f\*- - 2 A*Z x e'dx 'CxeX' d V = eYdx V= e x AJr cX* j s c X L = it-l 9 dt S 69 ' m Ax' ca)' 4 0, a
School: Michigan State University
@ yxe-*du (a) dJ=e (L-t U = P Ah= d % -X - =-* I / ' IBQ 8 \= - a- X + c ) - * . /'* -X -xe , c'L a = I+y a*= b(U 3 a = 3 3LJ'du 9=J3 4.-=-o,ds ? y c o 9K=I h=Z 4 u.29
School: Michigan State University
Winter 2008 FINAL EXAM 25 April 2008 Name _ Instructions. Show all work where it is normally expected. 1. [18] Evaluate the following. (a) sin t 1 cos 2 t dt (b) e y 1 4 ln y dy (c) x 9 3x 2 x 2 dx 2. [12] Set upbut do not evaluatethe integral(s) giving t
School: Michigan State University
Winter 2014 FINAL EXAM 25 April 2014 Name _ Instructions. Show all work where it is normally expected. 1. [32] Evaluate the following. (a) 9 y 8 3 y 2 y 1 dy (b) 4 sin 5 cos d (c) x 2 x e dx 2. [10] Set upbut do not evaluatethe integral(s) giving the a
School: Michigan State University
Winter 2011 FINAL EXAM 27 April 2011 Name _ Instructions. Show all work where it is normally expected. 1. [24] Evaluate the following. (a) 2 x x e dx (b) sin y 2 1 cos y dy (c) 20 3x 2 5 x dx 2. [12] Using horizontal cross sections (i.e., y cross sections
School: Michigan State University
School: Michigan State University
Fall 2010 FINAL EXAM 15 December 2010 Name _ Instructions. Show all work where it is normally expected. 1. [24] Evaluate the following. (a) xe x dx (b) 2 0 y2 1 y3 dy (c) 5 6 x 2 x 1 dx 2. [12] Set upbut do not evaluatethe integral(s) giving the area of t
School: Michigan State University
Fall 2004 FINAL EXAM 21 December 2004 Name _ Instructions. Show all work where it is normally expected. 1. [20] Evaluate the following. (a) e x dx (b) x 1 x 2 x 1 2 dx (c) cos 3t 5 sin 3t 4 dt 2. [14] Let R be the closed region which is shown to the righ
School: Michigan State University
HOMEWORK 7 5.10 For a point P on a variety X, let m be the maximal ideal of the local ring OP . We dene the Zariski tangent space TP (X) of X at P to be the dual k-vector space of m/m2 . a) For any point P X, dim TP (X) dim X, with equality if and only if
School: Michigan State University
HOMEWORK 5 SELECTED SOLUTIONS 4.3 a) Let f be the rational function on P2 given by f = x1 /x0 . Find the set of points where f is dened and describe the corresponding regular function. b) Now think of this function as a rational map from P2 to A1 . Embed
School: Michigan State University
HOMEWORK 4 SELECTED SOLUTIONS 2.8 If X is a classical variety, and U X open, then f : U k is regular in the above sense if and only if it is regular in the sense we dened for classical varieties. Solution: This problem is actually straightforward once one
School: Michigan State University
HOMEWORK 6 SELECTED SOLUTIONS 3.2 Show that if X is a variety, and for some P, Q X we have OP,X OQ,X K(X), then P = Q. Solution: Let U and V be ane open neighborhoods of P, Q, respectively. Then A(U ) OP,X and A(V ) OQ,X . As in Exercise I.4.7 of Hartshor
School: Michigan State University
HOMEWORK 1 SELECTED SOLUTIONS 1.1 a) Let Y be the plane curve y = x2 . Show that A(Y ) is isomorphic to a polynomial ring in one variable over k. b) Let Z be the plane curve xy = 1. Show that A(Z) is not isomorphic to a polynomial ring in one variable ove
School: Michigan State University
HOMEWORK 3 SELECTED SOLUTIONS 2.9 If Y An is an ane variety, we identify An with an open set U0 Pn by the homeomorphism 0 . Then we can speak of Y , the closure of Y in Pn , which is called the projective closure of Y. a) Show that I(Y ) is the ideal gene
School: Michigan State University
IMMERSE 2007 Algebra Exercises 2. Week 2 Unless otherwise noted, the letters R and A denote commutative rings with unity, the letters I and J denote ideals, the letter k denotes a eld, and letters like Xi denote variables. Denition 2.1. Let : Rn R be a li
School: Michigan State University
IMMERSE 2007 Algebra Exercises 1. Week 1 Unless noted otherwise, the letters R and A denote commutative rings with identity. The letters I and J denote ideals. Exercise 1.1. Let R be a ring (not necessarily commutative, not necessarily with identity). For
School: Michigan State University
IMMERSE 2007 Algebra Exercises 3. Week 3 Exercise 3.1. Let f and g be monomials in R = k[x1 , . . . , xn ]. a. b. c. d. Say f = xa1 xan and g = xb1 xbn . Show f | g if and only if for all 1 i n, we have ai bi . n n 1 1 Prove that if f (g)R, then deg(f ) d
School: Michigan State University
IMMERSE 2007 Algebra Exercises 4. Week 4 For the following exercise set, the Newton polytope (unbounded) of I is the UNBOUNDED convex hull of all the monomials in I. It is what Hbl denotes K(I). u Exercise 4.1. Show that if I and J are monomial ideals, th
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 2 Notes: Plane Conics 2.1 Plane conics A conic is a plane projective curve of degree 2. Such a curve has the form C=k : ax2 + by2 + cz 2 + dxy + exz + f yz with a; b; c; d; e; f 2 k. Assuming the characteristic of k is not 2, we can make d = e = f
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 7 Notes: Field Norms 7.1 o f g . Field norms and traces Let L=K be a _nite _eld extension of degree n = [L : K]. Then L is an ndimensional K-vector space, and each 2 L determines a linear operator T : L ! L T h i s corresponding to multiplication
School: Michigan State University
Course: Introduction To Arithmetic Geometry
r i n g Lecture 4 Notes: Inverse Limits 4.1 Inverse limits Z De_nition 4.1. An inverse system is a sequence of objects (e.g. sets/groups/rings) (An ) together with a sequence of morphisms p h a s (e.g. functions/homomorphisms) (fn ) _ ! An+1 A2 ! An ! _ !
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 3 Notes: Quadratic Reciprocity 3.1 Quadratic reciprocity Recall that for each odd prime p the Legendre symbol ( p ) is de_ned as 8if a is a _ nonzero _ quadratic residue < 1 modulo p; if p a is zero modulo p; = otherwise: : 0 1 The Legendre symbol
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 8 Notes: Completions of Q We already know that R is the completion of Q with respect to its archimedean absolute value j j1 . Now we consider the completion of Q with respect to any of its nonarchimedean absolute values j jp . Theorem 8.1. The com
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 5 Notes: P_ Adic Numbers As a fraction _eld, the elements of Qp are by de_nition all pairs (a; b) 2 Zp , typically written as a=b, modulo the equivalence relation a=b _ c=d whenever ad = bc. But we can represent elements of Qp more explicitly by e
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 11 Notes: Quadratic forms over Qp The Hasse-Minkowski theorem reduces the problem of determining whether a quadratic form f over Q represents 0 to the problem of determining whether f represents zero over Qp for all p _ 1. At _rst glance this migh
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 12 Notes: Field extensions Before beginning our introduction to algebraic geometry we recall some standard facts about _eld extensions. Most of these should be familiar to you and can be found in any standard introductory algebra text, such as We
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 10 Notes: The Hilbert symbol De_nition 10.1. For a; b 2 Qp the Hilbert symbol (a; b)p is de_ned by ax2 + by 2 = 1 has a solution in Qp ; ( (a; b)p = 1 otherwise: It is clear from the de_nition that the Hilbert symbol is symmetric, and that it only
School: Michigan State University
Course: Introduction To Arithmetic Geometry
Lecture 9 Notes: Quadratic forms We assume throughout k is a _eld of characteristic dierent from 2. De_nition 9.1. The four equivalent de_nitions below all de_ne a quadratic form on k. 1. A homogeneous quadratic polynomial f 2 k[x1 ; : : : ; xn ]. 2. Asso
School: Michigan State University
IMMERSE 2007 Algebra Exercises 6. Supplemental Exercises Exercise 6.1. This exercise goes back to review the denition of a ring, and explore an object that is almost a ring. It should be signicantly more elementary than most of the other exercises in the
School: Michigan State University
Course: Multivariable Calculus
IL Wed /4;. 3 LLwe i4YPa -_ 7G~ jizyio pccw,: On the YtJt9n .~ ~ the L/o/unc A>meI R b~r~%e. y~po4 f&v4.e /?t9) 44w,/ej f(Zy~1 I f1ot4 ic t4e flYe4 014d YeJ;n k / ~:1~ .tt;:~j. ~ Dake 46~ ~z~(.e en /tweprak lh CbeYtkd cfw_Y10) tthj c/ace A~ad~4 /ajeJ~k2 c
School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Transition To Formal Mathematics
Review Problems for Midterm Exam II MTH 299 Spring 2014 1. Use induction to prove that 1 + 3 + 6 + + n(n + 1)(n + 2) n(n + 1) = 2 6 for all n N. 2. Use induction to prove that 7|(9n 2n ) for every n N. 3. Use the Strong Principle of Mathematical Induction
School: Michigan State University
Course: Transition To Formal Mathematics
Math 299 Recitation 7: Existence Proofs and Mathematical Induction Existence proofs: To prove a statement of the form x S, P (x), we give either a constructive or a non-contructive proof. In a constructive proof, one proves the statement by exhibiting a s
School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Multivariable Calculus
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School: Michigan State University
Course: Transition To Formal Mathematics
Review Problems for Midterm Exam II MTH 299 Spring 2014 1. Use induction to prove that 1 + 3 + 6 + + n(n + 1) n(n + 1)(n + 2) = 2 6 for all n N. Solution: This statement is obviously true for n = 1 since So assume there is some k 1 for which 1 + 3 + 6 + +
School: Michigan State University
Course: Transition To Formal Mathematics
Induction (Solutions) Math 299 1. Prove 12 + 22 + . . . + n2 = n(n+1)(2n+1) . 6 Proof. We will prove this by induction on n N. Base Case: When n = 1, the formula reads 12 = obviously true. n(2)(3) , 6 Inductive Step: Let k 1, and suppose 12 + . . . + k 2
School: Michigan State University
Course: Transition To Formal Mathematics
Homework 17 Solutions Math 299 Fall 2013 Problem 0. We will prove by induction that n x2 = x=1 n(n + 1)(2n + 1) 6 for all positive n N. The base case is 1(2)(3) 6 which is obviously true. For the inductive step, assume 12 = k x2 = x=1 k (k + 1)(2k + 1) 6
School: Michigan State University
Course: Transition To Formal Mathematics
Recitation 7 (Solutions) Math 299 Exercise 1: Prove that there exists a natural number other than 6 that is the sum of its proper divisors. Proof. The number 28 is a sum of its proper divisors. Indeed, its proper divisors are 1, 2, 4, 7, 14, and 1 + 2 + 4
School: Michigan State University
Course: Transition To Formal Mathematics
Math 299 Homework 6 Solutions 1. (a) Apply A3 with c = a to get a2 ab. Since 0 a and a b, A1 implies that 0 b. This means that we can apply A3 again but with c = b to get ab b2 . By A1 we know that a2 ab and ab b2 implies a2 b2 , as desired. (b) We have a
School: Michigan State University
Course: Linear Algebra
Math 309 Exam 2 Review Exam 2 will be given in class, Friday Feb 21. The exam covers all the material done so far in the course, which corresponds to Sections 1.11.6, 2.1 and 2.2, and 3.4 in the textbook. What to study: 1. Denitions. You should be able to
School: Michigan State University
Course: Linear Algebra
Math 309 Exam 2 Review Exam 2 will be given in class, Friday Feb 21. The exam covers all the material done so far in the course, which corresponds to Sections 1.11.6, 2.1 and 2.2, and 3.4 in the textbook. What to study: 1. Denitions. You should be able to
School: Michigan State University
Course: Linear Algebra
Day 29 Review for Exam 3 Exam 3 will be given in class, Wednesday March 26. The exam covers all the material done so far in the course, which corresponds to Sections 1.11.6, 2.1 and 2.2, 3.4, 2.32.5, 3.2, 4.1-4.4, 5.1 and the rst page of 5.2 in the textbo
School: Michigan State University
Course: Linear Algebra
Matrix Multiplication * Solutions on pages 3-4.* Simplify. Write "undefined" for expressions that are undefined. 1) 0 2 2 6 5 3 3) 5 1 5 2 2 3 5) 0 3 5 5 4 1 2 1 7) 5 6 3 0 1 6 0 3 5 4 4 2) 6 5 3 4) 3 2 4 5 6 1 1 2 5 5 6) 1 3 5 4 5 3 0 3 3 2 2 3 4 5 5 1
School: Michigan State University
Course: Linear Algebra
Solutions for Exam 1 (1) (a) 1 2 2 3 3 1 4 5 (b) The solution is R22R1 1 0 2 1 3 7 x = 11z 2, y = 7 z 3 . 4 3 R1+2R2 1 0 0 1 11 7 2 . 3 Therefore the solution set is S = r(11, 7, 1) + (2, 3, 0) r R . (2) (a) False. (b) True. 00 a S . If A = 00 c a = d, b
School: Michigan State University
Course: Linear Algebra
Math 309 Exam 3 Review Exam 3 will be given in class, Wednesday March 26. The exam covers all the material done so far in the course, which corresponds to Sections 1.11.6, 2.1 and 2.2, 3.4, 2.32.5, 3.2, 4.1-4.4 in the textbook. What to study: 1. Denitions
School: Michigan State University
Course: Linear Algebra
SOLUTIONS FOR EXAM 3 (1) Suppose T : V ! W is one-to-one. For any v 2N (T ) we have T (v) = 0W = T (0V ). Thus v = 0V . Therefore N (T ) = f0V g. Suppose N (T ) = f0V g. If T (v1 ) = T (v2 ), then T (v1 v2 ) = T (v1 ) T (v2 ) = 0W , i.e. v1 v2 2N (T ). As
School: Michigan State University
Course: Linear Algebra
SOLUTIONS FOR EXAM 2 (1) Suppose r (u + 3v) + s (2u v) = 0. This implies (r + 2s) u + (3r 0. As fu; vg is linearly independent, we must have s) v = r + 2s = 0; 3r s = 0: This yields easily r = s = 0. Therefore fu + 3v;2u vg is linearly independent. (2) (a
School: Michigan State University
Course: Multivariable Calculus
TEST 1 MTH234-60 1/28/2014 Last Name: _ Student ID: First Name: _ _ Notice: 1. Clearly write each needed and important steps to get partial credit for workout problems. 2. Erase unnecessary writing and keep the paper clean. 3. Write as neatly as possible.
School: Michigan State University
Michael Wenstrup Mrs. Ann Dunayczan Character Analysis The Crucible 19 October 2009 Wenstrup 1 A Plethora of Lies To all stories comes an antagonist, in The Crucible by Arthur Miller, Abigail Williams is definitely the antagonist behind almost all the pro
School: Michigan State University
Wenstrup 1 Michael Wenstrup Mrs. Nott English 11 May 18, 2010 At The University of _, we are committed to building an academically superb and widely diverse educational community. What would you as an individual bring to our campus community? I could say
School: Michigan State University
Universities are committed to building an academically superb and widely diverse educational community. What would I as an individual bring to the campus community? I could say I am an inner city kid, that I lost my dad at a very young age to repeated dru
School: Michigan State University
Universities are committed to building an academically superb and widely diverse educational community. What would I as an individual bring to the campus community? I could say I am an inner city kid, that I lost my dad at a very young age to repeated dru
School: Michigan State University
While I live in a predominantly a white, Christian, middle-class community so typical of those found in Southwest Michigan, one of my best friends is Ranjeet Ghorpade who we effectively just call Jeet. Jeet is a first generation American of Hindu parents
School: Michigan State University
DescribetheuniquequalitiesthatattractyoutothespecificundergraduateCollegeorSchool(includingpreferredadmission anddualdegreeprograms)towhichyouareapplyingattheUniversityofMichigan.Howwouldthatcurriculumsupportyour interests?(500wordsmaximum) Mycareerambi
School: Michigan State University
All my life I have always dreamed of owning my own business. Not a big business, just two to three hundred million dollars in sales mind you. A business I built from the ground up into a thriving organization. The entirety of my life my dad, an engineerin
School: Michigan State University
Describe the unique qualities that attract you to the specific undergraduate College or School (including preferred admission and dual degree programs) to which you are applying at the University of Michigan. How would that curriculum support your interes
School: Michigan State University
I belong to various communities in Portage; the Portage Northern High School Community, the Baseball Community, and the DECA Community, but my closest community is the Elite Eight. This group began as a small collection of neighborhood kids, originally th
School: Michigan State University
I grew up in Portage, MI and belong to various communities; Portage Northern High School, the Baseball Team, and the DECA Team, but my closest community is my group of best friends the Elite Eight. Portage is not the most diverse community, although it bo
School: Michigan State University
Standardized Testing Michael Wenstrup English 12 Vaneenaanam November 23, 2010 Standardizedtestsaredesignedinawaythatthequestions,conditionsfor administering,scoringprocedures,andinterpretationsareconsistentandare administeredandscoredinapredetermined,s
School: Michigan State University
Bibliography C l o u d , J o h n . " S h o u l d S AT s M a t t e r . " T i m e 4 M a r. 2 0 0 1 : n . p a g . We b . 2 3 N o v 2 0 1 0 . < h t t p : / / w w w. t i m e . c om / t i m e / n a t i o n / a r t i c l e / 0 , 8 5 9 9 , 1 0 1 3 2 1 " C o l l e
School: Michigan State University
Lastname 1 Firstname Lastname WRA 135-003 12 September 2011 Name of Assignment Optional Title (Not in Bold or Italics) This is the required format for all papers you turn in. It is MLA or Modern Language Association style. The font is Times New Roman and
School: Michigan State University
Course: Transitions
Lectures 1 and 2: Chapter 1. Sets Math 299 Denition: A set is a welldened collection of distinct objects called the elements (or members). Sets are conventionally denoted with capital letters. 1. We enclose a set with braces (curly brackets) cfw_. 2. The
School: Michigan State University
Course: Transitions
Lectures 8 and 9: Chapter 3 Math 299 0. Ch3.1 A trivial proof and a vacuous proof (Reading assignment) 1. Ch3.2 Direct proofs 2. Ch3.3 Proof by contrapositive 3. Ch3.4 Proof by cases 4. Ch3.5 Proof evaluations (Reading assignment) Ch 3.2: Direct proofs A
School: Michigan State University
Course: Transitions
Lectures 6 and 7: Sec 2.9 and Sec 2.10 Math 299 2.9: Some Fundamental Properties of Logical Equivalence Theorem: For statements P , Q, and R, the following properties hold. 1. Commutative Laws (1) P Q Q P (2) P Q Q P 2. Associate Laws (1) P (Q R) (P Q) R
School: Michigan State University
Course: Transitions
Math 299 Lecture 5: Sections 2.6 and 2.7. Biconditional, Converse, Tautologies and Contradictions Converse The statement B A is the converse of the statement A B . Give an example for each of the following cases: The statement is true and its converse is
School: Michigan State University
Course: Transitions
Math 299 Lecture 3: Chapter 2. Logic Sec 2.1: Statements Mathematics is the business of proving mathematical statements to be true or false . Logic lays the foundation for rigorous mathematical proofs. Denition: A statement is a sentence that is either tr
School: Michigan State University
Course: Transitions
Math 299Lecture 4: Chapter 2. Logic. Sections 2.4 and 2.5: Implications If ., then . Statements Denition: Statements of the form If statement A is true, then statement B is true. are called implications. Mathematically this is denoted by A B . AB If A th
School: Michigan State University
Course: Transitions
Lecture 10: Sections 4.1 and 4.2 Math 299 Divisibility of Integers and Congruence of Integers Denition: Let a, b be non-zero integers. We say b is divisible by a (or a divides b, or b is a multiple of a) if there is an integer x such that a x = b. And if
School: Michigan State University
Course: Transitions
Lecture 11: Section 4.3 Math 299 Proofs Involving Real Numbers Properties: A1 For all real numbers a, b, c, if a b and b c then a c. A2 For all real numbers a, b, c, if a b then a + c b + c. A3 For all real numbers a, b, c, if a b and 0 c then ac bc. Prov
School: Michigan State University
Course: Transitions
Math 299 Lecture 19: Section 6.4 Strong Mathematical Induction. The Strong Principle of Mathematical Induction For each n N, let P (n) be a statement. If 1. Base step: P (1) is true and 2. Inductive step: k N, P (1) P (2) P (k ) = P (k + 1) is true, then
School: Michigan State University
Course: Transitions
Lectures 17 and 18: Sections 6.1 - 6.2 Math 299 Mathematical Induction. Denition: A nonempty set S of real numbers is well-ordered if every nonempty subset of S has a least element. Example of a well-ordered set: Example of a set which is not well-order
School: Michigan State University
Course: Transitions
Lecture 16: Sections 5.4 - 5.5 Math 299 Existence Proofs. Existence Proofs Our goal in this section is to prove a statement of the form There exists x for which P (x). (That is, 9x, P (x). I. A constructive proof of existence: The proof is to display a sp
School: Michigan State University
Course: Transitions
Math 299 Lectures 12 and 13: Sections 4.4 - 4.6 Proofs Involving Sets In general, to prove two sets A and B are equal, we need to show that both A B and B A are true. A B : x A, x B . EXAMPLES. (1H) A B = A B (2) (A B ) = A B (3H) Let A = cfw_x : x 1 (mod
School: Michigan State University
Course: Transitions
Lectures 14 and 15: Sections 5.1 - 5.2 Math 299 Counterexamples. Proof by Contradiction. Counterexamples (8x 2 S, P (x) 9x 2 S, P (x) If the statement, 8x 2 S, P (x), is false, there exists x 2 S satisfying P (x). Example 8x 2 Z, 9y 2 Z, ( x 1)y = y Is i
School: Michigan State University
Course: Differential Equations
The integrating factor method (Sect. 1.1) Overview of dierential equations. Linear Ordinary Dierential Equations. The integrating factor method. Constant coecients. The Initial Value Problem. Overview of dierential equations. Denition A dierential equatio
School: Michigan State University
Course: Differential Equations
On linear and Non-inear Equations (Sect. 1.6) Review: Linear Dierential Equations The Picard-Lindelf Theorem o The Picard Iteration Properties of Solutions to Non-Linear ODE Direction Fields Review: Linear Dierential Equations Theorem (Variable coecients)
School: Michigan State University
Course: Differential Equations
Modeling with rst order equations (Sect. 1.5). Radioactive decay. Carbon-14 dating. Salt in a water tank. The experimental device. The main equations. The equation for the salt mass. Predictions for particular situations. Radioactive decay Remarks: (a) Ra
School: Michigan State University
Course: Differential Equations
Exact equations (Sect. 1.4). Exact dierential equations. The Poincar Lemma. e Implicit solutions and the potential function. Generalization: The integrating factor method. Exact dierential equations. Denition The dierential equation for the unknown functi
School: Michigan State University
Course: Differential Equations
Linear Variable coecient equations (Sect. 1.2) Review: Linear constant coecient equations. The Initial Value Problem. Linear variable coecients equations. The Bernoulli equation: A nonlinear equation. Review: Linear constant coecient equations Denition Gi
School: Michigan State University
Course: Differential Equations
Separable dierential equations (Sect. 1.3). Separable ODE. Solutions to separable ODE. Explicit and implicit solutions. Euler homogeneous equations. Separable ODE. Denition A separable dierential equation on the function y has the form h(y ) y (t ) = g (t
School: Michigan State University
Course: Differential Equations
Special Second Order Equations (Sect. 2.2) Special Second Order Nonlinear Equations Function y Missing (Simpler) Variable t Missing (Harder) The Reduction Order Method Special Second Order Nonlinear Equations Denition Given a functions f : R3 R, a second
School: Michigan State University
Course: Differential Equations
Power Series Solutions Near Regular Points (Sect. 3.1) The Equation: P (x ) y + Q (x ) y + R (x ) y = 0 Review of Power Series Regular Point Equations Solutions Using Power Series Examples of the Power Series Method Power Series Solutions Near Regular Poi
School: Michigan State University
Course: Differential Equations
The Euler Equation (Sect. 3.2) We Study the Euler Equation: (x x0 )2 y + p0 (x x0 ) y + q0 y = 0 Solutions to the Euler Equation Near x0 The Roots of the Indicial Polynomial Dierent Real Roots Repeated Roots Dierent Complex Roots The Euler Equation Deniti
School: Michigan State University
Course: Differential Equations
Nonhomogeneous equations (Sect. 2.6-2.7) The Problem: L(y ) = f The General Solution Theorem The Undetermined Coecients Method The Variation of Parameters Method The Problem: L(y ) = f Problem: Given a constant coecients linear operator L(y ) = y + a1 y +
School: Michigan State University
Course: Differential Equations
Mechanical and Electrical Oscillations (Sect. 2.4) Review: On Solutions of y + a1 y + a0 y = 0 Application: Mechanical Oscillations Application: The RLC Electrical Circuit Remark: Dierent physical systems may be mathematically identical. Review: On Soluti
School: Michigan State University
Course: Differential Equations
Second Order Linear Equations (Sect. 2.3) Review: Second Order Linear Dierential Equations Idea: Solving Constant Coecients Equations The Characteristic Equation Main Result for Constant Coecients Equations Characteristic Polynomial with Complex Roots Rev
School: Michigan State University
Course: Differential Equations
Second Order Equations: Repeated Roots (Sect. 2.4) Review: On Solutions of y + a1 y + a0 y = 0 Main Result for Repeated Roots Proof Using the Reduction Order Method The Reduction Order Method Review: On Solutions of y + a1 y + a0 y = 0 Summary: Given cons
School: Michigan State University
Course: Ordinary Differential Equations
January 19, 2014 5-1 5. Exact Equations, Integrating Factors, and Homogeneous Equations Exact Equations A region D in the plane is a connected open set. That is, a subset which cannot be decomposed into two non-empty disjoint open subsets. The region D is
School: Michigan State University
Course: Introduction To Chaos And Fractals
Lecture 2 Some concepts from topological dynamics Let X be a metric space with metric d and let f : X X be a homeomorphism. The orbit o(x) of x is the set cfw_f n (x) : n Z. The forward orbit o+ (x) is the set cfw_f n (x) : n Z+ , and the backward orbit o
School: Michigan State University
Course: Introduction To Chaos And Fractals
Lecture 3 Let f : S1 S1 be an orientation preserving homeomorphism of S1 . Let : R S1 be the map (t) = exp(2it). There is a continuous map F : R R such that 1. F = f 2. F is monotone increasing 3. F id is periodic, with period 1 Moreover, any two such m
School: Michigan State University
Course: Introduction To Chaos And Fractals
April 26, 2012 20-1 Further Properties of Topological Entropy Let f : X X be a continuous self-map of the compact metric space X . Let x X and n be a positive integer. An n-orbit O(x, n) is a nite sequence x, f (x), f 2 (x), . . . , f n1 x. Let > 0. Two n
School: Michigan State University
Course: Introduction To Chaos And Fractals
4 Consider an orientation preserving homeomorphism f : S1 S1 with irrational rotation number (f ). Let E be the unique minimal set for f . Note that E equals the non-wandering set of f . Let R = R (f ) be the geometric rotation through angle (f ). Proposi
School: Michigan State University
Outline Review of Test Questions Annuities Summary Formulae Annuities 1 Lecture Math 384 Andrei Ordine [revised Karim Rahim] September 26, 2005 Andrei Ordine [revised Karim Rahim] Annuities 1 Lecture Math 384 Outline Review of Test Questions Annuities Sum
School: Michigan State University
Course: Calculus 3 And Multivariable Calculus
Chain rule for functions of 2, 3 variables (Sect. 14.4) Review: Chain rule for f : D R R. Chain rule for change of coordinates in a line. Functions of two variables, f : D R2 R. Chain rule for functions dened on a curve in a plane. Chain rule for change o
School: Michigan State University
School: Michigan State University
Math 110 FS 2008 CH.2.1-2.3, 3.3. Lines and Slopes. Linear Regression. Lectures #5- 6. Plotting Points in the Rectangular Coordinate System. Graphs of Equations. Each point in the rectangular coordinate system corresponds with an ordered pair of r
School: Michigan State University
Math 234 SS 2008 CH.15.5 Triple Integrals in Cylindrical and Spherical Coordinates. Lecture #28 When calculations in physics, engineering, or geometry involve a cylinder, cone, or sphere, we can simplify our work by using cylindrical or spherical c
School: Michigan State University
Math 234 SS 2008 CH. 15.4. Triple Integral in Rectangular Coordinates. Lectures #26-#27 Definition of Triple Integrals. Function F x, y, z is defined on a closed bounded region D in space. Using planes parallel to coordinate planes can be done pa
School: Michigan State University
Math 234 SS 2008 CH 14.2. Limits and Continuity in Higher dimensions. Lecture #13 Review (functions of one variable). Limits and Continuity. We begin with a review of the concepts of limits and continuity for real-valued functions of one variable
School: Michigan State University
Math 234 SS 2008 CH. 15.1 Double Integrals. Lectures #22-23 Definition of Double Integrals for Rectangular Planar Region. z f x, y is defined on R : a x b, c y d 1 Math 234 SS 2008 CH. 15.1 Double Integrals. Lectures #22-23 Definition. lim P
School: Michigan State University
School: Michigan State University
Course: CALCULUS II
Math 133 Dr. Kurtz Exam 1 Name_ Section No._ TA_ Instructions: Please show all of your work. Credit will not be given for answers with no supporting work. dy 1. (21 pts) Compute . dx 2 e- x y= 2 (a) x 2 2 x 2 e - x (-2 x) - e- x (2 x) = Quotient r
School: Michigan State University
School: Michigan State University
Name: ID Number: TA: Section Time: Math 20D. Exam 2 May 19, 2008 No calculators or any other devices are allowed on this exam. Read each question carefully. If any question is not clear, ask for clarication. Write your solutions clearly and legibly; no cr
School: Michigan State University
Name: ID Number: TA: Section Time: Math 20D. Exam 1 April 23, 2008 No calculators or any other devices are allowed on this exam. Read each question carefully. If any question is not clear, ask for clarication. Write your solutions clearly and legibly; no
School: Michigan State University
Name: Sect. Number: TA: Sect. Time: Math 20D. Quiz 2 April 18, 2008 Answer each question completely, and show your work. If you use extra paper, write your name on each extra page, and staple the question page and your own added pages together. 1. (30 poi
School: Michigan State University
Name: ID Number: TA: Section Time: Math 20D Exam 2. May 19, 2008 No calculators or any other devices are allowed on this exam. Read each question carefully. If any question is not clear, ask for clarication. Write your solutions clearly and legibly; no cr
School: Michigan State University
Name: Sect. Number: TA: Sect. Time: Math 20D. Quiz 3 May 2, 2008 Answer each question completely, and show your work. If you use extra paper, write your name on each extra page, and staple the question page and your own added pages together. 1. (30 points
School: Michigan State University
Name: Sec. Number: TA: Sec. Time: Math 20D. Quiz 1 April 11, 2008 Answer each question completely, and show your work. If you use extra paper, write your name on each extra page, and staple the question page and your own added pages together. 1. A radioac
School: Michigan State University
Name: Sect. Number: TA: Sect. Time: Math 20D. Quiz 4 May 9, 2008 Answer each question completely, and show your work. If you use extra paper, write your name on each extra page, and staple the question page and your own added pages together. 1. (30 points
School: Michigan State University
Name: Sect. Number: TA: Sect. Time: Math 20D. Quiz 5 May 30, 2008 Answer each question completely, and show your work. If you use extra paper, write your name on each extra page, and staple the question page and your own added pages together. 1. (a) (20 p
School: Michigan State University
Name: ID Number: TA: Section Time: Math 20D Exam 1. April 23, 2008 No calculators or any other devices are allowed on this exam. Read each question carefully. If any question is not clear, ask for clarication. Write your solutions clearly and legibly; no
School: Michigan State University
School: Michigan State University
School: Michigan State University
School: Michigan State University
School: Michigan State University
Course: Calculus I
MATH 132, SEC. 21, SAMPLE MIDTERM 4 1. Perform the following integrals and/or derivatives: (2x3 3) dx a. /2 b. cos(2t) dt 1 1 x2 dx c. 1 1 1 x2 dx x d. 1 d. e. 1 1 1 sin cos d 2 1 3 x dx x4 + 9 0 /4 tan2 (3t) sec2 (3t) dt f. 0 /2 g. 1 f (x)dx = 2 . 3f
School: Michigan State University
Course: Calculus I
MATH 132, SEC. 21, SAMPLE MIDTERM 4 ANSWERS 1 1. a. 2 x4 3x + C b. 0 c. 2 d. 0 d. 1 cos2 (1/) + C or 2 1 2 sin2 (1/) + C e. 103 2 f. 1 . (Actually, the integral doesnt converge, but if you noticed this, I dont think you 9 have to worry too much about the
School: Michigan State University
Course: Calculus I
MATH 132, SEC. 21, SAMPLE MIDTERM 3 1. Find the absolute maximum and minimum values of f (x) = x 4 x2 on the interval [0, 2]. 2. Show that at some instant during a 2-hour automobile trip the cars speedometer reading will equal the average speed for the tr
School: Michigan State University
Course: Calculus I
MATH 132, SEC. 21, SAMPLE MIDTERM 3 ANSWERS 1. Maximum ( 2, 2), Minimum (0, 0). 2. The cars position s(t) is a smooth function with respect to time, so we can apply the Mean Value Theorem. The Mean Value Theorem states that there is some time where the de
School: Michigan State University
Course: Calculus I
MATH 132, SEC. 21, SAMPLE MIDTERM 2 1. Find dy dx for the following: y = tan(cos( x) y = 17 cos(/10) y = (x 2)3 (x + 1)5 y = x3 sin(2x) x2 = y= xy x+y 3x2 4 x1 x(t) = t cos(t) y (t) = sin(t) 2. If x3 + y 3 = 16, nd 3. If y = x, calculate d2 y dx2 at the p
School: Michigan State University
Course: Calculus I
MATH 132, SEC. 21, SAMPLE MIDTERM 1 1. Are the following limits computed correctly? If not, identify where the mistake is made. x2 1 (x 1)(x + 1) = lim = lim (x + 1) x 1 x 1 x 1 x 1 x1 = lim x + lim 1 = 1 + 1 = 2 lim x1 x1 x3 4x2 = lim (x3 4x2 )(1/x) = li
School: Michigan State University
Course: Linear Algebra
TEST 2 REVIEW MATH 309, SECTION 6 You should remember the denitions and have a working knowledge of the following concepts already covered: subspaces, linear independence, span, basis, how to solve linear systems, parameterize solution spaces, nd a basis
School: Michigan State University
Course: Linear Algebra
Test 1 Practice Test Math 309, Section 6 1. Consider the following system of linear equations: 3y + z = 1 x + y 2z = 2 x 2y z = 3 Write the coecient matrix associated to the linear system. Use Gaussian elimination (and write what elementary row operations
School: Michigan State University
Course: Transition To Formal Mathematics
Recitation 4: Quantiers and basic proofs Math 299 1. Quantiers in sentences are one of the linguistic constructs that are hard for computers to handle in general. Here is a nice pair of example dialogues: 1. Student A: How was the birthday party after I l
School: Michigan State University
Course: Transition To Formal Mathematics
Quiz 4 Math 299 Please answer each question in the space provided. Use complete sentences and correct mathematical notation to write your answers. You have 20 minutes to complete this quiz. 1. (4 points) Rewrite each statement using English words instead
School: Michigan State University
Course: Calculus 1
Page 1 Name (Print Clearly): 1/22/2014 Student Number: MTH132 Section 1 & 12, Quiz 4 March 16, 2009 Instructor: Dr. W. Wu Instructions: Answer the following questions in the space provided. There is more than adequate space provided to answer each questio
School: Michigan State University
Course: Calculus 1
Page 1 Name (Print Clearly): 1/22/2014 Student Number: MTH132 Section 1 & 12, Quiz 5 March 272009 1 [3 pts each]. Find the limits. 3x 2 x + 2 x x 2 + 2 (a) lim 6x 1 (LH Rule) 2x 6 = lim = 3 x 2 = lim x sin x cos x (LH Rule) x / 4 x /4 (b) lim cos x + si
School: Michigan State University
Course: Calculus 1
Page 1 1/22/2014 Name (Print Clearly): Student Number: MTH132 Section 1 & 12, Quiz 6 April 13, 2009 Instructor: Dr. W. Wu Instructions: Answer the following questions in the space provided. There is more than adequate space provided to answer each questio
School: Michigan State University
Course: Calculus 1
Page 1 Name (Print Clearly): 1/22/2014 Student Number: MTH132 Section 1 & 12, Quiz 7 April 20, 2009 Instructor: Dr. W. Wu Instructions: Answer the following questions in the space provided. There is more than adequate space provided to answer each questio
School: Michigan State University
Course: Calculus 1
Page 1 1/22/2014 Name (Print Clearly): Student Number: MTH132 Section 1 & 12, Quiz 3 Feb 20, 2009 Instructor: Dr. W. Wu Instructions: Answer the following questions in the space provided. There is more than adequate space provided to answer each question.
School: Michigan State University
Course: Calculus 1
Page 1 Name (Print Clearly): 1/22/2014 Student Number: MTH132 Section 1 & 12, Quiz 2 Instructor: Dr. W. Wu Instructions: Answer the following questions in the space provided. There is more than adequate space provided to answer each question. The total ti
School: Michigan State University
Course: Calculus 1
MTH132 Section 1 & 12, Quiz 1 Jan 21, 2009 Instructor: Dr. W. Wu Name (Print Clearly): Student Number: Instructions: Answer the following questions in the space provided. There is more than adequate space provided to answer each question. The total time a
School: Michigan State University
Course: Analysis II
Math 421 Test II Solutions March 24, 2010 1. Determine whether each of the following statements is true or false. If a given statement is true, write the word TRUE (no explanation or proof is necessary). If a given statement is false, write the word FALSE
School: Michigan State University
Course: Analysis II
Math 421 Test I Solutions February 17, 2010 1. Determine whether each of the following statements is true or false. If a given statement is true, write the word TRUE (no explanation or proof is necessary). If a given statement is false, write the word FAL
School: Michigan State University
Course: Complex Analysis 1
Math 425 Test II Solutions April 11, 2011 1. Evaluate the following line integrals. (a) C z dz , where C is the straight-line segment connecting 0 to 2 + 2i. Answer. The curve C can be parametrized as z (t) = t(2 + 2i) t [0, 1]. We note that z (t) = 2 + 2
School: Michigan State University
Course: Complex Analysis 1
Math 425 1. Express Test I Solutions 3 2 + i 2 603 in the form a + ib. Simplify your answer as much as possible. (It may be convenient to express We rst write 3 2 February 25, 2011 + i 2 3 2 + i 2 in polar form.) in polar form. The modulus of 3 2 i 3 + =
School: Michigan State University
Course: College Algebra
MTH 103 College Algebra, Quiz 8 Solutions 1. (4 points) Suppose that $500 is invested in an account paying an annual interest rate of 6% compounded monthly. At the same time, another $500 is invested in an another account paying an annual interest rate of
School: Michigan State University
Course: College Algebra
MTH 103 College Algebra, Quiz 7 Solutions 1. (4 points) Determine the inverse function, f 1 (x), of the function f (x) shown below. 2x 3 f ( x) = x+7 Let y = 2y 3 2x 3 . Switch x and y and re-write: x = . x+7 y+7 Solve for y by multiplying by the LCD (y +
School: Michigan State University
Course: College Algebra
MTH 103 College Algebra, Quiz 9 Solutions 1. (3 points) Use factoring to solve the following equations. (Hint: Use a substitution.) 52x 7 5x 18 Let u = 5x . Since u2 = (5x )2 = 52x , the equation above reduces to the quadratic equation u2 7u 18 = 0. This
School: Michigan State University
Course: Trigonometry
MTH 114-201, Summer 2011 Quiz 5 Instructor: Cheryl Balm Score: Name: Problem 1. [9 points] Solve cos(2) + 6 sin2 = 4 for 0 < 2 . You must nd the exact solutions; no decimal approximations! Problem 2. [7 points] Find all possible sets of solutions for b, c
School: Michigan State University
Course: Trigonometry
MTH 114-201, Summer 2011 Quiz 4 Instructor: Cheryl Balm Score: Name: 1 for 0 < 2 . 2 Problem 1. [5 points] Solve 1 cos = Problem 2. [8 points] Solve tan(2) = 1 for 0 < 2 . 1 MTH 114-201, Summer 2011 Quiz 4 Instructor: Cheryl Balm Problem 3. [7 points] 1 F
School: Michigan State University
Course: Trigonometry
MTH 114-201, Summer 2011 Quiz 3 Instructor: Cheryl Balm 1 Score: Name: No calculators are permitted for this quiz. Problem 1. [5 points] Graph y = sec(2x) 1. Be sure to clearly indicate your nal graph and include dotted lines indicating x-values which are
School: Michigan State University
Course: Trigonometry
MTH 114-201, Summer 2011 Quiz 2 Instructor: Cheryl Balm Score: Name: No calculators are permitted for this quiz. Problem 1. [6+6=16 points] Verify the following equations. Remember to work with one side of the equation only and show all your work! (a) 1
School: Michigan State University
Course: Trigonometry
MTH 114-201, Summer 2011 Quiz 1 Instructor: Cheryl Balm Score: Name: No calculators are permitted for this quiz. Problem 1. [3+3=6 points] Factor the following expressions completely. (a) x2 3x 10 (b) 2x4 + 4x2 + 2 Problem 2. [6 points] Simply the express
School: Michigan State University
Course: Trigonometry
Math 114 Summer 2011 Exam III Review Sheet Instructor: Cheryl Balm Note: These are sample topics and problems to help you study for Exam III. This list is not meant to be exhaustive. Items 1-4 refer to the labels in the triangle below. c b A B C a 1. A +
School: Michigan State University
School: Michigan State University
Math 235 section 202 HW #1 Solution 1 dy - y = te2t , y(0) = 2 2 dt 1. (3 pts) Find the solution of the initial value problem: y - 2y = 2te2t , (t) = e e-2t y - 2e-2t y = 2t, -2dt = e-2t , (e-2t y) = 2t, e-2t y = t2 + c, c = 2, therefore y = (t2
School: Michigan State University
MAT 4200, eksamen 12.12.13, solutions Oppgave 1 La p1 , p2 , p3 vre forskjellige primtall, og la n = p1 p2 p3 . 2 3 1a. Hva er de maksimale idealene i Z/nZ? Hva er nilradikalet? Gi et eksempel p et a primrt ideal i Z/nZ som ikke er et primideal. (2pt) Sol
School: Michigan State University
Homework 1 (due: 10-7-11). All rings are commutative with identity! (1) [4pts] Let R be a nite ring. Show that R = NZD(R). (2) [8pts] Suppose that R is a subring of a ring S and that R is a direct summand of S as an R-module (i.e. S = R N as an R-module).
School: Michigan State University
Solutions Ark 1 From the book: Number 8, 9, 10 and 12 on page 11. Number 8 : Show that the set of prime ideals of A has a minimal element with respect to inclusion. Solution: By Zorns lemma it sucies to show that any descending chain of prime ideals conta
School: Michigan State University
Solutions Ark2 From the book: Number 10, 11 and 12 on page 32. Number 10 : Let A be a ring, a an ideal contained in the Jacobson radical of A; let M be an A-module and N a nitely generated A-module, and let u : M N be homomorphism. If the induced homomorp
School: Michigan State University
Solutions Ark5 From the book: Number 1,2 and 4 on page 78. Number 1: Let M be A-module and u : M M a module homomorphism. i) If M is Noetherian and u is surjective, then u is an isomprphism. ii) If M is Artinian and u is injective, then u is an isomorphis
School: Michigan State University
Solutions Ark4 From the book: Number 2, 4, 5, 10, and 11 on page 55 and 56. Number 2: If a = a, then a has no embedded prime ideals. Solution: Let a = q1 qr be a minimal primary decomposition end let pi = qi be the assoicated p arime ideals of a. As takin
School: Michigan State University
Solutions Ark3 From the book: Number 1, 2, 3, 5 and 6 on page 43 and 44. Number 1: Let S be a multiplicatively closed subset of a ring A, and let M be a nitely generated A-module. Then S 1 M = 0 if and only if there exists an element s S such that sM = 0.
School: Michigan State University
Homework 3 due: 11-18-11. (1) [6pts] For a polynomial P (t) Q show that the following conditions are equivalent: (a) P (n) Z for all integers n Z. (b) P (n) Z for all but nitely many integers n Z. n (c) P (t) = i=0 ai t with ai Z and n N suitable. i (2) [
School: Michigan State University
Homework 4 (due: 12-9-11). (1) [10pts] Let R be a semilocal Noetherian ring and I R an ideal of R. Show that the following conditions are equivalent: (a) (b) (c) (d) I is an ideal of denition of R. I Jrad(R) and R/I is an Artinian ring. I Jrad(R) and R/I
School: Michigan State University
Homework 2 (due: 10-28-11). (1) [4pts] Let R be a ring and Q R an ideal with radQ = P where P R is a prime ideal. Show that Q is P -primary if and only if for all a, b R with ab Q and a P we have that b Q. / (2) [10pts] Let R be a Noetherian ring, P R a p
School: Michigan State University
Homework 4 (due: 12-10-07). (1) Prove the Five Lemma: Consider a commutative diagram with exact rows: A1 A2 A3 A4 A5 t t t t t 1 2 3 4 f1 f2 f3 f4 5 B1 B2 B3 B4 B5 and prove: (a) If t2 and t4 are surjective and t5 is injective, then t3 is surjective. (b)
School: Michigan State University
Course: Ordinary Differential Equations
November 29, 2011 Supplemental ExercisesFall, 2011, Math 848 1. Construct a C vector eld X in the plane with exactly eight critical points such that half of them are hyperbolic saddle points and half of them are hyperbolic sinks. 2. Let X 1 (Rn ) be the s
School: Michigan State University
Course: Introduction To Chaos And Fractals
January 14, 2014 Math 840, Spring, 2014, HW-1 Instructions The Desire2Learn site for this course is at http:/d2l.msu.edu These exercises using Maxima are to be done, saved in a le called Your_Last_Name_HW_1.mac and uploaded to the D2L site for the course
School: Michigan State University
Course: Introduction To Chaos And Fractals
February 23, 2014 Math 840, Spring, 2014, HW-3 These exercises are to be done, typed up, and handed in. The programming part should be saved in a le called Your_Last_Name_HW_3.mac and sent to me by email. For example, if your name is Steven Jones, you wou
School: Michigan State University
Course: Introduction To Chaos And Fractals
February 17, 2014 Math 840, Spring, 2014, HW-2 Instructions 1. These exercises are to be done, typed up, and uploaded to the HW Dropbox at d2l.msu.edu. 2. The programming part should be saved in a le called Your_Last_Name_HW_2.mac and uploaded to the HW D
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 1 (due 01/13) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Section
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 6 (due 01/27) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Section
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 7 (due 01/29) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Section
School: Michigan State University
Course: Transitions
Homework Assignment 5 (due 01/24) Math 299, Section 003 Section 2.6 2.36 (2.25 in 2nd edition) Let P (x) : x is odd. and Q(x) : x2 is odd. be open sentences over the domain Z. State P (x) Q(x) in two ways: (1) using if and only if and (2) using necessary
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 4 (due 01/22) Section 2.4 2.20 (2.15 in 2nd edition) For statement P and Q, construct a truth table for (P = Q) = ( P ). 2.24 Two sets A and B are nonempty disjoint subsets of a set S . If x S , then which of the
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 2 (due 01/15) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Section
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 3 (due 01/17) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Section
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 8 (due 01/31) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Section
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 9 (due 02/03) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Section
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 14 (due 02/19) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Sectio
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 15 (due 02/21) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Sectio
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 13 (due 02/12) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Sectio
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 12 (due 02/10) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Sectio
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 10 (due 02/05) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Sectio
School: Michigan State University
Course: Transitions
Math 299, Section 003 Homework Assignment 11 (due 02/07) Please write your solutions on a dierent piece of paper. Please refer to the course syllabus for a detailed explanation of how you should write homework solutions and how they will be graded. Sectio
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 3 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA Appendix D #5, 8, 16, 17; 2.2 #2, 6, 10; 2.3 #3, 4, 6 Appendix D. 5. (a) Reexivity: Let (x, y ) be any point in the plane. Then (x, y ) (x, y ) because x = x. Symmetry: Suppose (x, y
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 4 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA 3.1 #1, 5, 6, 12, 14, 18, 27, 29; 3.2 #3, 4, 12, 13, 19 Section 3.1. 1. (a) Closure of addition, because 1 + 3 = 4, which is not odd. (b) The existence of additive inverses, because 1
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 6 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA 4.1 #7, 11, 13, 16, 17, 18, 19; 4.2 #1, 2, 3, 7, 10, 14 Note: When we take two arbitrary polynomials in R[x], we cannot assume that they have the same degree. However, the notation for
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 8 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA 4.4 #7, 12, 13, 15, 17, 24; 5.1 #3, 5, 7, 12 Section 4.4. 7. Let f (x) = x7 x Z7 [x]. Then, note that f (0) = 0, f (1) = 0, f (2) = 0, f (3) = 0, f (4) = 0, f (5) = 0, and f (6) = 0 mo
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 7 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA 4.3 #2, 4, 6, 10, 15, 19, 20; 4.4 #1, 2, 5, 6 Section 4.3. 2. Suppose g (x) = xn + cn1 xn1 + + c1 x + c0 and h(x) = xn + dn1 xn1 + + d1 x + d0 were both (monic) associates of f (x). Th
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 11 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA 6.1 #33, 38; 6.2 #4, 5, 8, 12, 13, 21, 23; 6.3 #3, 5, 11, 13 Section 6.1. 33. See solution in the back of the text. 38. Following the hint, suppose I is a non-zero ideal (since the ze
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 10 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA 6.1 #1, 3, 4, 6, 8, 10, 11, 12, 13 Section 6.1. 1. See solutions in the back of the text. 3. (a) First, we show that I is a subring of Z Z. First, observe that (0, 0) I , so I = . Sup
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 9 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA 5.2 #2, 3, 5, 11; 5.3 #1, 2, 3 Section 5.2. 3. See solutions in the back of the text. 5. For any two equivalence classes of polynomials of the form [ax + b] and [cx + d], their product
School: Michigan State University
Course: Abstract Algebra
HOMEWORK 2 SOLUTIONS MATT RATHBUN MATH 310, SECTION 3 ABSTRACT ALGEBRA 1.3 #2, 6, 13, 23; 2.1 #2, 8, 11, 27 Section 1.3. 2. Suppose that p is prime, and let a be an integer. Then (a, p) is a positive divisor of p, so it is either 1 or |p|. If it is 1, the
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 10 (1) Determine whether each of the following statements is true or false. Justify your answer. (a) Every bipartite graph is planar. (b) Every planar graph is bipartite. (c) A tree whose size is even must have a vertex of even degree.
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 4 (1) Suppose a0 = 0, a1 = 8, and ak = 2ak1 + 3ak2 for k 2. Write down the generating function G(x) as a quotient of two polynomials and nd an explicit formula for ak . (2) (a) Express the following function as the product of a polynomi
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 3 (1) Give two proofs (algebraic and combinatorial) of the following formula: n 2 2 =3 n n +3 3 4 (2) Let n be a positive integer. Prove the following formula for any real number x = 1: n1 1 xn xk = 1x k=0 (3) Find (and prove) a formula
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 5 (1) Give two proofs (algebraic and combinatorial) of the following formula: n n + 2 1 m m + 1 2 = n+m 2 (2) Consider a 4 4 chessboard (i.e. a square divided into 4 rows and 4 columns) with 4 chess pieces on it. There are 5 possible co
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 6 (1) Use the rst theorem of graph theory to nd the order and size of the complete bipartite graph Kn,m . (2) (30 points) The friendship graph is dened as follows: the vertices are people, and we draw an edge between two people if they
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 9 (1) Show that C6 is a polyhedron and draw its dual polyhedron. (2) Suppose a polyhedron is composed of 120 triangles and 160 squares. How many vertices and edges does it have? (3) Find the chromatic number of L(Kn,n ). Hint: Try an ex
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 8 (1) A saturated hydrocarbon is a connected graph where each vertex has degree 1 (hydrogen atom) or degree 4 (carbon atom) and the number of hydrogen atoms is two more than twice the number of carbon atoms. Prove the following statemen
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 7 (1) Let G be a non-empty graph of order n whose vertices have degrees d1 , . . . , dn . The line graph of G is dened as follows: the vertices of L(G) are the edges of G, and two vertices of L(G) are adjacent if they share an endpoint
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 2 (1) Let a, b, and c be positive integers. How many paths are there from (0, 0, 0) to (a, b, c) if we are only allowed to increase one of the coordinates by one at each step? (2) Let n be a positive integer. Give two proofs of the foll
School: Michigan State University
Course: Discrete Math I
MATH 481: HOMEWORK 1 (1) (a) How many possible outcomes are there if we roll a six-sided die two times or ip a coin three times? (b) How many ways are there to ip a coin ten times and get heads at least six times? (2) Sec. 2.1 #2. (3) Given a real number
School: Michigan State University
Course: Discrete Math I
MATH 310: HOMEWORK 1 (1) 1.1 # 6. (2) 1.1 # 7. (3) 1.2 # 10. (4) 1.2 # 32. (5) Prove that if n Z, then n(n + 1)(n + 2) is divisible by 6. (6) 2.1 # 12. (7) 2.2 # 2. (8) 2.3 # 7 (b),(d). Use the reverse Euclidean algorithm. (9) List all the positive diviso
School: Michigan State University
Course: Discrete Math I
MATH 310: EXTRA CREDIT PROBLEMS (1) (10 points, due 8/12) For which primes p is the polynomial x2 + 1 irreducible in Zp [x]? Equivalently, when does the equation x2 = 1 have no solutions mod p? You do not have to prove anything. Just do some examples and
School: Michigan State University
Course: Discrete Math I
MATH 310: HOMEWORK 2 (1) 1.1 # 8. (2) 1.2 # 24. (3) 1.3 # 23. (4) 1.3 # 28 (5) 2.1 # 16. (6) 2.2 # 8. (7) 2.2 # 11. (8) Use the Chinese remainder theorem to nd the 4-digit number x such that: x 4 mod 9 x 5 mod 11 x 27 mod 101 (9) Find all positive integer
School: Michigan State University
Course: Discrete Math I
MATH 310: HOMEWORK 3 (1) Determine if the following statement is true or false: If n is a positive integer not divisible by 41, then n2 + n + 41 is prime. Justify your answer. (2) 3.1 #9. (3) 3.1 #10. (4) 3.1 # 13 (b). (5) 3.1 # 24. Compare your answer to
School: Michigan State University
Course: Discrete Math I
MATH 310: HOMEWORK 6 (1) Solve 13x = 11 in Z83 . Can you solve 99x = 1 in Z110000 ? (2) Let m, n Z such that gcd(m, n) = 1. Use Theorem 1.5 to prove that (m) (n) = (mn) Is this formula still true if gcd(m, n) = 1? (3) Use the Chinese Remainder Theorem to
School: Michigan State University
Course: Discrete Math I
MATH 310: HOMEWORK 5 (1) Let R be a ring. Use the First Isomorphism Theorem to prove that R/(0) R = and R/R 0. Use these facts to describe the rings Z1 and Z0 . = (2) Let F be a eld and let f (x) F [x], f (x) = 0. Let I = (f (x) be the ideal generated by
School: Michigan State University
Course: Discrete Math I
MATH 310: HOMEWORK 4 (1) Let R be ring with identity. Prove that there exists exactly one ring homomorphism f : Z R such that f (1) = 1. (2) Let R be an integral domain of characteristic n > 0. Prove that n is prime. (3) Let R be a ring with identity. Pro
School: Michigan State University
Course: Discrete Math II
MATH 482: HOMEWORK 10 (1) Let n be a positive integer. Let S (n) be the number of self-conjugate partitions of n. Let O(n) be the number of partitions of n which consist of distinct odd numbers. Prove that S (n) = O(n). Hint: Give a combinatorial proof vi
School: Michigan State University
PNHSPhysics IBPhysicsDensityComparisonofDensityCubes MichaelWenstrup Date: 9/29/09 Codes: 1.2.11 Hours: 1 Evaluating: DPP & CE Discussion: Your task is to calculate the density of three different density cubes. One cube must be the wood block. You may use
School: Michigan State University
IBPhysicsHorizontalProjectileMotionNameMichaelWenstrup Codes: 9.1.1 Date: Materials: steel ball ramp Meter stick Carbon paper Hours: 1.5 Evaluating: DCP plumb line unlined white paper Procedure: Set up the apparatus as shown in the diagram. Use the appara
School: Michigan State University
PLANNINGLAB:HELICOPTERMOTION IBCRITERIA:TOPIC2MECHANICSHOURS:3 MICHAELWENSTRUP EVALUATING:DESIGN,DCP,CEDATE: _9/14/10_ ABSTRACT RESEARCHQUESTION Asthelength(independentvariable)ofthehelicoptersrotorsincrease,howistheamountoftime(dependent)the helicopterr
School: Michigan State University
PLANNINGLAB:HELICOPTERMOTION IBCRITERIA:TOPIC2MECHANICSHOURS:3 MICHAELWENSTRUP EVALUATING:DESIGN,DCP,CEDATE: _9/14/10_ ABSTRACT Theaimofthislabwastoinvestigateonefactorthataffectsthemotionofthepaperhelicopter.SoIdesignedthelabto measurehowthechangeinroto
School: Michigan State University
School: Michigan State University
1.8: 1. g(x)=x^2+2x+3 g(2+h)-g(2)= 2. f(x)=2x^2 and g(x)=x+3 f(g(x)= f(x)+k, move graph up k f(x+k), move graph left k cf(x), stretch graph vertically (c>1) cf(x), shrink graph vertically (0<c<1) -f(x), reflect graph 1,9 y=kx^p, find k and p 1.y=3/(x
School: Michigan State University
Course: Calc I
Version 1 FINAL EXAMINATION, M A T 2010 April 28, 2011 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possib
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, M A T 2010 April 29, 2010 Write your solutions in a blue book. To receive full credit you must show ull work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but us
School: Michigan State University
Course: Calc I
FINAL EXANIIIUATION, M A T 2010 April 26, 2012 Write your solutions ini a blue book. To receive full credit you must show all work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, M A T 2010 April 30, 2009 Write your solutions in a blue book. To receive full credit you must show all work. You are tl allowed t o use an u p p r o ~ ~ egraphing calculator unless otherwise indicated. Simplify your answer when possibl
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, MAT 2010 December 15, 2011 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed to use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but us
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, M A T 2010 December 16, 2010 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, MAT 2010 December 17, 2009 Write your solutions in a blue book. To receive full credit you must show a l l work. You are allowed t o use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, M A T 2010 December 18, 2006 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed t o use an approved graphing calculator unless otherwise indicated. There are 15 problems worth a total of
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, MAT 2010 April 24, 2008 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed to use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but use t
School: Michigan State University
Course: Calc I
FINAL EXAMINATION, MAT 2010 December 15, 2008 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed to use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but us
School: Michigan State University
School: Michigan State University
0 a @) Cc;t a = % ,iLexdx 2 = ~e - L . Y ,f\*- - 2 A*Z x e'dx 'CxeX' d V = eYdx V= e x AJr cX* j s c X L = it-l 9 dt S 69 ' m Ax' ca)' 4 0, a
School: Michigan State University
@ yxe-*du (a) dJ=e (L-t U = P Ah= d % -X - =-* I / ' IBQ 8 \= - a- X + c ) - * . /'* -X -xe , c'L a = I+y a*= b(U 3 a = 3 3LJ'du 9=J3 4.-=-o,ds ? y c o 9K=I h=Z 4 u.29
School: Michigan State University
Winter 2008 FINAL EXAM 25 April 2008 Name _ Instructions. Show all work where it is normally expected. 1. [18] Evaluate the following. (a) sin t 1 cos 2 t dt (b) e y 1 4 ln y dy (c) x 9 3x 2 x 2 dx 2. [12] Set upbut do not evaluatethe integral(s) giving t
School: Michigan State University
Winter 2014 FINAL EXAM 25 April 2014 Name _ Instructions. Show all work where it is normally expected. 1. [32] Evaluate the following. (a) 9 y 8 3 y 2 y 1 dy (b) 4 sin 5 cos d (c) x 2 x e dx 2. [10] Set upbut do not evaluatethe integral(s) giving the a
School: Michigan State University
Winter 2011 FINAL EXAM 27 April 2011 Name _ Instructions. Show all work where it is normally expected. 1. [24] Evaluate the following. (a) 2 x x e dx (b) sin y 2 1 cos y dy (c) 20 3x 2 5 x dx 2. [12] Using horizontal cross sections (i.e., y cross sections
School: Michigan State University
School: Michigan State University
Fall 2010 FINAL EXAM 15 December 2010 Name _ Instructions. Show all work where it is normally expected. 1. [24] Evaluate the following. (a) xe x dx (b) 2 0 y2 1 y3 dy (c) 5 6 x 2 x 1 dx 2. [12] Set upbut do not evaluatethe integral(s) giving the area of t
School: Michigan State University
Fall 2004 FINAL EXAM 21 December 2004 Name _ Instructions. Show all work where it is normally expected. 1. [20] Evaluate the following. (a) e x dx (b) x 1 x 2 x 1 2 dx (c) cos 3t 5 sin 3t 4 dt 2. [14] Let R be the closed region which is shown to the righ
School: Michigan State University
Physics study guide Amplitude Its position x as a function of time t is: where A is the amplitude of motion: the distance from the centre of motion to either extreme T is the period of motion: the time for one complete cycle of the motion Its position
School: Michigan State University
Pricing: Methodadoptedbyafirmtosetitssellingprice.Itusuallydependson thefirm'saveragecosts,andonthecustomer'sperceivedvalueoftheproduct incomparisontohisorherperceivedvalueofthecompetingproducts.Different pricingmethodsplacevaryingdegreeofemphasisonselect
School: Michigan State University
Course: Transition To Formal Mathematics
SYLLABUS TRANSITIONS: MATH 299, SECTION 6 SPRING 2014 Instructors Name: David Duncan Instructors Email: duncan42[at]math.msu.edu Instructors Oce: Wells Hall C-315 Oce Hours: TBD TAs Name: Alex A. Chandler Lecture Times and Location: M, W 5-6:20, Wells Hal
School: Michigan State University
Course: Intro To Algebra
Math 110. Finite Mathematics and Elements of College Algebra Fall 2007 Syllabus Students who do not satisfy the prerequisites for Math 110 will be dropped from this course by the Registrar's office. Course supervisor: Irina Kadyrova, Ph.D. Office Loc
School: Michigan State University
Course: Calc I
1 MTH 132 Calculus I SYLLABUS Spring 2008 Section: 18 TIME: MWF 3:00-3:50pm Room: C110 Wells Hall Instructor: Tsung-Lin Lee (Jules) Email: leetsung@msu.edu Office: D-316 Wells Hall Office Hours: M W F 13:50pm 14:50pm and by appointment Homepage