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ch04

Course: CHEM 3601, Spring 2009
School: Tulsa CC
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4 Chapter Alkanes: Nomenclature, Conformational Analysis, and an Introduction to Synthesis Shapes of Alkanes Straight-chain alkanes have a zig-zag orientation when they are in their most straight orientation 5 Straight chain alkanes are also called unbranched alkanes Chapter 4 2 Branched alkanes have at least one carbon which is attached to more than two other carbons Chapter 4 3 Constitutional isomers...

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4 Chapter Alkanes: Nomenclature, Conformational Analysis, and an Introduction to Synthesis Shapes of Alkanes Straight-chain alkanes have a zig-zag orientation when they are in their most straight orientation 5 Straight chain alkanes are also called unbranched alkanes Chapter 4 2 Branched alkanes have at least one carbon which is attached to more than two other carbons Chapter 4 3 Constitutional isomers have different physical properties (melting point, boiling point, densities etc.) 5 Constitutional isomers have the same molecular formula but different connectivity of atoms Chapter 4 4 The number of constitutional isomers possible for a given molecular formula increases rapidly with the number of carbons Chapter 4 5 IUPAC Nomenclature of Alkanes, Alkyl Halides and Alcohols Before the end of the 19th century compounds were named using nonsystematic nomenclature These common or trivial names were often based on the source of the compound or a physical property The International Union of Pure and Applied Chemistry (IUPAC) started devising a systematic approach to nomenclature in 1892 The fundamental principle in devising the system was that each different compound should have a unique unambiguous name The basis for all IUPAC nomenclature is the set of rules used for naming alkanes Chapter 4 6 Nomenclature of Unbranched Alkanes Chapter 4 7 Nomenclature of Unbranched Alkyl groups The unbranched alkyl groups are obtained by removing one hydrogen from the alkane and named by replacing the -ane of the corresponding alkane with -yl Chapter 4 8 Nomenclature of Branched-Chain Alkanes (IUPAC) Locate the longest continuous chain of carbons; this is the parent chain and determines the parent name. Number the longest chain beginning with the end of the chain nearer the substituent Designate the location of the substituent When two or more substituents are present, give each substituent C to its 9 a number corresponding hapter 4location on the longest chain 5 Substituents are listed alphabetically When two or more substituents are identical, use the prefixes di-, tri-, tetra- etc. 5 Commas are used to separate numbers from each other 5 The prefixes are used in alphabetical prioritization When two chains of equal length compete to be parent, choose the chain with the greatest number of substituents When branching first occurs at an equal distance from either end of the parent chain, choose the name that gives the lower number at the first point of difference Chapter 4 10 Nomenclature of Branched Alkyl Chains Two alkyl groups can be derived from propane Four groups can be derived from the butane isomers Chapter 4 11 The neopentyl group is a common branched alkyl group Examples Chapter 4 12 Classification of Hydrogen Atoms Hydrogens take their classification from the carbon they are attached to Chapter 4 13 Nomenclature of Alkyl Halides In IUPAC nomenclature halides are named as substituents on the parent chain 5 Halo and alkyl substituents are considered to be of equal ranking In common nomenclature the simple haloalkanes are named as alkyl halides 5 Common nomenclature of simple alkyl halides is accepted by IUPAC and still used Chapter 4 14 IUPAC Substitutive Nomenclature An IUPAC name may have up to 4 features: locants, prefixes, parent compound and suffixes Numbering generally starts from the end of the chain which is closest to the group named in the suffix IUPAC Nomenclature of Alcohols Select the longest chain containing the hydroxyl and change the suffix name of the corresponding parent alkane from -ane to -ol Number the parent to give the hydroxyl the lowest possible number The other substituents take their locations accordingly Chapter 4 15 Examples Common Names of simple alcohols are still often used and are approved by IUPAC Chapter 4 16 Alcohols with two hydroxyls are called diols in IUPAC nomenclature and glycols in common nomenclature Chapter 4 17 Nomenclature of Cycloalkanes The prefix cyclo- is added to the name of the alkane with the same number of carbons When one substituent is present it is assumed to be at position one and is not numbered When two alkyl substituents are present the one with alphabetical priority is given position 1 Numbering continues to give the other substituent the lowest number Hydroxyl has higher priority than alkyl and is given position 1 If a long chain is attached to a ring with fewer carbons, the cycloalkane is considered the substituent Chapter 4 18 Chapter 4 19 Bicyclic compounds Bicyloalkanes contain 2 fused or bridged rings The alkane with the same number of total carbons is used as the parent the and prefix bicyclo- is used The number of carbons in each bridge is included in the middle of the name in square brackets Chapter 4 20 Nomenclature of Alkenes and Cycloalkenes Alkenes are named by finding the longest chain containing the double bond and changing the name of the corresponding parent alkane from -ane to -ene The compound is numbered to give one of the alkene carbons the lowest number The double bond of a cylcoalkene must be in position 1 and 2 Chapter 4 21 Compounds with double bonds and alcohol hydroxyl groups are called alkenols 5 The hydroxyl is the group with higher priority and must be given the lowest possible number Two groups which contain double bonds are the vinyl and the allyl groups Chapter 4 22 If two identical groups occur on the same side of the double bond the compound is cis If they are on opposite sides the compound is trans Several alkenes have common names which are recognized by IUPAC Chapter 4 23 Physical Properties of Alkanes and Cycloalkanes Boiling points of unbranched alkanes increase smoothly with number of carbons Melting points increase in an alternating pattern according to whether the number of carbon atoms in the chain is even or odd Chapter 4 24 Sigma Bonds and Bond Rotation Ethane has relatively free rotation around the carbon-carbon bond The staggered conformation has C-H bonds on adjacent carbons as far apart from each other as possible 5 The drawing to the right is called a Newman projection The eclipsed conformation has all C-H bonds on adjacent carbons directly on top of each other Chapter 4 25 The potential energy diagram of the conformations of ethane shows that the staggered conformation is more stable than eclipsed by 12 kJ mol-1 Chapter 4 26 Conformational Analysis of Butane Rotation around C2-C3 of butane gives six important conformations 5 The gauche conformation is less stable than the anti conformation by 3.8 kJ mol-1 because of repulsive van der Waals forces between the two methyls Chapter 4 27 The Relative Stabilities of Cycloalkanes: Ring Strain Heats of combustion per CH2 unit reveal cyclohexane has no ring strain and other cycloalkanes have some ring strain Chapter 4 28 The Origin of Ring Strain in Cyclopropane and Cyclobutane : Angle Strain and Tortional Strain Angle strain is caused by bond angles different from 109.5o Tortional strain is caused by eclipsing C-H bonds on adjacent carbons Cyclopropane has both high angle and tortional strain Cyclobutane has considerable angle strain 5 It bends to relieve some tortional strain Cyclopentane has little angle strain in the planar form but bends to relieve some tortional strain Chapter 4 29 Conformations of Cyclohexane The chair conformation has no ring strain 5 All bond angles are 109.5o and all C-H bonds are perfectly staggered Chapter 4 30 The boat conformation is less stable because of flagpole interactions and tortional strain along the bottom of the boat The twist conformation is intermediate in stability between the Chapter 4 boat and the chair conformation 31 Substituted Cyclohexanes: Axial and Equatorial Hydrogen Atoms Axial hydrogens are perpendicular to the average plane of the ring Equatorial hydrogens lie around the perimeter of the ring The C-C bonds and equatorial C-H bonds are all drawn in sets of parallel lines 5 The axial hydrogens are drawn straight up and down Chapter 4 32 Methyl cyclohexane is more stable with the methyl equatorial 5 An axial methyl has an unfavorable 1,3-diaxial interaction with axial C-H bonds 2 carbons away 5 A 1,3-diaxial interaction is the equivalent of 2 gauche butane interactions Chapter 4 33 Disubstitued Cycloalkanes Can exist as pairs of cis-trans stereoisomers 5 Cis: groups on same side of ring 5 Trans: groups on opposite side of ring Chapter 4 34 Trans-1,4-dimethylcylohexane prefers a trans- diequatorial conformation Chapter 4 35 Cis-1,4-dimethylcyclohexane exists in an axial-equatorial conformation A very large tert-butyl group is required to be in the more stable equatorial position Chapter 4 36 Bicyclic and Polycyclic Alkanes The bicyclic decalin system exists in non-interconvertible cis and trans forms Chapter 4 37 Synthesis of Alkanes and Cycloalkanes Hydrogenation of Alkenes and Alkynes Chapter 4 38 Reduction of Alkyl Halides Chapter 4 39 Alkylation of Terminal Alkynes Alkynes can be subsequently hydrogenated to alkanes Chapter 4 40 Retrosynthetic Analysis-Planning Organic Synthesis The synthetic scheme is formulated working backward from the target molecule to a simple starting material Often several schemes are possible Chapter 4 41
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Tulsa CC - CHEM - 3601
Chapter 1 Carbon Compounds and Chemical Bonds Introduction Organic Chemistry The chemistry of the compounds of carbon The human body is largely composed of organic compounds Organic chemistry plays a central role in medicine, bioengineering etc. Vital
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Chapter 2 Representative Carbon Compounds: Functional Groups, Intermolecular Forces and Infrared (IR) Spectroscopy Carbon-carbon Covalent Bonds Carbon forms strong covalent bonds to other carbonsand to other elements such as hydrogen, oxygen, nitrogen
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U. Houston - MATH - 1431
Print Test3/6/10 1:56 PMPRINTABLE VERSIONQuiz 2You scored 100 out of 100Question 1Your answer is CORRECT.Given c = 2 and the graph of the function f below,Use the graph to finda) b) c) d)https:/www.casa.uh.edu/CourseWare/Root/UsersStudents/Print
U. Houston - MATH - 1431
Print Test3/6/10 1:57 PMPRINTABLE VERSIONQuiz 3You scored 100 out of 100Question 1Your answer is CORRECT.The graph of the function f is given below,On which of the following intervals is f continuous?a) b) c) d) e)( -1, 3] [5, 8) ( -4, -1] [ -1,
U. Houston - MATH - 1431
Print Test3/6/10 1:58 PMPRINTABLE VERSIONQuiz 4You scored 100 out of 100Question 1Your answer is CORRECT.Given thatand c = 5, find f ' ( c ) by forming the difference quotientand taking the limit as h0.a)b)c)d) e) Question 2Your answer is C
U. Houston - MATH - 1431
Print Test3/6/10 1:59 PMPRINTABLE VERSIONQuiz 5You scored 100 out of 100Question 1Your answer is CORRECT.Find .a) b) c) d) e) Question 2Your answer is CORRECT.Find .a)https:/www.casa.uh.edu/CourseWare/Root/UsersStudents/PrintTest.htm Page 1 of
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Print Test4/4/10 1:21 AMPRINTABLE VERSIONQuiz 6You scored 100 out of 100Question 1Your answer is CORRECT.Calculate the derivative of the given function:a) b) c) d) e) Question 2Your answer is CORRECT.Calculate the derivative of the given functio
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Print Test4/4/10 1:22 AMPRINTABLE VERSIONQuiz 7You scored 100 out of 100Question 1Your answer is CORRECT.Use differentials to estimate the value indicated.a)b)c) d)e) Question 2Your answer is CORRECT.Use differentials to estimate the value in
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Print Test4/4/10 1:23 AMPRINTABLE VERSIONQuiz 8You scored 100 out of 100Question 1Your answer is CORRECT.Find the critical numbers of f and classify all local extreme values.a) b) c) d) e)Critical nos. Critical nos.1; local max f (-1) = -19; loc
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Print Test4/26/10 10:20 PMPRINTABLE VERSIONQuiz 9You scored 100 out of 100Question 1Your answer is CORRECT.Describe the concavity of the graph of f and find the points of inflection (if any).a) b) c) d) e)concave down on (concave up on (concave u
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Print Test4/26/10 10:20 PMPRINTABLE VERSIONQuiz 11You scored 100 out of 100Question 1Your answer is CORRECT.Evaluate the definite integral:a)b)c)d) e) Question 2Your answer is CORRECT.Evaluate the definite integral:a)https:/www.casa.uh.edu/
U. Houston - MATH - 1431
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U. Houston - MATH - 1431
PRINTABLE VERSIONTest 1You scored 54 out of 72Question 1Your answer is INCORRECT.Computea) b) c) d) e) f ) None of the above. Question 2Your answer is CORRECT.Computea) b) c) d) e) f ) None of the above. Question 3Your answer is CORRECT.Compute
U. Houston - MATH - 1431
U. Houston - MATH - 1431
Print Test4/11/10 9:02 PMPRINTABLE VERSIONTest 2You scored 30 out of 30Question 1Your answer is CORRECT.Determine the interval(s) at which f ( x ) is concave up.a) b) c) d) e)( -5, (-, -5), (3, ) , -3), (5,))( -5, 3) (, 3)Question 2Your ans
U. Houston - MATH - 1431
PRINTABLE VERSIONTest 2You scored 50 out of 70Question 1Your answer is CORRECT.Use differentials to estimate the square root of 37. Hint: Do you know the square root of a number close to 37?a)b)c)d)e) f ) None of the above. Question 2Your answe
U. Houston - MATH - 1431
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PRINTABLE VERSIONTest 3You scored 50 out of 50Question 1Your answer is CORRECT.Find the average value of the function on the interval [-2,1]. a) b) c) d) e) f ) None of the above. Question 2Your answer is CORRECT.LetGive the derivative of F at x =
U. Houston - MATH - 1431
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U. Houston - MATH - 1431
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Answers to Odd-Numbered ProblemsCHAPTER 1 Exercises 1.1 1. (a) ordinary, rst order (c) partial, second order (e) ordinary, third order (g) ordinary, second order 3. Both y and z are solutions. 5. Both y and z are solutions. 7. Both u1 and u2 are solution
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U. Houston - MATH - 3321
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U. Houston - MATH - 3321
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U. Houston - MATH - 3321
CHAPTER 3Second Order Linear Dierential Equations3.1 Introduction; Basic Terminology and ResultsAny second order dierential equation can be written as F (x, y, y , y ) = 0 This chapter is concerned with special yet very important second order equations
U. Houston - MATH - 3321
The Laplace TransformLetfbe continuous function on The Laplace transform of[0, ).f , denoted by L[f (x)], or by F (s), is given byL[f (x)] = F (s) = sx e f (x) dx. 0The domain of F is the set of real numbers s for which the improper integral conve
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Systems of Linear Algebraic EquationsExample Solve the system x + 2y 5z = 1 3x 9y + 21z = 0 x + 6y 11z = 1 x + 2y 5z = 1 y 2z = 1 z = 1 Solution set: x = 4, y = 1, z = 11The Elementary OperationsThe operations that produce equivalent systems are calle
U. Houston - MATH - 3321
CHAPTER 5Linear Algebra5.1 IntroductionLinear algebra is an important mathematical subject independent of its connection to dierential equations. However, it is the relation between dierential equations and linear algebra that is important to us and so
U. Houston - MATH - 3321
Systems of Linear Dierential EquationsI. Converting a linear equation to a linear systemConsider the third-order equation y or y = r (t)y q (t)y p(t)y + f (t). + p (t )y + q (t )y + r (t )y = f (t )1Introduce new dependent variables x1, x2, x3, as fol
U. Houston - MATH - 3321
CHAPTER 6Linear Dierential Systems6.1 Higher-Order Linear Dierential EquationsThis section is a continuation of Chapter 3. As you will see, all of the theory that we developed for second-order linear dierential equations carries over, essentially verba
U. Houston - MATH - 3321
The Complex Number System The need for extending the real number system is evident when considering solutions of simple equations. For example, the equation x2 + 1 = 0 has no real number solutions, for if x is any real number, then x2 0 and so x2 + 1 1. W
U. Houston - MATH - 3321
MATH 3321 1. ln x dx = x (a) ln(ln x) + C (b) (d)1 2EMCF Quiz 11/20/10(ln x)2 (ln x)2 + C(c) (ln x) + C1 2(e) None of the above.2.dx = 4 + x2 (a) (b) (d)1 2 1 2ln(4 + x2) + C tan1 x + C tan1 (x/2) + C(c) tan1 (x/2) + C1 2(e) None of the abov
U. Houston - MATH - 3321
MATH 3321Quiz 21/26/101. If 2 + 3 = 10, 6 + 5 = 66, 8 + 4 = 96, and 3 + 6 = 27, then 9 + 6 =? (a) 122 (b) 108 (c) 135 (d) 152 (e) None of the above. 2. x dx = 9 x2 x +C 3 (b) (9 x2 )1/2 + C x (c) x sin1 +C 3 (d) 9 x2 + C (a) sin1 (e) None of the above.
U. Houston - MATH - 3321
MATH 3321 1. xy = x2y + 2x3y 2 is (a) a linear equation. (b) a separable equation. (c) a Bernoulli equation. (d) a homogeneous equation. (e) none of the above.Quiz 32/1/102. xy = x3 2x2y is (a) a linear equation. (b) a separable equation. (c) a Bernoul
U. Houston - MATH - 3321
MATH 3321Quiz 42/3/10 1 1 y = 2 + 2, x x y (1) = 0,1. If y = y (x) is the solution of the initial-value problem y + then y (e) =: (a) (b) (c) (d) (e) 1 +e e e 1 e2 1 e e None of the above. dy 3y = x3y 1/3 is: dx2. The general solution of x (a) y 1/3 =
U. Houston - MATH - 3321
MATH 3321Quiz 52/14/101. The general solution of y 2y 15y = 0 is: (a) y = C1 e5x + C2e3x (b) y = C1 e5x + C2 e3x (c) y = C1 e5x + C2e3x (d) y = C1 e5x + C2e3x (e) None of the above.2. The solution of the initial-value problem y + 8y + 16y = 0, y (0) =
U. Houston - MATH - 3321
MATH 3321Quiz 62/22/101. If y = y (x) is a solution of x2y + 6xy + 6y = 0, then lim y (x) = (Hint: the equation x has solutions of the form y = xr .) (a) (b) 1 (c) 0 (d) Does not exist. (e) None of the above. 2. If y = y (x) is the solution of the init
U. Houston - MATH - 3321
MATH 3321Quiz 73/4/101. If f (x) = 2e3x + cos 2x + 3x, then L[f (x)] = (a) (b) (c) (d) (e) 2 3 2 +2 +2 s3 s +4 s s 3 2 +2 +2 s+3 s +4 s s 3 2 +2 +3 s3 s +4 s 2 3 2 +2 + s+3 s +4 s None of the above.2. If f (x) = 2e2x sin 3x 2xe3x + 4, then L[f (x)] =
U. Houston - MATH - 3321
MATH 3321Quiz 810/15/091. Set f (x) = 2xex + 4 cos 2x + 2x. L[f (x)] = 2s 2 2 +2 +3 (s + 1) s + 4 s 4 2 2 +2 (b) +2 2 (s 1) s +4 s 2 2 4s +2 (c) +2 2 (s + 1) s +4 s 4 2 2s +2 (d) +2 (s + 1)3 s + 4 s (a) (e) None of the above.2. Set f (x) = 4 cosh x 3x
U. Houston - MATH - 3321
MATH 3321 1. Set f (x) = x2 + 2, x,Quiz 9 0x<1 x1 1 s2 1 s2 1 s2 1 s2 1 s 2 + s 3 s 2 s + . L[f (x)] =3/22/10(a) (b) (c) (d) (e)3 2 2 + es 3 3 s s s 2 2 2 + es 3 + 3 s s s 2 2 1 + es 3 + 3 s s s 2 2 2 + es 3 3 s s s None of the above.2. Set f (x) =
U. Houston - MATH - 3321
MATH 3321 1 3 1. If A = 2 2 , B = 4 0 then d23 = (a) 1 (b) 17 (c) 9 (d) 17 (e) None of the above. Quiz 10 3 2 2 0 4 24/7/10 2 3 0 , C = 4 2 3 , and D = AB 3C , 0 1 32. If A is 2 3, B is 3 4, C is 4 2, and D is 3 2, then ABC + 2D is: (a) 2 2 (b) 3 2 (c
U. Houston - MATH - 3321
MATH 3321Quiz 114/12/091. If a system of n linear equations in n unknowns is consistent, then the rank of the matrix of coecients is n. (a) Always true (b) Sometimes true (c) False (d) None of the above2. If the determinant of the matrix of coecients
U. Houston - MATH - 3321
MATH 3321Quiz 124/19/101. If a system of n linear equations in n unknowns has innitely many solutions, then the rank of the matrix of coecients is n 1. (a) Always true. (b) Sometimes true. (c) Never true. (d) None of the above.2. If 0 is an eigenvalue
U. Houston - MATH - 3321
MATH 3321Quiz 134/22/101. If 0 is not an eigenvalue of the matrix of coecients of a homogeneous system of n linear equations in n unknowns, then the system has innitely many solutions. (a) Always true. (b) Sometimes true. (c) Never true. (d) None of th
U. Houston - MATH - 3321
Exam 1 Review Show that y = y(x) is/is not a solution of the differential equation F (x, y, y , . . . y (n) = 0. Examples: 1. x2 y - 3x y + 4y = 0; 2. d3y dy + = ex ; 3 dx dx y1 (x) = x2 , y2 (x) = x2 ln xy(x) = 1 + sin x + 1 ex , z(x) = 2 cos x + 1 ex .
U. Houston - MATH - 3321
ANSWERS Exam 1 ReviewPart I. y is/is not a solution. 1. Both are solutions. 2. Both are solutions. 3. y1 is a solution, y2 is not. 4. Both are solutions. 5. y is a solution. Part II. Find the dierential equation. 1. 2xyy = 3y 2 + 6. 2. x2y xy 3y = 0. 3.
U. Houston - MATH - 3321
MATH 3321 Sample Questions for Exam 2 Linear Nonhomogeneous Dierential Equations 1. Find the general solution of y 4 6 4 y + 2 y = 2. x x x2. Find the general solution of y + 4 y = 2 tan 2x. 3. Find the general solution of y 6 y + 9 y = 4 e3x + 4. Find t
U. Houston - MATH - 3321
MATH 3321 Sample Questions for Exam 2Linear Nonhomogeneous Dierential Equations 1. Find the general solution of y Answer y = C1 x2 + C2 x3 + 2 . 3 2. Find the general solution of y + 4 y = 2 tan 2x. Answer y = C1 cos 2x + C2 sin 2x 1 24 6 4 y + 2 y = 2
U. Houston - MATH - 3321
MATH 3321 Sample Questions for Exam 3 2x 4 3 5x 3y 2 2 y 5 2 5 121. Find x and y so that+=.2. Let A =2 1 3 0 4 2,B=3 1 25,3 2 C = 0 1 . 1 2 (b) AB (c) B + AC (d) CBAPerform the indicated operations, if possible: (a) AC3. Show that the matrix
U. Houston - MATH - 3321
MATH 3321 Sample Questions for Exam 31. Find x and y so that2x 4 3 5x+3y 2 2 y=5 2 5 12.Answer: x = 31/17, y = 49/17 2 1 3 0 4 2 3 1 25 3 2 C = 0 1 . 1 2 (b) AB (c) B + AC (d) CBA 2. Let A =,B=,Perform the indicated operations, if possible:
U. Houston - MATH - 3321
MATH 3321 Final Exam Sample Questions 1. y 2 = Cx3 3 is the general solution of a dierential equation. Find the equation. Answer y = 3y 2 + 9 2xy cos 2x . x22. Find the general solution of x y + 3 y = Answer y = sin 2x C + 3. 2x3 x3. Find the general so
U. Houston - MATH - 3321
MATH 3321 ENGINEERING MATHEMATICS Homework Problems Chapter 1 Exercises 1.1: 1, 2, 3, 4, 7, 9, 10, 12, 13, 14, 16, 17, 18, 21, 22 Exercises 1.3: 2, 3, 6, 7, 8, 9, 11, 14, 15, 16, 17, 18, 20, 21, 24 Chapter 2 Exercises 2.1: 2, 3, 4, 7, 9, 11, 14, 17, 18, 2
U. Houston - MATH - 3321
POLYNOMIALSDenition 1: A real polynomial is an expression of the form P (x) = an xn + an1 xn1 + + a1 x + a0 where n is a nonnegative integer and a0 , a1, . . . , an1, an are real numbers with an = 0. The nonnegative integer n is called the degree of P .
U. Houston - MATH - 3321
Higher-Order Linear Differential Equationsnth order linear differential equation: (1) y(n) + pn1( x) y(n1) +L+ p1( x) y '+ p0 ( x) y = f ( x) where p0 , p1, K, pn1, and f are continuous functions on some interval I .(1) is homogeneous if f ( x) 0 on I :
U. Houston - MATH - 3321
Table of Integrals 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. ur du = ur+1 + C, r+1 r = 1 2. 4. 6. 8. 10. 12. 14. 16. 18. 1 u2 + C u2 1| + C 20. 1 du = ln |u| + C u sin u du = cos u + C tan u du = ln | sec u| + C sec u du = ln | sec u + tan u| + C sec u tan u
U. Houston - MATH - 3321
1.If a system of n linear equa-tions in n unknowns is consistent, then the rank of the matrix of coefcients is n.1. Always true2. Sometimes true3. False12.If the determinant of thematrix of coecients of a system of n linear equations in n unknown
U. Houston - POLS - 1336
1 .The US Constitution is the oldest and shortest constitution still in effect today.Student Response Value True 100%Score: 1.5/1.52 .Early Supreme Court decisions (to 1835) expanded the power of the national government at the expense of state govern
U. Houston - POLS - 1336
1 .Match 8th and 9th amendment cases with their proper descriptionStatement Response Value Griswold v. Connecticut 1963 Established Right to Privacy 11.11% Roe v. Wade 1973 Legalized 1st trimester abortions 11.11% Webster 1989 outlawed 1 abortion method