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School: UCSB
Course: Introduction To Partial Differential Equations
6.1 LAPLACES EQUATION MATH 124B Solution Key HW 03 6.1 LAPLACES EQUATION 4. Solve u x x + u y y + uzz = 0 in the spherical shell 0 < a < r < b with the boundary conditions u = A on r = a and u = B on r = b, where A and B are constants. Hint: Look for a so
School: UCSB
Course: Partial Differential Equations
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Answers Version A 1. Version B 4 5 1. 1 8 a) 1, 2 2. a = 1, b = 4 b) 0, 0 2. 3. 1 cos() 1, use squeeze thm. c) discontinuous at x = 0 3. lim f (x) = 0 = lim f (x) x0 4. x0+ 1 2 b) a) 5. f (0) = 2 and f (3) = 7, use I
School: UCSB
Course: Vector Calculus 2
SEAN CHEN WeBWorK assignment number HW 5 is due : 07/14/2010 at 04:00am PDT. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information. Math5C-A1-M10-Wirts This le is /conf/snippets/setH
School: UCSB
Course: Math 108A
Math 108A Winter Quarter 2014 Practice Problems for Midterm No notes or calculators are permitted on this exam. To obtain full credit you must show all work, provide complete proofs, and provide short but complete explanations for all questions in the Tru
School: UCSB
Course: CALC WITH APPLI 1
Math 3A Exam 2 (Winter 2009)-Practice Name: TAs Name: Lecture Time: Discussion Time: DIRECTIONS: Please do not open your exam until you are instructed to do so. Write your name and ll in the blanks corresponding to your lecture time, TAs name, and discuss
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 1 Copyright Daryl Cooper D.A.R.Y.L. January 5, 2015 Use your iclicker to respond. When did you take 34A ? A = Last quarter with me B = Last year with someone else C = A long time ago in a galaxy far, far away D = Never took it Lectures on
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
School: UCSB
School: UCSB
LJMHILHI.āSign.āAAI,.v;._m._u414;:H,L,-.A.g.ā4āāAāā1.val\Vā~ā.āā.A.A.-āy.,w.,.w nunā . . āW. . . ,A . ,LLLLLLU . nu H , A V ._.A_ā . . . a .u . yn ā,ll.ā.~āmām.l,y._.pppwv." V ā cĀ» Dfm/Vā I Ā© W CE M Mr āS cāQāCNJ "n W3 m .kgxwāļ¬wā MHMRBAX \V\ U Conslāw
School: UCSB
Ā® Linus Rn R A \{AQ ām Lovxshļ¬ oāg MM powĀ» {X \ U53 SMā 4x13} r\.% AX+V>NB+C=O x ļ¬nk Ewonufwuā Q Ā„O(M WM bQ Vāntļ¬ E. US? $sz Kris/r 0W \TWFG 39Mā; . Cw. 3} BQĀ„?OQ TR :<3,5Ā§ (MC! ānoLQ WON.ā Ā§Q= <Ā»\3\> mm m āPM \1AQ.7LUQāLĀ«Q (dwāwJ Janā) m WOWā (5499*
School: UCSB
Course: Calculus With Applications
Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
School: UCSB
Course: MATH 3A
Lecture 6 Continuity Part 1: What does it mean for a function to be continuous? o A function f is continuous at c if lim x c f ( x) both exist o Intuitively, this is meant to convey the idea that the graph of f near c leads to f (c) If f is not continuo
School: UCSB
Course: MATH 3A
Lecture 7 Continuity Part 2: Other functions Trigonometric functions: o We first consider f ( x )=sin ( x ) o Note that sin(x )< x for all x> 0 o o Since 0<sin ( x)< x + for 0<x < 2 the squeeze theorem tells us that x 0 sin ( x )=0 lim x 0 sin ( x )=0
School: UCSB
Course: MATH 3A
Implicit Differentiation: Chain rule means we can take the derivative of an equation, even if it isnt a function Example: Find the slope of the curve defined by x 2+ y 2 =4 Obviously one option is to solve for y Alternatively, we can take the derivative
School: UCSB
Course: MATH 3A
Product Rule: d d d fg=f g+ g f dx dx dx Example: Find d x ex dx Product Rule says that it is e ( x) ( 1 )=x e x +e x d d x e x +e x x=x e x + dx dx Proof of the Product Rule g ( x+ h )g ( x ) f ( x )f ( x ) g (x) d d lim h 0 f ( x+ h ) +g( x) =f ( x ) g
School: UCSB
Course: MATH 3A
Lecture 13 Our goal is to find rules for the derivative of as many functions as possible d cc c=l h 0 =l h 0 0=0 First we deal with the simplest functions dx h Thus, the derivative of any constant is zero ( x +h )x h =l x 0 =1 Next we do the identity f
School: UCSB
Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Answers Version A 1. Version B 4 5 1. 1 8 a) 1, 2 2. a = 1, b = 4 b) 0, 0 2. 3. 1 cos() 1, use squeeze thm. c) discontinuous at x = 0 3. lim f (x) = 0 = lim f (x) x0 4. x0+ 1 2 b) a) 5. f (0) = 2 and f (3) = 7, use I
School: UCSB
Course: Math 108A
Math 108A Winter Quarter 2014 Practice Problems for Midterm No notes or calculators are permitted on this exam. To obtain full credit you must show all work, provide complete proofs, and provide short but complete explanations for all questions in the Tru
School: UCSB
Course: CALC WITH APPLI 1
Math 3A Exam 2 (Winter 2009)-Practice Name: TAs Name: Lecture Time: Discussion Time: DIRECTIONS: Please do not open your exam until you are instructed to do so. Write your name and ll in the blanks corresponding to your lecture time, TAs name, and discuss
School: UCSB
Course: Calculus With Applications
version Math 34A Fall 2010 Prof D.A.R.Y.L. PRINT NANIE Quality SCORE 2 24 Bonus Put nal answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on this test we may investigate. Number your
School: UCSB
Course: Math 108A
Math 108A Winter Quarter 2014 Practice Problems for Final No notes or calculators are permitted on this exam. You must show all work, provide complete proofs, and provide short by complete explanations for all questions in the True/False section in your B
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: Introduction To Partial Differential Equations
6.1 LAPLACES EQUATION MATH 124B Solution Key HW 03 6.1 LAPLACES EQUATION 4. Solve u x x + u y y + uzz = 0 in the spherical shell 0 < a < r < b with the boundary conditions u = A on r = a and u = B on r = b, where A and B are constants. Hint: Look for a so
School: UCSB
Course: Partial Differential Equations
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: Vector Calculus 2
SEAN CHEN WeBWorK assignment number HW 5 is due : 07/14/2010 at 04:00am PDT. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information. Math5C-A1-M10-Wirts This le is /conf/snippets/setH
School: UCSB
Course: Intro To Linear Alg
Mathematics 700 Test #1 Name: Solution Key Show your work to get credit. An answer with no work will not get credit. 1. (15 Points) Dene the following: (a) Linear independence. The vectors v1 , . . . , vm in the vector space V are linearly independent i t
School: UCSB
Course: Transitions To Higher Mathematics
Math 8 - Solutions to Home Work 2 Due: October 11, 2007 1. Every/Only. Sometimes sentences with the words only and every can be conditional statements in disguise. For example, Every even number is a multiple of two. can be rephrased as If a number is eve
School: UCSB
Course: A Concise Introduction To Pure Mathematics
Homework 1 Solutions Kyle Chapman October 26, 2012 9.1 True, True, False, True, False, False, True, False, True 9.2 a D and E b You can only deduce that it is not D. 9.3 a Not valid, would be D C then D C . b Valid, this is contraposition. C D then D C c
School: UCSB
Course: Linear Algebra
Application Problems (Numerical Integration): l) The widths of a kidney shaped pool were measured at 2 meter intervals as indicated in the ļ¬gure below. Use Simpsonās Rule to estimate the area of the pool. 2) A radar gun was used to record the speed of
School: UCSB
Course: Differential Equations
chew cn CTGKGNS am '9' Gmle book a 3903; $Ā¢a17> RTeiew Maa In {Ramada- =eve13w has an assign a I. Go over an Iccker aueson, [new in 5 a 2' 1'? We WW; Chow. queous from and 10PTCTD make a {Lest Jaunuf, wa-h mm: Alwags check 86W answer? (like eignveerm' CM
School: UCSB
Course: Math 4A
Math 4A Midterm Review Prolbems Note: These problems are provided as review of the key ideas that well be tested on the midterm, but this is not a practice midterm. Because we are testing your understanding of concepts and not simply computational uency,
School: UCSB
Course: Differential Equations
Mathematics 4B Winter 2015: Review for Final March 12, 2015 Professor J. Douglas Moore YOU ARE ALLOWED ONE 3 x 5 CARD FOR THE FINAL EXAM. Recall that there is assigned seating for the nal exam. Please write your seat number on your card. You will need to
School: UCSB
Course: Numerical Analysis
University of California, Santa Barbara Department of Statistics & Applied Probability PSTAT 120B, Probability & Statistics, Spring 2010 Instructor: Jarad Niemi Email: niemi@pstat.ucsb.edu Course hours: MWF 10:00-10:50am in HFH 1104 TAs: Varvara Kulikova
School: UCSB
Course: Vector Calculus
Math 6A - Vector Calculus with Applications, First Course Instructor: Oce: E-mail: Oce Hours: Lectures: Classroom: Jon Tjun Seng Lo Kim Lin Graduate Tower, Oce 6431H jlokimlin@math.ucsb.edu TR 1:00 PM-2:00 PM, also available by appointment. MTWR 8:00 AM-9
School: UCSB
SPRING 2015, MATH 6 B, VECTOR CALCULUS 2 INSTRUCTOR : Gustavo Ponce (o. SH 6505 #8938365) SCHEDULE : TR 330 445 ROOM : HFH 1104 INSTRUCTOR OFFICE HOURS : T. 5 6, R. 11 12, 5 6. TEACHING ASSISTANT : Kathleen Hake (SH 6431 K), Garo Sarajian (SH 6432 F) TEAC
School: UCSB
Course: Vector Calculus
Math 6A - Vector Calculus with Applications, First Course Instructor: Oce: E-mail: Oce Hours: Lectures: Amanda R. Curtis Graduate Tower, Oce 6432Q arcurtis@math.ucsb.edu MTWR 1:00 PM-2:00 PM, also available by appointment. MTWR 3:30-4:35PM, 3515 Phelps Ha
School: UCSB
Course: Vector Calculus
Math 6A (Vector Calculus) Syllabus1 TR, 11:00 AM to 12:15 PM, Room 1701 in Theater/Dance West Instructor/E-mail: Jordan Schettler (jcs@math.ucsb.edu) Ofce Hours: Tuesday 2-3 pm, Friday 9-10 am, or by appointment, South Hall, Room 6721 Website: Use the cou
School: UCSB
Course: Homological Algebra
Math 236B, Spring 2015, MWF 9-9:50, HSSB 1223 Homological Algebra Instructor: Birge Huisgen-Zimmermann, SH 6518, Oce hours M, F 11 - 12, W 12:30 1:30. Accompanying texts, as for the winter quarter: The manuscript I will put on the board will again serve
School: UCSB
Course: Introduction To Partial Differential Equations
6.1 LAPLACES EQUATION MATH 124B Solution Key HW 03 6.1 LAPLACES EQUATION 4. Solve u x x + u y y + uzz = 0 in the spherical shell 0 < a < r < b with the boundary conditions u = A on r = a and u = B on r = b, where A and B are constants. Hint: Look for a so
School: UCSB
Course: Partial Differential Equations
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Answers Version A 1. Version B 4 5 1. 1 8 a) 1, 2 2. a = 1, b = 4 b) 0, 0 2. 3. 1 cos() 1, use squeeze thm. c) discontinuous at x = 0 3. lim f (x) = 0 = lim f (x) x0 4. x0+ 1 2 b) a) 5. f (0) = 2 and f (3) = 7, use I
School: UCSB
Course: Vector Calculus 2
SEAN CHEN WeBWorK assignment number HW 5 is due : 07/14/2010 at 04:00am PDT. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information. Math5C-A1-M10-Wirts This le is /conf/snippets/setH
School: UCSB
Course: Math 108A
Math 108A Winter Quarter 2014 Practice Problems for Midterm No notes or calculators are permitted on this exam. To obtain full credit you must show all work, provide complete proofs, and provide short but complete explanations for all questions in the Tru
School: UCSB
Course: CALC WITH APPLI 1
Math 3A Exam 2 (Winter 2009)-Practice Name: TAs Name: Lecture Time: Discussion Time: DIRECTIONS: Please do not open your exam until you are instructed to do so. Write your name and ll in the blanks corresponding to your lecture time, TAs name, and discuss
School: UCSB
Course: Intro To Linear Alg
Mathematics 700 Test #1 Name: Solution Key Show your work to get credit. An answer with no work will not get credit. 1. (15 Points) Dene the following: (a) Linear independence. The vectors v1 , . . . , vm in the vector space V are linearly independent i t
School: UCSB
Course: Transitions To Higher Mathematics
Math 8 - Solutions to Home Work 2 Due: October 11, 2007 1. Every/Only. Sometimes sentences with the words only and every can be conditional statements in disguise. For example, Every even number is a multiple of two. can be rephrased as If a number is eve
School: UCSB
Course: A Concise Introduction To Pure Mathematics
Homework 1 Solutions Kyle Chapman October 26, 2012 9.1 True, True, False, True, False, False, True, False, True 9.2 a D and E b You can only deduce that it is not D. 9.3 a Not valid, would be D C then D C . b Valid, this is contraposition. C D then D C c
School: UCSB
Course: Calculus With Applications
version Math 34A Fall 2010 Prof D.A.R.Y.L. PRINT NANIE Quality SCORE 2 24 Bonus Put nal answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on this test we may investigate. Number your
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 1 Copyright Daryl Cooper D.A.R.Y.L. January 5, 2015 Use your iclicker to respond. When did you take 34A ? A = Last quarter with me B = Last year with someone else C = A long time ago in a galaxy far, far away D = Never took it Lectures on
School: UCSB
Course: Numerical Analysis
University of California, Santa Barbara Department of Statistics & Applied Probability PSTAT 120B, Probability & Statistics, Spring 2010 Instructor: Jarad Niemi Email: niemi@pstat.ucsb.edu Course hours: MWF 10:00-10:50am in HFH 1104 TAs: Varvara Kulikova
School: UCSB
Course: Math 108A
Math 108A Winter Quarter 2014 Practice Problems for Final No notes or calculators are permitted on this exam. You must show all work, provide complete proofs, and provide short by complete explanations for all questions in the True/False section in your B
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: Calculus With Applications
Calculus and Mathematical Reasoning for Social and Life Sciences Final Exam, Math 34A, Spring 2015 Instructor: Jingrun Chen June 9, 2015 Answer the following 8 questions. Calculators are not allowed. The use of books of any kind is not allowed. A 3 5 note
School: UCSB
Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
School: UCSB
Course: INTRO NUM ANALYSIS
Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
School: UCSB
Course: Methods Of Analysis
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
School: UCSB
Course: Calculus With Applications
Calculus and Mathematical Reasoning for Social and Life Sciences Math 34A, Spring 2015 Instructor: Jingrun Chen April 23, 2015 Answer the following 6 questions. Calculators are not allowed. The use of books of any kind is not allowed. A 3x5 note card is a
School: UCSB
Course: Calculus
Being the final examination for Math 3B NO notes or calculators. READ all questions carefully. Make sure your answers are clearly marked and it is clear what work is relevant and should be graded. Each problem is worth 20 points. Note there is a blank pag
School: UCSB
Course: Differential Equations
-7 -3 -1 1 -2 26.(1 pt) 0 -2 -1 -2 -28 -12 -6 3 . 32.(1 pt) Let M = -2 3 Compute the rank of the above matrix -2 2 1 Find c1 , c2 , and c3 such that M 3 + c1 M 2 + c2 M + c3 I3 = 0, where 7 4 7 I3 is the identity 3 3 matrix. 27.(1 pt) 7 4 3 , c1 = 21 12
School: UCSB
Course: INTRO NUM ANALYSIS
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera March 17th, 2009 Answer the following 8 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presen
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
Course: Graph Theory
HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph Kn is a contraction of Kn,n . Solution: We describe the process for several small values of n. In this way, we can discern the inductive step. Clearly, K1 , which is just one vertex, is a
School: UCSB
Course: INTRO NUM ANALYSIS
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for pres
School: UCSB
Course: Calculus With Applications
Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
School: UCSB
Course: Differential Equations
Sample WeBWorK problems. 1.(1 pt) Which of the following are vectors in R2 ? A. (1,0,0) B. (1,0) C. x2 D. x E. 1 F. (0,1) G. None of the above Which of the following is the zero vector in R2 ? A. 0 B. (0,0,0) C. (0,0) D. None of the above Which pairs belo
School: UCSB
Course: MATH 3A
HOMEWORK ASSIGNMENT 5 MATH 10A LECTURE 002, FALL 2015, RICHARD BAMLER WRITTEN BY: JASON FERGUSON JMF@MATH.BERKELEY.EDU DUE: Tuesday, September 15th in your discussion section [unless your GSI says otherwise]. Unless otherwise stated, please include an exp
School: UCSB
Course: MATH 3A
HOMEWORK ASSIGNMENT 4 MATH 10A LECTURE 002, FALL 2015, RICHARD BAMLER WRITTEN BY: JASON FERGUSON JMF@MATH.BERKELEY.EDU DUE: Thursday, September 10th in your discussion section [unless your GSI says otherwise]. Unless otherwise stated, please include an ex
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 1 Copyright Daryl Cooper D.A.R.Y.L. January 5, 2015 Use your iclicker to respond. When did you take 34A ? A = Last quarter with me B = Last year with someone else C = A long time ago in a galaxy far, far away D = Never took it Lectures on
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
School: UCSB
School: UCSB
LJMHILHI.āSign.āAAI,.v;._m._u414;:H,L,-.A.g.ā4āāAāā1.val\Vā~ā.āā.A.A.-āy.,w.,.w nunā . . āW. . . ,A . ,LLLLLLU . nu H , A V ._.A_ā . . . a .u . yn ā,ll.ā.~āmām.l,y._.pppwv." V ā cĀ» Dfm/Vā I Ā© W CE M Mr āS cāQāCNJ "n W3 m .kgxwāļ¬wā MHMRBAX \V\ U Conslāw
School: UCSB
Ā® Linus Rn R A \{AQ ām Lovxshļ¬ oāg MM powĀ» {X \ U53 SMā 4x13} r\.% AX+V>NB+C=O x ļ¬nk Ewonufwuā Q Ā„O(M WM bQ Vāntļ¬ E. US? $sz Kris/r 0W \TWFG 39Mā; . Cw. 3} BQĀ„?OQ TR :<3,5Ā§ (MC! ānoLQ WON.ā Ā§Q= <Ā»\3\> mm m āPM \1AQ.7LUQāLĀ«Q (dwāwJ Janā) m WOWā (5499*
School: UCSB
W we vu~z pawwetb gvMaA ad :zļ¬ta W} m nonnlncvoxkvqi +0!ā 5 \ 'fo'm-āh CpanfanLt ham owL mom (Mitt/LR WLU 'rm:3a_+xāvĀ£ VDUULLā) WQV QāUDLJQA. Eiļ¬h 0Ā° .L D [Lech ļ¬ak i la dive/ā($03 āmummy/lag. L11 i) M ā¬UU\ Smlxmewxg SIāS\ā\S(M.A3 {X box/wad WA
School: UCSB
wnww-mnnwm. W'Inwup'lļ¬ww \ \ \nraz-m-m r. New \hāewEON on CNowLms We, now SQL mrM m OJQOVāIāQ/JVHC hehāonS Jennā deāur CHAch ā\-Ā« 9mm on Cm No {Q Wļ¬mhāg vie/w 0(1 M Ā£WE50/Vt' J IwLļ¬/Lā off 3: ; nah/v rān 110. J ch/ļ¬a (9 L 0ā4"}: QWffoļ¬S . RQCNU
School: UCSB
Course: Vector Calculus
3.2 07/08/15 Chain Rule 3.2 CONTENTS 07/08/15 Chain Rule We now begin to pursue the Fundamental Theorem of Calculus (FTC). Recall that the FTC states that derivatives and integrals are in some sense inverse operations. We rst explore the FTC and its relat
School: UCSB
Course: Vector Calculus
2.3 07/01/15 Critical Points & Extreme Values 2 VECTOR DIFFERENTIATION 2.3 2.3.1 07/01/15 Critical Points & Extreme Values Directional Derivatives and the Gradient Announcements Homework 02 is posted on Gauchospace and due next Wednesday July 8th at 4:20
School: UCSB
Course: Vector Calculus
3.1 07/07/15 Constrained Optimization: Lagrange Multipliers ONTENTS C 3.1 3.1.1 07/07/15 Constrained Optimization: Lagrange Multipliers The Self-Driving Car Let t = time and 0 t T = length of trip p : [0, T ] R2 is target route c : [0, T ] R2 is actual ro
School: UCSB
Course: Vector Calculus
1.2 06/23/15 Dot & Cross Product 1.2 CONTENTS 06/23/15 Dot & Cross Product Announcements: Homework 01 is due as soon as WeBWork is enabled. Jay Roberts oce hours are TR 2:00 PM-3:00 PM. The change is reected on the syllabus. Recall that Rn is the collec
School: UCSB
Course: Vector Calculus
2 VECTOR DIFFERENTIATION 2 Vector Dierentiation 2.1 2.1.1 06/29/15 Curves & Surfaces Review of Paths & Curves Announcements: The number of homework problems have been reduced to 50. The homework is due next Friday at 11:42AM You must take the midterm n
School: UCSB
Course: Vector Calculus
1.1 06/22/15 Vector Basics 1.1 CONTENTS 06/22/15 Vector Basics In calculus, we previously studied functions f: RR (1) with domain R (or a subset) which produce real numbers as output. For example f (x) = x2 (2) The study of functions of more than one vari
School: UCSB
Course: Vector Calculus
1.3 06/24/15 Lines & Planes 1.3 1.3.1 CONTENTS 06/24/15 Lines & Planes Lines in R2 through the origin Announcements: There was an typo in yesterdays lecture notes. The cross product is v w = (v2 w3 v3 w2 , v3 w1 v1 w3 , v1 w2 v2 w1 ). (62) Therefore, i j
School: UCSB
Course: Vector Calculus
1.4 06/25/15 Paths in Rn 1.4 1.4.1 CONTENTS 06/25/15 Paths in Rn Lines in R2 The line through x0 = (x0 , y0 ) and perpendicular (orthogonal) to n = (A, B) consists of the set of points x = (x, y) satisfying the expression Ax + By + C (standard form) (99)
School: UCSB
Course: Vector Calculus
3.3 07/09/15 Vector Fields 3.3 CONTENTS 07/09/15 Vector Fields Denition 2. A vector eld is a function F : Rn Rn (dimensions must match!) (40) Denition 3. Suppose the vector eld F : Rn Rn satises the equation F= V (41) for some real-valued function V : Rn
School: UCSB
Course: Vector Calculus
4.2 07/14/15 Double Integrals 4.2 4.2.1 CONTENTS 07/14/15 Double Integrals Volume beneath a Surface Denition 3. The rectangle R = [a, b] [c, d] in R2 consists of all points (x, y) satisfying y d axb c cxd a b x Question 2. Given a function f : R2 R (which
School: UCSB
Course: Vector Calculus
5.3 07/22/15 Flux Integrals 5.3 CONTENTS 07/22/15 Flux Integrals Since r and r are tangent to the surface (i.e. lie in the tangent plane to at each u v point on ), then their cross product r r is perpendicular to the tangent plane to the u v surface at ea
School: UCSB
Course: Vector Calculus
5.1 07/20/15 Path Integrals and Greens Theorem 5.1 5.1.1 CONTENTS 07/20/15 Path Integrals and Greens Theorem Path Independence f d r is independent of the path between any Theorem 1. In a region R, the line integral C f d r = 0 for every closed curve C wh
School: UCSB
Course: Vector Calculus
4.1 07/13/15 Path (Line) Integrals CONTENTS 4.1 07/13/15 Path (Line) Integrals 4.1.1 Intuitive idea behind Line Integrals In single-variable calculus you learned how to integrate a real-valued function f (x) over an interval [a, b] in R. This integral (us
School: UCSB
Course: Vector Calculus
4.3 07/15/15 Change of Variables CONTENTS 4.3 07/15/15 Change of Variables 4.3.1 Interchanging Order of Integration Example 10. Find the volume V under the plane z = 8x + 6y over the region R = cfw_(x, y) : 0 x 1, 0 y 2x2 . Using vertical slices we get: y
School: UCSB
Course: Vector Calculus
4.4 07/16/15 Triple Integrals 4.4 4.4.1 CONTENTS 07/16/15 Triple Integrals Double Integrals in PSTAT Example 17. A famous distribution function is given by the standard normal distribution, whose probability density function (PDF) f is 1 2 f (x) = ex /2 ,
School: UCSB
Course: Vector Calculus
6.1 07/27/15 6.1 6.1.1 CONTENTS 07/27/15 The parallelogram law Example 1. Find the area of the triangle that is determined by the points (2, 2, 0), (1, 0, 2), and (0, 4, 3). The area of the triangle, say A, is half the area of the parallelogram determined
School: UCSB
Course: Calculus With Applications
Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
School: UCSB
Course: MATH 3A
Lecture 6 Continuity Part 1: What does it mean for a function to be continuous? o A function f is continuous at c if lim x c f ( x) both exist o Intuitively, this is meant to convey the idea that the graph of f near c leads to f (c) If f is not continuo
School: UCSB
Course: MATH 3A
Lecture 7 Continuity Part 2: Other functions Trigonometric functions: o We first consider f ( x )=sin ( x ) o Note that sin(x )< x for all x> 0 o o Since 0<sin ( x)< x + for 0<x < 2 the squeeze theorem tells us that x 0 sin ( x )=0 lim x 0 sin ( x )=0
School: UCSB
Course: MATH 3A
Implicit Differentiation: Chain rule means we can take the derivative of an equation, even if it isnt a function Example: Find the slope of the curve defined by x 2+ y 2 =4 Obviously one option is to solve for y Alternatively, we can take the derivative
School: UCSB
Course: MATH 3A
Product Rule: d d d fg=f g+ g f dx dx dx Example: Find d x ex dx Product Rule says that it is e ( x) ( 1 )=x e x +e x d d x e x +e x x=x e x + dx dx Proof of the Product Rule g ( x+ h )g ( x ) f ( x )f ( x ) g (x) d d lim h 0 f ( x+ h ) +g( x) =f ( x ) g
School: UCSB
Course: MATH 3A
Lecture 13 Our goal is to find rules for the derivative of as many functions as possible d cc c=l h 0 =l h 0 0=0 First we deal with the simplest functions dx h Thus, the derivative of any constant is zero ( x +h )x h =l x 0 =1 Next we do the identity f
School: UCSB
Course: MATH 3A
Lecture 10 The Derivative of a Function lim f ( x +h ) f (x ) Given a function f(x), we define the derivative h 0 h ' f ( x )= This gives us a new function, since the defining limit depends only on x d f Another notation for f ' ( x ) is dx o This is call
School: UCSB
Course: MATH 3A
Slope of the tangent line The slope of the tangent line should be the limit of the slope of the secant lines Therefore, the slope of the tangent line is: f ( x )f ( a) M= lim x z xa where (a, f(a) is the point of tangency The expression inside the limi
School: UCSB
Course: MATH 3A
Lecture 5 The Precise Definition of the Limit: Recall, lim x c f ( x )=L if for every choice of output varience , there is a choice of input varience so that 0< xc implies that f ( x ) Example: Consider f (x)=5 x2 o Look at lim x 1 f ( x) o F(1)=52=3
School: UCSB
Course: MATH 3A
Lecture 4 The Limit Laws: Suppose lim x c f ( x ) and lim x c g ( x ) both exist. We then get the following: lim x c g ( x ) lim x c f ( x ) + o lim x c [ f ( x )+ g ( x ) ] = lim x c g ( x ) lim x c f ( x ) o lim x c [ f ( x )g ( x ) ] = lim x c g ( x )
School: UCSB
Course: MATH 3A
Lecture 3 Notation One Sided Limit + xc We write lim to indicate the limit from the right side, also called the limit as x decreases to c xc We write lim to indicate the limit from the left side, also called the limit as x increases to c Example: f ( x
School: UCSB
Course: MATH 3A
Lecture 1 What is a function? Input function output How do we represent functions? Verbally o Example: The temperature of water coming out a faucet in terms of time is a function o Example: A persons phone number over time may not be a function, since
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Course: MATH 3A
Lecture 2 Definition of a limit: lim x a f ( x )=L if for every error bound , there is a choice of error for the input, , so that 0<|xa|< guarantees |f ( x )L|< The idea is that no matter how close you need the answer to get to get L, that can be achieve
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School: UCSB
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School: UCSB
School: UCSB
School: UCSB
School: UCSB
,g M '-' - g) Find the Fourier series of x) = g for :c E (~7z, 7r]. Pme 2: x .33 eĀ§L E \ E} "W VJ TL, TL Q 'l \ ~ g 9 MEN wimĆ©jp war smgwĆ©x T'L " Kg m u i} N any GECN(. .j:"ll'*l"i (b) Compute 5320253 using (2) to nd the value of Ā°Ā°1 2152' k: _|
School: UCSB
Course: Differential Equations
Math 4B Lecture 9 February 3, 2015 Doug Moore Announcements. 1. Midterm this Thursday. You should know linear DEs (differential equations), exact dierentials, Cauchy-Euler polygon and homogeneous linear second order DEs with constant coefcients. No calcul
School: UCSB
Course: Differential Equations
Math 4B Lecture 16 February 26, 2015 Doug Moore Electrical circuits give rise to complicated systems of dierential equations. Circuits which include linear resistors, inductors, and capacitors give rise to linear systems of equations. Other circuit elemen
School: UCSB
Course: Differential Equations
Math 4B Lecture 14 February 19, 2015 Doug Moore . Our next goal is to consider the circuit given by a resistor of resistance R, an inductor of inductance L, and a capacitor of capacitance C, connected in series to a source of electromotive force E(t). Let
School: UCSB
Course: Differential Equations
Math 4B Lecture 15 February 24, 2015 Doug Moore Announcements. 1. i-Clicker scores are now posted on gauchospace. You get one point for attending when you answer most of the questions in a given lecture. (The number of questions you need to answer depends
School: UCSB
Course: Differential Equations
Math 4B Lecture 12 February 12, 2015 Doug Moore The method of variation of parameters, also known as variation of constants, was discovered by Euler and Lagrange, and used to understand perturbations of planetary motion from the exact elliptical motion pr
School: UCSB
Course: Differential Equations
Math 4B Lecture 13 February 17, 2015 Doug Moore In the last several lectures, we have studied how to solve certain nonhomogeneous second order linear dierential equations, which are of the form d2 x dx + Q(t)x = R(t), + P (t) dt2 dt where L is the linear
School: UCSB
Course: Differential Equations
Math 4B Lecture 17 March 3, 2015 Doug Moore Our rst goal today is to discuss phase portraits for homogeneous systems of dierential equations of the form dx1 /dt = a11 x1 + a12 x2 + . . . + a1n xn , dx2 /dt = a21 x1 + a22 x2 + . . . + a2n xn , dxn /dt = an
School: UCSB
Course: Differential Equations
Math 4B Lecture 20 March 12, 2015 Doug Moore A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. Henri Poincar (1854-1912) e Announcements. 1
School: UCSB
Course: Differential Equations
Math 4B Lecture 18 March 5, 2015 Doug Moore Linear systems of dierential equations of the form dx1 /dt = a11 x1 + a12 x2 + . . . + a1n xn + f1 (t), dx2 /dt = a21 x1 + a22 x2 + . . . + a2n xn + f2 (t), dxn /dt = an1 x1 + an2 x2 + . . . + ann xn + fn (t). c
School: UCSB
Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Answers Version A 1. Version B 4 5 1. 1 8 a) 1, 2 2. a = 1, b = 4 b) 0, 0 2. 3. 1 cos() 1, use squeeze thm. c) discontinuous at x = 0 3. lim f (x) = 0 = lim f (x) x0 4. x0+ 1 2 b) a) 5. f (0) = 2 and f (3) = 7, use I
School: UCSB
Course: Math 108A
Math 108A Winter Quarter 2014 Practice Problems for Midterm No notes or calculators are permitted on this exam. To obtain full credit you must show all work, provide complete proofs, and provide short but complete explanations for all questions in the Tru
School: UCSB
Course: CALC WITH APPLI 1
Math 3A Exam 2 (Winter 2009)-Practice Name: TAs Name: Lecture Time: Discussion Time: DIRECTIONS: Please do not open your exam until you are instructed to do so. Write your name and ll in the blanks corresponding to your lecture time, TAs name, and discuss
School: UCSB
Course: Calculus With Applications
version Math 34A Fall 2010 Prof D.A.R.Y.L. PRINT NANIE Quality SCORE 2 24 Bonus Put nal answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on this test we may investigate. Number your
School: UCSB
Course: Math 108A
Math 108A Winter Quarter 2014 Practice Problems for Final No notes or calculators are permitted on this exam. You must show all work, provide complete proofs, and provide short by complete explanations for all questions in the True/False section in your B
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: Calculus With Applications
Calculus and Mathematical Reasoning for Social and Life Sciences Final Exam, Math 34A, Spring 2015 Instructor: Jingrun Chen June 9, 2015 Answer the following 8 questions. Calculators are not allowed. The use of books of any kind is not allowed. A 3 5 note
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Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
School: UCSB
Course: Calculus With Applications
Calculus and Mathematical Reasoning for Social and Life Sciences Math 34A, Spring 2015 Instructor: Jingrun Chen April 23, 2015 Answer the following 6 questions. Calculators are not allowed. The use of books of any kind is not allowed. A 3x5 note card is a
School: UCSB
Course: Calculus
Being the final examination for Math 3B NO notes or calculators. READ all questions carefully. Make sure your answers are clearly marked and it is clear what work is relevant and should be graded. Each problem is worth 20 points. Note there is a blank pag
School: UCSB
Course: INTRO NUM ANALYSIS
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera March 17th, 2009 Answer the following 8 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presen
School: UCSB
Course: INTRO NUM ANALYSIS
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for pres
School: UCSB
Course: CACL WITH APPLI2
Calculus with applications Ninth 38, Winter 2013 Instructor: Baldvin Einarsson January 29, 2013 Answer the following 6 questions. You may bring one 3x5 inch notecard, but not the book. You may use a calculator (no phones), but you must show all your w
School: UCSB
Course: CACL WITH APPLI2
Calculus With applications Answer th phones) , notecards, but not the boo BUT YOU MUST SHOW YOUR FULL CREDIT. The grading Math 3B, Winter 2013 Instructor: Baldvin Einarsson e following 5 questions. You may bring two 3x5 inch k. You may use a calcu
School: UCSB
Course: Calculus With Applications
Tardis ire/[71ā Math 34A Fall 2009 l . _ version PRINT . NAME ā Midterm 3 Quality 1 ā no calculators 3Ā°ā: ' blue book for all now Put ļ¬nal answers in boxes on this page. Show high quality Work m ā1ā ļ¬ayyy Tļ¬anĆ©sgiving Points may be awarded for this. N
School: UCSB
Course: Calculus With Applications
\ version Tardis ' l r ' nam no calculators almamāha Bonus Math 34A Fa112012 - Prof DARYL. . . ' Ć©ā Signature Pm ļ¬nal answers in boxes on this Page. SHOW WORK for all questions in the blue bOOk- book If your work does not match your answer we may investi
School: UCSB
Course: Calculus With Applications
Tardis L! I version pRlNT W āM Green Math 34A Fall 2010 Quahty Z/z - .R.YL. . _1fthe wo xam lace Put ļ¬nal answers in boxes on this page- SHOW WORK m the blue in the blue book- At the end 0f the e P the answer You give on this test we may inVCSdgaāe' Numb
School: UCSB
Course: Calculus With Applications
no calculators Put ļ¬nal answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on this test we may investigate. Number your solutions in the blue book. At the end of the exam place this p
School: UCSB
Course: Calculus With Applications
Midterm #1 no calculators - 1 e book does not , match Put final answers in boxes on this page. SHOW WORK in the blue book. If the wogllc whiz: 1:ā the end of the: exam p1 ace the answer you give on this test we may investigate. Number your solutions in
School: UCSB
Course: Calculus With Applications
First Midterm, Math. 34A, Akemann, Winter 2012 BLUE VERSION PRINT FULL NAME _iĀ§g\g_ā¬3\_<5_\':1\5g3Ā§9_Ā£ _ w- PERM NUMBER 5195125 _ _ TA name (ļ¬rst, last or nickname) _Ā§_āQĀ£Ā£/_C _ _ Discussion section day and time 9-)};ā _ _ All answers must contain enough
School: UCSB
Course: Calculus With Applications
First Midt 1 m, Math. 34A, ā emann, Winter 2012 GREEN VE ; PRINT FULL NAME SQDJQāIXāKLQH; 1102?: _ _ TA name (ļ¬rst, last or nickname) _ _ Discussion section day and time _Wļ¬d_-_-U_ā_(_P_ā_5Q_ All answers-must contain enough explanation or calculation step
School: UCSB
Course: CACL WITH APPLI2
i :3ā f . ' - (NJ/hā? - - C: d , Martin Matt N AME: { 0W āll/LU. 1 TA (cu'cle one)@ā4 T5 T6 T7 SECTION (circle one): R4 R5 @ R7 Math 3B Midterm D February 11, 2013 You. have 50 minutes to complete this test. Ā«y 1. (5 pts) Which and integration? Circle all
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Course: CACL WITH APPLI2
@o Name Szczmeue Lmsļ¬n TARDIsmL Permmā TAās Nam l, mann, Second Midterm, Nov. 13, 2009 - 0 A ' v ' ORK AND DO NOT SIMPLIFY YOUR AN- SWERS. A11 problems are worth 3 points. (2 .< 1. Find the arc length of the curve y = 1n(1 ā x2), 0 g :c 5 l/x/i. 3- ,. L:
School: UCSB
Course: Calculus With Applications
version Math 343 Winter 2011 Midterm3 Prof DARYL. Putlļ¬nal answers in boxes on this page. . . . . te_ the answer you give on this test we may mvestiga this page INSIDE the blue book, and then put the blueboo k inside the enve C
School: UCSB
Course: Calculus With Applications
Problem 1 (19) A commuter railway has 800 p ges each one $2. For each cent the fare is ansport increased, 5 fewer peOple will go by train. It costs the r ' a passenger. What is the total proļ¬t (in cents) in terms of the number of tigketg 391; per day? Wri
School: UCSB
Course: Calculus With Applications
LolSl Nib (Q Cth wed SH Go 39 l; " '1 PW. Second Midterm, Math. 34A, Akemann, Winter 2012 GREEN VERSION -, PRINT FULL NAME _@Ā£{KlĀ§1lĀ§_ļ¬lĆ©ļ¬Ā£ļ¬ _ _ \ PERM NUMBER āfilllgfļ¬ _ _ TA name (ļ¬rst, last or nickname) _\4_āe_āļ¬ _ _ _E1J99.:g:_iļ¬w;r_ Discussion sectio
School: UCSB
Course: Calculus With Applications
Math 34B Midterm 2 Spring 2013 4. (10 points) A colony of bacteria is growing on a slice of iz_za. The initial mass oibanteriaJSJ grams. Initially, their rate of growth is approxim ely .02 times their mas After a. while the pepperoni pieces become covered
School: UCSB
Course: CACL WITH APPLI2
NAME: mā e MaYhn PermitN umber TA -ā Dz's.Tz'me : Put your ļ¬nal answers in the BOXES provided. Show your work as clearly as , possible. - Q 1. (10 points) i _ Q 2. (10 points) _'8:_ Q 3. (8 points) _āQ_ Q 4. (10 points) ifā Q 5. (12 points) FINAL 3ā7 _ M
School: UCSB
Course: CACL WITH APPLI2
Math 3B Midterm, Fall 2011 INSTRUCTIONS: Read each problem carefully. Write clearly and show all your work. Put the ļ¬nal answers, when appropriate, in BOXES. EXAMPLE: (For the instructor only) Q 1. (6 points) L Q 2. (6 points) L Q 3. (6 points) L Q 4.
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Course: CACL WITH APPLI2
Fiawāļ¬'is news.- - so that. the should be simplified enough to s! . z' ā ' 1. (You may draw the relevant diagram if it hel s ' 1 is justiļ¬ed by the picture above? (Write =1] it out; donāt try to remember what number I gave it.) 2. (a) Which property of
School: UCSB
Course: Introduction To Partial Differential Equations
6.1 LAPLACES EQUATION MATH 124B Solution Key HW 03 6.1 LAPLACES EQUATION 4. Solve u x x + u y y + uzz = 0 in the spherical shell 0 < a < r < b with the boundary conditions u = A on r = a and u = B on r = b, where A and B are constants. Hint: Look for a so
School: UCSB
Course: Partial Differential Equations
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: Vector Calculus 2
SEAN CHEN WeBWorK assignment number HW 5 is due : 07/14/2010 at 04:00am PDT. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information. Math5C-A1-M10-Wirts This le is /conf/snippets/setH
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Course: Intro To Linear Alg
Mathematics 700 Test #1 Name: Solution Key Show your work to get credit. An answer with no work will not get credit. 1. (15 Points) Dene the following: (a) Linear independence. The vectors v1 , . . . , vm in the vector space V are linearly independent i t
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Course: Transitions To Higher Mathematics
Math 8 - Solutions to Home Work 2 Due: October 11, 2007 1. Every/Only. Sometimes sentences with the words only and every can be conditional statements in disguise. For example, Every even number is a multiple of two. can be rephrased as If a number is eve
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Course: A Concise Introduction To Pure Mathematics
Homework 1 Solutions Kyle Chapman October 26, 2012 9.1 True, True, False, True, False, False, True, False, True 9.2 a D and E b You can only deduce that it is not D. 9.3 a Not valid, would be D C then D C . b Valid, this is contraposition. C D then D C c
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Course: INTRO NUM ANALYSIS
Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
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Course: Methods Of Analysis
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
School: UCSB
Course: Graph Theory
HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph Kn is a contraction of Kn,n . Solution: We describe the process for several small values of n. In this way, we can discern the inductive step. Clearly, K1 , which is just one vertex, is a
School: UCSB
Course: MATH 3A
HOMEWORK ASSIGNMENT 5 MATH 10A LECTURE 002, FALL 2015, RICHARD BAMLER WRITTEN BY: JASON FERGUSON JMF@MATH.BERKELEY.EDU DUE: Tuesday, September 15th in your discussion section [unless your GSI says otherwise]. Unless otherwise stated, please include an exp
School: UCSB
Course: MATH 3A
HOMEWORK ASSIGNMENT 4 MATH 10A LECTURE 002, FALL 2015, RICHARD BAMLER WRITTEN BY: JASON FERGUSON JMF@MATH.BERKELEY.EDU DUE: Thursday, September 10th in your discussion section [unless your GSI says otherwise]. Unless otherwise stated, please include an ex
School: UCSB
Course: MATH 3A
HOMEWORK ASSIGNMENT 3 MATH 10A LECTURE 002, FALL 2015, RICHARD BAMLER WRITTEN BY: JASON FERGUSON JMF@MATH.BERKELEY.EDU DUE: Tuesday, September 8th in your discussion section [unless your GSI says otherwise]. Unless otherwise stated, please include an expl
School: UCSB
Course: MATH 3A
HOMEWORK ASSIGNMENT 2 MATH 10A LECTURE 002, FALL 2015, RICHARD BAMLER WRITTEN BY: JASON FERGUSON JMF@MATH.BERKELEY.EDU DUE: Thursday, September 3rd in your discussion section [unless your GSI says otherwise]. Unless otherwise stated, please include an exp
School: UCSB
Course: MATH 3A
HOMEWORK ASSIGNMENT 1 MATH 10A LECTURE 002, FALL 2015, RICHARD BAMLER WRITTEN BY: JASON FERGUSON JMF@MATH.BERKELEY.EDU DUE: Tuesday, September 1st in your discussion section [unless your GSI says otherwise]. Unless otherwise stated, please include an expl
School: UCSB
Course: CACL WITH APPLI2
Huirui Liu Assignment Homework Set IV due 10/17/2015 at 11:59pm PDT 1. (1 pt) Let g(x) = shown below. Math3B-200-F15-Agboola 3. (1 pt) x 0 f (t) dt, where f is the function whose graph is x Let g(x) = (4 + t)dt. Find g (x). 6 g (x) = Answer(s) submitted:
School: UCSB
Course: Linear Algebra
L. Vandenberghe EE133A 12/8/2014 Homework 7 solutions 1. Exercise 6.2. (a) We apply Newtons method to the nonlinear equation n(t) 2 = 0. One Newton iteration is n(tk ) 2 n (tk ) 10tk exp(2tk ) + exp(tk ) 2 = tk . 10 exp(2tk ) 20tk exp(2tk ) exp(tk ) tk+1
School: UCSB
Course: Linear Algebra
L. Vandenberghe EE133A 12/12/2014 Homework 8 solutions 1. Exercise 9.10. (a) We use the inequalities Ax A , x A1 y , y A 1 which hold for all nonzero x and y, to nd lower bounds for A and A1 . Choosing x = (0, 1) and y = (1, 0), for example, gives A1 108
School: UCSB
Course: Linear Algebra
L. Vandenberghe EE133A 11/19/2014 Homework 5 solutions 1. Circulant Toeplitz matices. (a) For k = 1, the result is obvious because S k1 and diag(W e1 ) are the nn identity matrix. The columns of W S k1 are the columns of W , shifted circularly to the left
School: UCSB
Course: Linear Algebra
L. Vandenberghe EE133A 11/26/2014 Homework 6 solutions 1. The normal equations are (T (a)T T (a) + I)x = T (a)T b. We plug in the factorization T (a) = 1 H W diag(W a)W, n T (a)T = 1 H W diag(W a)H W, n and get 1 H 1 W diag(W a)H W W H diag(W a)W + I x =
School: UCSB
Course: Linear Algebra
L. Vandenberghe EE133A Fall 2014 Homework 5 Due: Wednesday 11/19/2014. Reading assignment: Chapter 8 in the course reader and the notes on Cholesky factorization on the course website. 1. A circulant Toeplitz matrix is a square matrix of the form a1 an an
School: UCSB
Course: Linear Algebra
L. Vandenberghe EE133A Fall 2014 Homework 3 Due: Wednesday 10/29/2014. Reading assignment: Sections 3.1, 3.2, and 3.3 in chapter 3. 1. Polynomial interpolation. In this problem we construct polynomials p(t) = x1 + x2 t + + xn1 tn2 + xn tn1 of degree 5, 10
School: UCSB
Course: Partial Differential Equations
2.1 FIRST-ORDER LINEAR EQUATIONS MATH 124A HW 02 You are encouraged to collaborate with your classmates and utilize internet resources provided you adhere to the following rules: (i) You read the problem carefully and make a serious effort to solve it bef
School: UCSB
Course: Introduction To Partial Differential Equations
7.3 GREENS FUNCTIONS MATH 124B Solution Key HW 07 PRELIMINARIES: All of our arguments rely on the fact that the function 1 for n = 1, 2 |x x 0 | 1 log | x x0 | for n = 2, v(x) = v(x; x0 ) = 2 1 | x x0 |2n for n 3 (2 n) An where x and x0 represent disti
School: UCSB
Course: Introduction To Partial Differential Equations
9.2 THE WAVE EQUATION IN SPACE-TIME MATH 124B Solution Key HW 08 9.2 THE WAVE EQUATION IN SPACE-TIME 1. Prove that (u) = (u) for any function; that is, the laplacian of the average is the average of the laplacian. Hint: Write u in spherical coordinates an
School: UCSB
Course: Introduction To Partial Differential Equations
7.2 GREENS SECOND IDENTITY MATH 124B Solution Key HW 06 7.2 GREENS SECOND IDENTITY 1. Derive the representation formula for harmonic functions (7.2.5) in two dimensions. u(x0 ) = 1 u(x) 2 bdy D n (log | x x0 |) u n log | x x0 | ds. SOLUTION. Let x0 be an
School: UCSB
Course: Introduction To Partial Differential Equations
5.3 ORTHOGONALITY AND GENERAL FOURIER SERIES MATH 124B Solution Key HW 02 5.3 ORTHOGONALITY AND GENERAL FOURIER SERIES 1. (a) Find the real vectors that are orthogonal to the given vectors (1, 1, 1) and (1, 1, 0). (b) Choosing an answer to (a), expand the
School: UCSB
Course: Introduction To Partial Differential Equations
5.1 THE COEFFICIENTS MATH 124B Solution Key HW 01 5.1 THE COEFFICIENTS 1. In the expansion 1 = the sum n odd (4/n) sin nx, 1 1 1 + 1 13 + + 5 9 1 3 valid for 0 < x < , put x = /4 to calculate 1 1 + 11 + 7 1 = 1 + 3 1 1 + 1 + 5 7 9 Hint: Since each of t
School: UCSB
Course: Introduction To Partial Differential Equations
6.3 POISSONS FORMULA MATH 124B Solution Key HW 04 6.3 POISSONS FORMULA 1. Suppose that u is a harmonic function in the disk D = cfw_r < 2 and that u = 3 sin 2 + 1 for r = 2. Without nding the solution, answer the following questions. (a) Find the maximum
School: UCSB
Course: Introduction To Partial Differential Equations
7.1 GREENS FIRST IDENTITY MATH 124B Solution Key HW 05 7.1 GREENS FIRST IDENTITY 1. Derive the 3-dimensional maximum principle from the mean value property. SOLUTION. We aim to prove that if u is harmonic in the bounded set D 3 and u is continuous on D =
School: UCSB
Course: Linear Algebra
Application Problems (Numerical Integration): l) The widths of a kidney shaped pool were measured at 2 meter intervals as indicated in the ļ¬gure below. Use Simpsonās Rule to estimate the area of the pool. 2) A radar gun was used to record the speed of
School: UCSB
Course: Differential Equations
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School: UCSB
Course: Math 4A
Math 4A Midterm Review Prolbems Note: These problems are provided as review of the key ideas that well be tested on the midterm, but this is not a practice midterm. Because we are testing your understanding of concepts and not simply computational uency,
School: UCSB
Course: Differential Equations
Mathematics 4B Winter 2015: Review for Final March 12, 2015 Professor J. Douglas Moore YOU ARE ALLOWED ONE 3 x 5 CARD FOR THE FINAL EXAM. Recall that there is assigned seating for the nal exam. Please write your seat number on your card. You will need to
School: UCSB
Course: Differential Equations
Math 3C Practice Word Problems Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1) A dangerous substance known as Chemical X is lethal if its concentration in the air is 100 parts per million by volume (ppmv). The half-life of che
School: UCSB
Math 6B Midterm Review #1 1 Greens Theorem (y + e 1. Compute the line integral x ) dx + (2x + cos(y 2 ) dy where C is the positively oriented boundary C of the region D enclosed by the parabolas y = x2 and x = y 2 . Solution: Here we use (curl form) Q P x
School: UCSB
Course: Calc With Appli 2
This is a study guide for the second Math 3B midterm. It indicates which types of problems you may be expected to answer on the midterm, with instructions on where to nd these topics in the Stewart calculus book. Plenty of examples can be found in the e
School: UCSB
Course: Numerical Analysis
University of California, Santa Barbara Department of Statistics & Applied Probability PSTAT 120B, Probability & Statistics, Spring 2010 Instructor: Jarad Niemi Email: niemi@pstat.ucsb.edu Course hours: MWF 10:00-10:50am in HFH 1104 TAs: Varvara Kulikova
School: UCSB
Course: Vector Calculus
Math 6A - Vector Calculus with Applications, First Course Instructor: Oce: E-mail: Oce Hours: Lectures: Classroom: Jon Tjun Seng Lo Kim Lin Graduate Tower, Oce 6431H jlokimlin@math.ucsb.edu TR 1:00 PM-2:00 PM, also available by appointment. MTWR 8:00 AM-9
School: UCSB
SPRING 2015, MATH 6 B, VECTOR CALCULUS 2 INSTRUCTOR : Gustavo Ponce (o. SH 6505 #8938365) SCHEDULE : TR 330 445 ROOM : HFH 1104 INSTRUCTOR OFFICE HOURS : T. 5 6, R. 11 12, 5 6. TEACHING ASSISTANT : Kathleen Hake (SH 6431 K), Garo Sarajian (SH 6432 F) TEAC
School: UCSB
Course: Vector Calculus
Math 6A - Vector Calculus with Applications, First Course Instructor: Oce: E-mail: Oce Hours: Lectures: Amanda R. Curtis Graduate Tower, Oce 6432Q arcurtis@math.ucsb.edu MTWR 1:00 PM-2:00 PM, also available by appointment. MTWR 3:30-4:35PM, 3515 Phelps Ha
School: UCSB
Course: Vector Calculus
Math 6A (Vector Calculus) Syllabus1 TR, 11:00 AM to 12:15 PM, Room 1701 in Theater/Dance West Instructor/E-mail: Jordan Schettler (jcs@math.ucsb.edu) Ofce Hours: Tuesday 2-3 pm, Friday 9-10 am, or by appointment, South Hall, Room 6721 Website: Use the cou
School: UCSB
Course: Homological Algebra
Math 236B, Spring 2015, MWF 9-9:50, HSSB 1223 Homological Algebra Instructor: Birge Huisgen-Zimmermann, SH 6518, Oce hours M, F 11 - 12, W 12:30 1:30. Accompanying texts, as for the winter quarter: The manuscript I will put on the board will again serve
School: UCSB
Course: Abstact Algebra
Math 111C, Spring 2015, MWF 10 10:50, Building 387, Room 103 Introduction to Abstract Algebra Instructor: Birge Huisgen-Zimmermann, SH 6518 Oce Hours: Mon, Fri 11-12, Wed 12:30 - 1:30 Teaching Assistant: Nathan Schley, SH 6431P, Oce hours Tue, Thur 4-5. E
School: UCSB
Course: Transitions To Higher Mathematics
Math 8: Transition to higher mathematics Syllabus Text: How to prove it (A structured approach) by D. J. Velleman, Second Edition Lectures: TR 12:30-1:45 ARTS 1353 Discussions : MW 5:00-5:50 HSSB 1223 MW 6:00-6:50 HSSB 1207 Instructor: Eleni Panagiotou Of
School: UCSB
Course: Math 4A
Math 4A - Jacob Course Information and Syllabus Spring Quarter 2015 Monday, Wednesday, Friday 1:00 - 1:50 Instructor: Bill Jacob Oce Hours: Monday and Wednesday 2:15 - 3:15, Friday 9:45 -10:45, and by appointment. Oce: South Hall 6719 Email: jacob@math.uc
School: UCSB
Course: Differential Equations
SYLLABUS FOR MATHEMATICS 4B CALCULUS WINTER 2015 Professor John Douglas Moore Office: South Hall 6714 Office hours: TuTh 3:30, W 1 Telephone: 893-3688 email: moore@math.ucsb.edu Lectures: Lotte Lehman Concert Hall TuTh 9:30-10:45 Text: Boyce, Diprima, Ele
School: UCSB
Course: Differential Equations
Math 4B UCSB, Spring 2015 Lecture: MWF 11:00-11:50, MUSICLLCH Textbook: The textbook for this course is optional. For those who would like a reference and extra practice problems the book Elementary Dierential Equations by Boyce and DiPrima will work. Ins
School: UCSB
Course: Math 4A
Math 4A Syllabus Winter 2014 Lecture: MWF 10:00 10:50am, MUSIC LLCH Text: Linear Algebra with applications by David C. Lay, Addison-Wesley, 4th Edition. Material to be covered: Chapters 1-6 in the text book. iClicker: You should purchase an iClicker and b
School: UCSB
Math 108a Professor: Padraic Bartlett Syllabus for Math 108a Weeks 1-10 UCSB 2013 Basic Course Information Professor: Padraic Bartlett. Class time/location: MWF 9-9:50, Phelps 3505. Oce hours/location: TTh 2-3pm, South Hall 6516. Additionally, I am tea
School: UCSB
Course: Methods Of Analysis
Analysis with an introduction to proof - Math 117 Spring2009 Monday, Wednesday, & Friday, 12:00-12:50pm, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.e
School: UCSB
Course: Methods Of Analysis
Analysis with an introduction to proof - Math 117 Spring2009 Monday, Wednesday, & Friday, 12:00-12:50pm, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.e
School: UCSB
Course: Numerical Analysis
Intro. to Numerical Analysis - Math 104B Winter 2011 Tuesday & Thursday, 8:00-9:15am, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia/Courses
School: UCSB
Course: INTRO NUM ANALYSIS
Math 104A, Fall 2010 Intro. to Numerical Analysis Tuesday & Thursday, 9:30-10:45am, 387 101 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: (805) 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia/Courses/Mat
School: UCSB
Course: INTRO NUM ANALYSIS
Math 104A, Winter 2009 Intro. to Numerical Analysis Monday, Wednesday, & Friday, 9:00-9:50am, Arts 1426 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: (805) 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia
School: UCSB
Math 108B Intro to Linear Algebra Winter 2010 Professor: Kenneth C. Millett Office: 6512 South Hall Office Hours: R 8:30 11:00 Email: millet@math.ucsb.edu Graduate Assistant: Tomas Kabbabe Office: 6432K South Hall Office Hour: W 10:00 11:00 Email: tomas@m