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School: Yale
Course: Fractional Geometry
Fractal and Multifractal Geometry A Gateway to Advanced Mathematics Michael Frame1 and Nial Neger c 2011 1 This material is based upon work supported by the National Science Foundation under Grant No. 0203203 ii T G C A Some shapes well see, and (at least
School: Yale
Course: None
1 Statistics for Decision Making final exam 1. Question: (TCO 9) The annual Salary of an electrical engineer is given in terms of the years of experience by the table below. Find the equation of linear regression for the above data and obtain the expected
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 1 SOLUTIONS DANIEL COREY Section 1.1 18. This system is not in echelon form because the leading variable in the third row is to the left of the leading variable in the second row. 20. This system is in echelon form. The leading variables
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 2 SOLUTIONS DANIEL COREY Section 1.4 2. This amounts to solving the system corresponding to the augmented matrix 1 0 0 1 1 1 0 0 55 0 1 0 1 0 1 1 0 55 which row reduces to 0 0 1 1 1 0 0 1 30 0 0 1 1 30 000 0 30 25 30 0 So the possible v
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 3 SOLUTIONS DANIEL COREY Section 2.2 16. 9 5 2 5 3 A= 1 x x= 1 x2 13 b = 9 2 30. The columns do not span R2 , for example e1 = (1, 0) is not in the span of the columns. 40. h can be any real number except h = 12/5 (this value of h is pre
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 4 SOLUTIONS DANIEL COREY Section 3.1 18. T is not linear since T (0, 0, 0) = (0, 4, 0) = (0, 0, 0) (see the solution to exercise 55 below). 22. The columns are linear dependent, so T is not one-to-one. The span of the columns is not all
School: Yale
Course: CalcOfFunctionsOfOneVariableII
uh 2 EJF W («wou- alpha-441:2, «9 3° 41%. ' 4 = _ ._.L E20 5 at 3" 3p ' 0:61 - How My Jaw; do You 14! 4° 151. a» W a? [0'3 WW 0.09°o? ' to: n V Wam? H .{évt L" N Jul»qu 1 MA-1"; 9" EN .4. luv-I 71°} l *4 _.L. ' ' r 1 (NH? 0 -. 4 to 5 (NH) ' 370 H?
School: Yale
Week 7 Important Discrete Distributions Expectation and Variance Lecture 17. Bernoulli and Binomial Distributions. Expectation Revisited. Discrete Uniform Distribution. We have seen some examples that all outcomes of an experiment are equally likely. Let
School: Yale
Week 5 Lecture 11. Conditional distribution and Conditional density. Review. Discrete random variables X1 ; X2 ; ; Xn are mutually independent if P (X1 = x1 ; X2 = x2 ; ; Xn = xn ) = P (X1 = x1 ) P (X2 = x2 ) : : : P (Xn = xn ) . Continuous random variabl
School: Yale
Week 4 Lecture 8. Discrete conditional distribution. Examples 1: A doctor gives a patient a test for a particular cancer. Before the results of the test, the only evidence the doctor has to go on is that 1 woman in 1000 has this cancer. Experience has sho
School: Yale
Week 6 Lecture 14 Random Walks Drunkard Walk. Imagine now a drunkard walking randomly in an ideals ized 1 dimensional city ( or 2 dimensional, or 3 and higher dimensional city). The city is eectively innite and arranged in a 1 dimensional equally-spaced g
School: Yale
Week 3 Lecture 5. Expectation Probability Density Function: Let f (x) 0 and P (E ) as following Z P (X 2 E ) = f (x) dx. R f (x) dx = 1. Dene E Are the probability axioms satised? It is important to observe that there a similar paradox in the calculus Za
School: Yale
Week 2 Lecture 3. Expectation and Probability axioms. Random variable. A random variable is a real-valued function dened on the sample space, i.e., X (! ) is a function from to R. For example, for = fBB; BG; GB; GGg, your X could be the number of boys, th
School: Yale
Course: None
1 Statistics for Decision Making final exam 1. Question: (TCO 9) The annual Salary of an electrical engineer is given in terms of the years of experience by the table below. Find the equation of linear regression for the above data and obtain the expected
School: Yale
Course: Calc:FunctionsSeveralVariables
u) Maq (10 MdJram 217:1 / L LTDPICB CoveveoL , ' Wf . V 3% Te :5 on chblocr (Mud) 7100 -8-$0Fm m OMLzo; 0. Choqt i7 Vaduz 2. The Geo nch/5 S?CLCQ Seahons cfw_11 r113 N6 calculqhHS ,1) Onawd 1% Ugdw Fundwns 2? ()S Lynda Sed'ums 13.1. - )3. 4 . _ Remew
School: Yale
Course: Calc:FunctionsSeveralVariables
3 mac/v, [10 Exam #1 M100 *0 Q 41" ' "rioowsnm 7%prszaj /\/O\/. I / . Tune " Loccdlow OML. 30.) i * Chaim 9? NquPle INGCJYOS 36% <c-~\kvrvlr ISAIMlaw/67 chap-m (g = WW CalculuS ( QCC V (mm is 1 (Amide Ilurah m3 ' Valuan xi double mic-3M9 funcdmn (Mum of
School: Yale
Course: Calc:FunctionsSeveralVariables
when 15. 6 542, Liv-(3 . :SCCUDH (6' 6 g! on). 2.41 (41417 1 1, 3 _ N 4 So , M 3) 3 K/thyma (LA-J, iIC'hQ h : a 2%) ) 1/ ha , 530 /[+(M)+(_/i(j)cm ,svjrc :ji / ,L ' " v S \j/JMH r (if 19- Z : (Rifkr) ll ' 0 0 v 3 lg _cfw__ 17 2'1. f 2 1 MT _L +
School: Yale
Course: Calc:FunctionsSeveralVariables
IVUJH'LLL VVULL Ha U H "TI 0 1/ Moq 111: cfw_4w 3 V H I -' -C . ' 4f / 3 17+. l '- :4, 23 I? 331+~+ 5 ~31. cfw_1+ g 18 D J IL?" 7 1' IO) '1; ,., 14g 1g, 13,29,451. T 2 TH Cuadrqn 53th \C on an IC '(y/L)[H(7X'l*9t32) :ECX,L)[ ?X1-?92>o /7. c
School: Yale
Course: Calc:FunctionsSeveralVariables
3 gm M pl L0: / / Wke pqu MC, Marni, l$~|- AlLb I - uie: ,l2,18,2.,3 (Lu) =t3.3 $5 Iii; +,S,'4,>L,2ov P(0,W,07 2 j X: 3; 5 r W 5+ C m. k C. x- - [04 P(2/l,b) (H3)>< (RH-[Q-(I/ l> '29 5(2/02) " h : 3) 1+ 7 "J WNW 3(X/2)+g+4(2~l7:o JHg
School: Yale
Course: Calc:FunctionsSeveralVariables
ngh :20 MN g I M, 16, 30 N19: 4, 10) I8 1 2:3 18114, s" i I 7. -Z ,2 H: 7 AhCOL) x zz 95 M29 .333 _2 [it Li) :" L) X1 . L L. -. - " . Ll, 6 I K v 2 W< o 14 p a!) z, cfw_x - LSIWX "-O ,3 (05 K ~ 0 The CHhCCLl P13 ave Hwy; ' (1+ (Etn) O) (LN/1510in S
School: Yale
STAT 241/541, Probability Theory with Applications Fall 2013 Instructor: Harrison H. Zhou (huibin.zhou@yale.edu) O ce hours: Wednesday 4:00-6:00pm (tentative) or by appointments, Room 204, 24 Hillhouse Ave., James Dwight Dana House. T.A.: Corey Brier <cor
School: Yale
Course: Fractional Geometry
Fractal and Multifractal Geometry A Gateway to Advanced Mathematics Michael Frame1 and Nial Neger c 2011 1 This material is based upon work supported by the National Science Foundation under Grant No. 0203203 ii T G C A Some shapes well see, and (at least
School: Yale
Course: None
1 Statistics for Decision Making final exam 1. Question: (TCO 9) The annual Salary of an electrical engineer is given in terms of the years of experience by the table below. Find the equation of linear regression for the above data and obtain the expected
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 1 SOLUTIONS DANIEL COREY Section 1.1 18. This system is not in echelon form because the leading variable in the third row is to the left of the leading variable in the second row. 20. This system is in echelon form. The leading variables
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 2 SOLUTIONS DANIEL COREY Section 1.4 2. This amounts to solving the system corresponding to the augmented matrix 1 0 0 1 1 1 0 0 55 0 1 0 1 0 1 1 0 55 which row reduces to 0 0 1 1 1 0 0 1 30 0 0 1 1 30 000 0 30 25 30 0 So the possible v
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 3 SOLUTIONS DANIEL COREY Section 2.2 16. 9 5 2 5 3 A= 1 x x= 1 x2 13 b = 9 2 30. The columns do not span R2 , for example e1 = (1, 0) is not in the span of the columns. 40. h can be any real number except h = 12/5 (this value of h is pre
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 4 SOLUTIONS DANIEL COREY Section 3.1 18. T is not linear since T (0, 0, 0) = (0, 4, 0) = (0, 0, 0) (see the solution to exercise 55 below). 22. The columns are linear dependent, so T is not one-to-one. The span of the columns is not all
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 5 SOLUTIONS DANIEL COREY Section 3.3 10. The augmented matrix 1 2 1 4 7 7 1 1 5 1 0 0 10 0 0 row reduces to 0 1 1 00 0 1 0 0 28 13 0 1 3 9 7 4 3 1 1 So the inverse of the matrix is 28 13 3 7 3 1 9 4 1 18. x1 1 x2 = 4 x3 1 1 1 2 28 7 1
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 6 SOLUTIONS DANIEL COREY Section 4.2 4. These vectors cannot form a basis since they are linearly dependent. (It looks like u1 and u3 are dependent. In any event, three vectors in R2 must be dependent) 8. Let A be the matrix whose rows a
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 7 SOLUTIONS DANIEL COREY Section 5.1 4. M23 0 = 0 3 2 2 6 0 0 M31 6 2 = 1 6 4 1 1 0 0 . 6 14. a) Expanding along the bottom row det A = (1)(4+2) 2 1 4 3 0 0 0 2 1 + (1)(4+3) (2) 1 1 4 1 2 2 0 1 = (12 3) 2(4 + 4 4 1) = 3 1 b) Expanding al
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 4 Due date - Sep 28 1. Section 2.6: 4, 6, 16, 18, 20, 24, 40(c), 44, 62 2. Section 2.7: 6, 10(a), 16, 22, 24 3. Section 2.8: 2, 4, 8, 12, 22, 24, 40, 46 1
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 3 Due date - Sep 21 1. Section 2.2: 2, 6, 8, 20, 26 2. Section 2.3: 2, 8, 10, 16, 18, 22, 28, 40, 42, 48 3. Section 2.5: 4, 6, 14, 40, 50, 52 4. Let f ( x) = ( 0 x0 1 x>0 g ( x) = x2 (a) Show that lim f x!0 g (x) 6= f ( lim (g (x).
School: Yale
Course: Calculus
Solutions to some limit convergence denition problems Dustin Cartwright October 2, 2006 These are some possible questions which I came up with involving the definition of a convergent series, and solutions to these questions. The solutions are probably mo
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 1 (due Friday 27 Jan 2012) Solutions 1. Graphs of Multivariable Functions a) Let f : R2 R be a function on R2 . Give R3 the standard (x, y, z ) coord
School: Yale
Homework 2 Due September 13. Chapter 2.2: 2, 4, 6, 8, 12. Problem 6: Let X U (0; 1). (i) Find the density of Y = 1=X and EY . X (ii) Find the density of Y = tan 2 and EY . 1
School: Yale
STAT 241/541 Homework 2 Solution Prepared by James Hu Problem 1: Ch2.2:2 (a) c = 1/48 . b 1 1 xdx = (b2 a2 ) . 96 a 48 10 1 75 (c) Pr(X > 5) = xdx = , Pr(X < 7) = 48 96 5 (b) Pr(E ) = 7 1 45 xdx = , 96 2 48 24 3 and Pr(X 2 12X + 35 > 0) = 1 Pr(5 < X < 7)
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 4 (Fri 17 Feb 2012) Selected Solutions 1. Let P be a point in R3 and let v be a direction vector at P . Find a parameterization of the line through P
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 3 (Fri 10 Feb 2012) Selected Solutions 1. CM 14.4 Exercise 14 Solution. z |(x,y) = 1 (yey (x+y )2 + ey (x2 + xy x) Exercise 24 Solution. f |(1,2)
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 2 (Fri 03 Feb 2012) Solutions 1. CM 14.2 Exercise 28 Solution. xy y 2 exy xy x ye ln(ye ) = = =y x yexy yexy Problem 40 1 Solution. Well, f (65, 16
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 7 Due date - Oct 19 1. Section 3.5: 6, 10, 14, 50, 54. 2. Section 3.6: 2, 10, 20, 28, 44, 48. 3. Section 3.8: 8, 18(a). 4. During the 1940s, scrolls which are now known as the Dead Sea Scrolls, were found in a cave in Khirbet Qumran
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 8 Due date - Nov 2 1. Section 3.9: 8, 12 (notice that the instructions for this problem appear just before quesiton 11), 16, 22, 30, 40. 2. Let f (x) = cos( x). (a) Find the linear approximation of f near x = 1 . 3 (b) Sketch the gr
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 9 Due date - Nov 9 1. Section 4.1: 4, 14, 22, 32, 40, 44, 56, 78. 2. Section 4.2: 2, 8, 12, 16, 18, 32. Practice problems (do but do not submit): 1. Section 4.1: 3, 13, 21, 33, 39, 47, 55. 2. Section 4.2: 1, 7, 9, 15, 19. 1
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 30, 2009 1. Write the solutions to the following system of equations in parametric vector form: 3x1 + 2x2 + 2x3 = 7 2x1 2x2 + 8x3 = 8 x1 + 4x2 6x3 = 1 The solutions are: x1 3 2 x2 = 1 + t 2 x3 0 1 2. Find bases for the row
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 21, 2009 1. Find the inverse of 1 A = 3 2 2 4 , 4 0 1 3 using both row reduction and the adjugate. Check that you get the same answer. A1 8 = 10 3 4 3 2 7 2 1 1 1 2 2. Use the adjugate to nd a formula for the inverse of a 2 2
School: Yale
Course: Calculus
Math 1B - Fall 2006 10/09/2006 Integrals sin2 x cos2 x dx Use 2 sin x cos x = sin 2x, then use Ex 1. Ex 2. 0 1 (1 cos 4x) = sin2 2x. 2 4 dx 1 1 dx Try x = tan , dx = sec2 d. 2+1 4x 2 2 Ex 3. Ex 4. 4 dx 4 + e2x x Use Weierstrass substitutions, let t = tan
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 14, 2009 1. Let T be the linear transformation from R3 to R3 which consists of rst rotating 30 degrees about the z axis and then rotating 90 degrees about the x axis. (I havent specied the directions of the rotations. Use whic
School: Yale
Course: CalcOfFunctionsOfOneVariableII
uh 2 EJF W («wou- alpha-441:2, «9 3° 41%. ' 4 = _ ._.L E20 5 at 3" 3p ' 0:61 - How My Jaw; do You 14! 4° 151. a» W a? [0'3 WW 0.09°o? ' to: n V Wam? H .{évt L" N Jul»qu 1 MA-1"; 9" EN .4. luv-I 71°} l *4 _.L. ' ' r 1 (NH? 0 -. 4 to 5 (NH) ' 370 H?
School: Yale
Course: CalcOfFunctionsOfOneVariableII
:0 01 Polar and; . (6-40: 3 Slorb I W vam: Wm WW x [X- rOOJG } _ . ' r . y? ruin? _ ,7 a, v (11 4" 153M9 - , ' . . x; ('2, P153140) c.pr _ Fw #1: in. A . _ Ye M + 1:130) m .9 7 "9 I. J .1 :c Qusame 4%? «Lumeo. NW 6x :0? '2, c.3161 r {2+ 23.3.3) Ji
School: Yale
Course: Integral Calculus
Yale University, Department of Mathematics Math 115 Calculus Fall 2013 Final Exam Review Guide When/Where. The Final Exam will take place during 7:00 10:30 pm on Sunday, December 15th, 2013 in Davies Auditorium, located underground between Dunham Labs and
School: Yale
Week 7 Important Discrete Distributions Expectation and Variance Lecture 17. Bernoulli and Binomial Distributions. Expectation Revisited. Discrete Uniform Distribution. We have seen some examples that all outcomes of an experiment are equally likely. Let
School: Yale
Week 5 Lecture 11. Conditional distribution and Conditional density. Review. Discrete random variables X1 ; X2 ; ; Xn are mutually independent if P (X1 = x1 ; X2 = x2 ; ; Xn = xn ) = P (X1 = x1 ) P (X2 = x2 ) : : : P (Xn = xn ) . Continuous random variabl
School: Yale
Week 4 Lecture 8. Discrete conditional distribution. Examples 1: A doctor gives a patient a test for a particular cancer. Before the results of the test, the only evidence the doctor has to go on is that 1 woman in 1000 has this cancer. Experience has sho
School: Yale
Week 6 Lecture 14 Random Walks Drunkard Walk. Imagine now a drunkard walking randomly in an ideals ized 1 dimensional city ( or 2 dimensional, or 3 and higher dimensional city). The city is eectively innite and arranged in a 1 dimensional equally-spaced g
School: Yale
Week 3 Lecture 5. Expectation Probability Density Function: Let f (x) 0 and P (E ) as following Z P (X 2 E ) = f (x) dx. R f (x) dx = 1. Dene E Are the probability axioms satised? It is important to observe that there a similar paradox in the calculus Za
School: Yale
Week 2 Lecture 3. Expectation and Probability axioms. Random variable. A random variable is a real-valued function dened on the sample space, i.e., X (! ) is a function from to R. For example, for = fBB; BG; GB; GGg, your X could be the number of boys, th
School: Yale
Course: IntroFunctionsSeveralVariables
Physics 181 To dos:! 1. Register online on classesv2! 2. If pretty sure of taking class, you need a clicker! !Get a clicker from Bass library !Register the clicker on classesv2 under ! !Tests & Quizzes by Wed. Jan 19 3.3. Read Ch. 22 for Fri. Jan. 14! 4.
School: Yale
Course: IntroFunctionsSeveralVariables
Physics 181 University Physics Natural continuation of Physics 180 Todays handout: syllabus Requirements: book, clickers, classesv2, online homeworks: due before lecture (classesv2) written homeworks: due before lecture each Wed. Grading, policies, etc.
School: Yale
Course: None
1 Statistics for Decision Making final exam 1. Question: (TCO 9) The annual Salary of an electrical engineer is given in terms of the years of experience by the table below. Find the equation of linear regression for the above data and obtain the expected
School: Yale
Course: Calc:FunctionsSeveralVariables
u) Maq (10 MdJram 217:1 / L LTDPICB CoveveoL , ' Wf . V 3% Te :5 on chblocr (Mud) 7100 -8-$0Fm m OMLzo; 0. Choqt i7 Vaduz 2. The Geo nch/5 S?CLCQ Seahons cfw_11 r113 N6 calculqhHS ,1) Onawd 1% Ugdw Fundwns 2? ()S Lynda Sed'ums 13.1. - )3. 4 . _ Remew
School: Yale
Course: Calc:FunctionsSeveralVariables
3 mac/v, [10 Exam #1 M100 *0 Q 41" ' "rioowsnm 7%prszaj /\/O\/. I / . Tune " Loccdlow OML. 30.) i * Chaim 9? NquPle INGCJYOS 36% <c-~\kvrvlr ISAIMlaw/67 chap-m (g = WW CalculuS ( QCC V (mm is 1 (Amide Ilurah m3 ' Valuan xi double mic-3M9 funcdmn (Mum of
School: Yale
Course: CalcOfFunctionsOfOneVariableII
SOLUTIONS TO EXAM 2, MATH 115, FALL 2012 1. (65 points) The graphs of f (x), g(x), and h(x) are shown below. You may assume that as x , the graphs of f , g, and h continue in a fashion similar to the trends observed in the graphs. 3 gx 2 1 hx f x 0 1 2 3
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 1 Math 115 October 3rd, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 April 10th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor w
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 November 13th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructo
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Mathematics 115b Mid-term # 1 2/19/08 Name: Section: Please answer all SIX questions; each question is worth TEN points. You may use the table of integrals supplied with the test. However, books, notes, calculators, computers, cellphones may NOT be used.
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Math 115 Exam 1 Practice This exam is meant to suggest the kind, diculty, and number of problems on the real exam. Note that topics covered in the course, but not included on this practice exam, still might appear on the real exam. Directions Set aside 90
School: Yale
Course: CalcOfFunctionsOfOneVariableII
1. Find the areas between these curves x = y 2 + 2y 4 and x = y + 2. 2. Find the volume of the solid formed by rotating the region between y = x2 and y = 2x about y = 2. 3. (a) Find the area enclosed by the curve r = 1 cos(), 0 2. (b) Find its arclength.
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Math 115 Final Practice This exam is meant to suggest the kind, diculty, and number of problems on the real exam. Note that topics covered in the course, but not included on this practice exam, still might appear on the real exam. Directions Set aside 3 1
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 1 Math 115 October 3rd, 2012 Name:_ Instructor:_Section:_ 1. D o not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instruct
School: Yale
Course: Linear Algebra With Applications
School: Yale
Course: Linear Algebra With Applications
School: Yale
Course: Calculus
2 x2 + 4 dx. 1. Find 0 e 2. Find (ln x)2 dx. 1 1 /2 arctan x dx. x 0 4. If the curve y = 2x x2 , 0 x 1 is rotated about the x-axis, nd the area of the resulting surface. 3. Find (1)n to within 0.01. (You do not n+1 n=1 need to carry out the computation, b
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 12, 2009 1 Dene the null space of a matrix (3 points). If A is an n m matrix, the null space of A is the set of vectors x in Rm such that Ax = 0. 2 Dene what it means for a set of vectors to be linearly dependent (3 points).
School: Yale
Course: Calculus Of Functions With One Variable
112b Midterm II Wed Apr 9, 2008 90 minutes Please answer all 7 parts below, in your blue books. Show all your work and justify clearly where appropriate. You may not use your books, notes or calculators on this exam. Good luck! 1. (16 pts.) Find the deriv
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 30, 2009 1 Dene eigenspace (2.5 points). For a matrix A and an eigenvalue of A, the -eigenspace is the set of all vectors x such that Ax = x. 2 Dene what it means for a matrix to be diagonalizable (2.5 points). A matrix A is
School: Yale
Course: Linear Algebra
Math 54 quiz solutions December 2, 2009 1 Find the Fourier cosine series expansion of f (x) = 3 + x dened on 0 < x < (7 points). For n > 0, we compute an using the formula: an = 2 (3 + x) cos nx dx 0 Use integration by parts with u = 3 + x, dv = cos nx,
School: Yale
Course: Linear Algebra
Math 54 quiz solutions November 23, 2009 1 Find a general solution to the system: (5 points) x (t) = 1 3 2 1 x(t) + . 2 1 The characteristic equation of the matrix is: 1 3 2 = 2 3 + 2 6 = 2 3 4 = ( 4)( + 1), 2 so the eigenvalues are 4 and 1. For = 4, 3 3
School: Yale
Course: Linear Algebra
Math 54 quiz solutions November 18, 2009 1 Find a general solution (6 points) for the system x (t) = Ax(t) A= 13 . 12 1 Find the solution with initial conditions x(0) = 2 (4 points). 0 The characteristic polynomial of A is: 1 12 3 = 1 2 + 2 36 = 2 2 35 =
School: Yale
Course: Calc:FunctionsSeveralVariables
when 15. 6 542, Liv-(3 . :SCCUDH (6' 6 g! on). 2.41 (41417 1 1, 3 _ N 4 So , M 3) 3 K/thyma (LA-J, iIC'hQ h : a 2%) ) 1/ ha , 530 /[+(M)+(_/i(j)cm ,svjrc :ji / ,L ' " v S \j/JMH r (if 19- Z : (Rifkr) ll ' 0 0 v 3 lg _cfw__ 17 2'1. f 2 1 MT _L +
School: Yale
Course: Calc:FunctionsSeveralVariables
IVUJH'LLL VVULL Ha U H "TI 0 1/ Moq 111: cfw_4w 3 V H I -' -C . ' 4f / 3 17+. l '- :4, 23 I? 331+~+ 5 ~31. cfw_1+ g 18 D J IL?" 7 1' IO) '1; ,., 14g 1g, 13,29,451. T 2 TH Cuadrqn 53th \C on an IC '(y/L)[H(7X'l*9t32) :ECX,L)[ ?X1-?92>o /7. c
School: Yale
Course: Calc:FunctionsSeveralVariables
3 gm M pl L0: / / Wke pqu MC, Marni, l$~|- AlLb I - uie: ,l2,18,2.,3 (Lu) =t3.3 $5 Iii; +,S,'4,>L,2ov P(0,W,07 2 j X: 3; 5 r W 5+ C m. k C. x- - [04 P(2/l,b) (H3)>< (RH-[Q-(I/ l> '29 5(2/02) " h : 3) 1+ 7 "J WNW 3(X/2)+g+4(2~l7:o JHg
School: Yale
Course: Calc:FunctionsSeveralVariables
ngh :20 MN g I M, 16, 30 N19: 4, 10) I8 1 2:3 18114, s" i I 7. -Z ,2 H: 7 AhCOL) x zz 95 M29 .333 _2 [it Li) :" L) X1 . L L. -. - " . Ll, 6 I K v 2 W< o 14 p a!) z, cfw_x - LSIWX "-O ,3 (05 K ~ 0 The CHhCCLl P13 ave Hwy; ' (1+ (Etn) O) (LN/1510in S
School: Yale
Course: Calc:FunctionsSeveralVariables
if |\M.AH[\l\J QLKLtUI|lvnI I) V] " "i lJU/ 3h Duet Odober [017.01% 2 a Hwt (w) 4 \J' rr Secnon IL 2 : l, 6, Uf- Mm. K)XH+U):f cfw_+039 5 gcoxi) ZCMO l (0. b 1- ., V ' g I: _[ 26 w 4, r - \D T is 089 zf 95:" y 19 3( L51" a . , 1 1 v? I cfw_m MNM ,
School: Yale
Course: Fractal Geometry
Firs! 111)111(\'()Ik 51! in 1 r V ' ' nnowork W1 )0 Univ m thv lwginning of (1:53 011 Ilnnmlay. .81 pt 11. Nu hm lu :lt'Hlel, l ("do ' ' ' no 011 t M 01151 . Fold \nm hmnkurk pupvr wrtu'ully and PRINT )(le Ildl Find tht IFS rules fur muh of thaw fruvtuls.
School: Yale
Course: Fractal Geometry
l v A} A / Seventh homework set Due at the beginning of class on Thursday; Nov 6. No late homework will be accepted. Fold your homework paper vertically and PRINT your name on the outside. 1. Suppose we build a randomized gasket this way: 7'1 = 1'2 = 1/2
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 14, 2009 1. Let T be the linear transformation from R3 to R3 which consists of rst rotating 30 degrees about the z axis and then rotating 90 degrees about the x axis. (I havent specied the directions of the rotations. Use whic
School: Yale
STAT 241/541, Probability Theory with Applications Fall 2013 Instructor: Harrison H. Zhou (huibin.zhou@yale.edu) O ce hours: Wednesday 4:00-6:00pm (tentative) or by appointments, Room 204, 24 Hillhouse Ave., James Dwight Dana House. T.A.: Corey Brier <cor