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Math 208 Midterm Exam 3 3/29/13 Key 1. Set up an integral that computes the volume of a hemisphere of radius 4 through which has been drilled a circular hole of radius 1 along the axis both orthogonal to and passing through the center of its base. This object is radially symmetric around the axis through the center of the drilled hole, but not spherically symmetric, so cylindrical coordinates would be best. Take the z-axis to be the one through the center of the hole, and recall that the equation for the top half of a sphere in the z-direction is z = p 16- x 2- y 2 = 16- r 2 . Thus, the integral is Z 2 Z 4 1 Z 16- r 2 d z d r d 2. Parameterize the circle of radius 3 in the ( x,y )-plane which is centered at the point (2,1), traversed clockwise. One clockwise parametrization of the unit circle is (cos( t ) ,- sin( t )). Here, we multiply by 3 to increase the radius and then shift the result. ~ r ( t ) = (3 cos( t ) + 2 ,- 3 sin( t ) + 1) , t 2 3. Consider the vector field ~ F = ( y 2 ,xy ). Is the parameterized curve ~ r ( t ) = (tan( t ) , sec( t )) a flow line of ~ F ?

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Berkeley - ARCH - 140

0.96 0.96 0.98 1.00 SE 0.79 0.76 0.76 0.79 0.83 0.91 0.96 0.96 0.98 1.00SummerS 1.16 0.94 0.81 0.78 0.77 0.78 0.81 0.86 0.92 0.96 SW 1.20 0.97 0.88 0.86 0.83 0.78 0.76 0.78 0.79 0.84 W 1.21 0.97 0.89 0.86 0.86 0.85 0.79 0.78 0.77 0.793 – Sola

CSU San Bernardino - ACCOUNTING - 212

Issuing Bonds at a DiscountIssuing Bonds at a DiscountQuestionDiscount on Bonds Payable: a. has a credit balance. b. is a contra account. c. is added to bonds payable on the balance sheet. d. increases over t

University of Florida - HUN - 4221

e) All statements are true.3. Your grandmother has just been diagnosed with gastric atrophy. Her doctor tells her tomake sure she is getting enough vitamin B12. Select the MOST APPROPRIATEdiagnosis and treatment based on her condition?a) Lack of

UNL - MATH - 208

Math 208Midterm Exam 44/19/13KeyDo seven of the following eight problems on the provided paper. Be sure to indicate whichproblem you did not do by crossing it out on this sheet. To receive full credit, show allrelevant work and clearly indicate whic

UNL - MATH - 208

Math 208Self Assessment Quiz 1Key1. Draw a contour diagram for the following function.f (x, y ) = 1 x2 + y 2Contour diagrams are given by choosing a starting number, say z0 = 2 and a constantincrement, say z = 1, and then drawing the curves zi = f (

UNL - MATH - 208

Math 208Self Assessment Quiz 2Name:1. Construct a function of three variables, f (x, y, z ), whose level surface at z = 3 is thebottom half of a cone.The equation of a (right circular) cone is x2 + y 2 z 2 = 0, and its bottom half isz = x2 + y 2 , o

UNL - MATH - 208

Math 208Self Assessment Quiz 3Name:1. Let u = (1, 3) and v = (2, 4). Decompose u as u = upar + uperp , where upar is parallelto v and uperp is orthogonal to it.The vector upar is just the projection of u onto v , and uperp can then be computed asu u

UNL - MATH - 208

Math 208Self Assessment Quiz 4Name:1. The quantity of beef Q purchased at a store is a function of b, the price of beef and c,the price of chicken. Describe in words what Q and Q mean and whether you expectbceach should be positive or negative.The

UNL - MATH - 208

Math 208Self Assessment Quiz 5Name:1. Find and classify all critical points of f (x, y ) = (x + y )(xy + 1). Does this function haveany global extrema?Since the function is a polynomial, the gradient exists everywhere. Thus, we can do thisby computi

UNL - MATH - 208

Math 208Self Assessment Quiz 6Name:1. Suppose f (x, y, z ) = x2 + 3x cos(y ) ln(z ), but x, y and z are themselves functions of tgiven by x(t) = 4t2 3, y (t) = t sin(t) and z (t) = tet . Compute df .dtThe chain rule tells us that we should build a t

UNL - MATH - 208

Math 208Self Assessment Quiz 7Name:1. Compute the mass of the solid bounded by z = x3 , z = x x + y = 1 and y = 10 in theregion x 0 if its density varies as (x, y, z ) = 2y + 1.Drawing a picture of this region is helpful, but we can think through it

UNL - MATH - 208

Math 208Self Assessment Quiz 8Name:1. Suppose a spherical, hollow metal casing is centered around the origin. Its outer surfaceof the casing is a sphere of radius 3 in, and the metal is 1/2 inch thick. Find the massof the shell if the density of the

UNL - MATH - 208

Math 208Self Assessment Quiz 9Key1. Compute the length of the curve r(t) = (cos(et ), sin(et ), 0 t 1.Recall that the length of a curve is given by1(et sin(et )2 + (et cos(et )2 dt|r (t)| dt =01et dt=0= e1Of course, we can also observe that

UNL - MATH - 208

Math 208Self Assessment Quiz 10Name:1. Compute the line integral along the quarter unit circle from (1,0) to (0,1) in the rstquadrant of the vector eld F = (x2/3 , e7y ).Clearly, the curl of F is zero, so there is a potential function to which we can

UNL - MATH - 208

Math 208Self Assessment Quiz 11Name:1. Parametrize the plane through the point (1, 2, -1) with normal vector (-2, 1, 1),We can write the equation of this plane using the normal vector and the point: rst,2x + y + z = d, then plugging in the point we g

UNL - MATH - 211h

Math 221HQuiz 1Thursday, August 23rd, 2012Key1. Check that u(t) = Aet sin(t) is a solution of the ODE y 2y cot(t) = 0.To solve this problem, we plug u(t) in for y in the given ODE.cos(t)sin(t)tttt= Ae sin(t) + Ae cos(t) + Ae cos(t) Ae sin(t)2

UNL - MATH - 211h

Math 221HQuiz 2Thursday, August 30th, 2012Keyt+41. Consider the ODE y = ty . What is the largest rectangle R on which this ODEsatises the conditions of the the E/U theorem for the IVP y (1) = 2?t+4The function f (y, t) = ty is continuous on its do

UNL - MATH - 211h

Math 221HQuiz 3Thursday, September 6th, 2012Key1. Solve the IVP.dy 3t2 y = t2y (0) = 1dtThis is a linear rst order ODE, so we can solve it by guessing an integrating factor. Weknow the proper guess is er(t) , where r(t) = 3t2 dt = t3 + C , and w

UNL - MATH - 211h

Math 221HQuiz 3Thursday, September 6th, 2012Name:1. Without solving, nd and classify the stability of all equilibrium solutions tody= y 3 4y 2 + 4ydtThe right hand side of the equation factors as f (y ) = y (y 2)2 , which has zeroes aty = 0 and y

UNL - MATH - 211h

Math 221HQuiz 5Thursday, October 4th, 2012Key1. Solve the IVP:y 2y + 5 = 0y (0) = 3y (0) = 5This is a homogeneous linear ODE, so we guess that the solution is of the form u = ert .Plugging this in, we obtain the characteristic equation0 = r2 2r

UNL - MATH - 211h

Math 221HQuiz 6Thursday, October 18th, 2012Name:1. Find a particular solution to the non-homogeneous ODE:y 2y + 5y = t + 1 etWe are going to apply the method of undetermined coecients. First, it is vital to besure that our naive guess doesnt involv

UNL - MATH - 211h

Math 221HQuiz 7Thursday, November 8th, 2012Name:1. Find the general solution to the ODE.x=866 4xBegin by nding the eigenvalues. That is, solvedet 8 66+4=00 = ( 8)( + 4) (6)(6)= 2 4 + 4= ( 2)2So, the matrix has one eigenvalue, = 2, of mu

UNL - MATH - 211h

Math 221HQuiz 8Thursday, November 29th, 2012Name:1. Compute from the denition L [e3t+1 ](s).L [e3t+1est e3t+1 dt](s) =0e(s3)t dt=e0e(s3)t= e(s 3)t=0e(s3)t1= e limt (s 3)(s 3)eif s > 3.=s32. Using the facts that L [cos(kt)](s) =L

UNL - MATH - 211h

Math 221HChallenge Problem 1 Due Tuesday, September 4th, 20121. A dog standing at point (c, 0) in the plane spots a rabbit standing at the origin. Therabbit bolts along the line y = ax (a < 0), traveling at a constant velocity VR . The dogimmediately

UNL - MATH - 211h

Math 221HChallenge Problem 2Due Tuesday, September 11th, 20121. Typically, the rst thing people will do in using a DE to model some part of the RealWorldTM is introduce dimensionless variables, which reduces the number of parambers inthe DE. Here, we

UNL - MATH - 211h

Math 221HChallenge Problem 3Due Tues, September 18th, 20121. Suppose a open-topped hemispherical tank of radius 10 feet full of water has a closed,one foot radius circular drain at the bottom. By Torricellis Law, water ows out of thehole with the sam

UNL - MATH - 211h

Math 221HChallenge Problem 4Due Tues, October 9th, 20121. Using the following scaold, sketch a proof of the existence and uniqueness theoremfor rst order IVPs:Suppose f (t, y ) is a continuous function on an interval I containing t0 . Thenthe IVPy

UNL - MATH - 211h

Math 221HChallenge Problem 5 Due Thursday, October 25th, 20121. An alternative way to solve (linear) ODEs. Assume that the solution to the ODEL[y ] = g (t) is analytic at the point t = 0. An analytic function is one which is equal toits power series e

UNL - MATH - 211h

Math 221HChallenge 6Due Tuesday, November 13th, 2012Here is a method for getting real solution more eciently by using complex numbers.Besides ei = cos + i sin , youll have to remember how to invert complex numbers, thatis,1 a iba1b==2i 2.2a

UNL - MATH - 211h

Math 221HChallenge Problem 7Due Tues, November 27th, 20121. Consider a matrixA=a11 a22.a21 a22The Determinant, which we denote D, is a11 a12 a12 a21 , and the Trace, denoted T , isa11 + a22 .1. Write the eigenvalues of A in terms of T and D. Hin

UNL - MATH - 428

WorkedExamplesforChapter11ExampleforSection11.1PawtucketUniversityisplanningtobuynewcopiermachinesforitslibrary.Three membersofitsOperationsResearchDepartmentareanalyzingwhattobuy.Theyare consideringtwodifferentmodels:ModelA,ahighspeedcopier,andModelB,a

UNL - MATH - 428

Worked Examples for Chapter 12Example for Section 12.5Consider the following unconstrained optimization problem:MaximizeThus,22f (x) = 2 12 2 1 2 .ff= 4 x1 + 2 x 2 ,= 2 x1 2 x 2 .x1x 2(a) Starting from the initial trial solution (x1, x2) = (

UNL - MATH - 428

Worked Examples for Chapter 14Example for Section 14.2For the game having the following payoff table, determine the optimal strategy foreach player by successively eliminating dominated strategies. (Indicate the order inwhich you eliminated strategies

UNL - MATH - 428

Worked Examples for Chapter 15Example for Section 15.2Consider a decision analysis problem whose payoffs (in units of thousands of dollars) aregiven by the following payoff table:State of NatureS225305060Prior probabilityS180Alternative400.

UNL - MATH - 428

Worked Examples for Chapter 16Example for Section 16.4Consider the Markov chain that has the following (one-step) transition matrix.StateP=012340 0 0.2 0.5 0.3 0 1 000102 0 0.2 0 0.1 0.73 010004 0.8 0.1 0 0.1 0 (a) Determine the cla

UNL - MATH - 428

Worked Examples for Chapter 17Example for Section 17.4A queueing system has two servers whose service times are independent randomvariables with an exponential distribution with a mean of 15 minutes. Customer Xarrives when both servers are idle. Five

BC - EN - 002

Chris AndersenTKM MWF 12Literature and SocietyBetray to LoveIn Harold Pinters Betrayal betrayal takes place inmany forms in the different relationships. By no means doesthis suggest that there is a lack of love in theserelationships. In Betrayal Em

UNL - MATH - 428

MATH 428 Final ExamSpring Semester, 2010Name:Closed book exam, but calculators are allowed.Score:1(40pts) The coach of an age group swim team needs to assign swimmers to a 200-yard medley relay team to send to the JuniorOlympics. Since most of his b

BC - EN - 002

Christopher AndersenProfessor Kaplan-MaxfieldLiterature and SocietyFebruary 29, 2012Socratic LoveBeauty of the mind is more honorable than the beauty of outward form(32).These words from Diotima were used to educate Socrates in her circular view of

UNL - MATH - 428

MATH 428 Exam ISpring Semester, 2010Name:Any 4 digits PIN:Score:1(15pts) A college textbook publishing company is introducing three new textbooks, Calculus, Biology, and Anatomy,at the same time. The expected prot is $10 per book on Calculus, $12 on

BC - EN - 002

Chris AndersenKPMJanuary 24, 2012Symposium AnalysisA symposium, a gathering of friendsAll with different opinionsTodays topic was that of loveSomething that no persons knowledge was abovePhaedrus was the first in lineWho claimed the God of Love w

UNL - MATH - 428

Spring 2010MATH 428 Exam IIOperations ResearchName:Score:1(25pts) A sports league does one drug testing of its players. Past data shows 10 percent of its plays used a type ofdrug the test is designed to catch. Lab research showed that the test is on

BC - EN - 002

Andersen 1Christopher AndersenDue November 1, 2011Art CritiqueForrest Gump for Pres.Forrest Gump is a movie about a man who lived from the 1950suntil the 1980s, crucial decades of American history. Forrest is a "slow" man, witha below average IQ. A

UNL - MATH - 221

BC - PH - 101

Philosophy of the Person PL070 MWF 1First Critical Analysis6250According to Socrates self-knowledge is the understanding of your currentposition and why you are in this position. Anaxagoras presented only physicalexplanations for why Socrates was in

UNL - MATH - 221

BC - PH - 101

Modern democracy and rights could not solve these conflicts.-Modern democracy lays out universal laws to govern over a large population-it is impossible to give each person exactly what they need/deserveit is very important to note that Prosperos actio

UNL - MATH - 221

BC - PH - 101

Christopher AndersenPL070 Philosophy of the PersonMartin Black1. Socrates produces a number of arguments in the Phaedo for the immortality of thesoul. Evaluate one (or more) and indicate how and why Socrates produces the argument.What is Socrates int

UNL - MATH - 221

BC - PH - 101

Christopher M. AndersenPL070Philosophy of the Person 1Martin BlackAugustines proof of GodWord count: 1017My parents did not force me to go to church and share their beliefs as a child.Because of this and many other reasons, I have hardly any connec

UNL - MATH - 221

BC - PH - 101

The Bible and associated material1. (Genesis, chapters 1-3): In Gods creation of the world the heaven and man are not said to be goodspecifically (although creation as a whole is said to be so). Why might this be the case?A) Man has free will, this cre

BC - PH - 101

Pl 070 Philosophy of the Human PersonMidterm Study QuestionsThe exam will feature roughly 10 short answer questions (from a few words to asentence) and roughly 5 paragraph length answers (2-5 sentences). The former willpredominately ask you to recall

UNL - MATH - 221

BC - PH - 101

Chris AndersenProfessor EnochTues/Thurs 9Pain in the neckI mean backBefore coming home from school Daniel knew that something wasnt right when Terrys(his grandmother) car was there instead of his mothers. Dans mother had a history ofback pain and wa

UNL - MATH - 221

UNL - MATH - 221

BC - EN - 002

Andersen 1Christopher AndersenDue November 1, 2011Art CritiqueForrest Gump for Pres.Forrest Gump is a movie about a man who lived from the 1950s untilthe 1980s, crucial decades of American history. Forrest is a "slow" man, with a belowaverage IQ. A

BC - EN - 002

1Christopher AndersenArgumentative essayYoure Fired!People face moral dilemmas every day, from whether or not you should tell onsomeone for cheating, or if you should take the extra five minutes to correctly cite aresource. Going out of your way to

UNL - MATH - 221