MTH 142 Solution Practice for Exam 2
|
|
- Veronica Watkins
- 6 years ago
- Views:
Transcription
1 MTH 4 Solution Pratie for Eam Updated /7/4, 8: a.m.. (a) = 4/, hene MID() = ( + + ) +/ +6/ +/ ( 4 ) =. ( LEFT = ( 4..). =.7 and RIGHT = (.. ). =.7. Hene TRAP =.7.. (a) MID = ( ). = ( TRAP = (.9 +.4)/ = 4.6 SIMP = ( )/ = (a) RIGHT, MID, TRAP, LEFT. Reason: sine the funtion is dereasing, we have RIGHT < LEFT. Also, TRAP is the average of RIGHT and LEFT. In addition, the funtion is onave up, whih implies MID < TRAP. ( Errors: eat - right() =., eat - mid() =., eat - trap()= -., eat - left () = -.4. () LEFT and RIGHT: the first deimals will be orret sine the error improves by deimal plae. MID: the first 4 deimals will be orret sine the error is improved by deimal plaes. TRAP: the first deimals will be orret sine the error is improved by deimal plaes. b 4. (a) Improper. = lim ( + ) / d = lim 6( + ) / b = + b b = lim 6 + b 6 =. Hene the integral diverges. b ( Improper. + d = lim + + d = lim ln + + = lim ln 6 ln + =. Hene the integral diverges. + () Improper. t + t = lim + ln t ln t + the integral diverges. (d) Improper. e t dt = + = lim + dt lim t(t + ) = + ln t t + = lim + e t dt + / t e t dt = (I) + (II). / t + dt = ln 8 ln + = Hene We now analyze (I) and (II) separately: (I) = lim e t et = lim e =. Hene (I) onverges. b (II) = lim e t b b b et = lim b eb =. Hene (II) diverges. We onlude that e t dt also diverges.. (a) Behaves-like analysis: = when is large (p=). Hene we suspet onvergene. We now ompare the integrand with a larger funtion whose integral onverges. We note that = for, whih implies that the following inequality is valid: + 6 =, for <. We onlude from the omparison test that d onverges.
2 t + ( Behaves-like analysis: t t = for large t (p=). Hene we suspet divergene. We now ompare the integrand with a smaller funtion whose integral diverges; t For this we note that t < t + and that t > t, whih imply that the following inequality is valid: = t t t + for t <. We onlude from the omparison test t that dt diverges. t + t 6. By taking setions perpendiular to the ais of rotation, we get washers. At the tikmark j the washer has inner radius r j = j, outer radius R j =, and thikness. The sum that approimates the volume is V (πrj πr j ) = n (π π( j ) ) j= j= The eat volume is obtained by taking limit as. We have, / (π π4 4 )d = π By taking setions perpendiular to the ais of rotation, we get disks. At the tikmark y j the radius is r j = j = yj / and the thikness is y The sum that approimates the volume is πrj y = π( y j /) y = πy j / y j= j= j=! The volume is obtained by taking limit as y. We have, π ydy = π 4 8. By taking setions perpendiular to the ais of rotation, we get washers. At the tikmark y j the washer has inner radius r j =, outer radius R j = + y j /, and thikness y. The sum that approimates the volume is V (πrj πrj ) y = (π( + j= j= The volume is obtained by taking limit as. We have, (π( + y j /) π() ) y y/) π)dy = π( 4 + ).747
3 9. a). a). ) = j= 7 A plot of y = ( )/( + ) produed with a graphing alulator shows that the plate has the shape shown in the figure. It is lear that the urve meets the X and Y aes at = and y = respetively. Sine the plate has onstant density, the formulas in page 6 of the tet apply. The total mass of the plate is Mass = is given by =.4 πr r + rj (.4)πr + r r ( +.) j= ( +.)d =.7 ( +.)d. = ( +.)d.7 =.666. d + Mass d The enter of mass (, y) + = By symmetry we have that y = = Hene (, y) = (.7686,.7686). Note: to obtain y with a alulation, it may be done as follows: Solve for in y = ( )/( + ) to obtain = ( y)/( + y). Then, y = y. y +y dy Mass = Slie the drum with horizontal, irular setions. Eah setion orresponds to a tikmark y j on the vertial ais. The number of bateria in a setion at tikmark y j is Number(Setion j ) density volume = (..y j ) π(4) y The total number of bateria is approimated by the Riemann Sum N umber(container) = (..y j ) π(4) y j=
4 The eat number is obtained by passing to the limit as y. It is, 6 (..y) π(4) dy = 697 million bateria. A ross-setion of the one (shown in the figure) is bounded by the lines y = ± and y =. Introdue tik marks in the y-ais. - The slab S j at height y j is a disk with radius R j = j = y j / and thikness y, so its volume is π(y j /) y, and its weight is 6.4 π(y j /) y. The work involved in raising the slab a distane of ( y j ) to the top of the one is The total work is approimated by The eat work is given by w j = ( y j ) 6.4 π(y j /) y W ( y j ) 6.4 π(y j /) y j= ( y) 6.4 π(y/) y = A sketh of the dam is shown in the figure below. 4 6 Note that the equation of the right hand, non-horizontal side is y =. Introdue tik marks y, y,..., y n, in the y ais. At height y j, the slab has area (y j + ) y, and the pressure at this height is 6.4( y j ). Therefore the fore on the slab is The total fore is approimated by F j = 6.4( y j )(y j + ) y F 6.4(y j + )( y j ) y j=
5 . a) The eat value of the total fore is obtained by taking the limit as y :.7.. T d =.69 d = F = 6.4(y j + )( y j )dy =,, 6. a) d =. = T =. = T + =. = T = 4 = d = d = ln() = T e d =. = e T =. = e T + =. = T = b e = b e d = lim e d = lim b b e + = lim ( b b e + b e ) ( b ) = ln a) The umulative distribution funtion is an antiderivative of the density, so P = d = +. We now find. We also know that P = when = (first value of ). Substituting we find =, so P () = +. Another way to solve it: P () = dt = t = +. t P = e d = e + C. We also know that P = when =. Substituting into P we have = + C, that is, C =. We onlude that P = e +. Another way to solve it: P () = e t dt = e t = e The inreasing funtion is the Cumulative Distribution Funtion, whih we know has range from to. This gives the vertial range, so the tik marks on the y ais are at {,.,.4,.6,.8,.}. Also, we know that the region under the density funtion (appro.. retangles) has area. Then eah retangle has area /. =.4. But we also know the height of the retangle is.. Then the base of the retangle is approimately.4/. =. Therefore the tik marks on the -ais are at {,, 4, 6, 8, }.
CHAPTER P Preparation for Calculus
PART I CHAPTER P Preparation for Calulus Setion P. Graphs and Models...................... Setion P. Linear Models and Rates of Change............. 7 Setion P. Funtions and Their Graphs.................
More information6x and find the. y=2x+5 and y=4 -x 2. gradient of the curve at this point. tangent to the curve y = 4 - x 2? Figure 5.11
Figure 5.11 EERCISE 5C 1 For eah part of this question, (a) find: (b) fmd the gradient of the urve at the given point. (il y = x- 2 ; (.25, 16) (iil y= x- 1 + x-4; (-1, ) (iiil y= 4x- 3 + 2x- 5 ; (1, 6)
More information12 th Maths Way to Success
th Maths Quarterly Eam-7-Answer Key Part - A Q.No Option Q.No Option Q.No Option Q.No Option 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 Part B. A adj A A adja..() adja A () A I () From (), (),() we get A adja adja
More information4.4 Solving Systems of Equations by Matrices
Setion 4.4 Solving Systems of Equations by Matries 1. A first number is 8 less than a seond number. Twie the first number is 11 more than the seond number. Find the numbers.. The sum of the measures of
More information10.2 The Occurrence of Critical Flow; Controls
10. The Ourrene of Critial Flow; Controls In addition to the type of problem in whih both q and E are initially presribed; there is a problem whih is of pratial interest: Given a value of q, what fators
More informationMathematics II. Tutorial 5 Basic mathematical modelling. Groups: B03 & B08. Ngo Quoc Anh Department of Mathematics National University of Singapore
Mathematis II Tutorial 5 Basi mathematial modelling Groups: B03 & B08 February 29, 2012 Mathematis II Ngo Quo Anh Ngo Quo Anh Department of Mathematis National University of Singapore 1/13 : The ost of
More informationRIEMANN S FIRST PROOF OF THE ANALYTIC CONTINUATION OF ζ(s) AND L(s, χ)
RIEMANN S FIRST PROOF OF THE ANALYTIC CONTINUATION OF ζ(s AND L(s, χ FELIX RUBIN SEMINAR ON MODULAR FORMS, WINTER TERM 6 Abstrat. In this hapter, we will see a proof of the analyti ontinuation of the Riemann
More informationTuesday, September 29, Page 453. Problem 5
Tuesday, September 9, 15 Page 5 Problem 5 Problem. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by y = x, y = x 5 about the x-axis. Solution.
More informationAnswer Key. ( 1) n (2x+3) n. n n=1. (2x+3) n. = lim. < 1 or 2x+3 < 4. ( 1) ( 1) 2n n
Math Midterm Eam #3 December, 3 Answer Key. [5 Points] Find the Interval and Radius of Convergence for the following power series. Analyze carefully and with full justification. Use Ratio Test. L lim a
More informationPHYSICS 212 FINAL EXAM 21 March 2003
PHYSIS INAL EXAM Marh 00 Eam is losed book, losed notes. Use only the provided formula sheet. Write all work and answers in eam booklets. The baks of pages will not be graded unless you so ruest on the
More informationSection 7.4 #1, 5, 6, 8, 12, 13, 44, 53; Section 7.5 #7, 10, 11, 20, 22; Section 7.7 #1, 4, 10, 15, 22, 44
Math B Prof. Audrey Terras HW #4 Solutions Due Tuesday, Oct. 9 Section 7.4 #, 5, 6, 8,, 3, 44, 53; Section 7.5 #7,,,, ; Section 7.7 #, 4,, 5,, 44 7.4. Since 5 = 5 )5 + ), start with So, 5 = A 5 + B 5 +.
More informationName: Answer Key David Arnold. Math 50B Integral Calculus May 13, Final Exam
Math 5B Integral Calculus May 3, 7 Final Exam Name: Answer Key David Arnold Instructions. (9 points) Follow the directions exactly! Whatever you are asked to do, you must do to receive full credit for
More informationMost results in this section are stated without proof.
Leture 8 Level 4 v2 he Expliit formula. Most results in this setion are stated without proof. Reall that we have shown that ζ (s has only one pole, a simple one at s =. It has trivial zeros at the negative
More information2. The Energy Principle in Open Channel Flows
. The Energy Priniple in Open Channel Flows. Basi Energy Equation In the one-dimensional analysis of steady open-hannel flow, the energy equation in the form of Bernoulli equation is used. Aording to this
More informationx+1 e 2t dt. h(x) := Find the equation of the tangent line to y = h(x) at x = 0.
Math Sample final problems Here are some problems that appeared on past Math exams. Note that you will be given a table of Z-scores for the standard normal distribution on the test. Don t forget to have
More informationOrdering Generalized Trapezoidal Fuzzy Numbers. Using Orthocentre of Centroids
International Journal of lgera Vol. 6 no. 69-85 Ordering Generalized Trapezoidal Fuzzy Numers Using Orthoentre of Centroids Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar Dept. of pplied Mathematis
More informationLesson 23: The Defining Equation of a Line
Student Outomes Students know that two equations in the form of ax + y = and a x + y = graph as the same line when a = = and at least one of a or is nonzero. a Students know that the graph of a linear
More informationChapter 2: Solution of First order ODE
0 Chapter : Solution of irst order ODE Se. Separable Equations The differential equation of the form that is is alled separable if f = h g; In order to solve it perform the following steps: Rewrite the
More informationExamining Applied Rational Functions
HiMAP Pull-Out Setion: Summer 1990 Eamining Applied Rational Funtions Flod Vest Referenes Environmental Protetion Agen. Gas Mileage Guide. (Copies an usuall e otained from a loal new ar dealer.) Information
More informationMicroeconomic Theory I Assignment #7 - Answer key
Miroeonomi Theory I Assignment #7 - Answer key. [Menu priing in monopoly] Consider the example on seond-degree prie disrimination (see slides 9-93). To failitate your alulations, assume H = 5, L =, and
More informationUNCORRECTED PAGE PROOFS
8 Kik off with CAS 8 Introdution to vetors 8 Operations on vetors Vetors 8 Magnitude, diretion and omponents of vetors 85 i, j notation 86 Appliations of vetors 87 Review 8 8 Kik off with CAS Eploring
More information5.5 Volumes: Tubes. The Tube Method. = (2π [radius]) (height) ( x k ) = (2πc k ) f (c k ) x k. 5.5 volumes: tubes 435
5.5 volumes: tubes 45 5.5 Volumes: Tubes In Section 5., we devised the disk method to find the volume swept out when a region is revolved about a line. To find the volume swept out when revolving a region
More informationEvaluation of effect of blade internal modes on sensitivity of Advanced LIGO
Evaluation of effet of blade internal modes on sensitivity of Advaned LIGO T0074-00-R Norna A Robertson 5 th Otober 00. Introdution The urrent model used to estimate the isolation ahieved by the quadruple
More informationA population of 50 flies is expected to double every week, leading to a function of the x
4 Setion 4.3 Logarithmi Funtions population of 50 flies is epeted to doule every week, leading to a funtion of the form f ( ) 50(), where represents the numer of weeks that have passed. When will this
More informationAP Calculus AB Free-Response Scoring Guidelines
Question pt The rate at which raw sewage enters a treatment tank is given by Et 85 75cos 9 gallons per hour for t 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per
More informationWhere as discussed previously we interpret solutions to this partial differential equation in the weak sense: b
Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential
More informationLECTURE 2 Geometrical Properties of Rod Cross Sections (Part 2) 1 Moments of Inertia Transformation with Parallel Transfer of Axes.
V. DEMENKO MECHNCS OF MTERLS 05 LECTURE Geometrial Properties of Rod Cross Setions (Part ) Moments of nertia Transformation with Parallel Transfer of xes. Parallel-xes Theorems S Given: a b = S = 0. z
More informationMODELLING THE POSTPEAK STRESS DISPLACEMENT RELATIONSHIP OF CONCRETE IN UNIAXIAL COMPRESSION
VIII International Conferene on Frature Mehanis of Conrete and Conrete Strutures FraMCoS-8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang Eds) MODELLING THE POSTPEAK STRESS DISPLACEMENT RELATIONSHIP
More informationFinal Review. A Puzzle... Special Relativity. Direction of the Force. Moving at the Speed of Light
Final Review A Puzzle... Diretion of the Fore A point harge q is loated a fixed height h above an infinite horizontal onduting plane. Another point harge q is loated a height z (with z > h) above the plane.
More informationMass Transfer (Stoffaustausch) Fall 2012
Mass Transfer (Stoffaustaush) Fall Examination 9. Januar Name: Legi-Nr.: Edition Diffusion by E. L. Cussler: none nd rd Test Duration: minutes The following materials are not permitted at your table and
More informationThe Hanging Chain. John McCuan. January 19, 2006
The Hanging Chain John MCuan January 19, 2006 1 Introdution We onsider a hain of length L attahed to two points (a, u a and (b, u b in the plane. It is assumed that the hain hangs in the plane under a
More informationAcoustic Waves in a Duct
Aousti Waves in a Dut 1 One-Dimensional Waves The one-dimensional wave approximation is valid when the wavelength λ is muh larger than the diameter of the dut D, λ D. The aousti pressure disturbane p is
More informationExperiment 03: Work and Energy
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.01 Purpose of the Experiment: Experiment 03: Work and Energy In this experiment you allow a art to roll down an inlined ramp and run into
More informationBuckling loads of columns of regular polygon cross-section with constant volume and clamped ends
76 Bukling loads of olumns of regular polygon ross-setion with onstant volume and lamped ends Byoung Koo Lee Dept. of Civil Engineering, Wonkwang University, Iksan, Junuk, 7-79, Korea Email: kleest@wonkwang.a.kr
More informationHeat exchangers: Heat exchanger types:
Heat exhangers: he proess of heat exhange between two fluids that are at different temperatures and separated by a solid wall ours in many engineering appliations. he devie used to implement this exhange
More informationChapter 3 Lecture 7. Drag polar 2. Topics. Chapter-3
hapter 3 eture 7 Drag polar Topis 3..3 Summary of lift oeffiient, drag oeffiient, pithing moment oeffiient, entre of pressure and aerodynami entre of an airfoil 3..4 Examples of pressure oeffiient distributions
More informationMath Exam III - Spring
Math 3 - Exam III - Spring 8 This exam contains 5 multiple choice questions and hand graded questions. The multiple choice questions are worth 5 points each and the hand graded questions are worth a total
More informationExamples of the Accumulation Function (ANSWERS) dy dx. This new function now passes through (0,2). Make a sketch of your new shifted graph.
Eamples of the Accumulation Function (ANSWERS) Eample. Find a function y=f() whose derivative is that f()=. dy d tan that satisfies the condition We can use the Fundamental Theorem to write a function
More informationName Solutions to Test 1 September 23, 2016
Name Solutions to Test 1 September 3, 016 This test onsists of three parts. Please note that in parts II and III, you an skip one question of those offered. Possibly useful formulas: F qequb x xvt E Evpx
More informationShear Force and Bending Moment
Shear Fore and Bending oent Shear Fore: is the algebrai su of the vertial fores ating to the left or right of a ut setion along the span of the bea Bending oent: is the algebrai su of the oent of the fores
More informationGLOBAL EDITION. Calculus. Briggs Cochran Gillett SECOND EDITION. William Briggs Lyle Cochran Bernard Gillett
GOBA EDITION Briggs Cohran Gillett Calulus SECOND EDITION William Briggs le Cohran Bernar Gillett ( (, ) (, ) (, Q ), Q ) (, ) ( Q, ) / 5 /4 5 5 /6 7 /6 ( Q, 5 5 /4 ) 4 4 / 7 / (, ) 9 / (, ) 6 / 5 / (Q,
More information1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
More informationTHEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE?
THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE? The stars are spheres of hot gas. Most of them shine beause they are fusing hydrogen into helium in their entral parts. In this problem we use onepts of
More information4x x dx. 3 3 x2 dx = x3 ln(x 2 )
Problem. a) Compute the definite integral 4 + d This can be done by a u-substitution. Take u = +, so that du = d, which menas that 4 d = du. Notice that u() = and u() = 6, so our integral becomes 6 u du
More informationUTC. Engineering 329. Proportional Controller Design. Speed System. John Beverly. Green Team. John Beverly Keith Skiles John Barker.
UTC Engineering 329 Proportional Controller Design for Speed System By John Beverly Green Team John Beverly Keith Skiles John Barker 24 Mar 2006 Introdution This experiment is intended test the variable
More informationBeams on Elastic Foundation
Professor Terje Haukaas University of British Columbia, Vanouver www.inrisk.ub.a Beams on Elasti Foundation Beams on elasti foundation, suh as that in Figure 1, appear in building foundations, floating
More informationMillennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six
More information1 sin 2 r = 1 n 2 sin 2 i
Physis 505 Fall 005 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.5, 7.8, 7.16 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with
More informationCalculus II - Fall 2013
Calculus II - Fall Midterm Exam II, November, In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit.. Find the area between
More informationCavity flow with surface tension past a flat plate
Proeedings of the 7 th International Symposium on Cavitation CAV9 Paper No. ## August 7-, 9, Ann Arbor, Mihigan, USA Cavity flow with surfae tension past a flat plate Yuriy Savhenko Institute of Hydromehanis
More informationA NORMALIZED EQUATION OF AXIALLY LOADED PILES IN ELASTO-PLASTIC SOIL
Journal of Geongineering, Vol. Yi-Chuan 4, No. 1, Chou pp. 1-7, and April Yun-Mei 009 Hsiung: A Normalized quation of Axially Loaded Piles in lasto-plasti Soil 1 A NORMALIZD QUATION OF AXIALLY LOADD PILS
More information6.1. The Derivative Definition (6.1.1). Let I R be an interval, f : I! R, and c 2 R. We say L is the derivative of f at c if
CHAPTER 6 Di erentiation 6.1. The Derivative Definition (6.1.1). Let I R be an interval, f : I! R, and 2 R. We say L is the derivative of f at if f(x) > 0 9 > 0 3 x 2 I and 0 < < =) L
More informationReview Topic 4: Cubic polynomials
Review Topi : ui polynomials Short answer Fatorise Px ( ) = x + 5x + x- 9 into linear fators. The polynomial Px ( ) = x - ax + x- leaves a remainer of when it is ivie y ( x - ) an a remainer of - when
More informationCasimir self-energy of a free electron
Casimir self-energy of a free eletron Allan Rosenwaig* Arist Instruments, In. Fremont, CA 94538 Abstrat We derive the eletromagneti self-energy and the radiative orretion to the gyromagneti ratio of a
More informationTORSION By Prof. Ahmed Amer
ORSION By Prof. Ahmed Amer orque wisting moments or torques are fores ating through distane so as to promote rotation. Example Using a wrenh to tighten a nut in a bolt. If the bolt, wrenh and fore are
More informationTorsion. Torsion is a moment that twists/deforms a member about its longitudinal axis
Mehanis of Solids I Torsion Torsional loads on Cirular Shafts Torsion is a moment that twists/deforms a member about its longitudinal axis 1 Shearing Stresses due to Torque o Net of the internal shearing
More informationErrata and changes for Lecture Note 1 (I would like to thank Tomasz Sulka for the following changes): ( ) ( ) lim = should be
Errata and hanges for Leture Note (I would like to thank Tomasz Sulka for the following hanges): Page 5 of LN: f f ' lim should be g g' f f ' lim lim g g ' Page 8 of LN: the following words (in RED) have
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More informationComparison of solution to FE. note: the distance from flange edge is x in these plots while it was y in the derivation!!!
Comparison of solution to FE note: the distane from flange edge is in these plots while it was y in the derivation!!! Comparison of solution to FE Comparison of solution to FE?? More elaborate solutions
More informationSample Final Questions: Solutions Math 21B, Winter y ( y 1)(1 + y)) = A y + B
Sample Final Questions: Solutions Math 2B, Winter 23. Evaluate the following integrals: tan a) y y dy; b) x dx; c) 3 x 2 + x dx. a) We use partial fractions: y y 3 = y y ) + y)) = A y + B y + C y +. Putting
More informationz k sin(φ)(x ı + y j + z k)da = R 1 3 cos3 (φ) π 2π dθ = div(z k)dv = E curl(e x ı + e x j + e z k) d S = S
Mathematis 2443-6H Name (please print) Final xamination May 7, 28 Instrutions: Give brief, lear answers. Use theorems whenever possible. I. Verify the Divergene Theorem for the vetor field F(x,y,z) z k
More informationMidterm Exam #1. (y 2, y) (y + 2, y) (1, 1)
Math 5B Integral Calculus March 7, 7 Midterm Eam # Name: Answer Key David Arnold Instructions. points) This eam is open notes, open book. This includes any supplementary tets or online documents. You are
More informationAP Calculus Review Assignment Answer Sheet 1. Name: Date: Per. Harton Spring Break Packet 2015
AP Calculus Review Assignment Answer Sheet 1 Name: Date: Per. Harton Spring Break Packet 015 This is an AP Calc Review packet. As we get closer to the eam, it is time to start reviewing old concepts. Use
More informationn n=1 (air) n 1 sin 2 r =
Physis 55 Fall 7 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.4, 7.6, 7.8 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with index
More informationSlenderness Effects for Concrete Columns in Sway Frame - Moment Magnification Method
Slenderness Effets for Conrete Columns in Sway Frame - Moment Magnifiation Method Slender Conrete Column Design in Sway Frame Buildings Evaluate slenderness effet for olumns in a sway frame multistory
More informationMath 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2
Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
More informationMat104 Fall 2002, Improper Integrals From Old Exams
Mat4 Fall 22, Improper Integrals From Old Eams For the following integrals, state whether they are convergent or divergent, and give your reasons. () (2) (3) (4) (5) converges. Break it up as 3 + 2 3 +
More informationREVISION SHEET FP2 (Edx) CALCULUS. x x 0.5. x x 1.5. π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + = + arcsin x = +
the Further Mathematis netwk www.fmnetwk.g.uk V 07 REVISION SHEET FP (Ed) CLCULUS The main ideas are: Calulus using inverse trig funtions & hperboli trig funtions and their inverses. Malaurin series Differentiating
More informationSlenderness Effects for Concrete Columns in Sway Frame - Moment Magnification Method
Slenderness Effets for Conrete Columns in Sway Frame - Moment Magnifiation Method Slender Conrete Column Design in Sway Frame Buildings Evaluate slenderness effet for olumns in a sway frame multistory
More information10Circular ONLINE PAGE PROOFS. functions
Cirular funtions. Kik off with CAS. Modelling with trigonometri funtions. Reiproal trigonometri funtions. Graphs of reiproal trigonometri funtions. Trigonometri identities.6 Compound- and doule-angle formulas.7
More informationMATH141: Calculus II Exam #1 review 6/8/2017 Page 1
MATH: Calculus II Eam # review /8/7 Page No review sheet can cover everything that is potentially fair game for an eam, but I tried to hit on all of the topics with these questions, as well as show you
More informationA Characterization of Wavelet Convergence in Sobolev Spaces
A Charaterization of Wavelet Convergene in Sobolev Spaes Mark A. Kon 1 oston University Louise Arakelian Raphael Howard University Dediated to Prof. Robert Carroll on the oasion of his 70th birthday. Abstrat
More informationEFFECT OF PITCH NUMBER IN OVERALL HEAT TRANSFER RATE IN DOUBLE PIPE HELICAL HEAT EXCHANGER
Volume 116 No. 5 2017, 1-6 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu EFFECT OF PITCH NUMBER IN OVERALL HEAT TRANSFER RATE IN DOUBLE PIPE HELICAL
More informationSolving Right Triangles Using Trigonometry Examples
Solving Right Triangles Using Trigonometry Eamples 1. To solve a triangle means to find all the missing measures of the triangle. The trigonometri ratios an be used to solve a triangle. The ratio used
More informationAstr 5465 Mar. 29, 2018 Galactic Dynamics I: Disks
Galati Dynamis Overview Astr 5465 Mar. 29, 2018 Subjet is omplex but we will hit the highlights Our goal is to develop an appreiation of the subjet whih we an use to interpret observational data See Binney
More informationNote: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.
997 AP Calculus BC: Section I, Part A 5 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number..
More informationVolumes of Solids of Revolution. We revolve this curve about the x-axis and create a solid of revolution.
Volumes of Solids of Revolution Consider the function ( ) from a = to b = 9. 5 6 7 8 9 We revolve this curve about the x-axis and create a solid of revolution. - 5 6 7 8 9 - - - We want to find the volume
More informationThe answers below are not comprehensive and are meant to indicate the correct way to solve the problem. sin
Math : Practice Final Answer Key Name: The answers below are not comprehensive and are meant to indicate the correct way to solve the problem. Problem : Consider the definite integral I = 5 sin ( ) d.
More informationON A PROCESS DERIVED FROM A FILTERED POISSON PROCESS
ON A PROCESS DERIVED FROM A FILTERED POISSON PROCESS MARIO LEFEBVRE and JEAN-LUC GUILBAULT A ontinuous-time and ontinuous-state stohasti proess, denoted by {Xt), t }, is defined from a proess known as
More informationA = (cosh x sinh x) dx = (sinh x cosh x) = sinh1 cosh1 sinh 0 + cosh 0 =
Calculus 7 Review Consider the region between curves y= cosh, y= sinh, =, =.. Find the area of the region. e + e e e Solution. Recall that cosh = and sinh =, whence sinh cosh. Therefore the area is given
More informationAtomic and Nuclear Physics
Atomi and Nulear Physis X-ray physis Compton effet and X-ray physis LD Physis Leaflets P6.3.7. Compton effet: Measuring the energy of the sattered photons as a funtion of the sattering angle Objets of
More informationLESSON 14: VOLUME OF SOLIDS OF REVOLUTION SEPTEMBER 27, 2017
LESSON 4: VOLUME OF SOLIDS OF REVOLUTION SEPTEMBER 27, 27 We continue to expand our understanding of solids of revolution. The key takeaway from today s lesson is that finding the volume of a solid of
More informationChapter 6 Some Applications of the Integral
Chapter 6 Some Applications of the Integral Section 6.1 More on Area a. Representative Rectangle b. Vertical Separation c. Example d. Integration with Respect to y e. Example Section 6.2 Volume by Parallel
More informationAtomic and Nuclear Physics
Atomi and Nulear Physis X-ray physis Compton effet and X-ray physis LD Physis Leaflets P6.3.7. Compton effet: Measuring the energy of the sattered photons as a funtion of the sattering angle Objets of
More informationAsymptotic non-degeneracy of the solution to the Liouville Gel fand problem in two dimensions
Comment. Math. Helv. 2 2007), 353 369 Commentarii Mathematii Helvetii Swiss Mathematial Soiety Asymptoti non-degeneray of the solution to the Liouville Gel fand problem in two dimensions Tomohio Sato and
More informationProblem 3 : Solution/marking scheme Large Hadron Collider (10 points)
Problem 3 : Solution/marking sheme Large Hadron Collider 10 points) Part A. LHC Aelerator 6 points) A1 0.7 pt) Find the exat expression for the final veloity v of the protons as a funtion of the aelerating
More informationChapter 8 Hypothesis Testing
Leture 5 for BST 63: Statistial Theory II Kui Zhang, Spring Chapter 8 Hypothesis Testing Setion 8 Introdution Definition 8 A hypothesis is a statement about a population parameter Definition 8 The two
More informationmax min z i i=1 x j k s.t. j=1 x j j:i T j
AM 221: Advaned Optimization Spring 2016 Prof. Yaron Singer Leture 22 April 18th 1 Overview In this leture, we will study the pipage rounding tehnique whih is a deterministi rounding proedure that an be
More informationarxiv:gr-qc/ v2 6 Feb 2004
Hubble Red Shift and the Anomalous Aeleration of Pioneer 0 and arxiv:gr-q/0402024v2 6 Feb 2004 Kostadin Trenčevski Faulty of Natural Sienes and Mathematis, P.O.Box 62, 000 Skopje, Maedonia Abstrat It this
More informationPh1c Analytic Quiz 2 Solution
Ph1 Analyti Quiz 2 olution Chefung Chan, pring 2007 Problem 1 (6 points total) A small loop of width w and height h falls with veloity v, under the influene of gravity, into a uniform magneti field B between
More informationWood Design. = theoretical allowed buckling stress
Wood Design Notation: a = name for width dimension A = name for area A req d-adj = area required at allowable stress when shear is adjusted to inlude self weight b = width of a retangle = name for height
More informationNatural Convection Experiment Measurements from a Vertical Surface
OBJECTIVE Natural Convetion Experiment Measurements from a Vertial Surfae 1. To demonstrate te basi priniples of natural onvetion eat transfer inluding determination of te onvetive eat transfer oeffiient.
More informationSPLINE ESTIMATION OF SINGLE-INDEX MODELS
SPLINE ESIMAION OF SINGLE-INDEX MODELS Li Wang and Lijian Yang University of Georgia and Mihigan State University Supplementary Material his note ontains proofs for the main results he following two propositions
More informationChapter 2 Lecture 8 Longitudinal stick fixed static stability and control 5 Topics
Flight dynamis II Stability and ontrol hapter 2 Leture 8 Longitudinal stik fied stati stability and ontrol 5 Topis 2.6 ontributions of power plant to mg and mα 2.6.1 Diret ontributions of powerplant to
More informationNUMERICAL SIMULATION OF ATOMIZATION WITH ADAPTIVE JET REFINEMENT
Paper ID ILASS8--7 ILASS 28 Sep. 8-, 28, Como Lake, Italy A44 NUMERICAL SIMULATION OF ATOMIZATION WITH ADAPTIVE JET REFINEMENT Anne Bagué, Daniel Fuster, Stéphane Popinet + & Stéphane Zaleski Université
More informationBending stress strain of bar exposed to bending moment
Elastiit and Plastiit Bending stress strain of ar eposed to ending moment Basi priniples and onditions of solution Calulation of ending (diret) stress Design of ar eposed to ending moment Comined stress
More informationDaily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012
The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,
More informationBeam Stresses Bending and Shear
Beam Stresses Bending and Shear Notation: A = name or area A web = area o the web o a wide lange setion b = width o a retangle = total width o material at a horizontal setion = largest distane rom the
More informationIN-PLANE VIBRATIONS OF CURVED BEAMS WITH VARIABLE CROSS-SECTIONS CARRYING ADDITIONAL MASS
11 th International Conferene on Vibration Problems Z. Dimitrovová et al. (eds.) Lisbon, Portugal, 9-1 September 013 IN-PLANE VIBRATIONS OF CURVED BEAMS WITH VARIABLE CROSS-SECTIONS CARRYING ADDITIONAL
More informationElectromagnetic radiation of the travelling spin wave propagating in an antiferromagnetic plate. Exact solution.
arxiv:physis/99536v1 [physis.lass-ph] 15 May 1999 Eletromagneti radiation of the travelling spin wave propagating in an antiferromagneti plate. Exat solution. A.A.Zhmudsky November 19, 16 Abstrat The exat
More information