3.6 Applied Optimization
|
|
- Suzanna Carroll
- 6 years ago
- Views:
Transcription
1 .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the deivative of something and setting it equal to zeo. EXAMPE: What is the smallest peimete possible fo a ectangle whose aea is 6 squae inches, and what ae the dimensions? Fo this one we can come up with two equations. We will use fo length and W fo width. The fist equation we have is W 6 since aea equals length times width. The second equation is fo the peimete, and this is W P. We want to get one equation with a single vaiable, so we can take the fist equation and solve fo eithe o W and put it into the second equation. I will solve the fist equation fo W: W 6. Now we 6 can put this into the second equation to eplace the W: P. This is the same as 7 P. Taking the deivative we will get: P 7. We need to set this equal to zeo and ewite this without 7 7 negative eponents: 0. Add the second tem to the othe side to get:. Coss multiplying will give us 7. You will get 6, so 6. Howeve we only want positive numbes, so we will just use = 6. We can use the equation W 6 6 to find W. We will get W 6. So = 6 in and W = 6 in. 6 To find the actual peimete, just put these numbes back into W P. So P (6 (6 4 in. EXAMPE: An open bo of maimum volume is to be made fom a squae piece of mateial, 4 inches on a side, by cutting equal squaes fom the cones and tuning up the sides (see figue. Fist wite an equation fo the volume, V, of the bo as a function of. Then find the maimum volume of the bo. To get the equation we notice that the height of this bo is. The length and width ae both 4. The fomula fo volume is V = WH. So we have V ( 4 (4 (, o V ( 4. We need to take the deivative and set it equal to zeo. Using the poduct ule we will get: V (4 ( (4 ( We can facto out 4 : V ( 4 ( 4 4 which is V ( 4 (4 6. Setting this equal to zeo you will get = 4 and =. Even though we get two answes, the value = will not be included because 4 ( = 0 which means that a bo will have a length o width of zeo which is impossible. Theefoe = 4 is the only value fo that makes sense. To find the maimum volume, put this in the ORIGINA equation fo volume, which is V ( 4. You will get V 4(4 (4 04. Theefoe the maimum aea is 04 squae inches.
2 Section.6 Notes Page EXAMPE: A ectangle is bounded by the -ais and the semicicle y 9 (see figue. What length and width should the ectangle have so that its aea is a maimum? What is the maimum aea? A A (9 9 9 ( (9 Fist we need to find the aea of the ectangle. The length is going to be since thee ae two s in the figue. The height of the bo depends on whee it hits the semicicle. The height will be equal to y, so y 9. So now we can find the aea fomula: A 9. We need to find this deivative and set it equal to zeo. This will involve the poduct ule. Fist I will ewite the poblem as: A (9. Now take the deivative: ( Poduct and chain ules used hee. Now simplify. 9 Now set the deivative equal to zeo. Move one tem to the left side solve fo. You will have 4 8 Then. Now coss multiply to get (9. Distibuting will give, so 9 8. Now. When you take the squae oot, just take the positive oot.. We know the length of the ectangle is, so the length =. The height of the ectangle is y 9. We will put ou answe fo into this fomula: y y 9. The maimum aea is these two answes multiplied togethe: A 9.. So EXAMPE: Two sides of a tiangle have lengths a and b, and the angle between them is. What value of will maimize the tiangle s aea? (Hint: A ab sin In this case a and b ae to be teated as constants, o odinay numbes. To find the maimum aea we need to take the deivative and set it equal to zeo. We will take the deivative with espect to. No poduct ule is needed hee because the only vaiable is : A ab cos. We now set this equal to zeo: 0 ab cos. Now solve fo cosine: 0 cos. To solve fo we need to take the invese cosine of both sides:. cos 0. We will get
3 Section.6 Notes Page EXAMPE: A anche has 00 feet of fencing which to enclose two adjacent coals. What dimensions should be used so that the enclosed aea will be a maimum? (See figue y In this poblem I labeled the width and the length y. It doesn t matte which vaiable you use. I can ceate two equations with this one. The fist equation is y 00. This deals with the peimete. The second equation is A y. You want to get one equation with a single vaiable, so we can solve fo eithe y o y. Solving fo y we get: y 00. You want to put this into A y fo y to get A 00 which is A 00. The deivative is: A 00. Setting it equal to zeo we will get 0 00, so. To find the y, we can use the equation y 00 and put in fo 00. You will get: y which simplifies to y = 50. So the dimensions ae by 50. EXAMPE: You ae designing a poste to contain 50 squae inches of pinting with a 4 inch magin at the top and bottom and a inch magin at each side. What oveall dimensions will minimize the amount of pape used? Fist we should daw a pictue and label sides. I will make be the height of the pinted mateial and y to epesent the? in the pictue below. Fom this pictue we can make two equations. Fist, the aea of the pinted mateial is 50 squae inches, so 50 = y. Anothe equation will involve the entie piece of pape including the magins. Hee is anothe equation: A = ( + 4(y + 8. We need to solve the fist equation fo eithe o y and 50 substitute it into the second equation. I will solve fo y in the fist equation: y. 50 Now substitute into the second equation: A ( 4 8. Instead of applying the poduct ule it may be easie to distibute fist: A This 00 simplifies to A 8 8. It is now time to take the deivative. We will ewite A as: 00 A The deivative is: A Now set it equal to zeo: 0 8. So Coss multiply: Divide to get: 5, so 5. We can t have a negative side, 50 so = 5 in. This means that y 0 in. Howeve the question asks us the find the oveall dimensions, so 5 the height will be = 8 inches and the width is = 9 inches. Theefoe the oveall dimensions of the pape ae 9 inches by 8 inches.
4 Section.6 Notes Page 4 EXAMPE: A 08 cubic foot bo with a squae base and an open top is to be constucted fom sheet metal of a given thickness. Find the dimensions of the tank with minimum weight. A bo with the minimum weight means we need to minimize the suface aea. To find the suface aea, we need to find the aea of each individual side. Thee is one bottom that is by. Then thee ae fou sides that have an aea of time y. So we y have S 4y. The volume of the bo would be V y. Since we ae give the volume, we have the fomula 08 y. We will solve this equation fo y and 08 substitute it into the othe equation. y. Now put this into the suface aea equation to get: S 4. This can be simplified to: S 4. Now we take the deivative: S 4. Setting this equal to zeo we get: 4 4 0, so. Cleaing the faction you will get 4, so 6 08 and = 6. Then we can put 6 in fo in y to get y =. So ou dimensions ae,6,6. EXAMPE: You have been asked to design a 000 cubic centimete can shaped like a ight cicula cylinde. What dimensions will use the least mateial. NOTE: The suface aea fomula fo a ight cicula cylinde is S h. The aea of a ight cicula cylinde is A h. We will ignoe the thickness of the mateial and the waste in manufactuing. Fist we ae given that the volume should be 000 cubic centimetes, so one equation we will have is 000 h. We want to eliminate one vaiable in the suface aea equation, so in ou volume equation we can solve fo eithe o h. It will be easie 000 to solve fo h since that avoids squae oots: h. Now we will substitute this into the suface aea fomula: S. This simplifies to S. Because we want to minimize the amount of mateial used, this equies us to take the deivative and set it equal to zeo. Fist we will ewite S as: S Now we take the deivative using the powe ule: S This can be 4000 ewitten as: S 4. We now want to set it equal to zeo and solve fo : , 4, , 000 0, so ou answe fo is: 6. 8 cm. Then 000 we also want to find h: h, so 000 h. This simplifies to h cm. So this means 0 we have a special elationship. We see that the height needs to be twice the adius in ode to use the least mateial.
5 Section.6 Notes Page 5 EXAMPE: The height above gound of an object moving vetically is given by s 6t 96t with s in feet and t in seconds. Find a. the object s velocity when t = 0, b. its maimum height and when it occus, and c. its velocity when s = 0. a. The deivative of position will give velocity, so s v t 96. Now we put in a zeo fo t: v (0 96. So the velocity is 96 ft/sec. b. We will take ou velocity and set it equal to zeo. This will occu when the object is at its maimum height: 0 t 96. Solving fo t will give us t = seconds. Now we put this into the oiginal equation fo s: s 6( 96(. So s = 56 feet. This is the height. c. When s = 0 we have 0 6t 96t. Factoing this will give: 0 6( t 6t 7, o 0 6( t ( t 7. Solving fo t gives t = - and t = 7. We will ignoe the negative time, so t = 7. Now that we know the time, we put this into the velocity fomula: v (7 96 = -8 ft/sec, which means it is falling back down.
Review Exercise Set 16
Review Execise Set 16 Execise 1: A ectangula plot of famland will be bounded on one side by a ive and on the othe thee sides by a fence. If the fame only has 600 feet of fence, what is the lagest aea that
More informationCALCULUS FOR TECHNOLOGY (BETU 1023)
CALCULUS FOR TECHNOLOGY (BETU 103) WEEK 7 APPLICATIONS OF DIFFERENTIATION 1 KHAIRUM BIN HAMZAH, IRIANTO, 3 ABDUL LATIFF BIN MD AHOOD, 4 MOHD FARIDUDDIN BIN MUKHTAR 1 khaium@utem.edu.my, iianto@utem.edu.my,
More informationCalculus I Section 4.7. Optimization Equation. Math 151 November 29, 2008
Calculus I Section 4.7 Optimization Solutions Math 151 Novembe 9, 008 The following poblems ae maimum/minimum optimization poblems. They illustate one of the most impotant applications of the fist deivative.
More informationRelated Rates - the Basics
Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the
More information11.2. Area of a Circle. Lesson Objective. Derive the formula for the area of a circle.
11.2 Aea of a Cicle Lesson Objective Use fomulas to calculate the aeas of cicles, semicicles, and quadants. Lean Deive the fomula fo the aea of a cicle. A diamete divides a cicle of adius into 2 semicicles.
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More information1. Show that the volume of the solid shown can be represented by the polynomial 6x x.
7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank depends
More informationPractice Integration Math 120 Calculus I Fall 2015
Pactice Integation Math 0 Calculus I Fall 05 Hee s a list of pactice eecises. Thee s a hint fo each one as well as an answe with intemediate steps... ( + d. Hint. Answe. ( 8 t + t + This fist set of indefinite
More informationPractice Integration Math 120 Calculus I D Joyce, Fall 2013
Pactice Integation Math 0 Calculus I D Joyce, Fall 0 This fist set of indefinite integals, that is, antideivatives, only depends on a few pinciples of integation, the fist being that integation is invese
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More informationChapter 1: Introduction to Polar Coordinates
Habeman MTH Section III: ola Coodinates and Comple Numbes Chapte : Intoduction to ola Coodinates We ae all comfotable using ectangula (i.e., Catesian coodinates to descibe points on the plane. Fo eample,
More informationMCV4U Final Exam Review. 1. Consider the function f (x) Find: f) lim. a) lim. c) lim. d) lim. 3. Consider the function: 4. Evaluate. lim. 5. Evaluate.
MCVU Final Eam Review Answe (o Solution) Pactice Questions Conside the function f () defined b the following gaph Find a) f ( ) c) f ( ) f ( ) d) f ( ) Evaluate the following its a) ( ) c) sin d) π / π
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More information2 Cut the circle along the fold lines to divide the circle into 16 equal wedges. radius. Think About It
Activity 8.7 Finding Aea of Cicles Question How do you find the aea of a cicle using the adius? Mateials compass staightedge scissos Exploe 1 Use a compass to daw a cicle on a piece of pape. Cut the cicle
More informationArea of Circles. Fold a paper plate in half four times to. divide it into 16 equal-sized sections. Label the radius r as shown.
-4 Aea of Cicles MAIN IDEA Find the aeas of cicles. Fold a pape plate in half fou times to New Vocabulay Label the adius as shown. Let C secto Math Online glencoe.com Exta Examples Pesonal Tuto Self-Check
More information612 MHR Principles of Mathematics 9 Solutions. Optimizing Measurements. Chapter 9 Get Ready. Chapter 9 Get Ready Question 1 Page 476.
Chapte 9 Optimizing Measuements Chapte 9 Get Ready Chapte 9 Get Ready Question Page 476 a) P = w+ l = 0 + 0 = 0 + 40 = 60 A= lw = 0 0 = 00 The peimete is 60 cm, and the aea is 00 cm. b) P = w+ l = 5. 8
More informationMath Section 4.2 Radians, Arc Length, and Area of a Sector
Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic
More informationTutorial Exercises: Central Forces
Tutoial Execises: Cental Foces. Tuning Points fo the Keple potential (a) Wite down the two fist integals fo cental motion in the Keple potential V () = µm/ using J fo the angula momentum and E fo the total
More informationMath 1105: Calculus I (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 3
Math : alculus I Math/Sci majos MWF am / pm, ampion Witten homewok Review: p 94, p 977,8,9,6, 6: p 46, 6: p 4964b,c,69, 6: p 47,6 p 94, Evaluate the following it by identifying the integal that it epesents:
More informationEuclidean Figures and Solids without Incircles or Inspheres
Foum Geometicoum Volume 16 (2016) 291 298. FOUM GEOM ISSN 1534-1178 Euclidean Figues and Solids without Incicles o Insphees Dimitis M. Chistodoulou bstact. ll classical convex plana Euclidean figues that
More informationKEPLER S LAWS AND PLANETARY ORBITS
KEPE S AWS AND PANETAY OBITS 1. Selected popeties of pola coodinates and ellipses Pola coodinates: I take a some what extended view of pola coodinates in that I allow fo a z diection (cylindical coodinates
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationOnline Mathematics Competition Wednesday, November 30, 2016
Math@Mac Online Mathematics Competition Wednesday, Novembe 0, 206 SOLUTIONS. Suppose that a bag contains the nine lettes of the wod OXOMOXO. If you take one lette out of the bag at a time and line them
More informationPDF Created with deskpdf PDF Writer - Trial ::
A APPENDIX D TRIGONOMETRY Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: http://www.docudesk.com D T i g o n o m e t FIGURE a A n g l e s Angles can be measued in degees
More informationMATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates
MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v
More information4.3 Area of a Sector. Area of a Sector Section
ea of a Secto Section 4. 9 4. ea of a Secto In geomety you leaned that the aea of a cicle of adius is π 2. We will now lean how to find the aea of a secto of a cicle. secto is the egion bounded by a cental
More informationRECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA
ECTIFYING THE CICUMFEENCE WITH GEOGEBA A. Matín Dinnbie, G. Matín González and Anthony C.M. O 1 Intoducction The elation between the cicumfeence and the adius of a cicle is one of the most impotant concepts
More informationK.S.E.E.B., Malleshwaram, Bangalore SSLC Model Question Paper-1 (2015) Mathematics
K.S.E.E.B., Malleshwaam, Bangaloe SSLC Model Question Pape-1 (015) Mathematics Max Maks: 80 No. of Questions: 40 Time: Hous 45 minutes Code No. : 81E Fou altenatives ae given fo the each question. Choose
More information2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8
5 CHAPTER Fundamentals When solving equations that involve absolute values, we usually take cases. EXAMPLE An Absolute Value Equation Solve the equation 0 x 5 0 3. SOLUTION By the definition of absolute
More informationExam 3, vers Physics Spring, 2003
1 of 9 Exam 3, ves. 0001 - Physics 1120 - Sping, 2003 NAME Signatue Student ID # TA s Name(Cicle one): Michael Scheffestein, Chis Kelle, Paisa Seelungsawat Stating time of you Tues ecitation (wite time
More informationCartesian Coordinate System and Vectors
Catesian Coodinate System and Vectos Coodinate System Coodinate system: used to descibe the position of a point in space and consists of 1. An oigin as the efeence point 2. A set of coodinate axes with
More informationBetween any two masses, there exists a mutual attractive force.
YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationPage 1 of 6 Physics II Exam 1 155 points Name Discussion day/time Pat I. Questions 110. 8 points each. Multiple choice: Fo full cedit, cicle only the coect answe. Fo half cedit, cicle the coect answe and
More information7.2.1 Basic relations for Torsion of Circular Members
Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationHW Solutions # MIT - Prof. Please study example 12.5 "from the earth to the moon". 2GmA v esc
HW Solutions # 11-8.01 MIT - Pof. Kowalski Univesal Gavity. 1) 12.23 Escaping Fom Asteoid Please study example 12.5 "fom the eath to the moon". a) The escape velocity deived in the example (fom enegy consevation)
More informationNo. 48. R.E. Woodrow. Mathematics Contest of the British Columbia Colleges written March 8, Senior High School Mathematics Contest
341 THE SKOLIAD CORNER No. 48 R.E. Woodow This issue we give the peliminay ound of the Senio High School Mathematics Contest of the Bitish Columbia Colleges witten Mach 8, 2000. My thanks go to Jim Totten,
More informationChapter Eight Notes N P U1C8S4-6
Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that
More informationPhysics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!
Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationMotithang Higher Secondary School Thimphu Thromde Mid Term Examination 2016 Subject: Mathematics Full Marks: 100
Motithang Highe Seconday School Thimphu Thomde Mid Tem Examination 016 Subject: Mathematics Full Maks: 100 Class: IX Witing Time: 3 Hous Read the following instuctions caefully In this pape, thee ae thee
More informationQualifying Examination Electricity and Magnetism Solutions January 12, 2006
1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and
More information, the tangent line is an approximation of the curve (and easier to deal with than the curve).
114 Tangent Planes and Linea Appoimations Back in-dimensions, what was the equation of the tangent line of f ( ) at point (, ) f ( )? (, ) ( )( ) = f Linea Appoimation (Tangent Line Appoimation) of f at
More informationTHE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2
THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace
More informationMAP4C1 Exam Review. 4. Juno makes and sells CDs for her band. The cost, C dollars, to produce n CDs is given by. Determine the cost of making 150 CDs.
MAP4C1 Exam Review Exam Date: Time: Room: Mak Beakdown: Answe these questions on a sepaate page: 1. Which equations model quadatic elations? i) ii) iii) 2. Expess as a adical and then evaluate: a) b) 3.
More information10.2 Parametric Calculus
10. Paametic Calculus Let s now tun ou attention to figuing out how to do all that good calculus stuff with a paametically defined function. As a woking eample, let s conside the cuve taced out by a point
More informationMODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...
MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationradians). Figure 2.1 Figure 2.2 (a) quadrant I angle (b) quadrant II angle is in standard position Terminal side Terminal side Terminal side
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationGraphs of Sine and Cosine Functions
Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the
More informationChapter 2: Basic Physics and Math Supplements
Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate
More information8.7 Circumference and Area
Page 1 of 8 8.7 Cicumfeence and Aea of Cicles Goal Find the cicumfeence and aea of cicles. Key Wods cicle cente adius diamete cicumfeence cental angle secto A cicle is the set of all points in a plane
More informationSMT 2013 Team Test Solutions February 2, 2013
1 Let f 1 (n) be the numbe of divisos that n has, and define f k (n) = f 1 (f k 1 (n)) Compute the smallest intege k such that f k (013 013 ) = Answe: 4 Solution: We know that 013 013 = 3 013 11 013 61
More information18.06 Problem Set 4 Solution
8.6 Poblem Set 4 Solution Total: points Section 3.5. Poblem 2: (Recommended) Find the lagest possible numbe of independent vectos among ) ) ) v = v 4 = v 5 = v 6 = v 2 = v 3 =. Solution (4 points): Since
More informationTest 2, ECON , Summer 2013
Test, ECON 6090-9, Summe 0 Instuctions: Answe all questions as completely as possible. If you cannot solve the poblem, explaining how you would solve the poblem may ean you some points. Point totals ae
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More informationSolutions to Problems : Chapter 19 Problems appeared on the end of chapter 19 of the Textbook
Solutions to Poblems Chapte 9 Poblems appeae on the en of chapte 9 of the Textbook 8. Pictue the Poblem Two point chages exet an electostatic foce on each othe. Stategy Solve Coulomb s law (equation 9-5)
More informationChapter 2: Introduction to Implicit Equations
Habeman MTH 11 Section V: Paametic and Implicit Equations Chapte : Intoduction to Implicit Equations When we descibe cuves on the coodinate plane with algebaic equations, we can define the elationship
More informationMath 259 Winter Handout 6: In-class Review for the Cumulative Final Exam
Math 259 Winte 2009 Handout 6: In-class Review fo the Cumulative Final Exam The topics coveed by the cumulative final exam include the following: Paametic cuves. Finding fomulas fo paametic cuves. Dawing
More informationMath 2263 Solutions for Spring 2003 Final Exam
Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate
More informationMath Notes on Kepler s first law 1. r(t) kp(t)
Math 7 - Notes on Keple s fist law Planetay motion and Keple s Laws We conside the motion of a single planet about the sun; fo simplicity, we assign coodinates in R 3 so that the position of the sun is
More informationNo. 39. R.E. Woodrow. This issue we give another example of a team competition with the problems
282 THE SKOLIAD CORNER No. 39 R.E. Woodow This issue we give anothe example of a team competition with the poblems of the 998 Floida Mathematics Olympiad, witten May 4, 998. The contest was oganized by
More informationGCSE: Volumes and Surface Area
GCSE: Volumes and Suface Aea D J Fost (jfost@tiffin.kingston.sc.uk) www.dfostmats.com GCSE Revision Pack Refeence:, 1, 1, 1, 1i, 1ii, 18 Last modified: 1 st August 01 GCSE Specification. Know and use fomulae
More informationRadian Measure CHAPTER 5 MODELLING PERIODIC FUNCTIONS
5.4 Radian Measue So fa, ou hae measued angles in degees, with 60 being one eolution aound a cicle. Thee is anothe wa to measue angles called adian measue. With adian measue, the ac length of a cicle is
More informationPhysics 107 TUTORIAL ASSIGNMENT #8
Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type
More information2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0
Ch : 4, 9,, 9,,, 4, 9,, 4, 8 4 (a) Fom the diagam in the textbook, we see that the flux outwad though the hemispheical suface is the same as the flux inwad though the cicula suface base of the hemisphee
More information5.8 Trigonometric Equations
5.8 Tigonometic Equations To calculate the angle at which a cuved section of highwa should be banked, an enginee uses the equation tan =, whee is the angle of the 224 000 bank and v is the speed limit
More informationPolar Coordinates. a) (2; 30 ) b) (5; 120 ) c) (6; 270 ) d) (9; 330 ) e) (4; 45 )
Pola Coodinates We now intoduce anothe method of labelling oints in a lane. We stat by xing a oint in the lane. It is called the ole. A standad choice fo the ole is the oigin (0; 0) fo the Catezian coodinate
More informationMuch that has already been said about changes of variable relates to transformations between different coordinate systems.
MULTIPLE INTEGRLS I P Calculus Cooinate Sstems Much that has alea been sai about changes of vaiable elates to tansfomations between iffeent cooinate sstems. The main cooinate sstems use in the solution
More informationPhysics 11 Chapter 3: Vectors and Motion in Two Dimensions. Problem Solving
Physics 11 Chapte 3: Vectos and Motion in Two Dimensions The only thing in life that is achieved without effot is failue. Souce unknown "We ae what we epeatedly do. Excellence, theefoe, is not an act,
More informationME 210 Applied Mathematics for Mechanical Engineers
Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the
More informationIn many engineering and other applications, the. variable) will often depend on several other quantities (independent variables).
II PARTIAL DIFFERENTIATION FUNCTIONS OF SEVERAL VARIABLES In man engineeing and othe applications, the behaviou o a cetain quantit dependent vaiable will oten depend on seveal othe quantities independent
More informationPhys 201A. Homework 5 Solutions
Phys 201A Homewok 5 Solutions 3. In each of the thee cases, you can find the changes in the velocity vectos by adding the second vecto to the additive invese of the fist and dawing the esultant, and by
More informationPhysics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism
Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up
More informationIntroduction and Vectors
SOLUTIONS TO PROBLEMS Intoduction and Vectos Section 1.1 Standads of Length, Mass, and Time *P1.4 Fo eithe sphee the volume is V = 4! and the mass is m =!V =! 4. We divide this equation fo the lage sphee
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More informationChapter 3 Optical Systems with Annular Pupils
Chapte 3 Optical Systems with Annula Pupils 3 INTRODUCTION In this chapte, we discuss the imaging popeties of a system with an annula pupil in a manne simila to those fo a system with a cicula pupil The
More informationGauss s Law Simulation Activities
Gauss s Law Simulation Activities Name: Backgound: The electic field aound a point chage is found by: = kq/ 2 If thee ae multiple chages, the net field at any point is the vecto sum of the fields. Fo a
More informationObjectives: After finishing this unit you should be able to:
lectic Field 7 Objectives: Afte finishing this unit you should be able to: Define the electic field and explain what detemines its magnitude and diection. Wite and apply fomulas fo the electic field intensity
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationProblem 1: Multiple Choice Questions
Mathematics 102 Review Questions Poblem 1: Multiple Choice Questions 1: Conside the function y = f(x) = 3e 2x 5e 4x (a) The function has a local maximum at x = (1/2)ln(10/3) (b) The function has a local
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More information1D2G - Numerical solution of the neutron diffusion equation
DG - Numeical solution of the neuton diffusion equation Y. Danon Daft: /6/09 Oveview A simple numeical solution of the neuton diffusion equation in one dimension and two enegy goups was implemented. Both
More informationThe geometric construction of Ewald sphere and Bragg condition:
The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.07: Electomagnetism II Septembe 5, 202 Pof. Alan Guth PROBLEM SET 2 DUE DATE: Monday, Septembe 24, 202. Eithe hand it in at the lectue,
More informationOn the Sun s Electric-Field
On the Sun s Electic-Field D. E. Scott, Ph.D. (EE) Intoduction Most investigatos who ae sympathetic to the Electic Sun Model have come to agee that the Sun is a body that acts much like a esisto with a
More informationProblem Set 10 Solutions
Chemisty 6 D. Jean M. Standad Poblem Set 0 Solutions. Give the explicit fom of the Hamiltonian opeato (in atomic units) fo the lithium atom. You expession should not include any summations (expand them
More informationSection 26 The Laws of Rotational Motion
Physics 24A Class Notes Section 26 The Laws of otational Motion What do objects do and why do they do it? They otate and we have established the quantities needed to descibe this motion. We now need to
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationMax/Min Word Problems (Additional Review) Solutions. =, for 2 x 5 1 x 1 x ( 1) 1+ ( ) ( ) ( ) 2 ( ) x = 1 + (2) 3 1 (2) (5) (5) 4 2
. a) Given Ma/Min Wod Poblems (Additional Review) Solutions + f, fo 5 ( ) + f (i) f 0 no solution ( ) (ii) f is undefined when (not pat of domain) Check endpoints: + () f () () + (5) 6 f (5) (5) 4 (min.
More informationAustralian Intermediate Mathematics Olympiad 2017
Austalian Intemediate Mathematics Olympiad 207 Questions. The numbe x is when witten in base b, but it is 22 when witten in base b 2. What is x in base 0? [2 maks] 2. A tiangle ABC is divided into fou
More informationStatic equilibrium requires a balance of forces and a balance of moments.
Static Equilibium Static equilibium equies a balance of foces and a balance of moments. ΣF 0 ΣF 0 ΣF 0 ΣM 0 ΣM 0 ΣM 0 Eample 1: painte stands on a ladde that leans against the wall of a house at an angle
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationPES 3950/PHYS 6950: Homework Assignment 6
PES 3950/PHYS 6950: Homewok Assignment 6 Handed out: Monday Apil 7 Due in: Wednesday May 6, at the stat of class at 3:05 pm shap Show all woking and easoning to eceive full points. Question 1 [5 points]
More informationPhysics 521. Math Review SCIENTIFIC NOTATION SIGNIFICANT FIGURES. Rules for Significant Figures
Physics 51 Math Review SCIENIFIC NOAION Scientific Notation is based on exponential notation (whee decimal places ae expessed as a powe of 10). he numeical pat of the measuement is expessed as a numbe
More informationVariables and Formulas
64 Vaiales and Fomulas Vaiales and Fomulas DEFINITIONS & BASICS 1) Vaiales: These symols, eing lettes, actually epesent numes, ut the numes can change fom time to time, o vay. Thus they ae called vaiales.
More information21 MAGNETIC FORCES AND MAGNETIC FIELDS
CHAPTER 1 MAGNETIC ORCES AND MAGNETIC IELDS ANSWERS TO OCUS ON CONCEPTS QUESTIONS 1. (d) Right-Hand Rule No. 1 gives the diection of the magnetic foce as x fo both dawings A and. In dawing C, the velocity
More information