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### M207-OptimizationGuidelines

Course: MATH 207, Spring 2012
School: City Colleges of Chicago
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Word Count: 248

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Ougouag Handout: GUIDELINES M207 Prof. FOR SOLVING OPTIMIZATION PROBLEMS 1. Read each problem slowly and carefully. Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly. 2. If appropriate, draw a sketch or diagram of...

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Ougouag Handout: GUIDELINES M207 Prof. FOR SOLVING OPTIMIZATION PROBLEMS 1. Read each problem slowly and carefully. Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly. 2. If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts. 3. Define variables to be used and carefully label your picture or diagram with these variables. This step is very important because it leads directly or indirectly to the creation of mathematical equations. 4. Write down all equations which are related to your problem or diagram. Clearly denote that you equation which are asked to maximize or minimize. Experience will show you that MOST optimization problems will begin with two equations. One equation is a "constraint" equation and the other is the "optimization" equation. The "constraint" equation is used to solve for one of the variables. This is then substituted into the "optimization" equation before differentiation occurs. Some problems may have NO constraint equation. Some problems may have two or more constraint equations. 5. Before differentiating, make sure that the optimization equation is a function of only one variable. Then differentiate using the well-known rules of differentiation. 6. Verify that your result is a maximum or minimum value using the first or second derivative test for extrema.
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City Colleges of Chicago - MATH - 207
MATH 207Graphs and Functions(Review)Distance FormulaExample: Find distance between (-1,4) and (-4,-2).Answer:6.71Midpoint FormulaExample: Find the midpoint from P1(-5,5) to P2(-3,1).Answer: (-4,3)Equations in two variables Example: Circle Equat
City Colleges of Chicago - MATH - 207
Math 141, Math 207Handout TrigDr. OugouagThe Unit circleAnglesinradiansanddegrees.Betweenparenthesesarethevalues(cos(),sin()ofeachangle.
City Colleges of Chicago - MATH - 207
Formulas and IdentitiesTrig Cheat SheetDefinition of the Trig FunctionsRight triangle definitionFor this definition we assume thatp0 &lt; q &lt; or 0 &lt; q &lt; 90 .2Unit circle definitionFor this definition q is any angle.y( x, y )hypotenuseyopposite
City Colleges of Chicago - MATH - 141
Conic Section FormulasParabola:y = a(x-h)2 + kIf a &gt; 0, opens upIf a &lt; 0, opens downVertex: (h, k)Focus: (h, k+p)Directrix: y = k-pAxis of Symmetry: x = hx = a(y-k)2 + hIf a &gt; 0, opens rightIf a &lt; 0, opens leftVertex: (h, k)Focus: (h+p, k)Di
City Colleges of Chicago - MATH - 141
MATH 141Chapter 1:Graphs and Functions(Review)1.1Distance FormulaExample: Find distance between (-1,4) and (-4,-2).Answer:6.71Midpoint FormulaExample: Find the midpoint from P1(-5,5) to P2(-3,1).Answer: (-4,3)1.2Equations in two variables Ex
City Colleges of Chicago - MATH - 141
Section 2.1Angles and Their Measure(a) Convert 4510'15&quot; to decimal in degrees.Round the answer to four decimal places.(b) Convert 21.256 to the DM'S&quot; form.Round the answer to the nearest second.Radians(a) 80(b) 140(c) -30(d) 1002(a)radians3
City Colleges of Chicago - MATH - 141
Section 2.2Trigonometric Functions(Unit Circle Approach)Recall:Arc length t = r. t (angle t in radians)Here r = 1 so t = ty1P(x,y)ytt-11xxBy definition, the coordinates of pointP are:X = cos tY-1= sin t 1 2 6Let t be a real number a
City Colleges of Chicago - MATH - 141
Section 2.3Properties of theTrigonometric FunctionsThe Range of the Six Trigonometric Functions(a) sin 39019(b) tan4(c) cos ( 7 )If sin &gt; 0 and cos &lt; 0, name the quadrant in which theangle lies.103 10Given sin =and cos =,1010find the val
City Colleges of Chicago - MATH - 141
Section 2.4 &amp; 2.5Graphs of Trig FunctionsAmplitude and PeriodDetermine the amplitude and period of y = 4 cos (3x)Graphy = 4 sin (2x)using key points.You practice!Graph y = 5cos 2x using key points.y54321x65432111234562345
City Colleges of Chicago - MATH - 141
Section 2.6Phase ShiftFind the amplitude, period and phase shift ofy = 5 sin ( 2 x + 5 ) and graph the function.Find the amplitude, period and phase shift ofy = 3 cos ( 4 x + ) and graph the function.
City Colleges of Chicago - MATH - 141
Section 3.1The Inverse Sine, Cosine,and Tangent FunctionsReview of Properties of Functions and Their InversesFind the exact value of: sin1322Find the exact value of: sin 21Find an approximate value of:3(a) sin51 2(b) sin 31(a) sin s
City Colleges of Chicago - MATH - 141
Section 3.2The Inverse TrigonometricFunctions (Continued)7 Find the exact value of: cos cos6-1 -1 3 Find the exact value of: cos tan 4 -1 2 Find the exact value of: tan sin 5 2 3csc 211(a) sec 5 4(b) csc 3-1 3(c) cot 3 2(d) c
City Colleges of Chicago - MATH - 141
Section 3.3Trigonometric Identitiestan by rewriting each trigonometric function in terms ofsec sine and cosine functions.(a) Simplifysin1 cos =by multiplying the numerator and1+cossin denominator by 1 cos (b) Show that11+by rewriting the
City Colleges of Chicago - MATH - 141
Section 3.4Sum and DifferenceFormulasFind the exact value of cos15.7Find the exact value of cos.1219Find the exact value of sin.12Find the exact value of cos 40 cos80 sin 40 sin 80.3If it is known that sin = , &lt; &lt; , and that5215 3sin =
City Colleges of Chicago - MATH - 141
Section 3.5Double-angle and Half-angleFormulas23If cos = , &lt; &lt;, find the exact value of:52(a) sin ( 2 )(b) cos ( 2 )Use a Half-angle Formula to find the exact value of:(a) sin 22.5(b) cos 5121If tan = , &lt; &lt; , find the exact value of:52(a
City Colleges of Chicago - MATH - 141
Section 3.6Product-to-Sum andSum-to-ProductFormulas(a) sin ( 3 ) sin ( 7 )(b) cos cos ( 5 )(c) sin ( 2 ) cos ( 7 )(a) sin ( 4 ) sin ( 6 )(b) cos ( 2 ) + cos ( 8 )
City Colleges of Chicago - MATH - 141
Sections 3.7 and 3.8TrigonometricEquationsDetermine whether = is a solution of the45equation 2sin + 2 = 0. Is =a solution?42 cos 3 = 0Solve the equation: 2 cos ( 2 ) 1 = 0, 0 &lt; 2Solve the equation:3 tan ( 3 ) + 1 = 0, 0 &lt; 2Solve the equation:
City Colleges of Chicago - MATH - 141
Section 4.1Right TriangleTrigonometry;ApplicationsFind the value of each of the six trigonometric functions ofthe angle .(a) sin 35 = cos ()?(b) tan = cot (3?(c) sec = csc (12?))(d) cos 2 25 + cos 2 65 = ?35 847
City Colleges of Chicago - MATH - 141
Section 4.2The Law of SinesSolve the triangle: A = 40, B = 60, a = 4Solve the triangle: B = 25, C= 85, a = 6SSA - The Ambiguous CaseSolve the triangle: a = 3, b = 5, B = 35Solve the triangle: a = 10, c = 6, C = 15Solve the triangle: a = 5, b = 8, A
City Colleges of Chicago - MATH - 141
Section 4.3The Law of CosinesSolve the triangle: b = 5, c = 8, A = 35Solve the triangle: a = 6, b = 8, c = 5
City Colleges of Chicago - MATH - 141
Section 5.1Polar Coordinates 3 (a) 2, 42 (b) 4, 3(c)( 5, 0 )(d) 3, 3 2 Consider the point with polar coordinates 4,. 3Find three other polar coordinates that represent the same point. 2, 4 (a) 5, 45 (b) 2, 6Find polar coordin
City Colleges of Chicago - MATH - 141
TRIGONOMETRYDEFINITIONRIGHT TRIANGLE DEFINITIONTRIG FUNCTIONS RANGEUNIT CIRCLE DEFINITIONTRIG FUNCTIONS DOMAINTRIG FUNCTIONS PERIODINVERSE TRIG FUNCTION NOTATIONINVERSE TRIG DOMAINeCalc.comINVERSE TRIG FUNCTION RANGEThe Best Online CalculatorU
City Colleges of Chicago - MATH - 141
Trigonometry on the TI-83/84Degrees and RadiansSetting the calculator's angle mode:1.Press MODE.2.The third option sets whether commands that returns angles return them in degreesor radians.3.Set this to Radians.The ANGLE menu:First, enter this
City Colleges of Chicago - MATH - 208
Chapter 6Applications of Integration6.1Velocity and Net ChangeSlide 6 - 3Slide 6 - 4Slide 6 - 5Slide 6 - 6Slide 6 - 7Slide 6 - 8Slide 6 - 9Slide 6 - 10Slide 6 - 116.2Areas Between CurvesSlide 6 - 13Slide 6 - 14Slide 6 - 15Slide 6 - 16Sl
City Colleges of Chicago - MATH - 208
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City Colleges of Chicago - MATH - 208
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Intel Architectural Support for Memory ManagementThe memory management facilities of the Intel Architecture are divided into two parts: segmentation and paging. Segmentation provides a mechanism of isolating individual code, data, and runtime stack memor
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Kernel Module Program ExamplesGet the file ModulePrograms.zip from the OS web sitehello_module.c#include&lt;linux/kernel.h&gt;#include&lt;linux/module.h&gt;inthello_module_init(void)cfw_printk(KERN_EMERG&quot;HelloModule~!I'minKernel\n&quot;);return0;voidhello_module
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1. Linux 1. Linux source tree structure Source tree root: /usr/src/kernels/ (from v2.6) kernel : Source code for task manager, scheduling, signal handling, arch/\$(ARCH)/kernel - hardware dependent task manager, context switching, thread management (e.g.
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Chapter 12. The Virtual FilesystemOne of Linux's keys to success is its ability to coexist comfortably with other systems.You can transparently mount disks or partitions that host file formats used byWindows , other Unix systems, or even systems with t
CUNY Lehman - CMP - 426
General overview of the Linux file systemFilesGeneralA simple description of the UNIX system, also applicable to Linux, is this: &quot;On a UNIX system, everything is a file; if something is not a file, it is a process.&quot; This statement is true because there
CUNY Lehman - CMP - 426
Chapter 2 Memory Addressing Programmers casually refer to a memory address as the way to access the contents of a memory cell. But when dealing with 80 x 86 microprocessors, we have to distinguish three kinds of addresses: Logical address Included in the
CUNY Lehman - CMP - 426
Chapter 10. System CallsOperating systems offer processes running in User Mode a set of interfaces to interact with hardware devices such as the CPU, disks, and printers. Putting an extra layer between the application and the hardware has several advanta
CUNY Lehman - CMP - 426
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CUNY Lehman - CMP - 426
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NPTL TGI concept POSIX thread specification specifies that threads created from a process should share same process id. Native LinuxThread does not comply with this specification. NPTL (Native POSIX Thread Library) does not comply with this specification
CUNY Lehman - CMP - 426
Chapter 1 IntroductionIntroduction to OS1What is an Operating System?!A program that acts as an intermediary between a user of a computer and the computer hardware. Various operating system goals:&quot; &quot;!&quot;Mainframe operating systems: to optimize util
CUNY Lehman - CMP - 426
Chapter 2: OS StructuresOS Structures1Objectives!To describe the services an operating system provides to users, processes of the system. To discuss the various ways of structuring an operating system. To explain how operating systems are installed a
CUNY Lehman - CMP - 426
Chapter 3: ProcessesProcesses1Outline! ! ! ! ! !Overview. Process Scheduling. Operations on Processes. Interprocess Communication. Examples of IPC Systems. Communication in Client-Server Systems.2Processes1Process Concept!The name of CPU activi
CUNY Lehman - CMP - 426
CUNY Lehman - CMP - 426
Chapter 5 CPU SchedulingCPU Scheduling1Outline! ! ! ! ! ! !Basic Concepts. Scheduling Criteria. Scheduling Algorithms. Multiple-Processor Scheduling. Thread Scheduling. Operating Systems Examples. Algorithm Evaluation.2CPU Scheduling1Basic Concep
CUNY Lehman - CMP - 426
Chapter 6 SynchronizationSynchronization1Outline! ! ! ! ! ! !Background The Critical-Section Problem Peterson's Solution Synchronization Hardware Semaphores Classic Problems of Synchronization Monitors2Synchronization1Background!A cooperating p
CUNY Lehman - CMP - 426
Virtual Doppelg nger: On the Performance, Isolation, and Scalability of a Para- and Paene- Virtualized SystemsStephen Soltesz*, Marc E. Fiuczynski*, Larry Peterson*, Michael McCabe+, Jeanna Matthews+ *Department of Computer Science, Princeton University
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Windows Anonymous Pipes Program Name: PipeParent.c #include &lt;stdio.h&gt; #include &lt;stdlib.h&gt; #include &lt;windows.h&gt; #define BUFFER_SIZE 25 int main(VOID) cfw_ HANDLE ReadHandle, WriteHandle; STARTUPINFO si; PROCESS_INFORMATION pi; char message[BUFFER_SIZE] = &quot;
CUNY Lehman - CMP - 426
CHAPTERProcess Synchronization6Practice Exercises6.1 In Section 6.4 we mentioned that disabling interrupts frequently could affect the systems clock. Explain why it could and how such effects could be minimized. Answer: The system clock is updated at
CUNY Lehman - CMP - 426
CHAPTER7DeadlocksPractice Exercises7.1 List three examples of deadlocks that are not related to a computersystem environment. Answer: Two cars crossing a single-lane bridge from opposite directions. A person going down a ladder while another person is
CUNY Lehman - CMP - 426
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fileio1.c program #include &lt;fcntl.h&gt; int fdrd, fdwt; char c; main(argc, argv) int argc; char *argv[]; cfw_ int PID; if (argc !=3) cfw_ printf(&quot;Usage: inputfile outfile\n&quot;, argv[0]); exit(1); PID = fork(); / child is created before opening/creating files
CUNY Lehman - CMP - 426
CMP 426 &amp; 697 (Operating Systems)Additional Reference onMulti-threading models and implementations in various Operating SystemsReferencesW. Stalling, Operating Systems: Internals and Design Principles, 7th edition, Prentice-Hall, 2012.F. Zabatta and
CUNY Lehman - CMP - 426
posix_thread.c /* * A pthread program illustrating how to * create a simple thread and some of the pthread API * This program implements the summation function where * the summation operation is run as a separate thread. * * Most Unix / L i nux / OS X use
CUNY Lehman - CMP - 426
Dealing With TLB Tags or I Want to Build a System, What Can L4 Do for Me?Gernot Heiser School of Computer Science and Engineering University of NSW, Sydney 2052, Australia gernot@unsw.edu.au October 11, 2001AbstractThis paper discusses TLB tags found o
CUNY Lehman - CMP - 426
CUNY Lehman - CMP - 426
Hardware based VirtualizationFull Virtualization:No modification to Guest OSVMM (VM monitor) or Hypervisor provides H/W resources transparently(e.g., VMWare ESX Server, HP Integrity ServerPara Virtualization:Guest OS needs to be modified since VMM p
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The Write Anywhere File Layout System (WAFL)Written by Sebastian Scholz, based on &quot;File System Design for an NFS File Server Appliance&quot; by Dave Hitz, James Lau &amp; Michael Malcolm, Network Appliance, Inc.1. AbstractThe patented Write Anywhere File Layout
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Answers to additional paper and pencil QuestionsPlease bring any errors to my attentionFall 2004Chapter 4Question 1bg bg bg b g.6 =.5 + Pb B g 0.1 = Pb B gP b A B g = P b Ag + P b B g P b A B g.6 =.5 + Pb B g.5 Pb B g.2 = Pb BgP b A B g = P b
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Screening of asymptomatic women using Mammograms Bayes Law and a cost/benefit analysis There are many misconceptions about the usefulness of mammograms, as well as a considerable amount of misleading information that has been propagated by advocates of ma