Free Essay

Calculus Quiz 1 Solved

In: Other Topics

Submitted By kkanniappank
Words 2358
Pages 10
Solutions to Chapter 1 Quiz: Functions

Problem 1 This problem tests your understanding of domains. Remember, the domain of a function is the largest portion of the real numbers where that function is actually defined. √ Question: What is the domain of the function f(x) = ln x? Answer: [1, +∞). We should recognize f as a composition: given an input number x, we first compute its logarithm, and then compute the square-root of that logarithm. The logarithm is only defined for positive x. So, we immediately see that we need x > 0 for ln(x) to make sense. But we are not done! The output of ln x for positive x can be any real number, and in order to make the output of ln x the input of the square root function, we must make sure that we restrict to only those values of x for which ln x 0. Since ln(1) = 0 and natural log is an increasing function, we see that the output of ln x is positive for any x 1. Thus, the domain of f is all x 1, or all x in [1, +∞). Problem 2 This problem tests your ability to compute Taylor series of a composition of functions if you know the Taylor series of each individual function. In short, you have to stuff Taylor series into Taylor series! Moreover, knowledge of big-O is extremely useful in this problem since it helps you figure out where to truncate the series.
1 Question: What is the Taylor series of ln 1−x about x = 0 up to and including terms of order 3? 1 Answer: ln 1−x = x + 1 x2 + 1 x3 + O(x4 ). 2 3 1 Let’s see: we should all at least know the Taylor series of 1−x about x = 0 since it is just the geometric series! That is,

1 = 1 + x + x2 + x3 + O(x4 ). 1−x Here we have stopped at O(x4 ) because we only need to compute up to order 3. So far so good, but this is not the answer we need. In fact we have to compute the natural log of this entire series. So, how do we approximate the expression ln(1 + x + x2 + x3 + O(x4 ))
1

2

VIDIT NANDA

up to terms of order 3? Since the expression in parentheses equals 1 for our chosen point x = 0, we are interested in the expansion of ln(1 + y) about y = 0, where y = x + x2 + x3 + O(x4 ). This is also a Taylor series that we have seen before, and so we know that: y2 y 3 + + O(y4 ). 2 3 Again, we stop at O(x4 ) because we only care about terms of order 3 and less! Now, let’s plug in our expression for y in terms of x into this series: ln(1 + y) = y − ln(1 + x + x2 + x3 + O(x4 )) = (x + x2 + x3 + O(x4 )) (x + x2 + x3 + O(x4 ))2 2 (x + x2 + x3 + O(x4 ))3 + + O(x4 ). 3 This looks horrible! But now we can use what we have learned about big-O to simplify things a lot. For instance, that last term on the right side contains a lot of junk: we only need to raise x to the third power, because everything else has a higher degree and is already in O(x4 ). Similarly, when we look at the second term, we only need to square up to the x2 term because again, all the other terms are already in O(x4 ). Let’s use these simplifications and see that (x + x2 )2 x3 ln(1 + x + x2 + x3 + O(x4 )) = (x + x2 + x3 ) − + + O(x4 ). 2 3 That’s much better! Expanding the middle term and collecting the powers of x (be careful about that middle term, it has a negative coefficient and I just got it wrong twice!), we get the final answer: 1 1 1 ln = x + x2 + x3 + O(x4 ), 1−x 2 3 − Problem 3 This problem tests the relation between derivatives and coefficients in Taylor series. Essentially, the only thing you need to know in order to solve the problem is this: the coefficient of the (x − a)n term in the Taylor series of f(x) about x = a is f(n) (a) n!

Question: Use your knowledge of Taylor series, find the sixth derivative f(6) (0) of f(x) = 2 e−x evaluated at x = 0. Answer: -120. Let’s start with the Taylor series of ey : ey = Or, if you are allergic to the yn . n! n=0


symbol we can just use ey = 1 + y + y2 y3 + + O(y4 ). 2! 3!

SOLUTIONS TO CHAPTER 1 QUIZ: FUNCTIONS

3

Plugging in y = −x2 , we get the alternating sum e−x = 1 − x2 +
2

x4 x6 − + O(x8 ). 2! 3!

Keeping the giant box above in mind, we examine the coefficient of the x6 term: it equals (6) 1 − 3! . We know that this coefficient equals f 6!(0) , so in fact f(6) (0) = − 6! = −(6 × 5 × 4) = −120. 3! Problem 4 This problem tests the application of Taylor series to compute complicated-looking limits. Again, understanding the big-O helps to make an otherwise nasty computation very easy. Question: Recall that the Taylor series for arctan is


arctan(x) = k=0 (−1)k

x2k+1 2k + 1

for |x| < 1. Using this, compute arctan x x→0 x3 + 7x lim
1 Answer: 7 . Let’s write out the first few terms of the arctan series:

arctan(x) = x − So, our answer is given by

x3 x5 x7 + − + O(x9 ). 3 5 7 x3 3
7 x5 −x 5 7 x3 + 7x

x− arctan x lim 3 = lim x→0 x + 7x x→0 Now let’s simplify using big-O:

+

+ O(x9 )

.

arctan x x + O(x3 ) lim = lim . x→0 x3 + 7x x→0 7x + O(x3 ) Note that there is an x that is a common factor of both numerator and denominator. Cancel it to get arctan x 1 + O(x2 ) 1 = lim = . lim 3 x→0 7 + O(x2 ) x→0 x + 7x 7 Another way of attacking this problem is by using l’Hopital’s rule (in Mathematics, there is usually more then one way of solving a problem!). Remember that the derivative of 1 arctan x is 1+x2 . Then, arctan x 1 1 2 lim = lim 1+x 2 = lim = x→0 x3 + 7x x→0 7 + 3x x→0 (1 + x2 )(7 + 3x2 ) 7
1

4

VIDIT NANDA

Problem 5 This problem tests the same concepts as the last one. You need to use the Taylor expansion for sines/cosines near 0. Question: Compute limx→0 cos(3x)−cos(5x) . x2 Answer: 8. The limit is taken as x → 0 so we can expand the cosines in the numerator at x = 0. 2 Remember cos(y) = 1 − x + O(x4 ) near y = 0. Plugging y = 3x gives 2! 9x2 cos(3x) = 1 − + O(x4 ), 2! and plugging y = 5x gives 25x2 + O(x4 ), cos(3x) = 1 − 2! So, the difference is 9 − 25 2 x + O(x4 ) = 8x2 + O(x4 ). 2 2 Now just divide by x and take the limit as x → 0 to get the answer! cos(3x) − cos(5x) = − Problem 6 This problem only tests your knowledge of standard Taylor series when the input variables have been changed slightly. Question: Determine which value is approximated by √ √ √ √ ( 2π)3 ( 2π)4 ( 2π)5 2 1 + 2π + π + + + + H.O.T. 3! 4! 5! √ Answer: e 2π . √ n Look at the terms of order 3 and above: they are all of the form y where y = 2π. n! Looking at the smaller terms, they also follow the same pattern: for example, consider the order 2 term: √ ( 2π)2 2π2 = = π2 . 2! 2 √ n Okay, so the series looks a lot like ∞ y = ey where y = 2π. Thus, this series is simply k=0 n! √ approximating e 2π . Problem 7 This problem only tests your understanding of the summation notation Question: Which of the following expressions describes the sum √ √ 2 2 3 3 2 −x + x − x + x4 + H.O.T. 4 9 16 Choose all that apply. Answer. The two correct answers were √ √ ∞ ∞ n n n + 1 n+1 (−1)n 2 x and (−1)n+1 x n (n + 1)2 n=1 n=0 .

SOLUTIONS TO CHAPTER 1 QUIZ: FUNCTIONS

5

These are equal: the index n of the first series corresponds to the index n − 1 of the second series. You can easily plug in low values of n to confirm that these give us the expression from the question. Any other series can be ruled out by plugging in values of n and comparing coefficients. √ 2n For example, when n = 2 the series ∞ (−1)n n2 xn gives us a coefficient of 1 for the x2 n=1 2 term, which is clearly wrong. Problem 8 This problem only tests your ability to recognize and manipulate the geometric series. All you had to know was ∞ 1 yk for |y| < 1. = 1 − y k=0 Question: Use the geometric series to evaluate the sum


3k+1 xk k=0 Don’t forget to indicate what restrictions there are on x. 3 Answer: 2−3x on |x| < 1 . 3 This is easy! The series looks like ∞ yk for y = 3x (but there is an extra factor of 3), k=0 3 so immediately we know that the answer equals 1−y , and plugging in y gives us 3 . 1 − 3x And what about the restriction on x? Well, the series only converges for |y| < 1, which for us means 3|x| < 1 or |x| < 1 . That’s all! 3 Problem 9 This problem tests your ability to compute the Taylor series of a polynomial. A polynomial written in the usual form a0 + a1 x + a2 x2 + . . . an xn is its own Taylor series about x = 0. But as we have seen in the lectures, things look very different when we expand about points that are not 0... Question: What is the Taylor series expansion about x = 2 of x3 − 2x2 + 3x − 4? Answer: 2 + 7(x − 2) + 4(x − 2)2 + (x − 2)3 + O((x − 2)4 ). Let’s write p(x) = x3 − 2x2 + 3x − 4. The formula for the first few terms of the Taylor series T (x) of p about the point x = a is given by p(2) (a) p(3) (a) (x − a)2 + (x − a)3 + O((x − a)4 ) 2! 3! The constant term is just p(2) = 23 − 23 + 6 − 4 = 2 , but now we should start computing the derivatives of p(x) at x = 2 to figure out the higher terms. Here they are: T (x) = p(a) + p (a) + p (x) = 3x2 − 4x + 3, so p (2) = 12 − 8 + 3 = 7 . Next, p (x) = 6x − 4, so p (2) = 12 − 4 = 8 .

6

VIDIT NANDA

Finally, p(3) (x) = 6 for all possible x, including x = 2 Of course, all higher derivatives are zero. So, plugging in the boxed numbers for the derivatives of p(x) at x = 2 into the Taylor series formula, we get T (x) = 2 + 7(x − 2) + 8 6 (x − 2)2 + (x − 2)3 . 2! 3!

Reduce the fractions to get the answer. Note: the Taylor series of any function about a point x = a must be expressed in terms of powers of (x − a). So, even if there were choices that were algebraically equivalent to this answer, you would not have received credit for selecting them. Problem 10 This problem tests your understanding of big-O. Remember, f(x) belongs to O(g(x)) as x → a if and only if we have some constant C so that |f(x)| C · |g(x)| when x is near a .

Question: Exactly two of the following statements are correct. Select the two correct statements. √ (1) 16x4 − 2 is in O(x2 ) as x → +∞. (2) ln(1 + x + x2 ) is in O(xn ) for all n 1 as x → +∞. 2 (3) ex is in O(x2 ) as x → +∞. (4) cosh 2x is in O(xn ) for all n 0 as x → +∞. √ (5) 7 x is in O(x4 ) as x → 0. (6) 3x4 − 14 is in O(x2 ) as x → +∞ (7) ex belongs to O(ln x) as x → +∞. (8) 7x3 is in O(x4 ) as x → 0. Answer: Only the first two statements are true. Note: the choices on Quizzes are randomized, so you may have seen these statements in some different order on your Quiz. I’ve shown why the first statement is true and why the fourth and fifth ones are false. The methods are very similar for the other statements, so try them out on your own. 1. To see why this statement is true, we will try to compute the limit √ 16x4 − 2 lim x→+∞ x2 Let’s slide the denominator into the square root of the numerator: the only difference this makes is that we have to square it, so we get: 16x4 − 2 2 = lim 16 − 4 4 x→+∞ x→+∞ x x √ and clearly, as x → +∞ this equals 16 = 4 since the limit exists and the statement is true. lim

2 x4

term vanishes. So, the

SOLUTIONS TO CHAPTER 1 QUIZ: FUNCTIONS

7

4. To see that this statement is false, remember the definition of cosh: e2x + e−2x cosh(2x) = 2 It is easy to see – by using the Taylor series of e2x and e−2x – that the Taylor series of cosh(2x) has strictly positive coefficients for all even powers of n, so in fact the series must grow faster than any xn . Thus, the statement is false. 5. To see why this statement is false, examine the limit √ 7 x lim , x→0 x4 √ 1 and note that since x = x 2 we may re-write this limit as 7 7 lim 4− 1 = lim 7 . x→0 x 2 x→0 x 2 For x positive but small, this goes to +∞. And for negative but small x it does not even exist!

Similar Documents

Premium Essay

Paper

...Post-Graduate Diploma in Management Managerial Economics Course Owner: Sadananda Prusty, Ph. D Name of Faculty Members to Teach this Course (To be mentioned after final course allocation) Institute of Management Technology Ghaziabad Course Background and Learning Objectives: “Economics is a study of mankind in the ordinary business of life.” So wrote Alfred Marshall, the great nineteenth-century economist, in his textbook, Principles of Economics. Although we have learned much about the economy since Marshall’s time, this definition of economics is as true today as it was in 1890, when the first edition of his text was published. In recent years, there are many questions about the economy that might spark any one’s curiosity. Why are apartments so hard to find in Mumbai City? Why do airlines charge less for a round-trip ticket? Why executive class airfare is costlier than economic class? Why are jobs easy to find in some years and hard to find in others? Why a package tour costs less as compared to individual booking of air ticket and hotel? Moreover, as one go about his/her life, he/she make many economic decisions. During student carrier one has to decide how many years to stay in school. After joining in a job, one has to decide how much of his/her income to spend, how much to save, and how to invest his/her savings. When running a small business or a large corporation, and one will......

Words: 2173 - Pages: 9

Free Essay

Nogotiation

...Artificial intelligence From Wikipedia, the free encyclopedia Jump to: navigation, search "AI" redirects here. For other uses, see Ai. For other uses, see Artificial intelligence (disambiguation). Artificial intelligence (AI) is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents"[1] where an intelligent agent is a system that perceives its environment and takes actions that maximize its chances of success.[2] John McCarthy, who coined the term in 1955,[3] defines it as "the science and engineering of making intelligent machines."[4] AI research is highly technical and specialized, deeply divided into subfields that often fail to communicate with each other.[5] Some of the division is due to social and cultural factors: subfields have grown up around particular institutions and the work of individual researchers. AI research is also divided by several technical issues. There are subfields which are focussed on the solution of specific problems, on one of several possible approaches, on the use of widely differing tools and towards the accomplishment of particular applications. The central problems of AI include such traits as reasoning, knowledge, planning, learning, communication, perception and the ability to move and manipulate objects.[6] General intelligence (or "strong AI") is still among the field's long term goals.[7] Currently popular approaches include......

Words: 7301 - Pages: 30

Premium Essay

Factors Associated to the Problem Solving Skills of Physical Science Majors Students in Physics

...Cooperative Problem Solving in Physics A User’s Manual Why? What? How? STEP 1 Recognize the Problem What's going on? STEP 2 Describe the problem in terms of the field What does this have to do with ...... ? STEP 3 Plan a solution How do I get out of this? STEP 4 Execute the plan Let's get an answer STEP 5 Evaluate the solution Can this be true? Kenneth Heller Patricia Heller University of Minnesota With support from the National Science Foundation, University of Minnesota, and U.S. Department of Education © Kenneth & Patricia Heller, 2010 Acknowledgments In reaching this stage in this work, we gratefully acknowledge the support of the University of Minnesota, the U.S. Department of Education FIPSE program, and the National Science Foundation. This work would not have existed without the close cooperation of the University of Minnesota School of Physics and Astronomy and Department of Curriculum and Instruction. We have incorporated the suggestions of many faculty members from both Physics and Education at the University of Minnesota and other institutions that have communicated with us at workshops, meetings, and by e-mail. This work has depended on the efforts and feedback of many graduate student teaching assistants in the School of Physics and Astronomy over the years. Much of this development is directly based on the research of the graduate students in the University of Minnesota Physics Education Program:......

Words: 40122 - Pages: 161

Free Essay

Accounting

...Click here to download the solutions manual / test bank INSTANTLY!! http://testbanksolutionsmanual.blogspot.com/2011/02/accounting-information-systems-romney.html ------------------------------------------------------------------------------------------------------------------------ Accounting Information Systems Romney 11th Edition Solutions Manual Accounting Information Systems Romney 11th Edition Solutions Manual Accounting Information Systems Romney 11th Edition Solutions Manual Accounting Information Systems Romney Steinbart 11th Edition Solutions Manual Accounting Information Systems Romney Steinbart 11th Edition Solutions Manual ------------------------------------------------------------------------------------------------------------------------ ***THIS IS NOT THE ACTUAL BOOK. YOU ARE BUYING the Solution Manual in e-version of the following book*** Name: Accounting Information Systems Author: Romney Steinbart Edition: 11th ISBN-10: 0136015182 Type: Solutions Manual - The file contains solutions and questions to all chapters and all questions. All the files are carefully checked and accuracy is ensured. - The file is either in .doc, .pdf, excel, or zipped in the package and can easily be read on PCs and Macs.  - Delivery is INSTANT. You can download the files IMMEDIATELY once payment is done. If you have any questions, please feel free to contact us. Our response is the fastest. All questions will always be answered in......

Words: 18533 - Pages: 75

Premium Essay

Organisatinal Behaviour

...BORICP07.doc - 1 Chapter 7 Motivation and Classroom Learning This chapter will help you answer the following questions about your learners: • How can I help my learners interpret their classroom setbacks in ways that elicit renewed effort? • What are some things teachers say to learners that can lower their motivation to succeed? • How can learning strategies improve my students’ motivation to learn? • How can I convey to my students the motivation to say “Yes, I can do what my teacher expects”? • How can I enhance my learners’ determination to learn what I teach? • How can I use project-based learning to motivate my learners? In this chapter you will also learn the meanings of these terms: antecedents attribution theory causal schemata deficiency/growth needs theory BORICP07.doc - 2 drive theory instinct theory intrinsic motivation locus of causality project-based learning self-determination theory self-efficacy theory situational cues Behavioral and cognitive psychologists agree that motivation is essential for learning. Yet how to motivate learners in the classroom continues to be one of the most puzzling problems confronting the teacher. Let’s look in on Professor Thomas’s learning seminar as his students discuss the topic of motivation. Betty: Well, it seems to me that motivation is becoming the scapegoat for all learning failures. That’s all I hear at school ... “These kids just aren’t motivated” or “This kid just hasn’t any motivation.” And all the......

Words: 13593 - Pages: 55

Premium Essay

Master in Business Management

...solution, Simplex ethod, artificial variable technique, Duality principle. Sensitivity analysis. Transportation Problem Formulation optimal solution. Unbalanced transportation problems, Degeneracy. Assignment problem, Formulation optimal solution, Variation i.e., Non-square (m x n) matrix restrictions. Sequencing Introduction, Terminology, notations and assumptions, problems with n-jobs and two machines, optimal sequence algorithm, problems with n-jobs and three machines, problems with n-jobs and m-machines, graphic solutions. Travelling salesman problem. Replacement Introduction, Replacement of items that deteriorate with time – value of money unchanging and changing, Replacement of items that fail completely. Queuing Models M.M.1 & M.M.S. system cost considerations. Theory of games introduction, Two-person zero-sum games, The Maximum –Minimax principle, Games without saddle points – Mixed Strategies, 2 x n and m x 2 Games – Graphical solutions, Dominance property, Use of L.P. to games, Algebraic solutions to rectangular games. Inventory Introduction, inventory costs, Independent demand systems: Deterministic models – Fixed order size systems – Economic order quantity (EOQ) – Single items, back ordering, Quantity discounts (all units quantity discounts), Batch – type production systems: Economic production quantity – Single items, Economic production quantity multiple items. Fixed order interval systems: Economic order interval (EOI) –Single items,......

Words: 30976 - Pages: 124

Free Essay

Physics

...Introductory Physics I Elementary Mechanics by Robert G. Brown Duke University Physics Department Durham, NC 27708-0305 rgb@phy.duke.edu Copyright Notice Copyright Robert G. Brown 1993, 2007, 2013 Notice This physics textbook is designed to support my personal teaching activities at Duke University, in particular teaching its Physics 141/142, 151/152, or 161/162 series (Introductory Physics for life science majors, engineers, or potential physics majors, respectively). It is freely available in its entirety in a downloadable PDF form or to be read online at: http://www.phy.duke.edu/∼rgb/Class/intro physics 1.php It is also available in an inexpensive (really!) print version via Lulu press here: http://www.lulu.com/shop/product-21186588.html where readers/users can voluntarily help support or reward the author by purchasing either this paper copy or one of the even more inexpensive electronic copies. By making the book available in these various media at a cost ranging from free to cheap, I enable the text can be used by students all over the world where each student can pay (or not) according to their means. Nevertheless, I am hoping that students who truly find this work useful will purchase a copy through Lulu or a bookseller (when the latter option becomes available), if only to help subsidize me while I continue to write inexpensive textbooks in physics or other subjects. This textbook is organized for ease of presentation and ease of learning. In particular, they......

Words: 224073 - Pages: 897

Premium Essay

Economy Help Exam

...Multiple Choice Quiz - Results You answered 0 out of 30 questions correctly, for a score of 0%. 1. Incorrect. You did not provide an answer. Which of the following is the best definition of managerial economics? Managerial economics is The correct answer was: b. a field that applies economic theory and the tools of decision science.. 2. Incorrect. You did not provide an answer. The value of an economic theory in practice is determined by The correct answer was: c. how well the theory can predict or explain.. 3. Incorrect. You did not provide an answer. Management decision problems are comprised of three elements. Which of the following is not one of them? The correct answer was: a. Profitability. 4. Incorrect. You did not provide an answer. Which of the following areas of economic theory is the single most important element of managerial economics? The correct answer was: d. Microeconomics. 5. Incorrect. You did not provide an answer. Which of the following is defined as the study of the aggregate economy studied as a whole? The correct answer was: c. Macroeconomics. 6. Incorrect. You did not provide an answer. Which of the following is the discipline that studies the use of statistical tools to estimate economic models? The correct answer was: b. Econometrics. 7. Incorrect. You did not provide an answer. Firms do not continue to grow without limit because of The correct answer was: a. managerial limitations.. 8. Incorrect. You did not provide an answer...

Words: 35982 - Pages: 144

Premium Essay

Mcgraw-Hill - Quality Management Demystified

...QUALITY MANAGEMENT DEMYSTIFIED Demystified Series Advanced Statistics Demystified Algebra Demystified Anatomy Demystified asp.net Demystified Astronomy Demystified Biology Demystified Business Calculus Demystified Business Statistics Demystified C++ Demystified Calculus Demystified Chemistry Demystified College Algebra Demystified Data Structures Demystified Databases Demystified Differential Equations Demystified Digital Electronics Demystified Earth Science Demystified Electricity Demystified Electronics Demystified Environmental Science Demystified Everyday Math Demystified Genetics Demystified Geometry Demystified Home Networking Demystified Investing Demystified Java Demystified JavaScript Demystified Linear Algebra Demystified Macroeconomics Demystified Math Proofs Demystified Math Word Problems Demystified Medical Terminology Demystified Meteorology Demystified Microbiology Demystified OOP Demystified Options Demystified Organic Chemistry Demystified Personal Computing Demystified Pharmacology Demystified Physics Demystified Physiology Demystified Pre-Algebra Demystified Precalculus Demystified Probability Demystified Project Management Demystified Quality Management Demystified Quantum Mechanics Demystified Relativity Demystified Robotics Demystified Six Sigma Demystified sql Demystified Statistics Demystified Trigonometry Demystified uml Demystified Visual Basic 2005 Demystified Visual C# 2005 Demystified xml Demystified QUALITY MANAGEMENT DEMYSTIFIED SID...

Words: 122414 - Pages: 490

Premium Essay

Ethics

...contents critical thinking thinking critically about ethical issues 1 ETHICAL REASONING 3 2 VALUES-BASED ETHICAL REASONING 9 3 RIGHTS-BASED ETHICAL REASONING 15 4 CONSEQUENCE-BASED ETHICAL REASONING 20 5 ERRORS IN ETHICAL REASONING 25 5.1 THE IS/OUGHT FALLACY 25 5.2 THE ARBITRARY LINE FALLACY 27 REVIEW OF TERMS 29 THINKING CRITICALLY ABOUT WHAT YOU SEE 29 THINKING CRITICALLY ABOUT WHAT YOU HEAR 30 THINKING CRITICALLY ABOUT WHAT YOU READ 30 THINKING CRITICALLY ABOUT WHAT YOU WRITE 35 THINKING CRITICALLY WHEN YOU DISCUSS 36 REASONING TEST QUESTIONS 36 T H I N K I N G C R I T I C A L L Y A B O U T E T H I C A L I S S U E S We don’t cover any particular step or steps in this section—you can, and should, apply all of your critical thinking skills when you think about ethical issues (hence, the entire template is bolded)! Template for critical analysis of arguments 1. What’s the point (claim/opinion/conclusion)? s Look for subconclusions as well. 2. What are the reasons/what is the evidence? s s Articulate all unstated premises. Articulate connections. 3. What exactly is meant by . . .? s s s Define terms. Clarify all imprecise language. Eliminate or replace “loaded” language and other manipulations. 4. Assess the reasoning/evidence: s s If deductive, check for truth/acceptability and validity. If inductive, check for truth/acceptability...

Words: 14346 - Pages: 58

Premium Essay

Hello

...70+ DVD’s FOR SALE & EXCHANGE www.traders-software.com www.forex-warez.com www.trading-software-collection.com www.tradestation-download-free.com Contacts andreybbrv@gmail.com andreybbrv@yandex.ru Skype: andreybbrv SCHAUM’S Easy OUTLINES PRINCIPLES OF ECONOMICS Other Books in Schaum’s Easy Outlines Series Include: Schaum’s Easy Outline: Calculus Schaum’s Easy Outline: College Algebra Schaum’s Easy Outline: College Mathematics Schaum’s Easy Outline: Discrete Mathematics Schaum’s Easy Outline: Differential Equations Schaum’s Easy Outline: Elementary Algebra Schaum’s Easy Outline: Geometry Schaum’s Easy Outline: Linear Algebra Schaum’s Easy Outline: Mathematical Handbook of Formulas and Tables Schaum’s Easy Outline: Precalculus Schaum’s Easy Outline: Probability and Statistics Schaum’s Easy Outline: Statistics Schaum’s Easy Outline: Trigonometry Schaum’s Easy Outline: Business Statistics Schaum’s Easy Outline: Principles of Accounting Schaum’s Easy Outline: Applied Physics Schaum’s Easy Outline: Biology Schaum’s Easy Outline: Biochemistry Schaum’s Easy Outline: Molecular and Cell Biology Schaum’s Easy Outline: College Chemistry Schaum’s Easy Outline: Genetics Schaum’s Easy Outline: Human Anatomy and Physiology Schaum’s Easy Outline: Organic Chemistry Schaum’s Easy Outline: Physics Schaum’s Easy Outline: Programming with C++ Schaum’s Easy Outline: Programming with Java Schaum’s Easy Outline: Basic Electricity Schaum’s Easy Outline: Electromagnetics Schaum’s Easy......

Words: 38596 - Pages: 155

Free Essay

Child Labour

...10000 quiz questions and answers www.cartiaz.ro 10000 general knowledge questions and answers 10000 general knowledge questions and answers www.cartiaz.ro No Questions Quiz 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Carl and the Passions changed band name to what How many rings on the Olympic flag What colour is vermilion a shade of King Zog ruled which country What colour is Spock's blood Where in your body is your patella Where can you find London bridge today What spirit is mixed with ginger beer in a Moscow mule Who was the first man in space What would you do with a Yashmak Who betrayed Jesus to the Romans Which animal lays eggs On television what was Flipper Who's band was The Quarrymen Which was the most successful Grand National horse Who starred as the Six Million Dollar Man In the song Waltzing Matilda - What is a Jumbuck Who was Dan Dare's greatest enemy in the Eagle What is Dick Grayson better known as What was given on the fourth day of Christmas What was Skippy ( on TV ) What does a funambulist do What is the name of Dennis the Menace's dog What are bactrians and dromedaries Who played The Fugitive Who was the King of Swing Who was the first man to fly across the channel Who starred as Rocky Balboa In which war was the charge of the Light Brigade Who invented the television Who would use a mashie niblick In the song who killed Cock Robin What do......

Words: 123102 - Pages: 493

Premium Essay

Normal Distribution

...C H A P T E R 6 The Normal Distribution Objectives After completing this chapter, you should be able to Outline Introduction 6–1 Normal Distributions 1 2 3 4 5 6 7 Identify distributions as symmetric or skewed. Identify the properties of a normal distribution. Find the area under the standard normal distribution, given various z values. Find probabilities for a normally distributed variable by transforming it into a standard normal variable. Find specific data values for given percentages, using the standard normal distribution. Use the central limit theorem to solve problems involving sample means for large samples. Use the normal approximation to compute probabilities for a binomial variable. 6–2 Applications of the Normal Distribution 6–3 The Central Limit Theorem 6–4 The Normal Approximation to the Binomial Distribution Summary 6–1 300 Chapter 6 The Normal Distribution Statistics Today What Is Normal? Medical researchers have determined so-called normal intervals for a person’s blood pressure, cholesterol, triglycerides, and the like. For example, the normal range of systolic blood pressure is 110 to 140. The normal interval for a person’s triglycerides is from 30 to 200 milligrams per deciliter (mg/dl). By measuring these variables, a physician can determine if a patient’s vital statistics are within the normal interval or if some type of treatment is needed to correct a condition and avoid future illnesses. The question then is,...

Words: 18901 - Pages: 76

Premium Essay

Mount Fuji

...Employers, job seekers, and puzzle lovers everywhere delight in William Poundstone's HOW WOULD YOU MOVE MOUNT FUJI? "Combines how-to with be-smart for an audience of job seekers, interviewers, Wired-style cognitive science hobbyists, and the onlooking curious. . . . How Would You Move Mount Fuji? gallops down entertaining sidepaths about the history of intelligence testing, the origins of Silicon Valley, and the brain-jockey heroics of Microsoft culture." — Michael Erard, Austin Chronicle "A charming Trojan Horse of a book While this slim book is ostensibly a guide to cracking the cult of the puzzle in Microsoft's hiring practices, Poundstone manages to sneak in a wealth of material on the crucial issue of how to hire in today's knowledge-based economy. How Would You Move Mount Fuji? delivers on the promise of revealing the tricks to Microsoft's notorious hiring challenges. But, more important, Poundstone, an accomplished science journalist, shows how puzzles can — and cannot — identify the potential stars of a competitive company.... Poundstone gives smart advice to candidates on how to 'pass' the puzzle game.... Of course, let's not forget the real fun of the book: the puzzles themselves." — Tom Ehrenfeld, Boston Globe "A dead-serious book about recruiting practices and abstract reasoning — presented as a puzzle game.... Very, very valuable to some job applicants — the concepts being more important than the answers. It would have usefulness as well to interviewers......

Words: 78201 - Pages: 313

Free Essay

Management

...Op"erations Research This page intentionally left blank Copyright © 2007, 2005 New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers All rights reserved. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to rights@newagepublishers.com ISBN (13) : 978-81-224-2944-2 PUBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com PREFACE I started my teaching career in the year 1964. I was teaching Production Engineering subjects till 1972. In the year 1972 I have registered my name for the Industrial Engineering examination at National Institution of Industrial Engineering, Bombay. Since then, I have shifted my field for interest to Industrial Engineering subjects and started teaching related subjects. One such subject is OPERATIONS RESEARCH. After teaching these subjects till my retirement in the year 2002, it is my responsibility to help the students with a book on Operations research. The first volume of the book is LINEAR PORGRAMMING MODELS. This was published in the year 2003. Now I am giving this book OPERATIONS RESEARCH, with other chapters to students, with a hope that it will help them to......

Words: 242596 - Pages: 971