PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY


 Brent McCormick
 1 years ago
 Views:
Transcription
1 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive Properties 8 4. Properties of Frctions 9 5. Properties of Exponents Properties of Rdicls The Slope nd Eqution of Line ACT Compss Prctice Tests If you hve Internet ccess, you cn ccess the online resources below through the pdf file by simply clicking on the links below.you cn use these resources to prctice with smple ACT Compss tests online or wtch video tutorils on Google Video. If you hve only printed copy of the pdf file, you cn still find these Internet resources by using the provided web links. (1) CUNY Compss Prctice Tests from Hostos Community College (2) Kentucky Erly Mth Testing Progrm Prctice Tests https://www.mthclss.org/wqs/k.sp?stte=1 (3) Google Video Tutoril on Order of Opertions (4) Google Video PreAlgebr Tutoril (5) Google Video Tutoril on Solving Equtions Dte: November
2 2 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY 2. Common Mistkes Common Mistke 1. A surprisingly common mistke is to incorrectly copy the problem in your exm booklet. Mke sure you re working on the correct problem! Common Mistke 2. Alwys put prenthesis round negtive number, especilly when you hve to multiply it by nother number, s in this cse: ( 5) = 5 10 = 5 Never drop the prenthesis round the negtive number becuse you will forget tht you hve multipliction nd you will get this insted: = 2 which is wrong nd hs nothing to do with the originl problem. Common Mistke 3. Be very creful with the order of opertions. The correct order of opertions is given below: (1) First do the opertions inside the Prenthesis. (2) Then tke cre of Exponents, (3) Multipliction, Division, (4) Addition, Subtrction. Consider s n exmple the lgebric expression ( 9). There re severl mistkes you cn mke. Firstly, if you don t put the prenthesis round the negtive number, you will get: = 4, which is wrong. Secondly, you cn get the order of opertion wrong: ( 9) = 5 ( 9) = 45 Wrong! fter first dding 2 nd 3 nd then multiplying the result by ( 9). This is wrong, becuse multipliction hs priority, so one should first multiply 3 nd ( 9) to get 27 nd only then dd 2 to the result. So, the correct thing to do is the following: ( 9) = 2 + ( 27) = 2 27 = 25 Correct!
3 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 3 It would be different if we hve the following expression (2+3) ( 9). The difference is tht 2 nd 3 re now inside prenthesis, so we would hve to do the opertion inside the prenthesis first nd then multiply: (2 + 3) ( 9) = 5 ( 9) = 45 Correct! Common Mistke 4. A few common mistkes re relted to the properties of exponents. For exmple, note tht (2 3) 2 becuse tking the exponent hs priority over multipliction. So, if one wnts to clculte then one should tke the exponent first 3 2 = 9 nd then multiply the result by 2 to get 18, tht is = 2 9 = 18. While for (2 3) 2, we first do the multipliction inside the prenthesis to get 6, which we then squre: (2 3) 2 = 6 2 = 6 6 = 36 Another common mistke relted to exponents is to write It s lso wrong to write insted of using the correct property = 3 10 Wrong! = 9 7 Wrong! = = 3 7 Correct! Keep in mind tht the generl property reds m n = m+n Correct! Finlly, if we hve to tke the power of power, it is wrong to write (x 2 ) 5 = x 7 Wrong! The correct ppliction of the power property reds (x 2 ) 5 = x 10 Correct! Remember the generl property hs the form: (x m ) n = x m n Correct!
4 4 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Exmple 1. Consider now the following exmple. Evlute the expression: (2 5) 3 One common mistke is to write the second term in this difference s 3 (2 5) 3 = (6 15) 3 Wrong! This is wrong becuse we re distributing 3 inside the prenthesis pretending tht we hve (2 5) nd ignoring the fct tht we ctully hve this difference rised to the power of 3. To do it correctly, we need to do the opertion inside the prenthesis first nd then rise the result to 3rd power: 3 (2 5) 3 = 3 ( 3) 3 = 3 ( 27) = 3 27 = 81 Correct! nd only then multiply the result by 3. In our cse, we hve ( 3) 3 = ( 3) ( 3) ( 3) = ( 1) = 27 Remember tht negtive number rised to n odd power must be negtive nd negtive number rised to n even power must be positive (negtive) odd = negtive (negtive) even = positive For exmple, ( 1) 2 = 1, ( 1) 3 = 1, ( 1) 4 = 1, ( 1) 5 = 1. Now, let s go bck to the originl exmple. In the first term, we must tke cre of the exponent in the numertor first, so write 2 3 = = 8 nd t the sme time simplify the second term s we did erlier: (2 5) 3 = ( 3) 3 = ( 27) Note here tht 4 2 = 4 2 = 2 (don t forget the minus sign): 2 3 ( 27) = 2 + ( 3) ( 27) = = = 79
5 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 5 We hve here the product of two negtive numbers 3 nd 27, which gives us positive number ( 3) ( 27) = 3 27 = 81. Finlly, if you re confused bout the sum , note tht this is relly the sme s the difference 81 2, we simply tke 2 wy from 81 to get 79. Common Mistke 5. Remember tht the frction A mens tht we divide A by B, B i.e. we hve A B. For tht reson, we cn express division by B in terms of multipliction by the reciprocl of B, which is 1, nmely B A B = A 1 B = A B Consider the division when A = 15b3 nd B = 5b 2, then we hve 2 ( ) 15b 3 (5b 2 ) 2 Sometimes, students ttempt to use the division rule bove but since they cnnot quite remember it, they would write something like this ( ) ( ) 15b 3 15b (5b 2 3 ) = 5b2 Wrong! This is wrong, of course, becuse the division is replced by multipliction but the reciprocl of 5b 2 is not tken. Insted, 5b 2 is divided by 1, which does not chnge nything since ny number divided by 1 is the number itself, tht is 5b2 = 5b 2. This 1 wy, the division is simply replced by multipliction while nothing else chnges nd this is wrong. The correct thing to do is to tke the reciprocl of 5b 2 when replcing division by multipliction, nmely: ( ) 15b 3 (5b 2 ) = 15b b = 15b3 2 10b = 3b 2 2 Correct! Let s recll the rule for multiplying two frctions tht is used bove. We multiply the nomintors nd the denomintors of both frctions b c d = c b d
6 6 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY In the exmple bove, we reduce the frction 15 = 5 3 = 3 by cnceling out the common fctor of 5. Remember tht if we hve the sme number bove nd below the br of frction then we cn cncel this number: b c = b c becuse the br of the frction represents division. The property of exponents, used bove to get the finl result, is b m b n = bm n which we pply to our exmple to conclude tht b 3 b 2 = b3 2 = b 1 = b Common Mistke 6. One of the most common mistkes when deling with frctions is the following rule tht students invent to dd unlike frctions: b + c d = + c b + d Wrong! It is very esy to see tht this rule is not correct by checking simple exmple. Tke = 2, b = 1, c = 3 nd d = 1 nd if this rule is correct we should get true sttement: ? = ? = 5 2 which is clerly flse sttement, since 5 5, so the rule cnnot be correct. 2 The correct rule we get by crossmultiplying numertors by denomintors nd the sum of the two products gives us the new numertor, while the new denomintor is just the product of the two denomintors: b + c d = d + c b b d
7 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 7 Of course, given specific numbers, one cn lso look for the LCD (lest common denomintor) but in generl mny students find the LCD concept more difficult. For exmple, let s dd the two frctions using the correct rule: = = = = = = = 7 4 Alterntively, it is esy, in this cse, to find the LCD, which is 12. The next step is to write the first frction s n equivlent frction hving denomintor of 12 nd then we cn esily dd the like frctions. Tht s why we multiply by 2 the numertor nd denomintor of the first frction: = = = = = 7 4 Remember the rule for dding like frctions (with the sme denomintors): b + c b = + c b which we use bove to dd the like frctions: = Common Mistke 7. Some common mistkes relted to rdicls re writing: x 16 = x 4 or x 9 = x 3 Wrong! We cn check esily if our guess is correct by simply using the definition of squre root, which in the first cse would men tht if we tke the squre of our guess x 4, we should get wht is inside the rdicl: (x 4 ) 2 should be equl to x 16. However, (x 4 ) 2 = x 4 2 = x 8 x 16, so our guess x 4 cnnot be correct. I cn only guess tht the logic tht leds to the wrong clims bove goes long these lines x 16 = x 16 = x 4 or x 9 = x 9 = x 3 Wrong! The correct rule to pply in the cse of n even power under the rdicl sign is: x 16 = x 16 2 = x 8 Correct!
8 8 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY It is esy to see tht x 8 is the correct nswer becuse if we squre it, we get (x 8 ) 2 = x 8 2 = x 16 nd x 16 is wht s inside the rdicl sign. In the second cse, when we don t hve n even power, 16 is n even power but 9 is not, we need to split the odd power in order to get n even power: x9 = x 8 x = x 8 x = x 8 2 x = x4 x Correct! Here for, b > 0, we used the product rule for rdicls b = b nd we cn generlize the rule we used bove for ny positive even power under the rdicl sign, like 16 or 8 x even = x even 2 3. Distributive Properties Let, b, c, d, e be ny numbers, positive (negtive) or terms contining vribles. If we hve to multiply two fctors, we must multiply every term in the first fctor by every term in the second fctor. It is useful to drw rrows indicting ll possible products. The following distributive properties re often used: (b + c) = b + c ( + b) (c + d) = c + d + b c + b d ( + b) (c + d + e) = c + d + e + b c + b d + b e Exmple 2. Multiply nd combine like terms in the following exmples: (1) 2x (4x 2 + 3) = 2x 4x 2 + 2x 3 = 8x 3 + 6x (2) 2 (4 3) = ( 3) = = 2 (3) (x y) (x + y) = x x + x y y x y y = x 2 y 2 (4) (x 4) (x 2 + 2x 2) = x x 2 + x 2x + x ( 2) 4 x 2 4 2x 4 ( 2) = = x 3 + }{{} 2x 2 2x 4x2 }{{} 8x + 8 = x3 2x 2 10x + 8
9 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 9 4. Properties of Frctions b c b = c Exmple 3. This pplies to numericl s well s lgebric frctions. Remember tht frction represents division, so if we hve the sme term (b in the formul) bove nd below the division line, being the br of the frction, then we cn cncel this term, s we re relly dividing the term by itself = 5 2 (x 1) (x+2) = (x 1) (x+2) (x+5) (x+5) b = b = b Exmple 4. We cn move the minus sign from the denomintor to the numertor or we cn plce it in front of the frction. If both the denomintor nd the numertor re negtive, we cn cncel out the minus. 5 7 = 5 7 = = 5 7 b c d = c b d Exmple 5. The product of two frctions is frction whose numertor is the product of the two numertors ( c bove) nd whose denomintor is the product of the two denomintors (b d bove). 5 2 = (x 4) (x 2) = (x 4) (x 2) (x+1) 3 3(x+1) b c d = b d c Exmple 6. We divide one frction (the divident) by nother (the divisor) by multiplying the divident by the reciprocl of the divisor.
10 10 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Remember, the reciprocl of c d is d c 5 2 = 5 3 = (x 3) (x+5) (x 2) (x 1) = (x 3) (x+5) (x 1) (x 2) = (x 3) (x 1) (x+5) (x 2) d + b d = + b d Exmple 7. We dd two like frctions (hving the sme denomintors) by dding the numertors nd keeping the common denomintor = x + 2 x = 7 x d b d = b d Exmple 8. We subtrct two like frctions (hving the sme denomintors) by subtrcting the corresponding numertors nd keeping the common denomintor = x 2 x = 3 x b + c d = d + c b b d Exmple 9. We dd two unlike frctions hving different denomintors b d, by trnsforming them first into equivlent like frctions, hving the sme denomintors: = = = = 2 x + 5 = 2x+5 x x x x = x = x = 2+5x x 2 x x 2 x x x 2 x 2 x 2 b c d = d c b b d
11 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 11 Exmple 10. We subtrct two unlike frctions hving different denomintors b d, by trnsforming them into equivlent like frctions, hving the sme denomintors: = = = x = 2 x x 5 x = 2x 5 x 2 5 = 2 5 x = 2 5 x x 2 x x 2 x x x 2 x 2 = 2 5x x 2 5. Properties of Exponents x n = n { }} { x x x Exmple 11. Rising number (or n lgebric expression) to some positive integer power is the sme s multiplying this number (or n lgebric expression) by itself s mny times s the positive integer power. ( 2) 3 = ( 2)( 2)( 2) = 8 (x 2) 3 = (x 2)(x 2)(x 2) ( 2) 4 = ( 2)( 2)( 2)( 2) = 16 x m x n = x m+n Exmple 12. Observe tht ll powers in the formul bove hve the sme bse x, which could be number or more generl lgebric expression. To multiply two powers hving the sme bse (x bove), we dd the exponents nd keep the bse = = 2 8 the bse here is 2 (x 2) 3 (x 2) 4 = (x 2) 3+4 = (x 2) 7 the bse here is (x 2) 4 5 = 4+5 = 9 the bse here is (2 3) 4+5 bses re different, 2 nd 3 (x m ) n = x m n Exmple 13. The power of power rule leds us to multipliction of exponents.
12 12 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY (2 3 ) 5 = = 2 15 ((x 2) 3 ) 4 = (x 2) 3 4 = (x 2) 12 ( 4 ) 5 = 4 5 = 20 (2 4 ) x m x n = xm n Exmple 14. The quotient of two powers hving the sme bse (x bove) nd ny exponents (m nd n bove) is the bse to the power tht is the difference of the two exponents (m n bove), where the exponent tht is below the br of the frction (n bove) is subtrcted from the exponent bove the br (m bove) = = 2 2 = 2 2 = 4 bse is 2 x5 x 3 = x 5 3 = x 2 bse is x = 10 4 ( 2) = = 10 6 bse is 10; used in Scientific Nottion = 10 4 ( 3) = = 10 1 = 1 10 used in Scientific Nottion x n = 1 x n, x0 = 1 Exmple 15. Used to convert negtive exponent into positive one. 2 3 = = (x y) n = x n y n Exmple 16. The power of product is the product of the powers. (2 5) 3 = (5x) 3 = 5 3 x 3 (2xy 2 ) 4 = 2 4 x 4 (y 2 ) 4 = 16x 4 y 8
13 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 13 ( ) n x = xn y y n Exmple 17. The power of quotient is the quotient of the powers. ( ) = ( ) x 3 5 = x Properties of Rdicls = ( ) 2 =, > 0 Exmple 18. Tking the squre of squre root leves us with the number inside the rdicl. We undo the squre root by tking the squre. 5 5 = 5 ( x) 2 = x, x > 0 2 =, > 0 Exmple 19. We cn undo the squre root by tking the squre inside the rdicl. 5 2 = 5 x 2 = x, x > 0 b = b, > 0, b > 0 Exmple 20. The squre root of the product is the product of the squre roots = = 5 4 = 20 x 2 y = x 2 y = x y 49 x y 2 z 2 = 49 x y 2 z 2 = 7yz x
14 14 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY b = b, > 0, b > 0 Exmple 21. The squre root of the quotient is the quotient of the squre roots. 5 2 = = 5 6 x 2 = x = x 8 2n = 2n 2 = n or even = even 2 Exmple 22. If we hve n even exponent inside the rdicl, 2n bove, we cn undo the rdicl by tking hlf of the even exponent, 2n 2 = n = = x 16 y 10 = 25 x 16 2 y 10 2 = 5x 8 y 5 x 8 y 17 = x 8 2 y 16 y = x 4 y 16 2 y = x 4 y 8 y 7. The Slope nd Eqution of Line The slope of line is number determined by the coordintes of ny two points on the line. If P (x 1, y 1 ) nd Q(x 2, y 2 ) re ny two points on the line, given by their coordintes, then the slope is the number m = y 2 y 1 x 2 x 1 = y x Note tht y, the difference in the ycoordintes, is on the top, while x, the difference in the xcoordintes, is on the bottom. It is common mistke to use x y y insted of x to compute the slope. Notice tht we chose bove to strt with the second point Q nd tht is why both differences begin with the coordintes of the second point (y 2 y 1 ) nd (x 2 x 1 ). One cn strt insted with the first point P, in which cse both differences should strt with the coordintes of the first point, nmely (y 1 y 2 ) nd (x 1 x 2 ), which gives the sme slope s bove
15 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 15 m = y 1 y 2 x 1 x 2 = (y 2 y 1 ) (x 2 x 1 ) = y 2 y 1 x 2 x 1 Exmple 23. Find the slope of the line pssing through the points P ( 1, 2) nd Q( 4, 5). If we choose to strt with the first point P, then both differences should strt with the coordintes of the first point, with y difference on the top Slope = m = 2 ( 5) 1 ( 4) = = 3 3 = 1 If we choose to strt with the second point Q, then both differences should strt with the coordintes of the second point, with y difference on the top Slope = m = 5 ( 2) 4 ( 1) = = 3 3 = 1 Exmple 24. The following results re useful to remember: Any horizontl line hs slope zero. For verticl lines the slope is not defined. If line hs positive slope, the line rises from left to right. If line hs negtive slope, the line flls down from left to right. The eqution of line pssing through two given points P (x 1, y 1 ) nd Q(x 2, y 2 ) is given in terms of the slope m nd the coordintes of one of the points: y = y 1 + m(x x 1 ) Note tht y nd x re vribles, representing the coordintes of n rbitrry point on the line. Here, we chose to use the coordintes of the first point P, nmely the given numbers x 1 nd y 1 but one cn lso use the coordintes of the second point Q nd still get the sme eqution for the line y = y 2 + m(x x 2 )
16 16 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Exmple 25. Tke the points bove P ( 1, 2) nd Q( 4, 5), for which we computed the slope m = 1. The eqution of the line pssing through these two points, using the coordintes of the first point, is then y = 2 + 1(x ( 1)) = 2 + x + 1 = x 1 We should get the sme eqution if we use the coordintes of the second point insted y = 5 + 1(x ( 4)) = 5 + x + 4 = x 1 Exmple 26. Let the eqution of line be given by 3x + 2y = 5 How cn we find the slope of the line from this eqution? We need to solve this eqution for y in terms of x: 2y = 5 3x subtrct from both sides 3x y = 5 3x 2 then divide both sides by 2 y = x this is wht we need to find the slope Once we hve the eqution in the form (for some numbers m nd b): y = mx + b slope = m nd yintercept = b the slope is simply the number m (including the sign) in front of the vrible x, while the number b is the yintercept. In our cse, The slope is the signed number in front of the vrible x, nmely slope = 3 2 nd the yintercept = 5 2 A common mistke is to include the vrible x in the nswer for the slope. Remember tht the slope is number. Another common mistke is to forget the sign nd write 3 for the slope, insted of 3. Remember tht the yintercept is the ycoordinte of 2 2 the point of intersection of the line nd the yxis, when x = 0. Emil ddress:
Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationSolving BAMO Problems
Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationQuick Reference Guide: Onetime Account Update
Quick Reference Guide: Onetime Account Updte How to complete The Quick Reference Guide shows wht existing SingPss users need to do when logging in to the enhnced SingPss service for the first time. 1)
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationMULTIPLYING OUT & FACTORING
igitl ircuit Engineering MULTIPLYING OUT & FTORING I IGITL SIGN Except for #$&@ fctoring st istributive X + X = X( + ) 2nd istributive (X + )(X + ) = X + (X + )(X + )(X + ) = X + Swp (X + )(X + ) = X +
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationSmall Business Cloud Services
Smll Business Cloud Services Summry. We re thick in the midst of historic sechnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationAP STATISTICS SUMMER MATH PACKET
AP STATISTICS SUMMER MATH PACKET This pcket is review of Algebr I, Algebr II, nd bsic probbility/counting. The problems re designed to help you review topics tht re importnt to your success in the clss.
More informationSINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1470  COLLEGE ALGEBRA (4 SEMESTER HOURS)
SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 470  COLLEGE ALGEBRA (4 SEMESTER HOURS). COURSE DESCRIPTION: Polynomil, rdicl, rtionl, exponentil, nd logrithmic functions
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationAAPT UNITED STATES PHYSICS TEAM AIP 2010
2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS  75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More informationQuick Reference Guide: Reset Password
Quick Reference Guide: Reset Pssword How to reset pssword This Quick Reference Guide shows you how to reset your pssword if you hve forgotten it. There re three wys to reset your SingPss pssword: 1) Online
More informationAntiSpyware Enterprise Module 8.5
AntiSpywre Enterprise Module 8.5 Product Guide Aout the AntiSpywre Enterprise Module The McAfee AntiSpywre Enterprise Module 8.5 is n ddon to the VirusScn Enterprise 8.5i product tht extends its ility
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
More informationMATH PLACEMENT REVIEW GUIDE
MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationData replication in mobile computing
Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL
More information#A12 INTEGERS 13 (2013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS
#A1 INTEGERS 13 (013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS Michel O. Rubinstein 1 Pure Mthemtics, University of Wterloo, Wterloo, Ontrio, Cnd mrubinst@uwterloo.c
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationUnderstanding Life Cycle Costs How a Northern Pump Saves You Money
Understnding Life Cycle Costs How Nrn Pump Sves You Money Reference: Hydrulic Institute (www.s.g) Introduction Wht Life Cycle Cost (LCC) Clculting Totl LCC LCC Components Wht Life Cycle Cost Life Cycle
More informationHillsborough Township Public Schools Mathematics Department Computer Programming 1
Essentil Unit 1 Introduction to Progrmming Pcing: 15 dys Common Unit Test Wht re the ethicl implictions for ming in tody s world? There re ethicl responsibilities to consider when writing computer s. Citizenship,
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This printout should hve 22 questions, check tht it is complete. Multiplechoice questions my continue on the next column or pge: find ll choices efore mking your
More informationRedistributing the Gains from Trade through Nonlinear. Lumpsum Transfers
Redistributing the Gins from Trde through Nonliner Lumpsum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lumpsum trnsfer rules to redistribute the gins from
More informationRecognition Scheme Forensic Science Content Within Educational Programmes
Recognition Scheme Forensic Science Content Within Eductionl Progrmmes one Introduction The Chrtered Society of Forensic Sciences (CSoFS) hs been ccrediting the forensic content of full degree courses
More informationNetwork Configuration Independence Mechanism
3GPP TSG SA WG3 Security S3#19 S3010323 36 July, 2001 Newbury, UK Source: Title: Document for: AT&T Wireless Network Configurtion Independence Mechnism Approvl 1 Introduction During the lst S3 meeting
More informationHealth insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
More informationCOMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, crossclssified
More informationbaby on the way, quit today
for mumstobe bby on the wy, quit tody WHAT YOU NEED TO KNOW bout smoking nd pregnncy uitting smoking is the best thing you cn do for your bby We know tht it cn be difficult to quit smoking. But we lso
More informationIntroduction to Integration Part 2: The Definite Integral
Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the
More informationSmall Business Networking
Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer
More informationtrademark and symbol guidelines FOR CORPORATE STATIONARY APPLICATIONS reviewed 01.02.2007
trdemrk nd symbol guidelines trdemrk guidelines The trdemrk Cn be plced in either of the two usul configurtions but horizontl usge is preferble. Wherever possible the trdemrk should be plced on blck bckground.
More informationVector differentiation. Chapters 6, 7
Chpter 2 Vectors Courtesy NASA/JPLCltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higherdimensionl counterprts
More informationWeek 11  Inductance
Week  Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n
More informationVersion 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This printout should he 35 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
More informationNumerical Methods of Approximating Definite Integrals
6 C H A P T E R Numericl Methods o Approimting Deinite Integrls 6. APPROXIMATING SUMS: L n, R n, T n, AND M n Introduction Not only cn we dierentite ll the bsic unctions we ve encountered, polynomils,
More informationAn Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process
An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct
More informationProject 6 Aircraft static stability and control
Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The
More informationCURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.
CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e
More informationHealth insurance marketplace What to expect in 2014
Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum
More informationCcrcs Cognitive  Counselling Research & Conference Services (eissn: 23012358)
Ccrcs Cognitive  Counselling Reserch & Conference Services (eissn: 23012358) Volume I Effects of Music Composition Intervention on Elementry School Children b M. Hogenes, B. Vn Oers, R. F. W. Diekstr,
More informationffiiii::#;#ltlti.*?*:j,'i#,rffi
5..1 EXPEDTNG A PROJECT. 187 700 6 o 'o' 600 E 500 17 18 19 20 Project durtion (dys) Figure 66 Project cost vs. project durtion for smple crsh problem. Using Excel@ to Crsh Project T" llt ffiiii::#;#ltlti.*?*:j,'i#,rffi
More informationPreApproval Application
PreApprovl Appliction In tody s rel estte mrket, PreApproved mortgge provides you the buyer with powerful tool in the home purchse process! Once you hve received your PreApprovl, you cn shop for your
More informationClearPeaks Customer Care Guide. Business as Usual (BaU) Services Peace of mind for your BI Investment
ClerPeks Customer Cre Guide Business s Usul (BU) Services Pece of mind for your BI Investment ClerPeks Customer Cre Business s Usul Services Tble of Contents 1. Overview...3 Benefits of Choosing ClerPeks
More informationProtocol Analysis. 17654/17764 Analysis of Software Artifacts Kevin Bierhoff
Protocol Anlysis 17654/17764 Anlysis of Softwre Artifcts Kevin Bierhoff TkeAwys Protocols define temporl ordering of events Cn often be cptured with stte mchines Protocol nlysis needs to py ttention
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
More informationEngineertoEngineer Note
EngineertoEngineer Note EE280 Technicl notes on using Anlog Devices DSPs, processors nd development tools Visit our Web resources http://www.nlog.com/eenotes nd http://www.nlog.com/processors or emil
More informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
More informationAccording to Webster s, the
dt modeling Universl Dt Models nd P tterns By Len Silversn According Webster s, term universl cn be defined s generlly pplicble s well s pplying whole. There re some very common ptterns tht cn be generlly
More informationGFI MilArchiver 6 vs C2C Archive One Policy Mnger GFI Softwre www.gfi.com GFI MilArchiver 6 vs C2C Archive One Policy Mnger GFI MilArchiver 6 C2C Archive One Policy Mnger Who we re Generl fetures Supports
More informationVectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
More informationEnterprise Risk Management Software Buyer s Guide
Enterprise Risk Mngement Softwre Buyer s Guide 1. Wht is Enterprise Risk Mngement? 2. Gols of n ERM Progrm 3. Why Implement ERM 4. Steps to Implementing Successful ERM Progrm 5. Key Performnce Indictors
More informationMorgan Stanley Ad Hoc Reporting Guide
spphire user guide Ferury 2015 Morgn Stnley Ad Hoc Reporting Guide An Overview For Spphire Users 1 Introduction The Ad Hoc Reporting tool is ville for your reporting needs outside of the Spphire stndrd
More informationbody.allowsidebar OR.nosidebar.homepage (if this is the home page).hascustombanner OR.nocustombanner .IR OR.noIR
body.llowsidebr OR.nosidebr.homepge (if this is the home pge).hscustombnner OR.nocustombnner.IR OR.noIR #IDENTIFIER_FOR_THIS_SITE div#pgecontiner.depends_on_page_ty PE llowsidebr mens tht there
More informationUNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE
UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10438 UNVERSTY OF NOTTNGHAM Discussion
More informationVectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics
Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higherdimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie threedimensionl spce nd
More informationRoudmup for Los Angeles Pierce College ADIV Program ancl csu Dominguez Hilk RltB^sr/ progrum
Roudmup for Los Angeles Pierce College ADIV Progrm ncl csu Dominguez Hilk RltB^sr/ progrum Admission Requirements for Los Angeles pierce (LApC) LAPC hs foursemester Associte Degree in Nursing (ADN) Progrm.
More informationHumana Critical Illness/Cancer
Humn Criticl Illness/Cncer Criticl illness/cncer voluntry coverges py benefits however you wnt With our criticl illness nd cncer plns, you'll receive benefit fter serious illness or condition such s hert
More informationThe 8 Essential Layers of SmallBusiness IT Security
The 8 Essentil Lyers of SmllBusiness IT Security While there is no technology tht cn gurntee your network is truly impenetrble, you cn significntly reduce your risk by deploying multiple lyers of defense.
More informationVectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics
Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higherdimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie threedimensionl spce nd
More informationSecondDegree Equations as Object of Learning
Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, 2008. Abstrct SecondDegree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson,
More informationTalent (or guardian) signature: Date:
Sesme Communictions Sesme Communictions Mkers of Ortho Sesme nd Dentl Sesme PRESENTS: Sesme Prctice Mrketing SAVE ALL IDEAS: Strt file to store the news items nd ides tht come up between your scheduled
More informationDeveloping Jazz Vocabulary
Developing Jzz Vocbulry For the Jr. High nd High School Jzz Plyer Your er is the finl judge s to wht sounds right nd wht sounds wrong Big Nic Nichols August 1994 Tim Price Jzz Lesson The Ply nd Lern Process
More informationSTATE OF MONTANA Developomental Disabilities Program Comprehensive Evaluation HiLine Home Programs, Inc Adult Services
Dtes of Review: FY '09 Evluttor(s): S. Crpenter DESK REVIEW: Accredittion: Acredittion is no longer required by the stte contrct. Significnt Events from the Agency: Developomentl Disbilities Progrm Comprehensive
More information5 a LAN 6 a gateway 7 a modem
STARTER With the help of this digrm, try to descrie the function of these components of typicl network system: 1 file server 2 ridge 3 router 4 ckone 5 LAN 6 gtewy 7 modem Another Novell LAN Router Internet
More informationHealth insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 11/12 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
More informationPay over time with low monthly payments. Types of Promotional Options that may be available: *, ** See Page 10 for details
With CreCredit... Strt cre immeditely Py over time with low monthly pyments For yourself nd your fmily Types of Promotionl Options tht my be vilble: Not ll enrolled helthcre prctices offer ll specil finncing
More informationThe invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism.
Instrutor: Longfei Li Mth 43 Leture Notes 16. Line Integrls The invention of line integrls is motivted by solving problems in fluid flow, fores, eletriity nd mgnetism. Line Integrls of Funtion We n integrte
More informationSpace Vector Pulse Width Modulation Based Induction Motor with V/F Control
Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:
More informationOn the Robustness of Most Probable Explanations
On the Robustness of Most Probble Explntions Hei Chn School of Electricl Engineering nd Computer Science Oregon Stte University Corvllis, OR 97330 chnhe@eecs.oregonstte.edu Adnn Drwiche Computer Science
More information