ab b. c 3. y 5x. a b 3ab. x xy. p q pq. a b. x y) + 2a. a ab. 6. Simplify the following expressions. (a) (b) (c) (4x

Size: px
Start display at page:

Download "ab b. c 3. y 5x. a b 3ab. x xy. p q pq. a b. x y) + 2a. a ab. 6. Simplify the following expressions. (a) (b) (c) (4x"

Transcription

1 . Simplif the following epessions. 8 c c d. Simplif the following epessions. 6b pq 0q. Simplif the following epessions. ( ) q( m n) 6q ( m n) 7 ( b c) ( b c) 6. Simplif the following epessions. b b b p p p 7. Simplif the following epessions. b b 6b 0pq p 8 p q 8. Simplif the following epessions. b b 9 c d 6c d cd c d 9. Simplif the following epessions. 6b p q p b 8q p p Simplif the following epessions. p 8q n n 9n 0. Simplif the following epessions. 8 b b b b c d c 8d b 6 b 8 b 6b. Simplif the following epessions. 6b 6b 6z z 0 n m m n. Simplif the following epessions. 0 c 8b c b 6 ( ) p 8 p pq p q p q pq Chung Ti Eductionl Pess. All ights eseved. 009 Chung Ti Eductionl Pess. All ights eseved.

2 . Fctoize b b. b b Fom the esult of, simplif. b. Fctoize b b. b b b b Fom the esult of, simplif. b b 7. Simplif the following epessions. h h b k k c c h h 8. Simplif the following epessions. 7 c c u u 8 6 v v 9. Fill in the bckets with suitble epessions.. Fill in the blnks with suitble numbes. = = z z z 7 6 = = p p p 0 ( = ) ( ) d = d e d e d e b ( ) b = c c c ( ) f g = f g f g f g 0. Simplif the following epessions.. Simplif the following epessions z 7z b b 8 7m n n 7m w w w w p 6 p 6 6 p. Simplif the following epessions. 6. Simplif the following epessions. b b b b b 7 ( ) 6 ( 6) ( b) b b b ( b) b Chung Ti Eductionl Pess. All ights eseved. 009 Chung Ti Eductionl Pess. All ights eseved.

3 7. Given tht =, find the vlue of if b = nd c =. b c 8. Given tht P = 0T, find the vlue of P if T = nd V =. V 9. Given tht E = mv, find the vlue of E if m = 0 nd v =.. Simplif the following epessions. ( ) ( ) h (h ) (h ) m (m ) (m ) ( ) 8( ). Simplif the following epessions. z ( z ) 6 b b 0. Given tht N = k ( t ), find the vlue of N if k = 0. nd t = 6.. (i) Convet into n lgebic fction with the denominto ( 8). (ii) Convet into n lgebic fction with the denominto ( 8). 8 Fom the esult of, simplif. 8. Given tht P = Q ( 0.n ), find the vlue of P if Q = 80 nd n =. z = bc d = ( b c) z = n ( A Fn ), find the vlue of S if n =, A = nd F = 6.. Given tht b c = 60, find the vlue of c if = 00 nd b = 60. d = bc d = b c 6. Given tht M = V, find the vlue of n if M = 0 nd V = 8. n 7. Given tht A = ( ), find the vlue of if A = 96 nd = Chung Ti Eductionl Pess. All ights eseved.. Given tht S = z = 6. Given tht =, b = nd c =, find the vlue of d in ech of the following epessions. d = T, find the vlue of if P =, T = nd U = 6. P U. Given tht P =, find the vlue of if P =.. Given tht = nd =, find the vlue of z in ech of the following epessions. z =. Given tht = Chung Ti Eductionl Pess. All ights eseved.

4 8. Given tht s = ut t, find the vlue of if s =., u = 0 nd t =. z 9. Given tht =, find the vlue of z if = 0 nd =.. Given tht ( n ) n ( n ) = n, whee n is positive intege. Find the vlue of Find the vlue of Fom the esults of nd, find the vlue of Given tht M = b c, find the vlue of if M = 0, b = nd c = 6.. Given tht =, find the vlue of if =. 6. The degee Celsius ( C) nd degee Fhenheit ( F) e two units of tempetue commonl used. The convesion between the two units cn be epessed in the following fomul: f =. 8c k whee f F is the degee Fhenheit, c C is the degee Celsius nd k is constnt. Given tht 0 C coesponds to F, find the vlue of k. (i) Given tht the boiling point of wte is 00 C, epess it in degee Fhenheit. (ii) Given tht the boiling point of lcohol is 7. F, epess it in degee Celsius. Suppose peson with the bod tempetue highe thn 7. C is egded s hving feve, is Jde hving feve if he bod tempetue is 00 F? Eplin biefl.. Simplif. It is given tht d = bc( ). Fom the esult of, if d =, = nd b = 6, find the vlue of c.. It is given tht the sufce e of clinde is A = πh π. If π =., = nd h = 0, find the vlue of A. If A = 6, π = nd =., find the vlue of A. 7. A sting of ( w ) cm long is evenl divided into sevel pts, the length cm of ech pt cn be w found b the fomul = whee w is n intege gete thn. w If sting is cm long oiginll, (i) find the vlue of w. (ii) find the length of ech pt. If ech pt of sting is 7 cm long, (i) find the vlue of w. (ii) find the oiginl length of the sting. h 7. Detemine whethe ech of the following sentences is coect. If es, put! in the bo; othewise put ". Mke the subject of the fomul =, we cn obtin the fomul =. Mke the subject of the fomul =, we cn obtin the fomul =. Mke the subject of the fomul =, we cn obtin the fomul =. Mke the subject of the fomul =, we cn obtin the fomul =. 8. Mke ech of the following lettes the subject of the fomul d = bc. b c 009 Chung Ti Eductionl Pess. All ights eseved Chung Ti Eductionl Pess. All ights eseved.

5 9. Mke the lettes in bckets the subjects of the coesponding fomule. z = 60 [z] = [] 0. Mke the lettes in bckets the subjects of the coesponding fomule. = [] = 6 8 [] 8. Mke the lettes in bckets the subjects of the coesponding fomule. d f b T = [] S = ( c)( b e) [e] c e e c 9. Mke the lettes in bckets the subjects of the coesponding fomule. m w = [] = [n] n m. Mke the lettes in bckets the subjects of the coesponding fomule. = b 7 [b] 6 q p = [q]. Mke the lettes in bckets the subjects of the coesponding fomule. n p 9 = [m] = 6 m q. Mke the lettes in bckets the subjects of the coesponding fomule. = b b c [c] [q] s t = [v] u v. Mke the lettes in bckets the subjects of the coesponding fomule. 8 = [] = [z] z. Mke the lettes in bckets the subjects of the coesponding fomule. P = mu mv [m] 9 f e = ef [e] 60. It is given tht t = ( t) nd = t. Epess t in tems of. Fom the esult of, epess in tems of. 6. It is given tht = nd =. (i) Epess in tems of. (ii) Epess in tems of. Fom the esult of, epess in tems of. Epess in tems of. 6. Simplif. It is given tht A =. (i) Fom the esult of, mke the subject of the bove fomul. (ii) If = nd A =, find the vlue of. 6. Mke the lettes in bckets the subjects of the coesponding fomule. p q = b( c de) [d] = ( ) 7. Mke the lettes in bckets the subjects of the coesponding fomule. = [] = b b [q] [] 6. Miss Cheung deposits sum of mone $P in bnk. The mount $A eceived fte T es is given b the following fomul: A = P( 0.0T ) If Miss Cheung deposits $ 000 in bnk, find the mount eceived fte es. Mke T the subject of the fomul. If Miss Cheung deposits $ 000 in the bnk, how long does it tke fo he to eceive n mount of $ 00? Chung Ti Eductionl Pess. All ights eseved. 009 Chung Ti Eductionl Pess. All ights eseved.

7.5-Determinants in Two Variables

7.5-Determinants in Two Variables 7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt

More information

Topics for Review for Final Exam in Calculus 16A

Topics for Review for Final Exam in Calculus 16A Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the

More information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque

More information

Optimization. x = 22 corresponds to local maximum by second derivative test

Optimization. x = 22 corresponds to local maximum by second derivative test Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible

More information

Mark Scheme (Results) January 2008

Mark Scheme (Results) January 2008 Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question

More information

Chapter Direct Method of Interpolation More Examples Mechanical Engineering

Chapter Direct Method of Interpolation More Examples Mechanical Engineering Chpte 5 iect Method o Intepoltion Moe Exmples Mechnicl Engineeing Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete o tunnion sht by cooling it though tempetue chnge o is given by

More information

Multiplying and Dividing Rational Expressions

Multiplying and Dividing Rational Expressions Lesson Peview Pt - Wht You ll Len To multipl tionl epessions To divide tionl epessions nd Wh To find lon pments, s in Eecises 0 Multipling nd Dividing Rtionl Epessions Multipling Rtionl Epessions Check

More information

1 Using Integration to Find Arc Lengths and Surface Areas

1 Using Integration to Find Arc Lengths and Surface Areas Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s

More information

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:

More information

Answers to test yourself questions

Answers to test yourself questions Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E

More information

STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0

STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0 STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed

More information

10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = =

10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = = Chpte 1 nivesl Gvittion 11 *P1. () The un-th distnce is 1.4 nd the th-moon 8 distnce is.84, so the distnce fom the un to the Moon duing sol eclipse is 11 8 11 1.4.84 = 1.4 The mss of the un, th, nd Moon

More information

EECE 260 Electrical Circuits Prof. Mark Fowler

EECE 260 Electrical Circuits Prof. Mark Fowler EECE 60 Electicl Cicuits Pof. Mk Fowle Complex Numbe Review /6 Complex Numbes Complex numbes ise s oots of polynomils. Definition of imginy # nd some esulting popeties: ( ( )( ) )( ) Recll tht the solution

More information

Language Processors F29LP2, Lecture 5

Language Processors F29LP2, Lecture 5 Lnguge Pocessos F29LP2, Lectue 5 Jmie Gy Feuy 2, 2014 1 / 1 Nondeteministic Finite Automt (NFA) NFA genelise deteministic finite utomt (DFA). They llow sevel (0, 1, o moe thn 1) outgoing tnsitions with

More information

CHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD

CHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD ollege Physics Student s Mnul hpte 8 HAPTR 8: LTRI HARG AD LTRI ILD 8. STATI LTRIITY AD HARG: OSRVATIO O HARG. ommon sttic electicity involves chges nging fom nnocoulombs to micocoulombs. () How mny electons

More information

10 Statistical Distributions Solutions

10 Statistical Distributions Solutions Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques

More information

This immediately suggests an inverse-square law for a "piece" of current along the line.

This immediately suggests an inverse-square law for a piece of current along the line. Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line

More information

Homework: Study 6.2 #1, 3, 5, 7, 11, 15, 55, 57

Homework: Study 6.2 #1, 3, 5, 7, 11, 15, 55, 57 Gols: 1. Undestnd volume s the sum of the es of n infinite nume of sufces. 2. Be le to identify: the ounded egion the efeence ectngle the sufce tht esults fom evolution of the ectngle ound n xis o foms

More information

Find this material useful? You can help our team to keep this site up and bring you even more content consider donating via the link on our site.

Find this material useful? You can help our team to keep this site up and bring you even more content consider donating via the link on our site. Find this mteil useful? You cn help ou tem to keep this site up nd bing you even moe content conside donting vi the link on ou site. Still hving touble undestnding the mteil? Check out ou Tutoing pge to

More information

AQA Maths M2. Topic Questions from Papers. Circular Motion. Answers

AQA Maths M2. Topic Questions from Papers. Circular Motion. Answers AQA Mths M Topic Questions fom Ppes Cicul Motion Answes PhysicsAndMthsTuto.com PhysicsAndMthsTuto.com Totl 6 () T cos30 = 9.8 Resolving veticlly with two tems Coect eqution 9.8 T = cos30 T =.6 N AG 3 Coect

More information

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s: Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,

More information

Week 8. Topic 2 Properties of Logarithms

Week 8. Topic 2 Properties of Logarithms Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e

More information

Quality control. Final exam: 2012/1/12 (Thur), 9:00-12:00 Q1 Q2 Q3 Q4 Q5 YOUR NAME

Quality control. Final exam: 2012/1/12 (Thur), 9:00-12:00 Q1 Q2 Q3 Q4 Q5 YOUR NAME Qulity contol Finl exm: // (Thu), 9:-: Q Q Q3 Q4 Q5 YOUR NAME NOTE: Plese wite down the deivtion of you nswe vey clely fo ll questions. The scoe will be educed when you only wite nswe. Also, the scoe will

More information

Homework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:

Homework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is: . Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo

More information

SSC TIER II (MATHS) MOCK TEST - 31 (SOLUTION)

SSC TIER II (MATHS) MOCK TEST - 31 (SOLUTION) 007, OUTRM LINES, ST FLOOR, OOSITE MUKHERJEE NGR OLIE STTION, DELHI-0009 SS TIER II (MTHS) MOK TEST - (SOLUTION). () We know tht x + y + z xyz (x + y + z) (x + y + z xy yz zx) (x + y + z)[(x + y + z) (xy

More information

On the Eötvös effect

On the Eötvös effect On the Eötvös effect Mugu B. Răuţ The im of this ppe is to popose new theoy bout the Eötvös effect. We develop mthemticl model which loud us bette undestnding of this effect. Fom the eqution of motion

More information

r a + r b a + ( r b + r c)

r a + r b a + ( r b + r c) AP Phsics C Unit 2 2.1 Nme Vectos Vectos e used to epesent quntities tht e chcteized b mgnitude ( numeicl vlue with ppopite units) nd diection. The usul emple is the displcement vecto. A quntit with onl

More information

Radial geodesics in Schwarzschild spacetime

Radial geodesics in Schwarzschild spacetime Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using

More information

1. Show that the volume of the solid shown can be represented by the polynomial 6x x.

1. Show that the volume of the solid shown can be represented by the polynomial 6x x. 7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank depends

More information

IDENTITIES FORMULA AND FACTORISATION

IDENTITIES FORMULA AND FACTORISATION SPECIAL PRODUCTS AS IDENTITIES FORMULA AND FACTORISATION. Find the product of : (i) (n + ) nd ((n + 5) ( + 0.) nd ( + 0.5) (iii) (y + 0.7) nd (y + 0.) (iv) + 3 nd + 3 (v) y + 5 nd 3 y + (iv) 5 + 7 nd +

More information

Electronic Supplementary Material

Electronic Supplementary Material Electonic Supplementy Mteil On the coevolution of socil esponsiveness nd behvioul consistency Mx Wolf, G Snde vn Doon & Fnz J Weissing Poc R Soc B 78, 440-448; 0 Bsic set-up of the model Conside the model

More information

B.A. (PROGRAMME) 1 YEAR MATHEMATICS

B.A. (PROGRAMME) 1 YEAR MATHEMATICS Gdute Couse B.A. (PROGRAMME) YEAR MATHEMATICS ALGEBRA & CALCULUS PART B : CALCULUS SM 4 CONTENTS Lesson Lesson Lesson Lesson Lesson Lesson Lesson : Tngents nd Nomls : Tngents nd Nomls (Pol Co-odintes)

More information

Solution of fuzzy multi-objective nonlinear programming problem using interval arithmetic based alpha-cut

Solution of fuzzy multi-objective nonlinear programming problem using interval arithmetic based alpha-cut Intentionl Jounl of Sttistics nd Applied Mthemtics 016; 1(3): 1-5 ISSN: 456-145 Mths 016; 1(3): 1-5 016 Stts & Mths www.mthsounl.com Received: 05-07-016 Accepted: 06-08-016 C Lognthn Dept of Mthemtics

More information

SPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018.

SPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018. SPA7U/SPA7P: THE GALAXY Solutions fo Cousewok Questions distibuted on: 25 Jnuy 28. Solution. Assessed question] We e told tht this is fint glxy, so essentilly we hve to ty to clssify it bsed on its spectl

More information

EXPANSION OF LIQUIDS

EXPANSION OF LIQUIDS EXPNSION OF LIQUIDS. block of woo is flotin on wte t C with cetin volume V bove wte level. The tempetue of wte is slowly ise fom C to C. How the volume V chne with the ise of tempetue ) V will emin unchne

More information

MATHEMATICS IV 2 MARKS. 5 2 = e 3, 4

MATHEMATICS IV 2 MARKS. 5 2 = e 3, 4 MATHEMATICS IV MARKS. If + + 6 + c epesents cicle with dius 6, find the vlue of c. R 9 f c ; g, f 6 9 c 6 c c. Find the eccenticit of the hpeol Eqution of the hpeol Hee, nd + e + e 5 e 5 e. Find the distnce

More information

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013 Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo

More information

Lecture 10. Solution of Nonlinear Equations - II

Lecture 10. Solution of Nonlinear Equations - II Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution

More information

Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.

Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2. Element Uniqueness Poblem Dt Stuctues Let x,..., xn < m Detemine whethe thee exist i j such tht x i =x j Sot Algoithm Bucket Sot Dn Shpi Hsh Tbles fo (i=;i

More information

Math Section 4.2 Radians, Arc Length, and Area of a Sector

Math Section 4.2 Radians, Arc Length, and Area of a Sector Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 3 Due on Sep. 14, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt

More information

THEORY OF EQUATIONS OBJECTIVE PROBLEMS. If the eqution x 6x 0 0 ) - ) 4) -. If the sum of two oots of the eqution k is -48 ) 6 ) 48 4) 4. If the poduct of two oots of 4 ) -4 ) 4) - 4. If one oot of is

More information

Cylindrical and Spherical Coordinate Systems

Cylindrical and Spherical Coordinate Systems Clindical and Spheical Coodinate Sstems APPENDIX A In Section 1.2, we leaned that the Catesian coodinate sstem is deined b a set o thee mtall othogonal saces, all o which ae planes. The clindical and spheical

More information

Radian and Degree Measure

Radian and Degree Measure CHAT Pe-Calculus Radian and Degee Measue *Tigonomety comes fom the Geek wod meaning measuement of tiangles. It pimaily dealt with angles and tiangles as it petained to navigation, astonomy, and suveying.

More information

COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT

COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit

More information

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

ELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy:

ELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy: LCTROSTATICS. Quntiztion of Chge: Any chged body, big o smll, hs totl chge which is n integl multile of e, i.e. = ± ne, whee n is n intege hving vlues,, etc, e is the chge of electon which is eul to.6

More information

FSK 116 Semester 1 Mathematics and Other Essentials. Priorities

FSK 116 Semester 1 Mathematics and Other Essentials. Priorities FSK 6 Semeste Mthemtics nd Othe Essentils Pioities Know how YOUR clculto woks nd lwys hve YOUR clculto with you. Alwys hve pencil (nd n ese) t hnd when doing Physics. Geek Alphbet Alph Et Nu Tu Bet Thet

More information

Important design issues and engineering applications of SDOF system Frequency response Functions

Important design issues and engineering applications of SDOF system Frequency response Functions Impotnt design issues nd engineeing pplictions of SDOF system Fequency esponse Functions The following desciptions show typicl questions elted to the design nd dynmic pefomnce of second-ode mechnicl system

More information

PDF Created with deskpdf PDF Writer - Trial ::

PDF Created with deskpdf PDF Writer - Trial :: A APPENDIX D TRIGONOMETRY Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: http://www.docudesk.com D T i g o n o m e t FIGURE a A n g l e s Angles can be measued in degees

More information

( ) ( )( ) ˆ. Homework #8. Chapter 27 Magnetic Fields II.

( ) ( )( ) ˆ. Homework #8. Chapter 27 Magnetic Fields II. Homewok #8. hapte 7 Magnetic ields. 6 Eplain how ou would modif Gauss s law if scientists discoveed that single, isolated magnetic poles actuall eisted. Detemine the oncept Gauss law fo magnetism now eads

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 4 Due on Sep. 1, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt

More information

Topic/Objective: Essential Question: How do solve problems involving radian and/or degree measure?

Topic/Objective: Essential Question: How do solve problems involving radian and/or degree measure? Topic/Objective: 4- RADIAN AND DEGREE MEASURE Name: Class/Peiod: Date: Essential Question: How do solve poblems involving adian and/o degee measue? Questions: TRIGONOMETRY. Tigonomety, as deived fom the

More information

Exponents and Logarithms Exam Questions

Exponents and Logarithms Exam Questions Eponents nd Logrithms Em Questions Nme: ANSWERS Multiple Choice 1. If 4, then is equl to:. 5 b. 8 c. 16 d.. Identify the vlue of the -intercept of the function ln y.. -1 b. 0 c. d.. Which eqution is represented

More information

1. A man pulls himself up the 15 incline by the method shown. If the combined mass of the man and cart is 100 kg, determine the acceleration of the

1. A man pulls himself up the 15 incline by the method shown. If the combined mass of the man and cart is 100 kg, determine the acceleration of the 1. n pulls hiself up the 15 incline b the ethod shown. If the cobined ss of the n nd ct is 100 g deteine the cceletion of the ct if the n eets pull of 50 on the ope. eglect ll fiction nd the ss of the

More information

1. The sphere P travels in a straight line with speed

1. The sphere P travels in a straight line with speed 1. The sphee P tels in stight line with speed = 10 m/s. Fo the instnt depicted, detemine the coesponding lues of,,,,, s mesued eltie to the fixed Oxy coodinte system. (/134) + 38.66 1.34 51.34 10sin 3.639

More information

Two dimensional polar coordinate system in airy stress functions

Two dimensional polar coordinate system in airy stress functions I J C T A, 9(9), 6, pp. 433-44 Intentionl Science Pess Two dimensionl pol coodinte system in iy stess functions S. Senthil nd P. Sek ABSTRACT Stisfy the given equtions, boundy conditions nd bihmonic eqution.in

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Test # Review Math (Pe -calculus) Name MULTIPLE CHOICE. Choose the one altenative that best completes the statement o answes the question. Use an identit to find the value of the epession. Do not use a

More information

Section 35 SHM and Circular Motion

Section 35 SHM and Circular Motion Section 35 SHM nd Cicul Motion Phsics 204A Clss Notes Wht do objects do? nd Wh do the do it? Objects sometimes oscillte in simple hmonic motion. In the lst section we looed t mss ibting t the end of sping.

More information

Physics 2A Chapter 10 - Moment of Inertia Fall 2018

Physics 2A Chapter 10 - Moment of Inertia Fall 2018 Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.

More information

x a y n + b = 1 0<b a, n > 0 (1.1) x 1 - a y = b 0<b a, n > 0 (1.1') b n sin 2 + cos 2 = 1 x n = = cos 2 6 Superellipse (Lamé curve)

x a y n + b = 1 0<b a, n > 0 (1.1) x 1 - a y = b 0<b a, n > 0 (1.1') b n sin 2 + cos 2 = 1 x n = = cos 2 6 Superellipse (Lamé curve) 6 Supeellipse (Lmé cuve) 6. Equtios of supeellipse A supeellipse (hoizotlly log) is epessed s follows. Implicit Equtio y + b 0 0 (.) Eplicit Equtio y b - 0 0 (.') Whe 3, b, the supeellipses fo

More information

Non-Linear Dynamics Homework Solutions Week 2

Non-Linear Dynamics Homework Solutions Week 2 Non-Linea Dynamics Homewok Solutions Week Chis Small Mach, 7 Please email me at smach9@evegeen.edu with any questions o concens eguading these solutions. Fo the ececises fom section., we sketch all qualitatively

More information

The Area of a Triangle

The Area of a Triangle The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest

More information

Electricity & Magnetism Lecture 6: Electric Potential

Electricity & Magnetism Lecture 6: Electric Potential Electicity & Mgnetism Lectue 6: Electic Potentil Tody s Concept: Electic Potenl (Defined in tems of Pth Integl of Electic Field) Electicity & Mgnesm Lectue 6, Slide Stuff you sked bout:! Explin moe why

More information

Chapter Eight Notes N P U1C8S4-6

Chapter Eight Notes N P U1C8S4-6 Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that

More information

4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for.

4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for. 4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX Some reliminries: Let A be rel symmetric mtrix. Let Cos θ ; (where we choose θ π for Cos θ 4 purposes of convergence of the scheme)

More information

9.4 The response of equilibrium to temperature (continued)

9.4 The response of equilibrium to temperature (continued) 9.4 The esponse of equilibium to tempetue (continued) In the lst lectue, we studied how the chemicl equilibium esponds to the vition of pessue nd tempetue. At the end, we deived the vn t off eqution: d

More information

A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW. factor and remainder theorems A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

More information

Fluids & Bernoulli s Equation. Group Problems 9

Fluids & Bernoulli s Equation. Group Problems 9 Goup Poblems 9 Fluids & Benoulli s Eqution Nme This is moe tutoil-like thn poblem nd leds you though conceptul development of Benoulli s eqution using the ides of Newton s 2 nd lw nd enegy. You e going

More information

When two numbers are written as the product of their prime factors, they are in factored form.

When two numbers are written as the product of their prime factors, they are in factored form. 10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The

More information

defined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z)

defined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z) 08 Tylo eie nd Mcluin eie A holomophic function f( z) defined on domin cn be expnded into the Tylo eie ound point except ingul point. Alo, f( z) cn be expnded into the Mcluin eie in the open dik with diu

More information

4.3 Right Triangle Trigonometry

4.3 Right Triangle Trigonometry Section. Right Tiangle Tigonomet 77. Right Tiangle Tigonomet The Si Tigonometic Functions Ou second look at the tigonometic functions is fom a ight tiangle pespective. Conside a ight tiangle, with one

More information

Electric Potential. and Equipotentials

Electric Potential. and Equipotentials Electic Potentil nd Euipotentils U Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil

More information

Ch 26 - Capacitance! What s Next! Review! Lab this week!

Ch 26 - Capacitance! What s Next! Review! Lab this week! Ch 26 - Cpcitnce! Wht s Next! Cpcitnce" One week unit tht hs oth theoeticl n pcticl pplictions! Cuent & Resistnce" Moving chges, finlly!! Diect Cuent Cicuits! Pcticl pplictions of ll the stuff tht we ve

More information

ITI Introduction to Computing II

ITI Introduction to Computing II ITI 1121. Intoduction to Computing II Mcel Tucotte School of Electicl Engineeing nd Compute Science Abstct dt type: Stck Stck-bsed lgoithms Vesion of Febuy 2, 2013 Abstct These lectue notes e ment to be

More information

Probabilistic Retrieval

Probabilistic Retrieval CS 630 Lectue 4: 02/07/2006 Lectue: Lillin Lee Scibes: Pete Bbinski, Dvid Lin Pobbilistic Retievl I. Nïve Beginnings. Motivtions b. Flse Stt : A Pobbilistic Model without Vition? II. Fomultion. Tems nd

More information

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3 DEPATMENT OF CIVIL AND ENVIONMENTAL ENGINEEING FLID MECHANICS III Solutions to Poblem Sheet 3 1. An tmospheic vote is moelle s combintion of viscous coe otting s soli boy with ngul velocity Ω n n iottionl

More information

SURFACE TENSION. e-edge Education Classes 1 of 7 website: , ,

SURFACE TENSION. e-edge Education Classes 1 of 7 website: , , SURFACE TENSION Definition Sufce tension is popety of liquid by which the fee sufce of liquid behves like stetched elstic membne, hving contctive tendency. The sufce tension is mesued by the foce cting

More information

AP Calculus AB Exam Review Sheet B - Session 1

AP Calculus AB Exam Review Sheet B - Session 1 AP Clcls AB Em Review Sheet B - Session Nme: AP 998 # Let e the nction given y e.. Find lim nd lim.. Find the solte minimm vle o. Jstiy tht yo nswe is n solte minimm. c. Wht is the nge o? d. Conside the

More information

Chapter 1: Introduction to Polar Coordinates

Chapter 1: Introduction to Polar Coordinates Habeman MTH Section III: ola Coodinates and Comple Numbes Chapte : Intoduction to ola Coodinates We ae all comfotable using ectangula (i.e., Catesian coodinates to descibe points on the plane. Fo eample,

More information

Physical Security Countermeasures. This entire sheet. I m going to put a heptadecagon into game.

Physical Security Countermeasures. This entire sheet. I m going to put a heptadecagon into game. Phsicl Secuit Countemesues This entie sheet Telmo, AHI I m going to put heptdecgon into gme. Cssie Hung Mechnicl lockpicking is mechnicked geometic constuctions with compss nd stightedge. Ech lock will

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Math Pecalculus Ch. 6 Review Name SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. Solve the tiangle. ) ) 6 7 0 Two sides and an angle (SSA) of a tiangle ae

More information

fractions Let s Learn to

fractions Let s Learn to 5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

More information

Continuous Charge Distributions

Continuous Charge Distributions Continuous Chge Distibutions Review Wht if we hve distibution of chge? ˆ Q chge of distibution. Q dq element of chge. d contibution to due to dq. Cn wite dq = ρ dv; ρ is the chge density. = 1 4πε 0 qi

More information

Physics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems.

Physics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems. Physics 55 Fll 5 Midtem Solutions This midtem is two hou open ook, open notes exm. Do ll thee polems. [35 pts] 1. A ectngul ox hs sides of lengths, nd c z x c [1] ) Fo the Diichlet polem in the inteio

More information

KR- 21 FOR FORMULA SCORED TESTS WITH. Robert L. Linn, Robert F. Boldt, Ronald L. Flaugher, and Donald A. Rock

KR- 21 FOR FORMULA SCORED TESTS WITH. Robert L. Linn, Robert F. Boldt, Ronald L. Flaugher, and Donald A. Rock RB-66-4D ~ E S [ B A U R L C L Ii E TI KR- 21 FOR FORMULA SCORED TESTS WITH OMITS SCORED AS WRONG Robet L. Linn, Robet F. Boldt, Ronald L. Flaughe, and Donald A. Rock N This Bulletin is a daft fo inteoffice

More information

u(r, θ) = 1 + 3a r n=1

u(r, θ) = 1 + 3a r n=1 Mth 45 / AMCS 55. etuck Assignment 8 ue Tuesdy, Apil, 6 Topics fo this week Convegence of Fouie seies; Lplce s eqution nd hmonic functions: bsic popeties, computions on ectngles nd cubes Fouie!, Poisson

More information

1. Twelve less than five times a number is thirty three. What is the number

1. Twelve less than five times a number is thirty three. What is the number Alger 00 Midterm Review Nme: Dte: Directions: For the following prolems, on SEPARATE PIECE OF PAPER; Define the unknown vrile Set up n eqution (Include sketch/chrt if necessr) Solve nd show work Answer

More information

NS-IBTS indices calculation procedure

NS-IBTS indices calculation procedure ICES Dt Cente DATRAS 1.1 NS-IBTS indices 2013 DATRAS Pocedue Document NS-IBTS indices clcultion pocedue Contents Genel... 2 I Rw ge dt CA -> Age-length key by RFA fo defined ge nge ALK... 4 II Rw length

More information

Chapter 21 The Electric Field I: Discrete Charge Distributions

Chapter 21 The Electric Field I: Discrete Charge Distributions Chpte The lectic ield I: Discete Chge Distibutions Conceptul oblems Objects e composed of toms which e composed of chged pticles (potons nd electons); howeve, we el obseve the effects of the electosttic

More information

4.2 Boussinesq s Theory. Contents

4.2 Boussinesq s Theory. Contents 00477 Pvement Stuctue 4. Stesses in Flexible vement Contents 4. Intoductions to concet of stess nd stin in continuum mechnics 4. Boussinesq s Theoy 4. Bumiste s Theoy 4.4 Thee Lye System Weekset Sung Chte

More information

NARAYANA I I T / P M T A C A D E M Y. C o m m o n Pr a c t i c e T e s t 0 9 XI-IC SPARK Date: PHYSICS CHEMISTRY MATHEMATICS

NARAYANA I I T / P M T A C A D E M Y. C o m m o n Pr a c t i c e T e s t 0 9 XI-IC SPARK Date: PHYSICS CHEMISTRY MATHEMATICS . (D). (B). (). (). (D). (A) 7. () 8. (B) 9. (B). (). (A). (D). (B). (). (B) NAAYANA I I T / T A A D E Y XIS-I-IIT-SA (..7) o m m o n c t i c e T e s t 9 XI-I SA Dte:..7 ANSWE YSIS EISTY ATEATIS. (B).

More information

FI 2201 Electromagnetism

FI 2201 Electromagnetism FI 1 Electomgnetism Alexnde A. Isknd, Ph.D. Physics of Mgnetism nd Photonics Resech Goup Electosttics ELECTRIC PTENTIALS 1 Recll tht we e inteested to clculte the electic field of some chge distiution.

More information

Chapter 25 Electric Potential

Chapter 25 Electric Potential Chpte 5 lectic Potentil consevtive foces -> potentil enegy - Wht is consevtive foce? lectic potentil = U / : the potentil enegy U pe unit chge is function of the position in spce Gol:. estblish the eltionship

More information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

arxiv: v1 [math.nt] 12 May 2017

arxiv: v1 [math.nt] 12 May 2017 SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

The 1958 musical Merry Andrew starred Danny Kaye as

The 1958 musical Merry Andrew starred Danny Kaye as The 1958 musical Me Andew staed Dann Kae as Andew Laabee, a teache with a flai fo using unconventional methods in his classes. He uses a musical numbe to teach the Pthagoean theoem, singing and dancing

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information