Mathematics for Physicists and Astronomers

Size: px
Start display at page:

Download "Mathematics for Physicists and Astronomers"

Transcription

1 PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester ASTRONOMY DEPARTMENT OF PHYSICS AND ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Spring 2017 Mthemtics for Physicists nd Astronomers Answer question ONE (Compulsory) nd TWO other questions, one ech from section A nd section B. Instructions: All questions Answer ALL re questions. mrked out of ten. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. There re 120 possible mrks for this pper. 2 hours The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. TURN OVER 1

2 Unit 7: Vectors nd Differentition - 50 mrks totl 1. () For the sclr field φ = x 2 y cosh z, clculte i. the grdient, φ, ii. the lplcin, 2 φ. (b) For the vector field F = e z cos (x y) i + e x cos (yz) j, clculte i. the divergence, F, ii. the curl, F, iii. the lplcin, 2 F. [3] [3] [4] 2. () By using plne polr coordintes, (r, θ), show tht the ccelertion ( = r) of [7] prticle in two dimensions is You my ssume the results = ( r r θ 2 ) ˆr + (2ṙ θ + r θ) ˆθ. d ˆr d t = θ ˆθ nd d ˆθ d t = θ ˆr. (b) Consider prticle moving on two dimensionl pth, so tht its position s function of time, t, is given by r(t) = e 2t θ(t) = t 2. i. Obtin expressions for the rdil (ˆr ) nd trnsverse (ˆθ ) components of the [4] ccelertion. ii. At wht time(s) is the force on the prticle purely trnsverse? 3. Consider the two dimensionl vector field F = 1 1 2x 2 i + y x j. () Find n eqution for the field lines of this field, giving n explicit expression for y [5] in terms of x. (b) Sketch the fields lines. CONTINUED 2

3 4. () Show tht, in sphericl polr coordintes, the unit vectors in the direction of in- [6] cresing r, θ nd φ re ˆr = sin θ cos φ i + sin θ sin φ j + cos θ k ˆθ = cos θ cos φ i + cos θ sin φ j sin θ k ˆφ = sin φ i + cos φ j. (b) Consider the vector field F = xz i + yz j (x 2 + y 2 ) k. i. Express this field in sphericl polr coordintes using the bsis vectors in (). [3] ii. Obtin the curl of the field in sphericl polr coordintes. [5] iii. Convert your nswer bck to crtesin coordintes. Note: You my wish to use the formul sheets provided on pges 6 onwrds. TURN OVER 3

4 Unit 8: Vector Integrtion - 50 mrks totl 5. A prticle moves through force field F = x 3 i + x y j + xz 2 k. Find the work done on the prticle if it moves from the origin O = (0, 0, 0) to the point P = (1, 1, 1) long: () the curve described by the prmeteristion x = t, y = t 2, z = t 3, [3] (b) the stright line OP. 6. The mss density of thick sphericl shell is given by ρ(r) = ρ 0 where r is the distnce [8] r3 from the centre of the sphere nd ρ 0 is constnt. The inner nd outer rdii of the shell re nd b respectively. Clculte the moment of inerti of the shell when centred on the origin nd rotting bout the z-xis. 7. Evlute the double integrl I = x 2 y dx dy where R is the region described by 0 x 2 nd x 2 y 4. R [7] 8. () Stte Stokes Theorem, explining the mening of ll terms. [6] (b) Clculte the circultion of the vector field V = 2 yi + 3x j + 0k round the circle [9] x 2 + y 2 = 2, z = 0 9. A force field F = x yi + y 2 j moves prticle round closed loop in the x y plne. [8] The prticle strts t the origin, moves long the x-xis to (1, 0), then follows the curve y = 1 x 2 to (0, 1) nd finlly returns to the origin long the y-xis. Use Green s theorem in the plne to clculte the work done by the force on the prticle. 10. Show tht the vector field V = (x + y + 3z)i + (x 2 y + 2z)j + (3x + 2 y 4z)k is [7] conservtive, nd find the sclr potentil φ such tht V = φ. CONTINUED 4

5 Unit 9: Probbility nd Sttistics - 20 mrks totl 11. You re presented with three coins, two of which re fir nd one counterfeit tht lwys lnds heds. You choose coin t rndom nd flip it three times nd ech time it lnds heds. Use Byes theorem to find the probbility tht the chosen coin is the counterfeit one. [5] 12. Consider rndom vrible A tht cn tke three vlues A = { 1, 0, 1} with probbilities P A ( 1) = 0.2, P A (0) = 0.3, nd P A (1) = 0.5. A function of the rndom vrible A is defined s F() =. Find the probbility distribution of F(). [5] 13. The probbility distribution function p(x) of rndom vrible is positive definite, such tht the probbility of its vlue lying between x nd x + d x is given by p(x)d x. The rndom vrible is constrined between x = 0 nd x = 1 nd p(x) = ke x. () Sketch the probbility distribution function p(x). (b) Find the constnt k. (c) Find the men vlue of x. (d) Find the vrince of x. [4] TURN OVER 5

6 Cylindricl Polr Coordintes For sclr field V (r, θ, z), For vector field F(r, θ, z), Supplementry Formule V = V r ˆr + 1 V r θ ˆθ + V z ẑ 2 V = 1 r V V r r r r 2 θ + 2 V 2 z. 2 F = 1 r r (r F r) + 1 F θ r θ + F z z F = 1 ˆr r ˆθ ẑ r / r / θ / z F r r F θ F z 1 F z = r θ F θ Fr ˆr + z z F z ˆθ + 1 r r r (r F θ) F r ẑ θ Sphericl Polr Coordintes For sclr field V (r, θ, φ), V = V r ˆr + 1 V r θ ˆθ + 1 V r sin θ φ ˆφ. 2 V = 1 r 2 V 1 + r 2 r r r 2 sin θ θ For vector field F(r, θ, φ), r (r2 F r ) + 1 r sin θ θ (sin θ F θ) + 1 sin θ V θ F φ F = 1 r 2 r sin θ φ 1/(r 2 sin θ) ˆr 1/(r sin θ) ˆθ 1/r ˆφ F = / r / θ / φ F r r F θ r sin θ F φ = 1 r sin θ θ (sin θ F φ) F θ ˆr φ r sin θ + 1 r 2 sin 2 θ 2 V φ 2. F r φ r (r F φ) ˆθ + 1 r r (r F θ) F r ˆφ. θ CONTINUED 6

7 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c 2 proton mss m p = kg = MeV c 2 neutron mss m n = kg = MeV c 2 Plnck s constnt h = J s Dirc s constnt ( = h/2π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = N m 2 kg 2 Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m 2 K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 2 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 2 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

8 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x x 2 2 +x 2 1 x x x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx

9 Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A

10 Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk

11 END OF EXAMINATION PAPER 11

Data Provided: A formula sheet and table of physical constants is attached to this paper.

Data Provided: A formula sheet and table of physical constants is attached to this paper. PHY106 PHY47 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 009-010 ASTRONOMY DEPARTMENT

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. DEPARTMENT OF PHYSICS & Autumn Semester ASTRONOMY

Data Provided: A formula sheet and table of physical constants is attached to this paper. DEPARTMENT OF PHYSICS & Autumn Semester ASTRONOMY PHY221 PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper.

Data Provided: A formula sheet and table of physical constants is attached to this paper. PHY15-B PHY47 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 009-010 ASTRONOMY DEPARTMENT

More information

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available.

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available. DEPARTMENT OF PHYSICS AND ASTRONOMY Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner grph pper is vilble. Spring Semester 2015-2016 PHYSICS 1 HOUR Answer questions

More information

Data Provided: A formula sheet and table of physical constants are attached to this paper.

Data Provided: A formula sheet and table of physical constants are attached to this paper. PHY40 Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester 015-016 PHYSICS OF MUSIC 1.5 HOURS ANSWER ANY QUESTIONS. All questions

More information

Introduction to Astrophysics

Introduction to Astrophysics PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

Introduction to Astrophysics

Introduction to Astrophysics PHY104 PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS

Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn (2014) SOLID STATE PHYSICS 2 HOURS The pper is divided into 5 questions. Answer

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. Answer question ONE (Compulsory) and TWO other questions.

Data Provided: A formula sheet and table of physical constants is attached to this paper. Answer question ONE (Compulsory) and TWO other questions. Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (2014-2015) NUCLEAR PHYSICS 2 HOURS Answer question ONE (Compulsory)

More information

Data Provided: A formula sheet and table of physical constants are attached to this paper. DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester

Data Provided: A formula sheet and table of physical constants are attached to this paper. DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. Ancillry Mteril: None DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester 2016-2017 MEDICAL PHYSICS: Physics of Living Systems

More information

ADVANCED QUANTUM MECHANICS

ADVANCED QUANTUM MECHANICS PHY216 PHY472 Dt Provided: Dt Provided: Formul Formul sheet ndsheet physicl nd constnts physicl constnts DEPARTMENT DEPARTMENT OF PHYSICS OF PHYSICS & Spring& Semester 2015-2016 Autumn Semester 2009-2010

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear-linear graph paper is required.

Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear-linear graph paper is required. Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner-liner grph pper is required. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (2015-2016) SEMICONDUCTOR PHYSICS

More information

Electricity and Magnetism

Electricity and Magnetism PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear-linear graph paper is required.

Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear-linear graph paper is required. Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner-liner grph pper is required. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (015) SEMICONDUCTOR PHYSICS AND TECHNOLOGY

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS

Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn (015) SOLID STATE PHYSICS HOURS The pper is divided into 5 questions. Answer compulsory

More information

Mechanics, Oscillations and Waves

Mechanics, Oscillations and Waves PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

Introduction to Astrophysics

Introduction to Astrophysics PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear-linear graph paper is required.

Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear-linear graph paper is required. Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner-liner grph pper is required. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (2016-2017) SEMICONDUCTOR PHYSICS

More information

Data Provided: A formula sheet and table of physical constants are attached to this paper.

Data Provided: A formula sheet and table of physical constants are attached to this paper. PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

DARK MATTER AND THE UNIVERSE. Answer question ONE (Compulsory) and TWO other questions.

DARK MATTER AND THE UNIVERSE. Answer question ONE (Compulsory) and TWO other questions. Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Picture of glxy cluster Abell 2218 required for question 4(c) is ttched. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn Semester

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. MEDICAL PHYSICS: Aspects of Medical Imaging and Technology

Data Provided: A formula sheet and table of physical constants is attached to this paper. MEDICAL PHYSICS: Aspects of Medical Imaging and Technology Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Ancillry Mteril: None DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn semester exm (Acdemic yer 2014-15) MEDICAL PHYSICS: Aspects

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS

Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn (2016) SOLID STATE PHYSICS 2 HOURS Instructions: The pper is divided into 5 questions.

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. MEDICAL PHYSICS: Physics of Living Systems 2 2 HOURS

Data Provided: A formula sheet and table of physical constants is attached to this paper. MEDICAL PHYSICS: Physics of Living Systems 2 2 HOURS Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Ancillry Mteril: None DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (2015/2016) MEDICAL PHYSICS: Physics of Living

More information

Data Provided: A formula sheet and table of physical constants are attached to this paper.

Data Provided: A formula sheet and table of physical constants are attached to this paper. PHY51A Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (015-016) From Thermodynmics to Atomic nd Nucler Physics Pper

More information

Data Provided: A formula sheet and table of physical constants are attached to this paper.

Data Provided: A formula sheet and table of physical constants are attached to this paper. PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available.

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available. DEPARTMENT OF PHYSICS AND ASTRONOMY Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner grph pper is vilble. Autumn Semester 2015-2016 PHYSICS : 3 HOURS Answer questions

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available.

Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available. DEPARTMENT OF PHYSICS AND ASTRONOMY Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner grph pper is vilble. Autumn Semester 2014-2015 PHYSICS : Elements of Physics 2

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. THE PHYSICS OF SOFT CONDENSED MATTER

Data Provided: A formula sheet and table of physical constants is attached to this paper. THE PHYSICS OF SOFT CONDENSED MATTER Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Ancillry Mteril: Grph pper (liner) DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn Semester (2015) THE PHYSICS OF SOFT CONDENSED

More information

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available.

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available. DEPARTMENT OF PHYSICS AND ASTRONOMY Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner grph pper is vilble. Spring Semester 2016-2017 PHYSICS PHY009: 3 HOURS Answer questions

More information

A formula sheet and table of physical constants is attached to this paper.

A formula sheet and table of physical constants is attached to this paper. Dt Provided: A formul heet nd tble of phyicl contnt i ttched to thi pper. Ancillry Mteril: DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn 2014 The phyic of mteril 2 hour Intruction: Anwer quetion ONE (Compulory)

More information

Year 12 Mathematics Extension 2 HSC Trial Examination 2014

Year 12 Mathematics Extension 2 HSC Trial Examination 2014 Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of

More information

Candidates must show on each answer book the type of calculator used.

Candidates must show on each answer book the type of calculator used. UNIVERSITY OF EAST ANGLIA School of Mthemtics My/June UG Exmintion 2007 2008 ELECTRICITY AND MAGNETISM Time llowed: 3 hours Attempt FIVE questions. Cndidtes must show on ech nswer book the type of clcultor

More information

Phys 4321 Final Exam December 14, 2009

Phys 4321 Final Exam December 14, 2009 Phys 4321 Finl Exm December 14, 2009 You my NOT use the text book or notes to complete this exm. You nd my not receive ny id from nyone other tht the instructor. You will hve 3 hours to finish. DO YOUR

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions re of equl vlue

More information

Mathematics Extension 2

Mathematics Extension 2 S Y D N E Y B O Y S H I G H S C H O O L M O O R E P A R K, S U R R Y H I L L S 005 HIGHER SCHOOL CERTIFICATE TRIAL PAPER Mthemtics Extension Generl Instructions Totl Mrks 0 Reding Time 5 Minutes Attempt

More information

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015 Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, 15 1. (i) Eight eul chrges re locted t corners of cube of side s, s shown in Fig. 1. Find electric potentil t one corner, tking zero potentil to be

More information

Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas

Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

Student Handbook for MATH 3300

Student Handbook for MATH 3300 Student Hndbook for MATH 3300 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 0.5 0 0.5 0.5 0 0.5 If people do not believe tht mthemtics is simple, it is only becuse they do not relize how complicted life is. John Louis

More information

Phys 6321 Final Exam - Solutions May 3, 2013

Phys 6321 Final Exam - Solutions May 3, 2013 Phys 6321 Finl Exm - Solutions My 3, 2013 You my NOT use ny book or notes other thn tht supplied with this test. You will hve 3 hours to finish. DO YOUR OWN WORK. Express your nswers clerly nd concisely

More information

SAINT IGNATIUS COLLEGE

SAINT IGNATIUS COLLEGE SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

Week 10: Line Integrals

Week 10: Line Integrals Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.

More information

Math 100 Review Sheet

Math 100 Review Sheet Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

More information

2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm

2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm 2.57/2.570 Midterm Exm No. 1 Mrch 31, 2010 11:00 m -12:30 pm Instructions: (1) 2.57 students: try ll problems (2) 2.570 students: Problem 1 plus one of two long problems. You cn lso do both long problems,

More information

Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016

Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016 Physics 333, Fll 16 Problem Set 7 due Oct 14, 16 Reding: Griffiths 4.1 through 4.4.1 1. Electric dipole An electric dipole with p = p ẑ is locted t the origin nd is sitting in n otherwise uniform electric

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

Final Exam Solutions, MAC 3474 Calculus 3 Honors, Fall 2018

Final Exam Solutions, MAC 3474 Calculus 3 Honors, Fall 2018 Finl xm olutions, MA 3474 lculus 3 Honors, Fll 28. Find the re of the prt of the sddle surfce z xy/ tht lies inside the cylinder x 2 + y 2 2 in the first positive) octnt; is positive constnt. olution:

More information

In Mathematics for Construction, we learnt that

In Mathematics for Construction, we learnt that III DOUBLE INTEGATION THE ANTIDEIVATIVE OF FUNCTIONS OF VAIABLES In Mthemtics or Construction, we lernt tht the indeinite integrl is the ntiderivtive o ( d ( Double Integrtion Pge Hence d d ( d ( The ntiderivtive

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS Victorin CertiÞcte of Euction 007 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Wors SPECIALIST MATHEMATICS Written exmintion Mony 5 November 007 Reing time: 3.00 pm to 3.5 pm

More information

Electromagnetism Answers to Problem Set 10 Spring 2006

Electromagnetism Answers to Problem Set 10 Spring 2006 Electromgnetism 76 Answers to Problem Set 1 Spring 6 1. Jckson Prob. 5.15: Shielded Bifilr Circuit: Two wires crrying oppositely directed currents re surrounded by cylindricl shell of inner rdius, outer

More information

CHM Physical Chemistry I Chapter 1 - Supplementary Material

CHM Physical Chemistry I Chapter 1 - Supplementary Material CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion

More information

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) = Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?

More information

UNIVERSITY OF BOLTON SCHOOL OF ENGINEERING B.ENG (HONS) ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATION SEMESTER /2018

UNIVERSITY OF BOLTON SCHOOL OF ENGINEERING B.ENG (HONS) ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATION SEMESTER /2018 ENG005 B.ENG (HONS) ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATION SEMESTER 1-017/018 MODULE NO: EEE4001 Dte: 19Jnury 018 Time:.00 4.00 INSTRUCTIONS TO CANDIDATES: There re SIX questions. Answer ANY

More information

Math 113 Exam 1-Review

Math 113 Exam 1-Review Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =. Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts

More information

ragsdale (zdr82) HW2 ditmire (58335) 1

ragsdale (zdr82) HW2 ditmire (58335) 1 rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc

More information

Mathematics Extension 2

Mathematics Extension 2 00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd

More information

Session Trimester 2. Module Code: MATH08001 MATHEMATICS FOR DESIGN

Session Trimester 2. Module Code: MATH08001 MATHEMATICS FOR DESIGN School of Science & Sport Pisley Cmpus Session 05-6 Trimester Module Code: MATH0800 MATHEMATICS FOR DESIGN Dte: 0 th My 06 Time: 0.00.00 Instructions to Cndidtes:. Answer ALL questions in Section A. Section

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

MATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2

MATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2 MATH 53 WORKSHEET MORE INTEGRATION IN POLAR COORDINATES ) Find the volume of the solid lying bove the xy-plne, below the prboloid x + y nd inside the cylinder x ) + y. ) We found lst time the set of points

More information

Total Score Maximum

Total Score Maximum Lst Nme: Mth 8: Honours Clculus II Dr. J. Bowmn 9: : April 5, 7 Finl Em First Nme: Student ID: Question 4 5 6 7 Totl Score Mimum 6 4 8 9 4 No clcultors or formul sheets. Check tht you hve 6 pges.. Find

More information

Mathematics Extension Two

Mathematics Extension Two Student Number 04 HSC TRIAL EXAMINATION Mthemtics Etension Two Generl Instructions Reding time 5 minutes Working time - hours Write using blck or blue pen Bord-pproved clcultors my be used Write your Student

More information

l 2 p2 n 4n 2, the total surface area of the

l 2 p2 n 4n 2, the total surface area of the Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

More information

This final is a three hour open book, open notes exam. Do all four problems.

This final is a three hour open book, open notes exam. Do all four problems. Physics 55 Fll 27 Finl Exm Solutions This finl is three hour open book, open notes exm. Do ll four problems. [25 pts] 1. A point electric dipole with dipole moment p is locted in vcuum pointing wy from

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

US01CMTH02 UNIT Curvature

US01CMTH02 UNIT Curvature Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently

More information

Mathematics Extension 2

Mathematics Extension 2 00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Extension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors m be used A tble of stndrd

More information

Reference. Vector Analysis Chapter 2

Reference. Vector Analysis Chapter 2 Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter

More information

Unit 5. Integration techniques

Unit 5. Integration techniques 18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A-1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd

More information

Summary: Method of Separation of Variables

Summary: Method of Separation of Variables Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section

More information

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

More information

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students.

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students. - 5 - TEST 2 This test is on the finl sections of this session's syllbus nd should be ttempted by ll students. Anything written here will not be mrked. - 6 - QUESTION 1 [Mrks 22] A thin non-conducting

More information

SULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING

SULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS

More information

Physics 241 Exam 1 February 19, 2004

Physics 241 Exam 1 February 19, 2004 Phsics 241 Em 1 Februr 19, 24 One (both sides) 8 1/2 11 crib sheet is llowed. It must be of our own cretion. k = 1 = 9 1 9 N m2 4p 2 2 = 8.85 1-12 N m 2 e =1.62 1-19 c = 2.99792458 1 8 m/s (speed of light)

More information

Final Review, Math 1860 Thomas Calculus Early Transcendentals, 12 ed

Final Review, Math 1860 Thomas Calculus Early Transcendentals, 12 ed Finl Review, Mth 860 Thoms Clculus Erly Trnscendentls, 2 ed 6. Applictions of Integrtion: 5.6 (Review Section 5.6) Are between curves y = f(x) nd y = g(x), x b is f(x) g(x) dx nd similrly for x = f(y)

More information

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

Math 116 Final Exam April 26, 2013

Math 116 Final Exam April 26, 2013 Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems

More information

Problem Solving 7: Faraday s Law Solution

Problem Solving 7: Faraday s Law Solution MASSACHUSETTS NSTTUTE OF TECHNOLOGY Deprtment of Physics: 8.02 Prolem Solving 7: Frdy s Lw Solution Ojectives 1. To explore prticulr sitution tht cn led to chnging mgnetic flux through the open surfce

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS Victorin Certificte of Euction 08 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Tuesy 5 June 08 Reing time:.00 pm to.5 pm (5 minutes) Writing

More information

MAT187H1F Lec0101 Burbulla

MAT187H1F Lec0101 Burbulla Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is

More information

4. Calculus of Variations

4. Calculus of Variations 4. Clculus of Vritions Introduction - Typicl Problems The clculus of vritions generlises the theory of mxim nd minim. Exmple (): Shortest distnce between two points. On given surfce (e.g. plne), nd the

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS Victorin Certificte of Euction 08 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Friy 9 November 08 Reing time: 9.00 m to 9.5 m (5 minutes) Writing

More information

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know. Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must

More information

Indefinite Integral. Chapter Integration - reverse of differentiation

Indefinite Integral. Chapter Integration - reverse of differentiation Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the

More information

(6.5) Length and area in polar coordinates

(6.5) Length and area in polar coordinates 86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx + 6 6 x x dx Centre of mss +

More information

MATH 13 FINAL STUDY GUIDE, WINTER 2012

MATH 13 FINAL STUDY GUIDE, WINTER 2012 MATH 13 FINAL TUY GUI, WINTR 2012 This is ment to be quick reference guide for the topics you might wnt to know for the finl. It probbly isn t comprehensive, but should cover most of wht we studied in

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS Victorin Certificte of Euction 04 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Friy 7 November 04 Reing time: 9.00 m to 9.5 m (5 minutes) Writing

More information

HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016

HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016 HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M

More information

CBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0

CBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0 CBSE-XII- EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section

More information

df dx There is an infinite number of different paths from

df dx There is an infinite number of different paths from Integrl clculus line integrls Feb 7, 18 From clculus, in the cse of single vrible x1 F F x F x f x dx, where f x 1 x df dx Now, consider the cse tht two vribles re t ply. Suppose,, df M x y dx N x y dy

More information

Final Exam - Review MATH Spring 2017

Final Exam - Review MATH Spring 2017 Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.

More information

Today in Physics 122: work, energy and potential in electrostatics

Today in Physics 122: work, energy and potential in electrostatics Tody in Physics 1: work, energy nd potentil in electrosttics Leftovers Perfect conductors Fields from chrges distriuted on perfect conductors Guss s lw for grvity Work nd energy Electrosttic potentil energy,

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

Improper Integrals, and Differential Equations

Improper Integrals, and Differential Equations Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

MATH , Calculus 2, Fall 2018

MATH , Calculus 2, Fall 2018 MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly

More information