MTH Assignment 1 : Real Numbers, Sequences
|
|
- Edward Newton
- 6 years ago
- Views:
Transcription
1 MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 3. Let y (, ) ad (, ). Evaluate lim (2) y. 4. For a R, let = a ad + = 4 (2 + 3) for all N. Show that ( ) coverges if ad oly if a 3. Moreover, fid the limit of the sequece whe it coverges. 5. Show that the sequece ( ) defied by = 2 ad + = ( ) for N satisfies the Cauchy criterio. 6. Let = for N. Show that 2 2 sequece ( ) satisfy the Cauchy criterio? for every N. Does the 7. Let ( ) be defied by =, 2 = 2 ad +2 = for. Show that ( ) coverges. Further, by observig that = + + 2, fid the limit of ( ). Assigmet 2 : Cotiuity, Eistece of miimum, Itermediate Value Property. Let [] deote the iteger part of the real umber. Suppose f() = g()h() where g() = [ 2 ] ad h() = si 2π. Discuss the cotiuity/discotiuity of f, g ad h at = 2 ad = Determie the poits of cotiuity for the fuctio f : R R defied by f() = { 2 if is ratioal + 3 if is irratioal. 3. Let f : R R be a cotiuous fuctio ad let, c R. Show that if f( ) > c, the there eists a δ > such that f() > c for all ( δ, + δ). 4. Let f : [, ] (, ) be a o-to fuctio. Show that f is ot cotiuous o [, ]. 5. Let f : [a, b] R ad for every [a, b] there eists y [a, b] such that f(y) < 2 f(). Fid if{ f() : [a, b]}. Show that f is ot cotiuous o [a, b]. 6. Let f : [, 2] R be a cotiuous fuctio ad f() = f(2). Prove that there eist real umbers, 2 [, 2] such that 2 = ad f( 2 ) = f( ). 7. Let p be a odd degree polyomial ad g : R R be a bouded cotiuous fuctio. Show that there eists R such that p( ) = g( ). Further show that the equatio si = + cos 2 has a solutio i R. + 2 Assigmet 3 : Derivatives, Maima ad Miima, Rolle s Theorem. Show that the fuctio f() = is differetiable at. More geerally, if f is cotiuous at, the g() = f() is differetiable at.
2 2. Prove that if f : R R is a eve fuctio (i.e., f( ) = f() for all R) ad has a derivative at every poit, the the derivative f is a odd fuctio (i.e.,f( ) = f() for all R). 3. Show that amog all triagles with give base ad the correspodig verte agle, the isosceles triagle has the maimum area. 4. Show that eactly two real values of satisfy the equatio 2 = si + cos. 5. Suppose f is cotiuous o [a, b], differetiable o (a, b) ad satisfies f 2 (a) f 2 (b) = a 2 b 2. The show that the equatio f ()f() = has at least oe root i (a, b). 6. Let f : (, ) R be twice differetiable. Suppose f( ) = for all N. Show that f () = f () =. 7. Let f : (, ) R be a twice differetiable fuctio such that f () >. Show that there eists N such that f( ). Assigmet 4 : Mea Value Theorem, Taylor s Theorem, Curve Sketchig. Show that y ( y) y ( y) if < y, N. 2. Let f : [, ] R be differetiable, f( 2 ) = 2 ad < α <. Suppose f () α for all [, ]. Show that f() < for all [, ]. 3. Let f : [a, b] R be cotiuous o [a, b] ad differetiable o (a, b). Suppose that f(a) = a ad f(b) = b. Show that there is c (a, b) such that f (c) =. Further, show that there are distict c, c 2 (a, b) such that f (c ) + f (c 2 ) = Usig Cauchy Mea Value Theorem, show that (a) 2 2! < cos for. (b) 3 3! < si for >. 5. Fid lim (6 ) 5 ad lim 5 +( + ). 6. Sketch the graphs of f() = ad f() = (a) Let f : [a, b] R be such that f () for all [a, b]. Suppose [a, b]. Show that for ay [a, b] f() f( ) + f ( )( ) i.e., the graph of f lies above the taget lie to the graph at (, f( )). (b) Show that cos y cos ( y) si for all, y [ π 2, 3π 2 ]. 8. Suppose f is a three times differetiable fuctio o [, ] such that f( ) =, f() = ad f () =. Usig Taylor s theorem show that f (c) 3 for some c (, ). Assigmet 5 : Series, Power Series, Taylor Series. Let f : [, ] R ad a = f( ) f( + ). Show that if f is cotiuous the = a coverges ad if f is differetiable ad f () < for all [, ] the = a coverges. 2. I each of the followig cases, discuss the covergece/divergece of the series = a where a equals:
3 (a) + (e) (b) cos (c) 2 ( ) (d) l 2 (f) log, (p > ) p (g) e (cos ) 2 si 3. Let a ad = that if = b be series of positive terms satisfyig a + a b + b = b coverges the a also coverges. Test the series = 4. Show that the series coverges. 5. Show that the series = ( ) si 6. Determie the values of for which the series 7. Show that cos = ( ) = (2)! 2, R. coverges but ot absolutely. = ( ) = coverges. ( ) + (+) for all N. Show 2 e! for covergece. Assigmet 6: Itegratio. Usig Riema s criterio for the itegrability, show that f() = is itegrable o [, 2]. 2. If f ad g are cotiuous fuctios o [a, b] ad if g() for a b, the show the mea value theorem for itegrals : there eists c [a, b] such that (a) Show that there is o cotiuous fuctio f o [, ] such that N. b a f()g()d = f(c) b a f()d = g()d. for all (b) If f is cotiuuous o [a, b] the show that there eists c [a, b] such that b a f()d = f(c)(b a). (c) If f ad g are cotiuous o [a, b] ad b a f()d = b a g()d the show that there eists c [a, b] such that f(c) = g(c). 3. Let f : [, 2] R be a cotiuous fuctio such that 2 f()d = 2. Fid the value of 2 [f() + f(t)dt]d. u 4. Show that ( f(t)dt)du = f(u)( u)du, assumig f to be cotiuous. 5. Let f : [, ] R be a positive cotiuous fuctio. Show that lim (f( )f( 2 ) f( )) = e lf(). Assigmet 7: Improper Itegrals. Test the covergece/divergece of the followig improper itegrals: (a) (e) d log(+ ) (b) si(/) d (f) d log(+) e 2 d (g) (c) log si 2 d, 2. Determie all those values of p for which the improper itegral (d) (h) si(/)d. π/2 cot d. e d coverges. p
4 3. Show that the itegrals si d. 4. Show that log (+ 2 ) 2 d =. 5. Prove the followig statemets. si 2 d ad 2 si d coverge. Further, prove that si 2 2 d = (a) Let f be a icreasig fuctio o (,) ad the improper itegral f() eist. The i. f()d f( )+f( 2 ii. lim f( )+f( 2 (b) lim l +l 2 (c) lim! = e. )+ +f( ) f()d. )+ +f( ) = f()d. + +l =. Assigmet 8: Applicatios of Itegratio, Pappus Theorem. Sketch the graphs of r = cos(2θ) ad r = si(2θ). Also, fid their poits of itersectio. 2. A curved wedge is cut from a cylider of radius 3 by two plaes. Oe plae is perpedicular to the ais of the cylider. The secod plae crosses the first plae at a 45 agle at the ceter of the cylider. Fid the volume of the wedge. 3. Let C deote the circular disc of radius b cetered at (a, ) where < b < a. Fid the volume of the torus that is geerated by revolvig C aroud the y-ais usig (a) the Washer Method (b) the Shell Method. 4. Cosider the curve C defied by (t) = cos 3 (t), y(t) = si 3 t, t π 2. (a) Fid the legth of the curve. (b) Fid the area of the surface geerated by revolvig C about the -ais. (c) If (, y) is the cetroid of C the fid y. 5. A square is rotated about a ais lyig i the plae of the square, which itersects the square oly at oe of its vertices. For what positio of the ais, is the volume of the resultig solid of revolutio the largest? 6. Fid the cetroid of the semicircular arc ( r) 2 + y 2 = r 2, r > described i the first quadrat. If this arc is rotated about the lie y + m =, m >, determie the geerated surface area A ad show that A is maimum whe m = π/2. Assigmet 9: Vectors, Curves, Surfaces, Vector Fuctios. Cosider the plaes y + z =, + ay 2z + = ad 2 3y + z + b =, where a ad b are parameters. Determie the values of a ad b such that the three plaes (a) itersect at a sigle poit,
5 (b) itersect i a lie, (c) itersect (take two at a time) i three distict parallel lies. 2. Determie the equatio of a coe with verte (, a, ) geerated by a lie passig through the curve 2 = 2y, z = h. 3. The velocity of a particle movig i space is d dt c(t) = (cos t) i (si t) j + k. Fid the particle s positio as a fuctio of t if c() = 2 i + k. Also fid the agle betwee its positio vector ad the velocity vector. 4. Show that c(t) = si t 2 i + cos t 2 j + 5 k has costat magitude ad is orthogoal to its derivative. Is the velocity vector of costat magitude? 5. Fid the poit o the curve c(t) = (5 si t) i + (5 cos t) j + 2t k at a distace 26π uits alog the curve from (, 5, ) i the directio of icreasig arc legth. 6. Reparametrize the curves (a) c(t) = t2 2 i + t3 3 k, t 2, (b) c(t) = 2 cos t i + 2 si t j, t 2π i terms of arc legth. 7. Show that the parabola y = a 2, a has its largest curvature at its verte ad has o miimum curvature. Assigmet : Fuctios of several variables (Cotiuity ad Differetiability). Idetify the poits, if ay, where the followig fuctios fail to be cotiuous: (i) f(, y) = { y if y y if y < { y if y is ratioal (ii) f(, y) = y if y is irratioal. 2. Cosider the fuctio f : R 2 R defied by { 2 y 2 if (, y) (, ) f(, y) = 2 y 2 +( y) 2 if(, y) = (, ) Show that the fuctio satisfy the followig: [ ] [ ] (a) The iterated limits lim lim f(, y) ad lim lim f(, y) eist ad equals ; y y (b) lim f(, y) does ot eist; (,y) (,) (c) f(, y) is ot cotiuous at (, ); (d) the partial derivatives eist at (, ). 3. Let f(, y) = ( 2 +y 2 ) si 2 +y 2 if (, y) (, ) ad, otherwise. Show that f is differetiable at every poit of R 2 but the partial derivatives are ot cotiuous at (, ). 4. Let f(, y) = y for all (, y) R 2. Show that (a) f is differetiable at (,.)
6 (b) f (, y ) does ot eist if y. 5. Suppose f is a fuctio with f (, y) = f y (, y) = for all (, y). The show that f(, y) = c, a costat. Assigmet : Directioal derivatives, Maima, Miima, Lagrage Multipliers. Let f(, y) = 2 ( y y ). Is f cotiuous at (, )? Which directioal derivatives of f eist at (, )? Is f differetiable at (, )? 2. Let f(, y) = 2 y for (, y) (, ) ad f(, ) =. Show that the directioal derivative 2 +y 2 of f at (, ) i all directios eist but f is ot differetiable at (, ). 3. Let f(, y) = 2 e y + cos(y). Fid the directioal derivative of f at (, 2) i the directio ( 3 5, 4 5 ). 4. Fid the equatio of the surface geerated by the ormals to the surface + 2yz + yz 2 = at all poits o the z-ais. 5. Eamie the followig fuctios for local maima, local miima ad saddle poits: i) 4y 4 y 4 ii) 3 3y 2 6. Fid the absolute maima of f(, y) = y o the uit disc {(, y) : 2 + y 2 }.. Evaluate the followig itegrals: Assigmet 2 : Double Itegrals i) 2 y 2 dyd ii) π π si y y dyd iii) y 2 ep y ddy. 2. Evaluate ddy where R is the regio ( y) 2 ad y 2. R 3. Usig double itegral, fid the area eclosed by the curve r = si3θ give i polar cordiates. 4. Compute lim ep (2 +y 2) ddy, where a D(a) i) D(a) = {(, y) : 2 + y 2 a 2 } ad ii) D(a) = {(, y) : a, y a}. Hece prove that (i) e 2 d = π 2 (ii) 2 e 2 d = π Fid the volume of the solid which is commo to the cylider 2 + y 2 = ad 2 + z 2 =. Assigmet 3 : Triple Itegrals, Surface Itegrals, Lie itegrals. Evaluate the itegral W dzdyd + 2 +y 2 +z 2 ; where W is the ball 2 + y 2 + z 2.
7 2. What is the itegral of the fuctio 2 z take over the etire surface of a right circular cylider of height h which stads o the circle 2 + y 2 = a 2. What is the itegral of the give fuctio take throughout the volume of the cylider. 3. Fid the lie itegral of the vector field F (, y, z) = y i j + k alog the path c(t) = t (cos t, si t, 2π ), t 2π joiig (,, ) to (,, ). 4. Evaluate T dr, where C is the circle 2 + y 2 = ad T is the uit taget vector. C 5. Show that the itegral yzd+(z+)dy+ydz is idepedet of the path C joiig (,, ) C ad (2,, 4). Assigmet 4 : Gree s /Stokes /Gauss Theorems. Use Gree s Theorem to compute C (2 2 y 2 ) d + ( 2 + y 2 ) dy where C is the boudary of the regio {(, y) :, y & 2 + y 2 }. 2. Use Stokes Theorem to evaluate the lie itegral C y 3 d + 3 dy z 3 dz, where C is the itersectio of the cylider 2 + y 2 = ad the plae + y + z = ad the orietatio of C correspods to couterclockwise motio i the y-plae. 3. Let F = r r 3 where r = i + y j + z k ad let S be ay surface that surrouds the origi. Prove that S F. dσ = 4π. 4. Let D be the domai iside the cylider 2 +y 2 = cut off by the plaes z = ad z = +2. If F = ( 2 + ye z, y 2 + ze, z + e y ), use the divergece theorem to evaluate F dσ. D
Assignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1
Assigmet : Real Numbers, Sequeces. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate
More informationIndian Institute of Information Technology, Allahabad. End Semester Examination - Tentative Marking Scheme
Idia Istitute of Iformatio Techology, Allahabad Ed Semester Examiatio - Tetative Markig Scheme Course Name: Mathematics-I Course Code: SMAT3C MM: 75 Program: B.Tech st year (IT+ECE) ate of Exam:..7 ( st
More informationCalculus. Ramanasri. Previous year Questions from 2016 to
++++++++++ Calculus Previous ear Questios from 6 to 99 Ramaasri 7 S H O P NO- 4, S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N E W D E L H I. W E B S I T E :
More informationHonors Calculus Homework 13 Solutions, due 12/8/5
Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y
More information(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More information1988 AP Calculus BC: Section I
988 AP Calculus BC: Sectio I 9 Miutes No Calculator Notes: () I this eamiatio, l deotes the atural logarithm of (that is, logarithm to the base e). () Uless otherwise specified, the domai of a fuctio f
More informationWBJEE Answer Keys by Aakash Institute, Kolkata Centre
WBJEE - 7 Aswer Keys by, Kolkata Cetre MATHEMATICS Q.No. B A C B A C A B 3 D C B B 4 B C D D 5 D A B B 6 C D B B 7 B C C A 8 B B A A 9 A * B D C C B B D A A D B B C B 3 A D D D 4 C B A A 5 C B B B 6 C
More informationFINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,
More informationCalculus 2 Test File Spring Test #1
Calculus Test File Sprig 009 Test #.) Without usig your calculator, fid the eact area betwee the curves f() = - ad g() = +..) Without usig your calculator, fid the eact area betwee the curves f() = ad
More informationNBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?
NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily
More informationJEE(Advanced) 2018 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 20 th MAY, 2018)
JEE(Advaced) 08 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 0 th MAY, 08) PART- : JEE(Advaced) 08/Paper- SECTION. For ay positive iteger, defie ƒ : (0, ) as ƒ () j ta j j for all (0, ). (Here, the iverse
More informationCalculus 2 Test File Fall 2013
Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to
More information7.) Consider the region bounded by y = x 2, y = x - 1, x = -1 and x = 1. Find the volume of the solid produced by revolving the region around x = 3.
Calculus Eam File Fall 07 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = 3 + 4. Produce a solid by revolvig the regio aroud
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationMath 5C Discussion Problems 2 Selected Solutions
Math 5 iscussio Problems 2 elected olutios Path Idepedece. Let be the striaght-lie path i 2 from the origi to (3, ). efie f(x, y) = xye xy. (a) Evaluate f dr. olutio. (b) Evaluate olutio. (c) Evaluate
More informationMath 122 Test 3 - Review 1
I. Sequeces ad Series Math Test 3 - Review A) Sequeces Fid the limit of the followig sequeces:. a = +. a = l 3. a = π 4 4. a = ta( ) 5. a = + 6. a = + 3 B) Geometric ad Telescopig Series For the followig
More informationReview Problems Math 122 Midterm Exam Midterm covers App. G, B, H1, H2, Sec , 8.9,
Review Problems Math Midterm Exam Midterm covers App. G, B, H, H, Sec 8. - 8.7, 8.9, 9.-9.7 Review the Cocept Check problems: Page 6/ -, Page 690/- 0 PART I: True-False Problems Ch. 8. Page 6 True-False
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationWe will conclude the chapter with the study a few methods and techniques which are useful
Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs
More informationMath 21C Brian Osserman Practice Exam 2
Math 1C Bria Osserma Practice Exam 1 (15 pts.) Determie the radius ad iterval of covergece of the power series (x ) +1. First we use the root test to determie for which values of x the series coverges
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
More informationMATHEMATICS Code No. 13 INSTRUCTIONS
DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO COMBINED COMPETITIVE (PRELIMINARY) EXAMINATION, 0 Serial No. MATHEMATICS Code No. A Time Allowed : Two Hours Maimum Marks : 00 INSTRUCTIONS. IMMEDIATELY
More information6.) Find the y-coordinate of the centroid (use your calculator for any integrations) of the region bounded by y = cos x, y = 0, x = - /2 and x = /2.
Calculus Test File Sprig 06 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = +. Produce a solid by revolvig the regio aroud
More informationFundamental Concepts: Surfaces and Curves
UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat
More informationMath 5C Discussion Problems 2
Math iscussio Problems Path Idepedece. Let be the striaght-lie path i R from the origi to (3, ). efie f(x, y) = xye xy. (a) Evaluate f dr. (b) Evaluate ((, 0) + f) dr. (c) Evaluate ((y, 0) + f) dr.. Let
More informationx x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,
Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative
More informationMaximum and Minimum Values
Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f
More informationMATH 1A FINAL (7:00 PM VERSION) SOLUTION. (Last edited December 25, 2013 at 9:14pm.)
MATH A FINAL (7: PM VERSION) SOLUTION (Last edited December 5, 3 at 9:4pm.) Problem. (i) Give the precise defiitio of the defiite itegral usig Riema sums. (ii) Write a epressio for the defiite itegral
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas 1 to compute
Math, Calculus II Fial Eam Solutios. 5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute 4 d. The check your aswer usig the Evaluatio Theorem. ) ) Solutio: I this itegral,
More informationAP Calculus BC 2011 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The College Board The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success ad opportuity. Fouded i 9, the College
More informationReview Problems for the Final
Review Problems for the Fial Math - 3 7 These problems are provided to help you study The presece of a problem o this hadout does ot imply that there will be a similar problem o the test Ad the absece
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationPRACTICE FINAL/STUDY GUIDE SOLUTIONS
Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)
More informationMathematics Extension 2
004 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard
More informationRegn. No. North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: ,
. Sectio-A cotais 30 Multiple Choice Questios (MCQ). Each questio has 4 choices (a), (b), (c) ad (d), for its aswer, out of which ONLY ONE is correct. From Q. to Q.0 carries Marks ad Q. to Q.30 carries
More informationMath 142, Final Exam. 5/2/11.
Math 4, Fial Exam 5// No otes, calculator, or text There are poits total Partial credit may be give Write your full ame i the upper right corer of page Number the pages i the upper right corer Do problem
More informationMathematics Extension 1
016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationAIEEE 2004 (MATHEMATICS)
AIEEE 00 (MATHEMATICS) Importat Istructios: i) The test is of hours duratio. ii) The test cosists of 75 questios. iii) The maimum marks are 5. iv) For each correct aswer you will get marks ad for a wrog
More informationIIT JAM Mathematical Statistics (MS) 2006 SECTION A
IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim
More informationBITSAT MATHEMATICS PAPER III. For the followig liear programmig problem : miimize z = + y subject to the costraits + y, + y 8, y, 0, the solutio is (0, ) ad (, ) (0, ) ad ( /, ) (0, ) ad (, ) (d) (0, )
More informationCalculus with Analytic Geometry 2
Calculus with Aalytic Geometry Fial Eam Study Guide ad Sample Problems Solutios The date for the fial eam is December, 7, 4-6:3p.m. BU Note. The fial eam will cosist of eercises, ad some theoretical questios,
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationAssignment ( ) Class-XI. = iii. v. A B= A B '
Assigmet (8-9) Class-XI. Proe that: ( A B)' = A' B ' i A ( BAC) = ( A B) ( A C) ii A ( B C) = ( A B) ( A C) iv. A B= A B= φ v. A B= A B ' v A B B ' A'. A relatio R is dified o the set z of itegers as:
More information(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)
Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More information2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)
Math PracTest Be sure to review Lab (ad all labs) There are lots of good questios o it a) State the Mea Value Theorem ad draw a graph that illustrates b) Name a importat theorem where the Mea Value Theorem
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationn 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent.
06 微甲 0-04 06-0 班期中考解答和評分標準. ( poits) Determie whether the series is absolutely coverget, coditioally coverget, or diverget. Please state the tests which you use. (a) ( poits) (b) ( poits) (c) ( poits)
More information+ {JEE Advace 03} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks: 00. If A (α, β) = (a) A( α, β) = A( α, β) (c) Adj (A ( α, β)) = Sol : We
More informationSolving equations (incl. radical equations) involving these skills, but ultimately solvable by factoring/quadratic formula (no complex roots)
Evet A: Fuctios ad Algebraic Maipulatio Factorig Square of a sum: ( a + b) = a + ab + b Square of a differece: ( a b) = a ab + b Differece of squares: a b = ( a b )(a + b ) Differece of cubes: a 3 b 3
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a
More informationSolutions to Homework 1
Solutios to Homework MATH 36. Describe geometrically the sets of poits z i the complex plae defied by the followig relatios /z = z () Re(az + b) >, where a, b (2) Im(z) = c, with c (3) () = = z z = z 2.
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationSubstitute these values into the first equation to get ( z + 6) + ( z + 3) + z = 27. Then solve to get
Problem ) The sum of three umbers is 7. The largest mius the smallest is 6. The secod largest mius the smallest is. What are the three umbers? [Problem submitted by Vi Lee, LCC Professor of Mathematics.
More informationGULF MATHEMATICS OLYMPIAD 2014 CLASS : XII
GULF MATHEMATICS OLYMPIAD 04 CLASS : XII Date of Eamiatio: Maimum Marks : 50 Time : 0:30 a.m. to :30 p.m. Duratio: Hours Istructios to cadidates. This questio paper cosists of 50 questios. All questios
More informationCalculus I Practice Test Problems for Chapter 5 Page 1 of 9
Calculus I Practice Test Problems for Chapter 5 Page of 9 This is a set of practice test problems for Chapter 5. This is i o way a iclusive set of problems there ca be other types of problems o the actual
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationThe type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE.
NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Iitial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to fid TANGENTS ad VELOCITIES
More informationChapter 8. Uniform Convergence and Differentiation.
Chapter 8 Uiform Covergece ad Differetiatio This chapter cotiues the study of the cosequece of uiform covergece of a series of fuctio I Chapter 7 we have observed that the uiform limit of a sequece of
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationSolutions to quizzes Math Spring 2007
to quizzes Math 4- Sprig 7 Name: Sectio:. Quiz a) x + x dx b) l x dx a) x + dx x x / + x / dx (/3)x 3/ + x / + c. b) Set u l x, dv dx. The du /x ad v x. By Itegratio by Parts, x(/x)dx x l x x + c. l x
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More informationMATH Exam 1 Solutions February 24, 2016
MATH 7.57 Exam Solutios February, 6. Evaluate (A) l(6) (B) l(7) (C) l(8) (D) l(9) (E) l() 6x x 3 + dx. Solutio: D We perform a substitutio. Let u = x 3 +, so du = 3x dx. Therefore, 6x u() x 3 + dx = [
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationCALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
AP Calculus AB Portfolio Project Multiple Choice Practice Name: CALCULUS AB SECTION I, Part A Time 60 miutes Number of questios 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directios: Solve
More informationThe volume V of the solid of revolution obtained by revolving the region of Fig about the x axis is given by. (disk formula)
CHAPTER Applicatios of Itegratio II: Volume A solid of revolutio is obtaied by revolvig a regio i a plae about a lie that does ot itersect the regio. The lie about which the rotatio takes place is called
More information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More informationName: Math 10550, Final Exam: December 15, 2007
Math 55, Fial Exam: December 5, 7 Name: Be sure that you have all pages of the test. No calculators are to be used. The exam lasts for two hours. Whe told to begi, remove this aswer sheet ad keep it uder
More informationExample 2. Find the upper bound for the remainder for the approximation from Example 1.
Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute
More informationn n 2 + 4i = lim 2 n lim 1 + 4x 2 dx = 1 2 tan ( 2i 2 x x dx = 1 2 tan 1 2 = 2 n, x i = a + i x = 2i
. ( poits) Fid the limits. (a) (6 poits) lim ( + + + 3 (6 poits) lim h h h 6 微甲 - 班期末考解答和評分標準 +h + + + t3 dt. + 3 +... + 5 ) = lim + i= + i. Solutio: (a) lim i= + i = lim i= + ( i ) = lim x i= + x i =
More informationAH Checklist (Unit 3) AH Checklist (Unit 3) Matrices
AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow
More informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More informationMathematics Extension 2
009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard
More informationMATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of
MATHEMATICS 6 The differetial equatio represetig the family of curves where c is a positive parameter, is of Order Order Degree (d) Degree (a,c) Give curve is y c ( c) Differetiate wrt, y c c y Hece differetial
More informationPower Series: A power series about the center, x = 0, is a function of x of the form
You are familiar with polyomial fuctios, polyomial that has ifiitely may terms. 2 p ( ) a0 a a 2 a. A power series is just a Power Series: A power series about the ceter, = 0, is a fuctio of of the form
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationConsortium of Medical Engineering and Dental Colleges of Karnataka (COMEDK) Undergraduate Entrance Test(UGET) Maths-2012
Cosortium of Medical Egieerig ad Detal Colleges of Karataka (COMEDK) Udergraduate Etrace Test(UGET) Maths-0. If the area of the circle 7 7 7 k 0 is sq. uits, the the value of k is As: (b) b) 0 7 K 0 c)
More informationFourier Series and their Applications
Fourier Series ad their Applicatios The fuctios, cos x, si x, cos x, si x, are orthogoal over (, ). m cos mx cos xdx = m = m = = cos mx si xdx = for all m, { m si mx si xdx = m = I fact the fuctios satisfy
More informationFooling Newton s Method
Foolig Newto s Method You might thik that if the Newto sequece of a fuctio coverges to a umber, that the umber must be a zero of the fuctio. Let s look at the Newto iteratio ad see what might go wrog:
More informationMath 21B-B - Homework Set 2
Math B-B - Homework Set Sectio 5.:. a) lim P k= c k c k ) x k, where P is a partitio of [, 5. x x ) dx b) lim P k= 4 ck x k, where P is a partitio of [,. 4 x dx c) lim P k= ta c k ) x k, where P is a partitio
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..
More informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More information18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016
18th Bay Area Mathematical Olympiad February 3, 016 Problems ad Solutios BAMO-8 ad BAMO-1 are each 5-questio essay-proof exams, for middle- ad high-school studets, respectively. The problems i each exam
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
QUESTION BANK 8 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayaavaam Road 5758 QUESTION BANK (DESCRIPTIVE Subject with Code : (6HS6 Course & Brach: B.Tech AG Year & Sem: II-B.Tech& I-Sem
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationTHE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.
THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of
More informationNotes 8 Singularities
ECE 6382 Fall 27 David R. Jackso Notes 8 Sigularities Notes are from D. R. Wilto, Dept. of ECE Sigularity A poit s is a sigularity of the fuctio f () if the fuctio is ot aalytic at s. (The fuctio does
More informationWBJEE MATHEMATICS
WBJEE - 06 MATHEMATICS Q.No. 0 A C B B 0 B B A B 0 C A C C 0 A B C C 05 A A B C 06 B C B C 07 B C A D 08 C C C A 09 D D C C 0 A C A B B C B A A C A B D A A A B B D C 5 B C C C 6 C A B B 7 C A A B 8 C B
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More information