INEQUALITIES THROUGH PROBLEMS. 1. Heuristics of problem solving
|
|
|
- Elvin Murphy
- 6 years ago
- Views:
Transcription
1 INEQUALITIES THROUGH PROBLEMS Hojoo Lee. Heuristis of prolem solving Strtegy or ttis in prolem solving is lled heuristis. Here is summry tken from Prolem-Solving Through Prolems y Loren C. Lrson.. Serh for pttern.. Drw figure.. Formulte n equivlent prolem. 4. Modify the prolem. 5. Choose effetive nottion. 6. Exploit symmetry. 7. Divide into ses. 8. Work kwrds. 9. Argue y ontrdition. 0. Chek for prity.. Consider extreme ses.. Generlize.. Clssil Theorems Theorem. Shur) Let x, y, z e nonnegtive rel numers. For ny r > 0, we hve x r x y)x z) 0. yli Theorem. Muirhed) Let,,,,, e rel numers suh tht 0, 0,, + +, + + = + +. Let x, y, z e positive rel numers. Then, we hve sym x y z sym x y z.
2 INEQUALITIES THROUGH PROBLEMS Theorem. The Cuhy-Shwrz inequlity) Let,, n,,, n e rel numers. Then, + + n ) + + n ) + + n n ). Theorem 4. AM-GM inequlity) Let,, n Then, we hve + + n n n n. e positive rel numers. Theorem 5. Weighted AM-GM inequlity) Let ω,, ω n > 0 with ω + + ω n =. For ll x,, x n > 0, we hve ω x + + ω n x n x ω x n ω n. Theorem 6. Hölder s inequlity) Let x ij i =,, m, j =, n) e positive rel numers. Suppose tht ω,, ω n re positive rel numers stisfying ω + + ω n =. Then, we hve n m ) ωj m n x ij ω x j ij. j= i= i= j= Theorem 7. Power Men inequlity) Let x,, x n > 0. The power men of order r is defined y ) M x,,x n )0) = n x r r r x x n, M x,,x n )r) = + + x n r 0). n Then, M x,,x n ) : R R is ontinuous nd monotone inresing. Theorem 8. Mjoriztion inequlity) Let f : [, ] R e onvex funtion. Suppose tht x,, x n ) mjorizes y,, y n ), where x,, x n, y,, y n [, ]. Then, we otin fx ) + + fx n ) fy ) + + fy n ). Theorem 9. Bernoulli s inequlity) For ll r nd x, we hve + x) r + rx. Definition. Symmetri Mens) For given ritrry rel numers x,, x n, the oeffiient of t n i in the polynomil t + x ) t + x n ) is lled the i-th elementry symmetri funtion σ i. This mens tht t + x ) t + x n ) = σ 0 t n + σ t n + + σ n t + σ n. For i {0,,, n}, the i-th elementry symmetri men S i is defined y S i = σ i n i).
3 INEQUALITIES THROUGH PROBLEMS Theorem 0. Let x,..., x n > 0. For i {,, n}, we hve ) Newton s inequlity) S i S i+ S i S i, ) Mlurin s inequlity) S i i Si+ i+. Theorem. Rerrngement inequlity) Let x x n nd y y n e rel numers. For ny permuttion σ of {,..., n}, we hve n n n x i y i x i y σi) x i y n+ i. i= i= i= Theorem. Cheyshev s inequlity) Let x x n nd y y n e rel numers. We hve ) ) x y + + x n y n x + + x n y + + y n. n n n Theorem. Hölder s inequlity) Let x,, x n, y,, y n e positive rel numers. Suppose tht p > nd q > stisfy p + q =. Then, we hve n n ) ) p n q p q x i y i x i y i i= i= i= Theorem 4. Minkowski s inequlity) If x,, x n, y,, y n > 0 nd p >, then n ) p n ) p n ) p p p x i + y i x i + y i ) p i= i= i=. Yers Eh prolem tht I solved eme rule, whih served fterwrds to solve other prolems. Rene Desrtes. BMO 005, Proposed y Seri nd Montenegro),, > 0) ) + +. Romni 005, Cezr Lupu),, > 0)
4 4 INEQUALITIES THROUGH PROBLEMS. Romni 005, Trin Tmin),, > 0) + + d + + d + + d d Romni 005, Cezr Lupu) ,,, > 0) Romni 005, Cezr Lupu) = + ) + ) + ),,, > 0) Romni 005, Roert Szsz) + + =,,, > 0) ) ) ) 7. Romni 005),,, > 0) Romni 005, Unused) =,,, > 0) + ) + + ) + + ) 9. Romni 005, Unused) ,,, > 0) + ) + + ) + + ) 0. Romni 005, Unused) + + =,,, > 0) Romni 005, Unused) =,,, > 0) + +. Chzeh nd Solvk 005) =,,, > 0) + ) + ) + + ) + ) + + ) + ) 4
5 INEQUALITIES THROUGH PROBLEMS 5. Jpn 005) + + =,,, > 0) + ) + + ) + + ) 4. Germny 005) + + =,,, > 0) + + ) Vietnm 005),, > 0) ) ) ) Chin 005) + + =,,, > 0) ) ) 7. Chin 005) d =,,,, d > 0) + ) + + ) + + ) + + d) 8. Chin 005) + + =,,, 0) Polnd 005) 0,, ) Polnd 005) + + =,,, > 0) Blti Wy 005) =,,, > 0) Seri nd Montenegro 005),, > 0) )
6 6 INEQUALITIES THROUGH PROBLEMS. Seri nd Montenegro 005) + + =,,, > 0) Bosni nd Heregovin 005) + + =,,, > 0) Irn 005),, > 0) + + ) + + ) + + ) 6. Austri 005),,, d > 0) d d d 7. Moldov 005) =,,, > 0) APMO 005) = 8,,, > 0) + ) + ) + + ) + ) + + ) + ) 4 9. IMO 005) xyz, x, y, z > 0) x 5 x x 5 + y + z + y5 y y 5 + z + x + z5 z z 5 + x + y 0 0. Polnd 004) + + = 0,,, R) Blti Wy 004) =,,, > 0, n N) n + n + + n + n + + n + n +. Junior Blkn 004) x, y) R {0, 0)}) x + y x + y x xy + y
7 INEQUALITIES THROUGH PROBLEMS 7. IMO Short List 004) + + =,,, > 0) APMO 004),, > 0) + ) + ) + ) ) 5. USA 004),, > 0) 5 + ) 5 + ) 5 + ) + + ) 6. Junior BMO 00) x, y, z > ) + x + y + z + + y + z + x + + z + x + y 7. USA 00),, > 0) + + ) + + ) + + ) ) ) + + ) 8 8. Russi 00) x + y + z =, x, y, z > 0) x + y + z xy + yz + zx ) 9. Ltvi 00) d =,,,, d > 0 4 d 40. Alni 00),, > 0) ) + + ) Belrus 00),,, d > 0) + ) + + d) d + + ) + + d) d + ) + + d) 4. Cnd 00),, > 0)
8 8 INEQUALITIES THROUGH PROBLEMS 4. Vietnm 00, Dung Trn Nm) + + = 9,,, R) + + ) Bosni nd Heregovin 00) + + =,,, R) Junior BMO 00),, > 0) + ) + + ) + + ) ) 46. Greee 00) + + =,,, > 0) ) 47. Greee 00) 0, 0,,, R) ) Tiwn 00),,, d ]) 0, d ) ) ) d) d 4 ) 4 + ) 4 + ) 4 + d) APMO 00) x + y + z =, x, y, z > 0) x + yz + y + zx + z + xy xyz + x + y + z 50. Irelnd 00) x + y =, x, y 0) x y x + y ). 5. BMO 00) + +,,, 0) USA 00) = 4,,, 0) Columi 00) x, y R) x + y + ) + xy
9 INEQUALITIES THROUGH PROBLEMS KMO Winter Progrm Test 00),, > 0) + + ) + + ) + + ) + ) + ) 55. IMO 00),, > 0) Yers Life is good for only two things, disovering mthemtis nd tehing mthemtis. Simeon Poisson 56. IMO 000, Titu Andreesu) =,,, > 0) + ) + ) + ) 57. Czeh nd Slovki 000), > 0) + ) + ) Hong Kong 000) =,,, > 0) Czeh Repuli 000) m, n N, x [0, ]) x n ) m + x) m ) n 60. Medoni 000) x, y, z > 0) x + y + z xy + yz) 6. Russi 999),, > 0) > 6. Belrus 999) + + =,,, > 0)
10 0 INEQUALITIES THROUGH PROBLEMS 6. Czeh-Slovk Mth 999),, > 0) Moldov 999),, > 0) + ) + + ) + + ) United Kingdom 999) p + q + r =, p, q, r > 0) 7pq + qr + rp) + 9pqr 66. Cnd 999) x + y + z =, x, y, z 0) x y + y z + z x Proposed for 999 USAMO, [AB, pp.5]) x, y, z > ) x x +yz y y +zx z z +xy xyz) xy+yz+zx 68. Turkey, 999) 0) + ) + 4) + ) Medoni 999) + + =,,, > 0) Polnd 999) + + =,,, > 0) Cnd 999) x + y + z =, x, y, z 0) 7. Irn 998) x y + y z + z x 4 7 x + y + z =, x, y, z > ) x + y + z x + y + z 7. Belrus 998, I. Gorodnin),, > 0)
11 INEQUALITIES THROUGH PROBLEMS 74. APMO 998),, > 0) + ) + ) + ) ) 75. Polnd 998) d + e + f =, e + df 08,,, d, e, f > 0) + d + de + def + ef + f Kore 998) x + y + z = xyz, x, y, z > 0) x + y + z 77. Hong Kong 998),, ) ) 78. IMO Short List 998) xyz =, x, y, z > 0) x + y) + z) + y + z) + x) + z + x) + y) Belrus 997), x, y, z > 0) + y + x x + + z + x y + + x + y z x + y + z + z + z x + + x + y y + + y + z z 80. Irelnd 997) + +,,, 0) Irn 997) x x x x 4 =, x, x, x, x 4 > 0) ) x + x + x + x 4 mx x + x + x + x 4, x + x + x + x4 8. Hong Kong 997) x, y, z > 0) + 9 xyzx + y + z + x + y + z ) x + y + z )xy + yz + zx) 8. Belrus 997),, > 0)
12 INEQUALITIES THROUGH PROBLEMS 84. Bulgri 997) =,,, > 0) Romni 997) xyz =, x, y, z > 0) x 9 + y 9 x 6 + x y + y 6 + y 9 + z 9 y 6 + y z + z 6 + z 9 + x 9 z 6 + z x + x Romni 997),, > 0) USA 997),, > 0) Jpn 997),, > 0) + ) + ) + ) + ) ) ) Estoni 997) x, y R) x + y + > x y + + y x APMC 996) x + y + z + t = 0, x + y + z + t =, x, y, z, t R) xy + yz + zt + tx 0 9. Spin 996),, > 0) + + ) ) 9. IMO Short List 996) =,,, > 0) Polnd 996) ) + + =,,,
13 INEQUALITIES THROUGH PROBLEMS 94. Hungry 996) + =,, > 0) Vietnm 996),, R) + ) ) ) ) 96. Berus 996) x + y + z = xyz, x, y, z > 0) xy + yz + zx 9x + y + z) 97. Irn 996),, > 0) ) + + ) + ) + + ) + + ) Vietnm 996) + + d + + d + d) + + d + d + d = 6,,,, d 0) d + + d + + d + d) 5. Yers Any good ide n e stted in fifty words or less. S. M. Ulm 99. Blti Wy 995),,, d > 0) d d + d + d Cnd 995),, > 0) IMO 995, Nzr Agkhnov) =,,, > 0) + ) + + ) + + ) 0. Russi 995) x, y > 0) xy x x 4 + y + y y 4 + x
14 4 INEQUALITIES THROUGH PROBLEMS 0. Medoni 995),, > 0) APMC 995) m, n N, x, y > 0) n )m )x n+m +y n+m )+n+m )x n y m +x m y n ) nmx n+m y+xy n+m ) 05. Hong Kong 994) xy + yz + zx =, x, y, z > 0) x y ) z ) + y z ) x ) + z x ) y ) IMO Short List 99),,, d > 0) + + d + + d + + d d APMC 99), 0) ) ) 08. Polnd 99) x, y, u, v > 0) xy + xv + uy + uv x + y + u + v xy x + y + uv u + v 09. IMO Short List 99) d =,,,, d > 0) + d + d + d d 0. Itly 99) 0,, ) Polnd 99),, R) + ) + ) + ) + ) + ) + ). Vietnm 99) x y z > 0) x y z + y z x + z x y x + y + z
15 INEQUALITIES THROUGH PROBLEMS 5. Polnd 99) x + y + z =, x, y, z R) x + y + z + xyz 4. Mongoli 99) + + =,,, R) IMO Short List 990) + + d + d =,,,, d > 0) + + d + + d + + d d Supplementry Prolems Every Mthemtiin Hs Only Few Triks. A long time go n older nd well-known numer theorist mde some disprging remrks out Pul Erdös s work. You dmire Erdos s ontriutions to mthemtis s muh s I do, nd I felt nnoyed when the older mthemtiin fltly nd definitively stted tht ll of Erdos s work ould e redued to few triks whih Erdös repetedly relied on in his proofs. Wht the numer theorist did not relize is tht other mthemtiins, even the very est, lso rely on few triks whih they use over nd over. Tke Hilert. The seond volume of Hilert s olleted ppers ontins Hilert s ppers in invrint theory. I hve mde point of reding some of these ppers with re. It is sd to note tht some of Hilert s eutiful results hve een ompletely forgotten. But on reding the proofs of Hilert s striking nd deep theorems in invrint theory, it ws surprising to verify tht Hilert s proofs relied on the sme few triks. Even Hilert hd only few triks! Gin-Crlo Rot, Ten Lessons I Wish I Hd Been Tught, Noties of the AMS, Jnury Lithuni 987) x, y, z > 0) x x + xy + y + y y + yz + z + z z + zx + x x + y + z 7. Yugoslvi 987), > 0) + ) ) + 8. Yugoslvi 984),,, d > 0) d + d + + d + 9. IMO 984) x + y + z =, x, y, z 0) 0 xy + yz + zx xyz 7 7
16 6 INEQUALITIES THROUGH PROBLEMS 0. USA 980),, [0, ]) ) ) ) USA 979) x + y + z =, x, y, z > 0) x + y + z + 6xyz 4.. IMO 974),,, d > 0) < + + d d + d + + d <. IMO 968) x, x > 0, y, y, z, z R, x y > z, x y > z ) x y z + x y z 8 x + x )y + y ) z + z ) 4. Nesitt s inequlity),, > 0) Poly s inequlity),, > 0) + + ) ln ln 6. Klmkin s inequlity) < x, y, z < ) x) y) z) + + x) + y) + z) 7. Crlson s inequlity),, > 0) + ) + ) + ) [ONI], Vsile Cirtoje),, > 0) + ) + ) + + ) + ) + + ) + ) 9. [ONI], Vsile Cirtoje),,, d > 0) d + d d + + d + 0
17 INEQUALITIES THROUGH PROBLEMS 7 0. Elemente der Mthemtik, Prolem 07, Sefket Arslngić) x, y, z > 0) x y + y z + z x x + y + z xyz. W URZEL, Wlther Jnous) x + y + z =, x, y, z > 0) + x) + y) + z) x ) + y ) + z ). W URZEL, Heinz-Jürgen Seiffert) xy > 0, x, y R) xy x + y + x + y xy + x + y. W URZEL, Šefket Arslngić),, > 0) x + y ) z x + y + z) 4. W URZEL, Šefket Arslngić) =,,, > 0) + ) + + ) + + ). 5. W URZEL, Peter Strek, Donuwörth) =,,, > 0) ) + ) + ). 6. W URZEL, Peter Strek, Donuwörth) x + y + z =, x + y + z = 7, x, y, z > 0) + 6 xyz x z + y x + z ) y 7. W URZEL, Šefket Arslngić),, > 0) ) [ONI], Griel Dospinesu, Mire Lsu, Mrin Tetiv),, > 0) ) + ) + ) 9. Gzet Mtemtiã),, > 0)
18 8 INEQUALITIES THROUGH PROBLEMS 40. C 6, Mohmmed Assil),, > 0) C580),, > 0) C58),, > 0) C5) + + =,,, > 0) ) 44. C0, Vsile Cirtoje) + + =,,, > 0) C645),, > 0) + + ) ) + + ) 46. x, y R) x + y) xy) + x ) + y ) < x, y < ) x y + y x > 48. x, y, z > 0) xyz + x y + y z + z x x + y + z 49.,,, x, y, z > 0) + x) + y) + z) + xyz CRUX with MAYHEM
19 INEQUALITIES THROUGH PROBLEMS x, y, z > 0) x x + x + y)x + z) + y y + y + z)y + x) + z z + z + x)z + y) 5. x + y + z =, x, y, z > 0) x x + y y + z z 5.,, R) + ) + + ) + + ) 5.,, > 0) xy + yz + zx =, x, y, z > 0) x + x + y + y + z + z x x ) + x ) + y y ) + y ) + z z ) + z ) 55. x, y, z 0) xyz y + z x)z + x y)x + y z) 56.,, > 0) + ) + + ) + + ) ) + ) + ) 57. Drij Grinerg) x, y, z 0) x y + z) + y z + x) + z x + y) ) x + y + z y + z) z + x) x + y). 58. Drij Grinerg) x, y, z > 0) y + z z + x x + y + + x y z 4 x + y + z) y + z) z + x) x + y). 59. Drij Grinerg),, > 0) + ) + ) + + ) + + ) + ) + + ) + + ) + ) + + ) >.
20 0 INEQUALITIES THROUGH PROBLEMS 60. Drij Grinerg),, > 0) + + ) ) ) <. 6. Vsile Cirtoje),, R) + + ) + + )
INEQUALITIES. Ozgur Kircak
INEQUALITIES Ozgur Kirk Septemer 27, 2009 2 Contents MEANS INEQUALITIES 5. EXERCISES............................. 7 2 CAUCHY-SCHWARZ INEQUALITY 2. EXERCISES............................. REARRANGEMENT INEQUALITY
TOPICS IN INEQUALITIES. Hojoo Lee
TOPICS IN INEQUALITIES Hojoo Lee Version 0.5 [005/08/5] Introdution Inequlities re useful in ll fields of Mthemtis. The purpose in this ook is to present stndrd tehniques in the theory of inequlities.
How many proofs of the Nebitt s Inequality?
How mny roofs of the Neitt s Inequlity? Co Minh Qung. Introdution On Mrh, 90, Nesitt roosed the following rolem () The ove inequlity is fmous rolem. There re mny eole interested in nd solved (). In this
Topics in Inequalities - Theorems and Techniques. Hojoo Lee
Topis in Inequlities - Theorems nd Tehniques Hojoo Lee Introdution Inequlities re useful in ll fields of Mthemtis The im of this problem-oriented book is to present elementry tehniques in the theory of
Introduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
Topics in Inequalities - Theorems and Techniques
Topics in Inequlities - Theorems nd Techniques Hojoo Lee The st edition 8th August, 007 Introduction Inequlities re useful in ll fields of Mthemtics The im of this problem-oriented book is to present elementry
Basics of Olympiad Inequalities. Samin Riasat
Bsics of Olympid Inequlities Smin Rist ii Introduction The im of this note is to cquint students, who wnt to prticipte in mthemticl Olympids, to Olympid level inequlities from the sics Inequlities re used
Inequalities of Olympiad Caliber. RSME Olympiad Committee BARCELONA TECH
Ineulities of Olymid Clier José Luis Díz-Brrero RSME Olymid Committee BARCELONA TECH José Luis Díz-Brrero RSME Olymi Committee UPC BARCELONA TECH jose.luis.diz@u.edu Bsi fts to rove ineulities Herefter,
Inequalities Marathon version 1.00
MthLinks.ro Inequlities Mrthon version.00 Author: MthLink ers http://www.mthlinks.ro/viewtopic.php?t=99899 Editor: Hssn Al-Siyni Octoer 0, 009 Introduction Hello everyone, this file contin 00 prolem in
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths
Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t
MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx,
![MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx,](/thumbs/89/98261542.jpg "MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx,")
MATH3403: Green s Funtions, Integrl Equtions nd the Clulus of Vritions 1 Exmples 5 Qu.1 Show tht the extreml funtion of the funtionl I[y] = 1 0 [(y ) + yy + y ] dx, where y(0) = 0 nd y(1) = 1, is y(x)
Quadratic reciprocity
Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition
Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
A Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
![T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.](/thumbs/78/77085178.jpg "T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.")
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
Part 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
Co-ordinated s-convex Function in the First Sense with Some Hadamard-Type Inequalities
Int. J. Contemp. Mth. Sienes, Vol. 3, 008, no. 3, 557-567 Co-ordinted s-convex Funtion in the First Sense with Some Hdmrd-Type Inequlities Mohmmd Alomri nd Mslin Drus Shool o Mthemtil Sienes Fulty o Siene
Factorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
Algorithm Design and Analysis
Algorithm Design nd Anlysis LECTURE 8 Mx. lteness ont d Optiml Ching Adm Smith 9/12/2008 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov, K. Wyne Sheduling to Minimizing Lteness Minimizing
Probability. b a b. a b 32.
Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,
QUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
Algorithm Design and Analysis
Algorithm Design nd Anlysis LECTURE 5 Supplement Greedy Algorithms Cont d Minimizing lteness Ching (NOT overed in leture) Adm Smith 9/8/10 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov,
Quadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
Chapter 4 State-Space Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version
A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment
Arrow s Impossibility Theorem
Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties
(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
![(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.](/thumbs/72/66745617.jpg "(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.")
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
Number 12 Spring 2019
Number 2 Spring 209 ROMANIAN MATHEMATICAL MAGAZINE Avilble online www.ssmrmh.ro Founding Editor DANIEL SITARU ISSN-L 250-0099 ROMANIAN MATHEMATICAL MAGAZINE PROBLEMS FOR JUNIORS JP.66. If, b [0; + ) nd
Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs
Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if
PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles
Logic Synthesis and Verification
Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions Jie-Hong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 2010 Reding: Logi Synthesis in Nutshell Setion 2 most
Discrete Structures Lecture 11
Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.
April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
Bisimulation, Games & Hennessy Milner logic
Bisimultion, Gmes & Hennessy Milner logi Leture 1 of Modelli Mtemtii dei Proessi Conorrenti Pweł Soboiński Univeristy of Southmpton, UK Bisimultion, Gmes & Hennessy Milner logi p.1/32 Clssil lnguge theory
Linearly Similar Polynomials
Linerly Similr Polynomils rthur Holshouser 3600 Bullrd St. Chrlotte, NC, US Hrold Reiter Deprtment of Mthemticl Sciences University of North Crolin Chrlotte, Chrlotte, NC 28223, US hbreiter@uncc.edu stndrd
MATH 423 Linear Algebra II Lecture 28: Inner product spaces.
MATH 423 Liner Algebr II Lecture 28: Inner product spces. Norm The notion of norm generlizes the notion of length of vector in R 3. Definition. Let V be vector spce over F, where F = R or C. A function
ON AN INEQUALITY FOR THE MEDIANS OF A TRIANGLE
Journl of Siene nd Arts Yer, No. (9), pp. 7-6, OIGINAL PAPE ON AN INEQUALITY FO THE MEDIANS OF A TIANGLE JIAN LIU Mnusript reeived:.5.; Aepted pper:.5.; Pulished online: 5.6.. Astrt. In this pper, we give
#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS
#A42 INTEGERS 11 (2011 ON THE CONDITIONED BINOMIAL COEFFICIENTS Liqun To Shool of Mthemtil Sienes, Luoyng Norml University, Luoyng, Chin lqto@lynuedun Reeived: 12/24/10, Revised: 5/11/11, Aepted: 5/16/11,
6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
The Modified Heinz s Inequality
Journl of Applied Mthemtics nd Physics, 03,, 65-70 Pulished Online Novemer 03 (http://wwwscirporg/journl/jmp) http://dxdoiorg/0436/jmp03500 The Modified Heinz s Inequlity Tkshi Yoshino Mthemticl Institute,
12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
Can one hear the shape of a drum?
Cn one her the shpe of drum? After M. K, C. Gordon, D. We, nd S. Wolpert Corentin Lén Università Degli Studi di Torino Diprtimento di Mtemti Giuseppe Peno UNITO Mthemtis Ph.D Seminrs Mondy 23 My 2016 Motivtion:
RIEMANN INTEGRATION. Throughout our discussion of Riemann integration. B = B [a; b] = B ([a; b] ; R)
![RIEMANN INTEGRATION. Throughout our discussion of Riemann integration. B = B [a; b] = B ([a; b] ; R)](/thumbs/92/109668177.jpg "RIEMANN INTEGRATION. Throughout our discussion of Riemann integration. B = B [a; b] = B ([a; b] ; R)")
RIEMANN INTEGRATION Throughout our disussion of Riemnn integrtion B = B [; b] = B ([; b] ; R) is the set of ll bounded rel-vlued funtons on lose, bounded, nondegenerte intervl [; b] : 1. DEF. A nite set
MATH Final Review
MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out
Surface maps into free groups
Surfce mps into free groups lden Wlker Novemer 10, 2014 Free groups wedge X of two circles: Set F = π 1 (X ) =,. We write cpitl letters for inverse, so = 1. e.g. () 1 = Commuttors Let x nd y e loops. The
Hyers-Ulam stability of Pielou logistic difference equation
vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo
Lecture notes. Fundamental inequalities: techniques and applications
Lecture notes Fundmentl inequlities: techniques nd pplictions Mnh Hong Duong Mthemtics Institute, University of Wrwick Emil: m.h.duong@wrwick.c.uk Februry 8, 207 2 Abstrct Inequlities re ubiquitous in
Electromagnetism Notes, NYU Spring 2018
Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system
Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
Lecture 1 - Introduction and Basic Facts about PDEs
* 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
Theoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
CS 491G Combinatorial Optimization Lecture Notes
CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,
MATH1050 Cauchy-Schwarz Inequality and Triangle Inequality
MATH050 Cuchy-Schwrz Inequlity nd Tringle Inequlity 0 Refer to the Hndout Qudrtic polynomils Definition (Asolute extrem for rel-vlued functions of one rel vrile) Let I e n intervl, nd h : D R e rel-vlued
Final Exam Review. [Top Bottom]dx =
![Final Exam Review. [Top Bottom]dx =](/thumbs/90/103300886.jpg "Final Exam Review. [Top Bottom]dx =")
Finl Exm Review Are Between Curves See 7.1 exmples 1, 2, 4, 5 nd exerises 1-33 (odd) The re of the region bounded by the urves y = f(x), y = g(x), nd the lines x = nd x = b, where f nd g re ontinuous nd
Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:
MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES
MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These
Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
Chapter 8 Roots and Radicals
Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi
440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam
440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP
Project 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}
Symmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent
CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6
CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized
CS 573 Automata Theory and Formal Languages
Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple
SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE
RGMIA Reserch Report Collection, Vol., No., 998 http://sci.vut.edu.u/ rgmi SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE S.S. DRAGOMIR Astrct. Some clssicl nd new integrl inequlities of Grüss type re presented.
CIRCULAR COLOURING THE PLANE
CIRCULAR COLOURING THE PLANE MATT DEVOS, JAVAD EBRAHIMI, MOHAMMAD GHEBLEH, LUIS GODDYN, BOJAN MOHAR, AND REZA NASERASR Astrct. The unit distnce grph R is the grph with vertex set R 2 in which two vertices
Unit 4. Combinational Circuits
Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute
Arrow s Impossibility Theorem
Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep
Trigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
Study Guide and Intervention
- Stud Guide nd Intervention with the Sme Sign The quotient of two integers with the sme sign is positive. Emple. 7 The dividend nd the divisor hve the sme sign. b. () The dividend nd divisor hve the sme
Ch. 2.3 Counting Sample Points. Cardinality of a Set
Ch..3 Counting Smple Points CH 8 Crdinlity of Set Let S e set. If there re extly n distint elements in S, where n is nonnegtive integer, we sy S is finite set nd n is the rdinlity of S. The rdinlity of
The Riemann-Stieltjes Integral
Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0
y z A left-handed system can be rotated to look like the following. z
Chpter 2 Crtesin Coördintes The djetive Crtesin bove refers to René Desrtes (1596 1650), who ws the first to oördintise the plne s ordered pirs of rel numbers, whih provided the first sstemti link between
AP Calculus AB Unit 4 Assessment
Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope
Linear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
Advanced Algebra & Trigonometry Midterm Review Packet
Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.
p n m q m s m. (p q) n
Int. J. Nonliner Anl. Appl. (0 No., 6 74 ISSN: 008-68 (electronic http://www.ijn.com ON ABSOLUTE GENEALIZED NÖLUND SUMMABILITY OF DOUBLE OTHOGONAL SEIES XHEVAT Z. ASNIQI Abstrct. In the pper Y. Ouym, On
Algebraic Expressions
Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly
Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
University of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours
Mi-Term Exmintion - Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into
I. INTEGRAL THEOREMS. A. Introduction
1 U Deprtment of Physics 301A Mechnics - I. INTEGRAL THEOREM A. Introduction The integrl theorems of mthemticl physics ll hve their origin in the ordinry fundmentl theorem of clculus, i.e. xb x df dx dx
Algebra 2 Semester 1 Practice Final
Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers
More Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
Now we must transform the original model so we can use the new parameters. = S max. Recruits
MODEL FOR VARIABLE RECRUITMENT (ontinue) Alterntive Prmeteriztions of the pwner-reruit Moels We n write ny moel in numerous ifferent ut equivlent forms. Uner ertin irumstnes it is onvenient to work with
Polynomials. Polynomials. Curriculum Ready ACMNA:
Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression
Chapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
Chapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
Dong-Myung Lee, Jeong-Gon Lee, and Ming-Gen Cui. 1. introduction
J. Kore So. Mth. Edu. Ser. B: Pure Appl. Mth. ISSN 16-0657 Volume 11, Number My 004), Pges 133 138 REPRESENTATION OF SOLUTIONS OF FREDHOLM EQUATIONS IN W Ω) OF REPRODUCING KERNELS Dong-Myung Lee, Jeong-Gon
TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS
TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity
The Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral
DOI 763/s4956-6-4- Moroccn J Pure nd Appl AnlMJPAA) Volume ), 6, Pges 34 46 ISSN: 35-87 RESEARCH ARTICLE Generlized Hermite-Hdmrd-Fejer type inequlities for GA-conve functions vi Frctionl integrl I mdt
Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,
A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES. 1. Introduction
Ttr Mt. Mth. Publ. 44 (29), 159 168 DOI: 1.2478/v1127-9-56-z t m Mthemticl Publictions A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES Miloslv Duchoň Peter Mličký ABSTRACT. We present Helly
MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.
MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].