Handout I - Mathematics II

Size: px
Start display at page:

Download "Handout I - Mathematics II"

Transcription

1 Hndout I - Mthemtics II The im of this hndout is to briefly summrize the most importnt definitions nd theorems, nd to provide some smple exercises. The topics re discussed in detil t the lectures nd seminrs. Students should lso consult the suggested redings... Theory.. Derivtives of rel functions Definition.. Let f : D R be function, where D is n intervl, nd let x D. If the limit f(x) f(x ) lim x x x x exists then we sy tht f is differentible t x. In this cse we write f (x ) = lim x x f(x) f(x ) x x for the derivte of f t x. If f is differentible for ll x D then we sy tht f is differentible. We write f for the derivtive of f. Theorem.. Suppose tht f(x) nd g(x) re differentible functions for ll x R. Then we hve (f(x) + g(x)) = f (x) + g (x), (f(x) g(x)) = f (x) g (x), (cf(x)) = cf (x) (c R), (f(x)g(x)) = f (x)g(x) + f(x)g (x), ( ) f(x) = f (x)g(x) f(x)g (x), (f(g(x))) = f (g(x)) g (x). g(x) g 2 (x) Theorem.2. All the elementry functions re differentible. Further, we hve c = (c R), (x α ) = αx α (α R), ( x ) = x ln() ( > ), (e x ) = e x, (log (x)) = x ln(), (ln(x)) = x, (sin(x)) = cos(x), (cos(x)) = sin(x), (ctg(x)) = sin 2 (x), (tg(x)) = cos 2 (x), (rctg(x)) = + x. 2 Here rctg(x) is the inverse of the function tg(x) : ( π/2, π/2) R.

2 2.2. Smple exercises. Exercise.. Differentite the following functions: 2(x 5 5 x cos x x), e x sin(x), x 2 + ln x, ln(sin(x)). Exercise.2. Differentite the following functions: ( ) 3 2 x cos(cos(x 2 )) x 2 + x ln( x) +, e x2 sin(3x). ln(x) Throughout this mteril ll functions f re ssumed to be elementry functions, with f : D R, where D is n intervl. 2.. Theory. 2. Anlysis of rel functions Definition 2.. Let f be function nd x D. If there exists n ε > such tht for ll x D (x ε, x + ε) we hve f(x ) f(x), then f hs locl minimum t x, f(x ) < f(x), then f hs strict locl minimum t x, f(x ) f(x), then f hs locl mximum t x, f(x ) > f(x), then f hs strict locl mximum t x. In ny of the bove cses we sy tht f hs locl extremum t x. Definition 2.2. Let f be function. If for ll x, y D with x < y we hve f(x) f(y), then we sy tht f is monotone incresing. If for ll x, y D with x < y we hve f(x) f(y), then we sy tht f is monotone decresing. Definition 2.3. Let f be function. If for ll x, y D nd λ [, ] we hve λf(x) + ( λ)f(y) f(λx + ( λ)y), then we sy tht f is convex on D. If f is convex on D, then we sy tht f is concve on D. Let x D. If for some ε >, f is convex on (x ε, x ) nd concve on (x, x + ε), or vice vers, then x is clled n inflection point of f. Theorem 2.. Suppose tht f hs locl extremum t some x D. Then we hve f (x ) =. Theorem 2.2. Suppose tht x D, nd f (x ) = f (x ) = = f (n) (x ) =, but f (n+) (x ). If n + is even, then f hs locl extremum t x. Further, if f (n+) (x ) < then f hs locl mximum t x, nd if f (n+) (x ) > then f hs locl minimum t x, respectively. On the other hnd, if n + is odd, then f hs no locl extremum t x.

3 Theorem 2.3. Suppose tht f (x) for ll x D. monotone incresing on D. Theorem 2.4. Suppose tht f (x) for ll x D. monotone decresing on D. 3 Then f is Then f is Theorem 2.5. Suppose tht f (x) for ll x D. Then f is convex on D. Theorem 2.6. Suppose tht f (x) for ll x D. concve on D Smple exercise. Then f is Exercise 2.. Let f : R R be defined by f(x) = x 3 3x. Give the complete nlysis of f, tht is: determine the zeroes of f; clculte the limits lim f(x) nd lim f(x); determine the locl extrem of f; x x determine the intervls where f is monotone; determine the intervls where f is convex/concve, nd give the inflection points of f. 3.. Theory. 3. L Hospitl s rule Theorem 3.. Let f, g be two functions, such tht g(x) (x D). Further, ssume tht either lim f(x) = lim g(x) =, or lim f(x) = x x x x x x ±, lim g(x) = ±, where x D, or x is n endpoint of D x x (possibly x = ± ). Then we hve f(x) lim x x g(x) = lim f (x) x x g (x), if the ltter limit exists Smple exercises. Exercise 3.. Clculte the following limits: sin x lim x x, lim x x e, lim x ln x. x x 4.. Theory. 4. Tylor polynomil, Tylor series Definition 4.. Let f : [, b] R be function differentible n times. Then the Tylor polynomil of order n of f(x) round point x (, b) is given by n f (k) (x ) T n (x) = (x x ) k. k! k= If x = then the bove polynomil is clled the McLurin polynomil of f(x) of order n.

4 4 Theorem 4.. Let f : [, b] R be function differentible n + times. Then we hve R n (x) = f (n+) (ξ) (n + )! (x x ) n+, where ξ is number between x nd x, depending on x. Definition 4.2. Let f : [, b] R be function differentible infinitely mny times. Then the Tylor series of f(x) round point x (, b) is given by f (k) (x ) T (x) = (x x ) k. k! k= If x = then the bove series is clled the McLurin series of f(x) of order n Smple exercises. Exercise 4.. Give the Tylor-series of the functions 5.. Theory. x 3 + 2x 2 x + 3, e x, sin x, cos x. 5. Primitive functions, indefinite integrl Definition 5.. Suppose tht we hve F (x) = f(x). Then we sy tht F (x) is primitive function of f(x). Theorem 5.. Let F (x) be primitive function of f(x). Then G(x) is primitive function of f(x) if nd only if G(x) = F (x) + c is vlid with some c R. Definition 5.2. The set of primitive functions of f(x) is clled the indefinite integrl of f(x). Nottion: f(x)dx = F (x) + c (c R), where F (x) is primitive function of f(x). Theorem 5.2. We hve (f(x) ± g(x))dx = f(x)dx ± g(x)dx nd λf(x)dx = λ f(x)dx (λ R). Theorem 5.3. We hve x α dx = xα+ + c (α ), α + cos xdx = sin x + c, e x dx = e x + c, dx = ln x + c, x sin xdx = cos x + c, dx = rctgx + c. + x2

5 5 Theorem 5.4. We hve f (x)f(x) α dx = f(x)α+ α + nd f (x) dx = lnf(x) + c. f(x) + c (α ) Theorem 5.5. (Prtil integrtion (integrtion by prts)) We hve f (x)g(x)dx = f(x)g(x) f(x)g (x)dx. Theorem 5.6. (Prtil frctions) Suppose tht x 2 + x + b hs two distinct rel roots, α nd β. Then there exist rel numbers A nd B such tht x 2 + x + b = (x α)(x β) = A (x α) + B (x β). Thus we hve dx = A ln(x α) + B ln(x β) + c. x 2 + x + b Theorem 5.7. (Integrl with substitution) We hve ( (g f(x)dx = f(g(t))g (t)dt) (x) ), where g(x) is n invertible function Smple exercises. Exercise 5.. Determine the following indefinite integrls: ( ) x + dx, (e x cos x)dx, x + x dx. 2 Exercise 5.2. Determine the following indefinite integrls: cos x sin 3 x xdx, ( + x 2 ) dx, e x ctgxdx, 4 2 e. x Exercise 5.3. Determine the following indefinite integrls: x cos xdx, x ln xdx, x 3 sin xdx. Exercise 5.4. Determine the following indefinite integrls: x + x + 2 x 2 +, x 2 dx, 2x x 2 x 2 dx. Exercise 5.5. Determine the following indefinite integrls:, e 2x xe x ( x + 2) 3 e x dx, x dx.

6 6 6.. Theory. 6. Riemnn (definite) integrl Definition 6.. Suppose tht f is continuous on D = [, b], with, b R, < b. For ny positive integer n put δ n := b nd x (n) 2 n i := +iδ n for i =,,..., 2 n. Further, let t (n) i [x (n) i, x (n) i+ ] such tht f(t(n) i ) f(x) for ll x [x (n) i, x (n) i+ ] (i =,,..., 2n ). Put S n := 2 n i= f(t (n) i )δ n. Then the Riemnn integrl of f(x) between nd b is defined s b f(x)dx = lim n S n. Definition 6.2. We use the following nottion: b f(x)dx = f(x)dx, f(x)dx =. Theorem 6.. We hve b b b b (f(x) ± g(x))dx = f(x)dx ± g(x)dx nd b b λf(x)dx = λ f(x)dx (λ R). Theorem 6.2. (Newton-Leibniz formul) Let F (x) be primitive function of f(x). Then we hve b f(x)dx = [F (x)] b = F (b) F (). Theorem 6.3. (Prtil integrtion (integrtion by prts) for definite integrls) We hve b f (x)g(x)dx = [f(x)g(x)] b b f(x)g (x)dx.

7 Theorem 6.4. (Integrl with substitution for definite integrls) We hve b f(x)dx = β α f(g(t))g (t)dt, where g(x) is n invertible function, nd α = g () nd β = g (b) Smple exercises. Exercise 6.. Clculte the following integrls: 2 ( ) x + dx, x π (e x + 2 cos x)dx, Exercise 6.2. Clculte the following integrls: π cos x sin 3 xdx, 2 2 x ( + x 2 ) 4 dx, π/2 π/4 Exercise 6.3. Clculte the following integrls: 2π x cos xdx, 3 xe x dx, π/2 Exercise 6.4. Clculte the following integrls: x + x 2 +, 4 x + 2 x 2 dx, Exercise 6.5. Clculte the following integrls: Theory. 2, x ( x + 2) e 2x e x dx, ctgxdx, x 3 sin xdx. 3 + x 2 dx. 2x x 2 x 2 dx Improprius (improper) integrl e x 2 e x. xe x dx. Definition 7.. Let f be bounded on the intervl [, ). Then the improprius integrl of f(x) between nd is defined by if the limit exists. b f(x)dx = lim f(x)dx, b 7

8 8 Definition 7.2. Let f be bounded on the intervl (, b]. Then the improprius integrl of f(x) between nd b is defined by if the limit exists. b f(x)dx = b lim f(x)dx, Definition 7.3. Let f : R R be bounded. Then the improprius integrl of f(x) between nd is defined by f(x)dx = c f(x)dx + f(x)dx with n rbitrry c R, if the ltter improprius integrls exist. Definition 7.4. Let f[, b] R such tht f is bounded on [, c] for ny c with < c < b. Then the improprius integrl of f(x) between nd b is defined by c if the limit exists. b c f(x)dx = lim f(x)dx, c b Definition 7.5. Let f : [, b] R such tht f is bounded on [c, b] for ny c with < c < b. Then the improprius integrl of f(x) between nd b is defined by if the limit exists. b b f(x)dx = lim f(x)dx, c c 7.2. Smple exercises. Exercise 7.. Clculte the following improprius integrls: 2 x 3 dx, 3 x dx. 8.. Theory. 8. Applictions of the integrl

9 Theorem 8.. Let f, g : [, b] R be two functions, such tht f(x) g(x) for x b. Then the re of the domin enclosed by f(x) nd g(x) over [, b] is given by b (f(x) g(x))dx. Theorem 8.2. Let T be domin in R 3 (i.e. solid), nd let A(x) denote the re of the intersection of T nd the plin {(u, v, w) : u = x} for x R. Then the volume of T is given by if the bove integrl exists. V (T ) = A(x)dx, Theorem 8.3. Let f : [, b] R be function. Then the length of the curve of f(x) over [, b] is given by if the integrl exists. L(f) = b + (f (x)) 2 dx Theorem 8.4. Let f : [, b] R be function nd let T be the domin (solid) obtined by rotting the curve of f round he intervl [, b] in R 3. Then the volume of T nd the surfce of T re given by V (T ) = b πf 2 (x)dx 9 nd b S(T ) = 2πf(x) + (f (x)) 2 dx respectively, if the integrls exist. Definition 8.. Let f : [, 2π] R be function. Then the Fourierseries of f(x) is given by + ( k cos kx + b k sin kx), with k= = 2π 2π f(x)dx

10 nd k = π 2π f(x) cos kxdx, b k = π 2π f(x) sin kxdx (k =, 2,... ) Smple exercises. Exercise 8.. Clculte the re enclosed by the curves of the functions f(x) = x 2 nd g(x) = x + 2 over the intervl [, 2]. Exercise 8.2. Clculte the length of the curve of f(x) = 2x + 3 over the intervl [, 3], nd the volume nd surfce of the solid T obtined by rotting f(x) over this intervl. Exercise 8.3. Give the Fourier trnsform of the function f(x) = x.

Calculus II: Integrations and Series

Calculus II: Integrations and Series Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]

More information

Chapter 6. Riemann Integral

Chapter 6. Riemann Integral Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

More information

Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.

Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones. Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description

More information

Mathematics 1. (Integration)

Mathematics 1. (Integration) Mthemtics 1. (Integrtion) University of Debrecen 2018-2019 fll Definition Let I R be n open, non-empty intervl, f : I R be function. F : I R is primitive function of f if F is differentible nd F = f on

More information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

More information

0.1 Chapters 1: Limits and continuity

0.1 Chapters 1: Limits and continuity 1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97

More information

Overview of Calculus I

Overview of Calculus I Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

More information

IMPORTANT THEOREMS CHEAT SHEET

IMPORTANT THEOREMS CHEAT SHEET IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop.

More information

f(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all

f(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all 3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the

More information

Euler-Maclaurin Summation Formula 1

Euler-Maclaurin Summation Formula 1 Jnury 9, Euler-Mclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,

More information

Definite integral. Mathematics FRDIS MENDELU

Definite integral. Mathematics FRDIS MENDELU Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

More information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30 Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

More information

Advanced Calculus I (Math 4209) Martin Bohner

Advanced Calculus I (Math 4209) Martin Bohner Advnced Clculus I (Mth 4209) Spring 2018 Lecture Notes Mrtin Bohner Version from My 4, 2018 Author ddress: Deprtment of Mthemtics nd Sttistics, Missouri University of Science nd Technology, Roll, Missouri

More information

Math Bootcamp 2012 Calculus Refresher

Math Bootcamp 2012 Calculus Refresher Mth Bootcmp 0 Clculus Refresher Exponents For ny rel number x, the powers of x re : x 0 =, x = x, x = x x, etc. Powers re lso clled exponents. Remrk: 0 0 is indeterminte. Frctionl exponents re lso clled

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

More information

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

More information

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer. Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points

More information

Section Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?

Section Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled? Section 5. - Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles

More information

Section 7.1 Integration by Substitution

Section 7.1 Integration by Substitution Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find

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

The Riemann Integral

The Riemann Integral Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function

More information

Calculus MATH 172-Fall 2017 Lecture Notes

Calculus MATH 172-Fall 2017 Lecture Notes Clculus MATH 172-Fll 2017 Lecture Notes These notes re concise summry of wht hs been covered so fr during the lectures. All the definitions must be memorized nd understood. Sttements of importnt theorems

More information

FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I FINALTERM EXAMINATION 011 Clculus &. Anlyticl Geometry-I Question No: 1 { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...

More information

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1 Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

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

Calculus I-II Review Sheet

Calculus I-II Review Sheet Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing

More information

Week 10: Riemann integral and its properties

Week 10: Riemann integral and its properties Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart. MATH 1080: Clculus of One Vrile II Fll 2017 Textook: Single Vrile Clculus: Erly Trnscendentls, 7e, y Jmes Stewrt Unit 2 Skill Set Importnt: Students should expect test questions tht require synthesis of

More information

The Riemann-Lebesgue Lemma

The Riemann-Lebesgue Lemma Physics 215 Winter 218 The Riemnn-Lebesgue Lemm The Riemnn Lebesgue Lemm is one of the most importnt results of Fourier nlysis nd symptotic nlysis. It hs mny physics pplictions, especilly in studies of

More information

Main topics for the First Midterm

Main topics for the First Midterm Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

More information

Big idea in Calculus: approximation

Big idea in Calculus: approximation Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

More information

The Product Rule state that if f and g are differentiable functions, then

The Product Rule state that if f and g are differentiable functions, then Chpter 6 Techniques of Integrtion 6. Integrtion by Prts Every differentition rule hs corresponding integrtion rule. For instnce, the Substitution Rule for integrtion corresponds to the Chin Rule for differentition.

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

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

Chapter 6. Infinite series

Chapter 6. Infinite series Chpter 6 Infinite series We briefly review this chpter in order to study series of functions in chpter 7. We cover from the beginning to Theorem 6.7 in the text excluding Theorem 6.6 nd Rbbe s test (Theorem

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

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8 Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite

More information

x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx . Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

More information

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1 Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

More information

1 The Riemann Integral

1 The Riemann Integral The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

More information

Integration Techniques

Integration Techniques Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

Analysis-2 lecture schemes

Analysis-2 lecture schemes Anlysis-2 lecture schemes (with Homeworks) Csörgő István November, 204 A jegyzet z ELTE Informtiki Kr 204. évi Jegyzetpályáztánk támogtásávl készült Contents. Lesson 4.. Continuity of functions.........................

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

7.2 Riemann Integrable Functions

7.2 Riemann Integrable Functions 7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous

More information

Calculus in R. Chapter Di erentiation

Calculus in R. Chapter Di erentiation Chpter 3 Clculus in R 3.1 Di erentition Definition 3.1. Suppose U R is open. A function f : U! R is di erentible t x 2 U if there exists number m such tht lim y!0 pple f(x + y) f(x) my y =0. If f is di

More information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

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

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), ) Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s

More information

MA 124 January 18, Derivatives are. Integrals are.

MA 124 January 18, Derivatives are. Integrals are. MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,

More information

1 The fundamental theorems of calculus.

1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

More information

Math 554 Integration

Math 554 Integration Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

More information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More information

1 The fundamental theorems of calculus.

1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

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

MATH 144: Business Calculus Final Review

MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

More information

DEFINITE INTEGRALS. f(x)dx exists. Note that, together with the definition of definite integrals, definitions (2) and (3) define b

DEFINITE INTEGRALS. f(x)dx exists. Note that, together with the definition of definite integrals, definitions (2) and (3) define b DEFINITE INTEGRALS JOHN D. MCCARTHY Astrct. These re lecture notes for Sections 5.3 nd 5.4. 1. Section 5.3 Definition 1. f is integrle on [, ] if f(x)dx exists. Definition 2. If f() is defined, then f(x)dx.

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

Example Sheet 6. Infinite and Improper Integrals

Example Sheet 6. Infinite and Improper Integrals Sivkumr Exmple Sheet 6 Infinite nd Improper Integrls MATH 5H Mteril presented here is extrcted from Stewrt s text s well s from R. G. Brtle s The elements of rel nlysis. Infinite Integrls: These integrls

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

F (x) dx = F (x)+c = u + C = du,

F (x) dx = F (x)+c = u + C = du, 35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil

More information

Notes on Calculus. Dinakar Ramakrishnan Caltech Pasadena, CA Fall 2001

Notes on Calculus. Dinakar Ramakrishnan Caltech Pasadena, CA Fall 2001 Notes on Clculus by Dinkr Rmkrishnn 253-37 Cltech Psden, CA 91125 Fll 21 1 Contents Logicl Bckground 2.1 Sets... 2.2 Functions... 3.3 Crdinlity... 3.4 EquivlenceReltions... 4 1 Rel nd Complex Numbers 6

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

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

1. On some properties of definite integrals. We prove

1. On some properties of definite integrals. We prove This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.

More information

Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).

Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher). Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested

More information

Chapters 4 & 5 Integrals & Applications

Chapters 4 & 5 Integrals & Applications Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

More information

We know that if f is a continuous nonnegative function on the interval [a, b], then b

We know that if f is a continuous nonnegative function on the interval [a, b], then b 1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

More information

MATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral.

MATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral. MATH 409 Advnced Clculus I Lecture 18: Drboux sums. The Riemnn integrl. Prtitions of n intervl Definition. A prtition of closed bounded intervl [, b] is finite subset P [,b] tht includes the endpoints

More information

Math 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that

Math 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that Mth 118: Honours Clculus II Winter, 2005 List of Theorems Lemm 5.1 (Prtition Refinement): If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound

More information

Lecture 19: Continuous Least Squares Approximation

Lecture 19: Continuous Least Squares Approximation Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for

More information

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

More information

11 An introduction to Riemann Integration

11 An introduction to Riemann Integration 11 An introduction to Riemnn Integrtion The PROOFS of the stndrd lemms nd theorems concerning the Riemnn Integrl re NEB, nd you will not be sked to reproduce proofs of these in full in the exmintion in

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

f(a+h) f(a) x a h 0. This is the rate at which

f(a+h) f(a) x a h 0. This is the rate at which M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the

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

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

Mathematics 19A; Fall 2001; V. Ginzburg Practice Final Solutions

Mathematics 19A; Fall 2001; V. Ginzburg Practice Final Solutions Mthemtics 9A; Fll 200; V Ginzburg Prctice Finl Solutions For ech of the ten questions below, stte whether the ssertion is true or flse ) Let fx) be continuous t x Then x fx) f) Answer: T b) Let f be differentible

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

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

Advanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015

Advanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015 Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n

More information

Principles of Real Analysis I Fall VI. Riemann Integration

Principles of Real Analysis I Fall VI. Riemann Integration 21-355 Principles of Rel Anlysis I Fll 2004 A. Definitions VI. Riemnn Integrtion Let, b R with < b be given. By prtition of [, b] we men finite set P [, b] with, b P. The set of ll prtitions of [, b] will

More information

Review. April 12, Definition 1.2 (Closed Set). A set S is closed if it contains all of its limit points. S := S S

Review. April 12, Definition 1.2 (Closed Set). A set S is closed if it contains all of its limit points. S := S S Review April 12, 2017 1 Definitions nd Some Theorems 1.1 Topology Definition 1.1 (Limit Point). Let S R nd x R. Then x is limit point of S if, for ll ɛ > 0, V ɛ (x) = (x ɛ, x + ɛ) contins n element s S

More information

More Properties of the Riemann Integral

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

More information

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus The Fundmentl Theorem of Clculus Professor Richrd Blecksmith richrd@mth.niu.edu Dept. of Mthemticl Sciences Northern Illinois University http://mth.niu.edu/ richrd/mth229. The Definite Integrl We define

More information

Functions of Several Variables

Functions of Several Variables Functions of Severl Vribles Sketching Level Curves Sections Prtil Derivtives on every open set on which f nd the prtils, 2 f y = 2 f y re continuous. Norml Vector x, y, 2 f y, 2 f y n = ± (x 0,y 0) (x

More information

Introduction and Review

Introduction and Review Chpter 6A Notes Pge of Introuction n Review Derivtives y = f(x) y x = f (x) Evlute erivtive t x = : y = x x= f f(+h) f() () = lim h h Geometric Interprettion: see figure slope of the line tngent to f t

More information

a n+2 a n+1 M n a 2 a 1. (2)

a n+2 a n+1 M n a 2 a 1. (2) Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside

More information

The Regulated and Riemann Integrals

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

More information

Review of Riemann Integral

Review of Riemann Integral 1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals. MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded

More information

Improper Integrals. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Improper Integrals. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics Improper Integrls MATH 2, Clculus II J. Robert Buchnn Deprtment of Mthemtics Spring 28 Definite Integrls Theorem (Fundmentl Theorem of Clculus (Prt I)) If f is continuous on [, b] then b f (x) dx = [F(x)]

More information

MAA 4212 Improper Integrals

MAA 4212 Improper Integrals Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which

More information

MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008

MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008 MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul

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

FUNCTIONS OF α-slow INCREASE

FUNCTIONS OF α-slow INCREASE Bulletin of Mthemticl Anlysis nd Applictions ISSN: 1821-1291, URL: http://www.bmth.org Volume 4 Issue 1 (2012), Pges 226-230. FUNCTIONS OF α-slow INCREASE (COMMUNICATED BY HÜSEYIN BOR) YILUN SHANG Abstrct.

More information