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1 INTRODUCTION Ho Chi Mih City OF Uiversity ENVIRONMENTAL of Techology DESIGN Faculty Chapter of Civil 1: Orietatio. Egieerig Evaluatio Departmet of mathematical of Water Resources skill Egieerig & Maagemet INTRODUCTION Ho Chi Mih City OF Uiversity ENVIRONMENTAL of Techology DESIGN Faculty Chapter of Civil 1: Orietatio. Egieerig Evaluatio Departmet of mathematical of Water Resources skill Egieerig & Maagemet Lecturer: Associ. Prof. Dr. NGUYỄN Thốg or Web: /14/16 1 Tél. (8) /14/16 CONTENTS. Chapter : Taylor series (1). Partial derivatives. Chapter 3: Taylor series (). Directioal derivatives. Chapter 4: Gradiet vector. Egieerig applicatio. Chapter 5: Mea, variace ad stadard deviatio. Normal distributio. Chapter 6: Least square method. Chapter 7: Correlatio coefficiet. Chapter 8: Egieerig applicatios. INTRODUCTION Ho Chi Mih City OF Uiversity ENVIRONMENTAL of Techology DESIGN Faculty Chapter of Civil 1: Orietatio. Egieerig Evaluatio Departmet of mathematical of Water Resources skill Egieerig & Maagemet Course Goals Uderstad ad describe the physical cocepts ad mathematical treatmet of Taylor series, partial derivatives, directioal derivatives, ad gradiet vector. Uderstad ad describe the physical cocepts ad mathematical treatmet of mea, variace, stadard deviatio, least square method ad correlatio coefficiet. Apply the obtaied skill to fudametal egieerig problems. /14/16 3 INTRODUCTION Ho Chi Mih City OF Uiversity ENVIRONMENTAL of Techology DESIGN Faculty Chapter of Civil 1: Orietatio. Egieerig Evaluatio Departmet of mathematical of Water Resources skill Egieerig & Maagemet Course Materials Lecture Notes: [1 ] M.YUHI AND Y. MAENO. Physical mathematics as a egieerig tool. Nakaishiya Pub. Co., 4 Referece books: [1] RAYMOND A. BERNETT et al. Applied Mathematics. DELLEN PUBLISHING COMPANY, [] ROBERT WREDE, Ph.D et al. Theory ad problems of advaced calculus. Schaum oulie. McGRAW-HILL,. [3] JAMES T. McCLAVE et al. Statistics for Busiessad Ecoomics. MAXWELL MACMILLAN INTERNATIONAL EDITIONS. /14/
2 /14/16 5 Chapter 1: Geeralities Excercise to test the mathematical skill 1. Series Arithmetic progressios Defiitio: Cosider two costats a, d: a (a+d) (a+d)... (a+d) a: iitial term d: commo differece The sum of the members of a fiite arithmetic progressio is called a arithmetic series: S /14/16 6 a a [1] 1 Geometric progressios Cosider a serie where each term after the first is foud by multiplyig the previous oe by a fixed, ozero umber called the commo ratio: Example: commo ratio = commo ratio =.5 The geeral form of a geometric sequece is: a ar where r is the commo ratio ad a is a scale /14/16 factor, equal to the sequece's start value. 7 ar ar 3...ar (1) Elemetary properties The -th term of a geometric sequece with iitial value a ad commo ratio r is 1 give by a a r Such a geometric sequece also follows the recursive relatio a for ra 1 every iteger /14/16 8 1
3 A geometric series is the sum of the umbers i a geometric progressio: For example: first term (here ), m be the umber of terms (here 4), ad r be the costat that each term is multiplied by to get the ext term (here 5), the sum is give by: m a(1 r ) Sm [] 1 r k1 1 1 Geerally: S /14/16 ar ar ar ar... ar 9 k1 1. Demostrate the formula [1] ad [].. Calculate the sum of members with : S= /1 +1/ /14/16 1 DEFINITION OF A SEQUENCE A sequece is a set of umbers u 1 ; u ; u 3 ;... i a defiite order of arragemet (i.e., a correspodece with the atural umbers) ad formed accordig to a defiite rule. Each umber i the sequece is called a term; u is called the th term. The sequece is called fiite or ifiite accordig as there are or are ot a fiite umber of terms. The sequece u1; u; u3;... is also desigated briefly by {u }. DEFINITION OF A SEQUENCE EXAMPLES 1. The set of umbers ; 7; 1; 17;... ; 3 is a fiite sequece; the th term is give by u =+5(-1) EXAMPLES. The set of umbers 1; 1=3; 1=5; 1=7;... is a ifiite sequece with th term u =1/(-1), Note: Uless otherwise specified, we shall cosider ifiite sequeces oly. /14/16 11 /14/16 1 3
4 LIMIT OF A SEQUENCE A umber l is called the limit of a ifiite sequece u 1 ; u ; u 3 ;... if for ay positive umber we ca fid a positive umber N depedig o such that u l for all itegers > N. I such case we write lim u l EXAMPLE. u 31/ 4; 7/ ;1/3;... we ca show that lim u 3 If the limit of a sequece exists, the sequece is called /14/16 13 coverget; otherwise, it is called diverget Write the first five terms of each of the followig sequeces. 1 1 ( 1) ; ; 3 3 /14/ ( 1) 1 ; ( 1) x ( 1)! ; 8 Two studets were asked to write a th term for the sequece 1; 16; 81; 56;... ad to write the 5th term of the sequece. Oe studet gave the th term as u = 4. The other studet, who did ot recogize this simple law of formatio, wrote u = Which studet gave the correct 5th term? Coclusio? /14/16 15 : LIMIT OF A SEQUENCE 1. Prouve that 1.1 lim /14/
5 /14/16 17 DERIVATIVES Let P (x ) be a poit o the graph of y=f(x). Let P(x) be a earby poit o this same graph of the the futio f. The the lie through these two poits is called a secat lie. Its slope, ms, is the differece quotiet: f (x) f (x) y ms x x x where x ad y are called the icremets i x ad y, respectively. Also this slope may be writte: f (x h) f (x ) h m with h x x /14/16 s 18 x f (x h) f (x ) (x) lim h f /14/16 19 called the derivative of the fuctio f at its domai value x RIGHT- AND LEFT-HAND DERIVATIVES The status of the derivative at ed poits of the domai of f, ad i other special circumstaces, is clarified by the followig defiitios. The right-had derivative of f(x) at x=x is defied as: ' f (x h) f (x) f(x) lim h If this limit exists. Similarly, the left-had derivative of f(x) at x=x is defied as: ' f(x) lim /14/16 f (x h) f (x ) h 5
6 A fuctio f has a derivative at x=x if ad oly if: ' ' f(x) f(x) DIFFERENTIALS Let x= be a icremet give to x. The: y f(x x) f(x) [1] is called the icremet i y=f(x). If f(x) is cotiuous ad has a cotiuous first derivative i a iterval, the: y f(x) x x f(x) Where as x. The expressio dy f(x). [] is called the differetial of y or f(x) or the pricipal part of y /14/16 1 Because of the defiitios [1] ad [], we ofte write: dy ' f (x x) f (x) y f (x) lim lim x x /14/16 THE DIFFERENTIATION OF COMPOSITE FUNCTION May fuctios are a compositio of simpler oes. For example, if f ad g have the rules of correspodece u=x 3 ad y=siu respectively, the y= si(x 3 ) is the rule for a composite fuctio F=g(f). /14/16 3 The domai of F is that subset of the domai of F whose correspodig rage values are i the domai of g. The rule of composite fuctio differetiatio is called the chai rule ad is represeted by: dy I this example: /14/16 4 dy. du du or F (x) g (u).f (x) 3 dy d(si x ) 3 cos x.(3x ) 6
7 IMPLICIT DIFFERENTIATION The rule of correspodece for a fuctio may ot be explicit. For example, the rule y=f(x) is implicit to the equatio x + 4xy 5 + 7xy + 8 =. Furthermore, there is o reaso to believe that this equatio ca be solved for y i terms of x. 5 4 dy dy x 4 y 5xy 7 y x Observe that this equatio ca be solved for (dy/) as a fuctio of x ad y (but ot of x aloe). HIGHER ORDER DERIVATIVES If f(x) is differetiable i a iterval, its derivative is give by f (x), y or dy=, where y=f(x). If f (x) is also differetiable i the iterval, its derivative is deoted by f x), y or dy d (y) d Similarly, the th derivative of f(x), if it exists, is deoted by f () (x) or y () /14/16 5 /14/16 6 : DIFFERENTIALE 1. If y=f(x)= x 3-6x, fid (a) y; (b) dy; (c) y- dy.. Prove the formula: d d g(x) df (x) f (x).g(x) f (x). g(x). assumig f ad g are differetiable. /14/16 7 L HOSPITAL S RULES If lim f(x)=a ad lim g(x=b whe x x where A ad B f (x) are either both zero or both ifiite, lim whe x x g(x) is ofte called a idetermiate of the form / or /, respectively. The followig theorems, called L Hospital s rules, facilitate evaluatio of such limits. If f(x) ad g(x) are differetiable i the iterval [a;b] except possibly at a poit x i this iterval, ad if, g(x) for x the: f (x) f(x) lim lim g(x) g (x) xx xx /14/16 8 7
8 Evaluate: x 1. e 1 1 cos x lim & lim x x x1 x x 1. x 3x x 5 lim x e & lim x x 5x 6x 3 1/ x 3. limcox & lim x.l x x x /14/16 9 : DERIVATIVES 1. Show directly from defiitio that the derivative of f(x)=x 3 is 3x.. f(x)=x (1/) 3. Let x.si(1/ x) if x f (x) if x (a) Is f(x) cotiuous at x=? (b) Does f(x) have a derivative at x=? 4. Let x.si(1/ x) if x f (x) if x (a) Is f(x) differetiable at x=? (b) Is f (x) cotiuous at at /14/16 x=? 3 5. Prove that f= x is differetiable i <= x <= Fid a equatio for the taget lie to y= x at the poit where (a) x = 1/3; (b) x = Calculate dy/ if (a) xy 3-3x =xy + 5, (b) e xy = y l x + cos x. /14/16 31 /14/16 3 8
9 END OF CHAPTER 1 /14/
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