《高等数学》课程电子教案（PPT课件）Chapter 1 Functions and Limits §1.1 Preliminaries for Calculus

AdvancedMathematicsWanYuanSchool of Science, WHUTMymailbox:wanwanyuan@sina.comMy phone: 13971622102Class mailbox:2010qiiy@sina.com

Advanced Mathematics Wan Yuan School of Science, WHUT My mailbox: wanwanyuan@sina.com My phone: 13971622102 Class mailbox: 2010gjjy@sina.com

Some rules for our classPlease take notesPlease turn off your cell-phonesIf you are late, just come in and join usAsk questions as soon as possibleDo homework by yourselfHand up your homework in time

Some rules for our class ◼ Please take notes ◼ Please turn off your cell-phones ◼ If you are late, just come in and join us ◼ Ask questions as soon as possible ◼ Do homework by yourself ◼ Hand up your homework in time

IntroductionAdvanced mathematics is a primary course for collagestudents.Inthisclass,wewill introduce somebasicconceptsand some useful tools of mathematics foryourfurtherstudy or research.Part oneTheoreticalbasisofCalculusThedifferentialCalculusanditsApplicationTheintegralCalculusand itsApplication

Introduction Advanced mathematics is a primary course for collage students. In this class, we will introduce some basic concepts and some useful tools of mathematics for your further study or research. Part one ◼ Theoretical basis of Calculus ◼ The differential Calculus and its Application ◼ The integral Calculus and its Application

时代教育·国外高校优秀教材精选普通高等教育“十一五国家级规划教材微积分Calculus（英文版·原书第8版）高等数学第六版上册同济大学数学系编高等教育出版社Prntiey虹械工业出票社GMENEOVCATION PAESHall

S 1.1 Preliminaries for CalculusI. TheIntegersandtheRational Numbers1.Set (集合)Def: A collection of numbers which share the same property自然数natural number（正、负）整数(positive, negative) integer有（无）理数rational (irrational) number负数positive number 正数negative number实数复数real numbercomplexnumber实轴原点real lineorigin

I. The Integers and the Rational Numbers 1. Set (集合) Def: A collection of numbers which share the same property. natural number rational (irrational) number (positive, negative) integer 自然数 （正、负）整数 有(无）理数 real number 实数 real line 实轴 origin 原点 complex number 复数 positive number 正数 negative number 负数 §1.1 Preliminaries for Calculus

S 1.1 Preliminaries for CalculusI. The Integers and the Rational Numbers（四则运算）2. The Four Arithmetic Operations（加减乘除）addsubtractdividemultiply（和差积商）differenceproductquotientsum

2. The Four Arithmetic Operations （四则运算） add sum subtract difference multiply divide product quotient （加减乘除） （和差积商） I. The Integers and the Rational Numbers §1.1 Preliminaries for Calculus

S 1.1 Preliminaries for CalculusI. Inequalities(区间)1. Intervals(xa<x<b)=(a,b) Open Intervalb0ax[xa ≤ x ≤b] =[a,b] Closed Intervalxb0a(x|a<x≤b)=(a,b]bxa0(xx<b) =(-00,b)bx0(x|x ≥a} =[a,+)InfiniteIntervalx0aR=(-00,+0)

II. Inequalities {x a  x  b}= (a,b) Open Interval o a b x {x a  x  b}= [a,b] o a b x {x a  x  b}= (a,b] o a b x Closed Interval {x x  b}= (−,b) o b x {x x  a}=[a,+) o a x R = (−,+) x Infinite Interval 1. Intervals （区间） §1.1 Preliminaries for Calculus

S 1.1 Preliminaries for CalculusII.Inequalities2.Neighborhood of point a （点 a的邻域)Def : U(a,8)=(x x-a <8) =(x|a-8<x<a+8)a is the center of this neighborhood, and S is the radiusa-8aa+8xGeometriclnterpretation:The set of point x whose distance from point a is less than S.Def : U(a,8)=(x0<|x-a <8)

2. Neighborhood of point a (点 a 的邻域） a −  a a +  x Def : U(a, ) ={x 0  x − a  } o Geometric Interpretation: The set of point x whose distance from point a is less than . a is the center of this neighborhood, and is the radius.  Def : U(a, ) ={x x − a  } ={x | a −  x  a +} II. Inequalities §1.1 Preliminaries for Calculus

S 1.1 Preliminaries for CalculusII.Inequalities3.Solve lInegualitiesExample1Solve the inequality [3x + 1 (x +13)(5x -11)<0Then the solution set is (-13

II. Inequalities 3. Solve Inequalities Solution 3x +1  2 x −6  3x +1  2x −12 2 2  (3x +1)  (2x −12) 5 54 143 0 2  x + x −   (x +13)(5x −11)  0 Then the solution set is . ) 5 11 (−13, Example1 Solve the inequality . 3x +1  2 x −6 §1.1 Preliminaries for Calculus

S 1.1 Preliminaries for Calculus（绝对值）Ill.AbsoluteValuesx, x≥01.Absolute value-x,x<0Example 2 Let be a positive number. Find a positivenumberssuchthatx-5<8=|6x-30 <8Solution6x-30< =6x-5<8E68Thenwe choose=6

III. Absolute Values （绝对值） 1. Absolute value    −   = , 0 0 x x x, x x Solution Example 2 Let be a positive number. Find a positive  x − 5    6x − 30   number such that  6x − 30    6 x − 5   6 5   x −  Then we choose . 6   = §1.1 Preliminaries for Calculus