《微积分》课程教学课件（Calculus）01. Preliminaries

Some rules for ourclassCalculus Please preview lessons before class.可时 Please take notes.551Pleaseturn off your cell-phones.Prof.Dr.GAOFeiDDon't be late.15 minutes late is the equal of missing one class.可治and the percentage ofabsence wllinfluence your grade ofthiscourse whichconsists ofthree parts, the scoresofthe finalhttp://feigao.weebly.comexamination, homework scoresand the attendance.@rgaofeie@gmail.com Ask questions as soonas possibleDo homeworkby yourself Hand up your homework in time.IntroductionIntroductionCalculus is a primary course for collegeLearning calculus is a process; it does not come allstudents.atonce.In this class, we will introduce some basic concepts andBepatient,persevere,ask questions,discuss ideassome useful tools of mathematics for yourfurther studyand work with classmates,and seek help when youorresearch.need it, right away.PartoneTherewardsoflearningcalculuswill beveryTheoretical basis of Calculussatisfying,both intellectually and professionally.The Differential Calculus and its ApplicationThe Integral Calculus and its ApplicationCALCULUSTHOMAS'INTALCALCULUSEARLYTRANSCENDENTALChapter 1Prelimiaries

2016/11/15 1 Calculus Prof. Dr. GAO Fei gaofeie@gmail.com http://feigao.weebly.com Some rules for our class  Please preview lessons before class.  Please take notes.  Please turn off your cell-phones.  Don’t be late. 15 minutes late is the equal of missing one class, and the percentage of absence will influence your grade of this course which consists of three parts, the scores of the final examination, homework scores and the attendance.  Ask questions as soon as possible.  Do homework by yourself.  Hand up your homework in time. Introduction Calculus is a primary course for college 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  Learning calculus is a process; it does not come all at once.  Be patient, persevere, ask questions, discuss ideas and work with classmates, and seek help when you need it, right away.  The rewards of learning calculus will be very satisfying, both intellectually and professionally. Introduction Chapter 1 Prelimiaries

2Whatis Calculus?1.1Howtolearn Calculus?LinesDefinitionIncrementsIfaparticlemoves fromthe1.Incrementspoint (x,Ji) to the point(,Js)The increments in itscoordinates are3Ar=x, x, and Ay=)2-414.=)--FIGUR(Example1)DefinitionSlopeLet P() and P(x2.y2)bepoints onanonvertical2.Slopeof a Lineline L.The slope of LisriseAy_M-m=rmArx-xLLine going uphill - positive slopeLine going downhil -negative slopeFIGURE1.8Triangles POP,anP'OrP'are similar,so the tatio ef tHorizontal line -zero slopesides has the same value for amy tuo pointon the line. This coon valoe is the line?Vertical line no slopedope

2016/11/15 2 What is Calculus? How to learn Calculus? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.1 Lines Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1. Increments Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Increments If a particle moves from the point to the point The increments in its coordinates are 1 1 ( , ) x y 2 2 ( , ) x y 2 1 2 1       x x x and y y y Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2. Slope of a Line Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Slope Let and be points on a nonvertical line L. The slope of L is 1 1 1 P x y ( , ) 2 2 2 P x y ( , ) 2 1 2 1 rise y y y m run x x x        Line going uphill – positive slope Line going downhill – negative slope Horizontal line – zero slope Vertical line – no slope

Fot2 1Lten6,=0,d3.ParallelandPerpendicularLines=m..Comcly.ifm=mthen8=6,andL,LParallel lines ~m, = m,Perpendicular lines ~ m, =mAC.acinCDS.hronthesidesoraCDewadtasa/hThe equationJ= +mx=x)is the point-slope equation of the line that passes through the point (xi,yt) andhas slope m4.Equations ofLinesThe equationy=mr+bis called the slope-intereept equation of the line with slope m and y-interceptoThe equationAr+By=C(.4and Bnot both 0)iscalled thc geieral linearequatioa inxandy because its gnaphahaaysrepesents a lincand every line has an equation in this form (including lines with undefimed slope)1.DefinitionofFunctionsand1.2Their Domains and RangesFunctions and Their Graphs

2016/11/15 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3. Parallel and Perpendicular Lines Parallel lines ～ Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley m m 1 2  1 2 1 m m Perpendicular lines ～  Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4. Equations of Lines Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.2 Functions and Their Graphs Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1. Definition of Functions and Their Domains and Ranges

1f(x)DEFINITIONOutputA function from a set D to a set Yis a rule that assigns a untiqueInput(domain)(range)(single)element f(x)e Y to each element xe D.FIGURE1.1Adiagram showing afunction as a kind of machine.f(a)y(x)D=domain setY= set containingthe range(a) + y= 1a yVi(e) y--VrFIGURE1.2AfunctionfromasetDtoasct Yassigns a uniquc element of Yto eachFIGURE1,11(a)The circle is aot the graphofafunction itfails the vertical line test. (b) Theupper semicircle is the graph of a function f(x) VI x. (c) The lower semicircle is the graphelement in Dofa fiunction g(r) = V1 -FunctionDomainFormulaFunctionDomain ()Range ()1+g(f +gx)=Vi+VI-x[0,1] =D(D(g)U-g)=V-Vi-[0, 1]1-gJar[0, 00](00, 00)(g-JKx)-V---V[0,1]g-Jy= 1/x(00,0)U(0, 00)(-00.0)U(0,00)f-g[0, 1](f-g)(x) = f(x)g(x) = Vx(1 x)y-Ve[0, 0][0, 00]20Sig[0, 1) (r = 1 excluded)g("V-xy-VA-x(00,4)[0, 00]g(x)9(0)-y=V-r[-1, 1] [0, 1]g/f(0, I) (r - 0 excluded)fx)

2016/11/15 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

=f+8Ju-Vxg(x)-V1-x(f+exg(xf(a)+g(a)g(a)=f(x)f(a)FIGURE 1.26 The domain ofthefunctionf+g isthe intersection ofthe domains of f and g, theFIGURE 1.25 Graphical addition of twointerval [0, 1] on the x-axis where these domainsoverlap. This interval is also the domain of thefunctions.function / -g (Example 1)DEFINITIONTwo sariables y and x are proportional (to one another) if one is2.Graphs of Functionsahways a constant multiple ofthe other that is ify = kx forsome nonzeroconstantkTeuEr.niphseff) r,n1.23.4.50FIGURE 1.12(a) Lines through theorigin with slope m. (b) A constantfunction with slope m = 0.h

2016/11/15 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2. Graphs of Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Domain:x*(Range:y+0Dcx±0Ranee:y>0(a)(b)FIGURE 1.14 Graphs of the power functions f(x) =x"for part (a)a = 1andforpart (b)a =2UF11(a) ) =sin(b) (x) = cos.xFIGURE 1.19'Graphs of the sine and cosine functions(b)eNFIGURE-1.18Graphs of thre

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y=log2x=1og±3105y= logiot3.y-2-J210.500.50.500.5(a) y = 2', y = 3',y= 10(b) y-2.y 3-f.y 10~FIGURE1.21Graphs of four logarithmicFIGURE 1.20 Graphs of exponential fiunctions.functions.FunctionWhere inereasingWhere decreasingJ=x05x<810050JNowhere00X:00y = 1/xNowhere00≤x<0and0<x<00J = 1/x200<x<00<x<003.Increasing versus Decreasingy=Vi0≤X<00NowhereJ=00<X50Functions0x<00if the graph ofafunction climbs orrises as you movefrom left to right, the function is increasingIfthegraphofa functiondescends orfalls asyoumovefrom left to right, the function is decreas ing.DEFINITIONSAfunction yf(x) is aneven funetion ofx if j() - J(t)4.Even Functions and Oddodd funetion ofx if f(x)= f(x),Functions:Symmetryfor every x in the function's domain

2016/11/15 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3. Increasing versus Decreasing Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley If the graph of a function climbs or rises as you move from left to right, the function is increasing. If the graph of a function descends or falls as you move from left to right, the function is decreasing. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4. Even Functions and Odd Functions: Symmetry Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The graph ofan evenfunction is symmetric aboutthe y-axisThe graph of anodd fiunctionis symmetric abouthe origin.(a)(a)ihFIGURE1.24(a) When we add the constant term 1 to the functionFIGURE 1.23 (a) The graph of y = x2y x2, the resulting function y x? + 1 is still even and its graph is(an even function) is symmetric about thestll symmetric about the y-axis. (b) When we add the constant term 1 toy-axis (b) Thegraphof y =+ (an oddthe function y = x, the resulting function y = x + 1 is no longer odd.function) is symmetric about the origin.bThe symmetry about the origin is lost (Example 7)f(x)5.FunctionsDefinedinPiecesFIGURE 1.8To graph thefunction y = f(x) shown here,we apply different formulas todifferent parts of its domain(Example 4).Absolute Value Properties1.|-aHa2. [a·b/-[al-(bl3210-/al3./g[b]FIGURE1.7The absolute valuefunction has domain (oo, oo)4.la+b|≤al+[b]and range [0, oo]

2016/11/15 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The graph of an even function is symmetric about the y-axis. The graph of an odd function is symmetric about the origin. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5. Functions Defined in Pieces Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute Value Properties 1.| | | |   a a 2. |a b|=|a||b|   | | 3. | |= | | a a b b 4. |a+b| |a|+|b| 

=x2 +2-26.Tansformations of GraphsofFunctions=x22FIGURE1.29To shift thegraphof f(x) = x up (or down), we addpositive (ornegative)constants to2unitsthe formula for f (Example 3aand b).Add a positiveAdd a negativeconstantto.x.constant to.x.Shift Formulas= (x 2)2Vertical Shitsy=fx) +kShifsthe graph ofupkunits ifk> 0Shifts it down|| units ifk 0Shifts it righr|a/units ifh 1,y"cfo)Stretches the graph of f vertically by a factor of ey(x)Compresses the graph of f vertically bya factor ofc.y = f(ex)Compressesthegraphofhorizontallybyafactorof.y =f(x/c)Stretches the graph of f horizontally by a factor of e.Fore = 1,y = f(x)Refletsthe graphoffacross the-axisFIGURE 1.31 Shifing the graph ofy =J(x)Reflects the graph of J across the y-axisy = x|2 units to the right and 1 unitdown (Example 3d)

2016/11/15 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6. Tansformations of Graphs of Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

3VV3xcompressy-Vstretchy=Vxstretch3=Vy-Vx/3compress0FIGURE1.32Vertically stretching andFIGURE1.33 Horizontally stretching andcompressing the graph y = Vx by acompressing the graph y =Vby a factor offactor of 3 (Example 4a).3 (Example 4b),RVofy=nliwttwahesenis (Exrple.5)FIGURE1.34 Reflections of the graphy =Vx across the coordinate axes(Example 4c),Major axisCenter(btipc,6rFIGURE1.37Graph of the ellipset + g = 1,a >b, where the major02axis is horizontal

2016/11/15 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley