1、Chapter TwoOverview2In this chapter,we will introduce two very important concepts in calculus,the derivative 导数导数 and the differential 微分微分.l Concept of derivativesl Rules of finding derivativesl Derivation rules for inverse functions and composite functionsl Higher-order derivativesl Derivatives of
2、 implicit and parametric equationsl DifferentialsConcept of DerivativesDefinition of derivatives4Assume that a rigid body moving along a straight line.00,t tt svt 0()v tExample(Instantaneous Velocity)And this can be seen as an approximate value of the instantaneous velocity t If the increment of tim
3、eis very small,then the average velocity of the body over the time intervalcan be calculated by|t Moreover,as becomes smaller,00()()s tts tt .of the body.0t and hence,when ,the limit of average velocity is defined to be the instantaneous velocity.That is,0000()()lim()ts tts tv tt .Definition of deri
4、vatives5and choose a small part,00,xxx xOl0 x0 xx Example(The linear density of a thin bar)We consider a thin bar with continuous nonuniform mass distribution.Its length is l;how can we find the linear density at any point on the bar?It is well known that the mass distribution can be seen as uniform
5、 if the length of bar is very small.Thus,if we select a coordinate axis as following Therefore,the average density can be calculated bythen the mass of this part ison the bar,00()()mm xxm x ;00()()m xxm xmxx .Definition of derivatives6 While tends to zero,x 00000()()()limlimxxm xxm xmxxx 0 xmx will
6、approaches the value of the linear density at ,00()()m xxm xmxx 0 xis the linear density of the thin bar at the point ,that is the limitprovided that this limit exists0 x as .Example(The linear density of a thin bar)Definition of derivatives7AB0 x0 xx x y Oxy()yf x Just like the following figure,it
7、is easy to see that the slope of a tangent line can be defined by000()()tanlimxf xxf xx Example(The slope of a tangent line to a plane curve)Definition of derivatives80(),fx Suppose that the function000()()limxf xxf xx (Derivative导数导数)If the limit0().U xexists,then the function f is said to be deriv
8、able可导可导 at x0,and the limit is called the derivative 导数导数of f at x0,denoted by 0()x xdf xdx,0 x xy ,or0 x xdydx.y=f(x)is defined in a neighborhood of x0,Definition of derivatives900()()yf xxf x 0 xxx If we writethen the formula can be also written as0000()()limlimxxf xxf xyxx .and00000()()()limlimx
9、xf xxf xyfxxx Examples of derivatives10 Let f(x)=13x-6.Find f (4).Solution00(4)(4)(4)lim13(4)13 4lim13.xxfxffxxx Finish.Definition of derivatives110000()()()lim,hhf xf xfxh0000()()()lim.xxf xf xfxxx Note The derivative of f at x0 can be also written in the followingformsorNote If the limit in the de
10、finition does not exist,f is said to be non-If the limit is infinite,f is also non-derivable at x0,but for convenience,we say that the derivative of f at x0 is infinitederivable at x0.().fxand writeDefinition of derivatives12and be denoted by In the same way,we can define the left derivative左导左导数数 o
11、f the function f at x0,0().fx It is easy to prove that if f is derivable at x0,then00()().fxfx 0()fx and are both exist and0()fx 000()()limxf xxf xx Note If the right limit exists,then the limit is calledthe right derivative 右导数右导数 ofand be denoted by0().fx the function f at x0,Definition of derivat
12、ives13Note If the function is derivable at every point of the interval I(when I contains the left end point a of I,f is right derivable at a;when I contains the right end point b of I,f is left derivable at b),:fIRis called derivable on I.then the function funique correspondingIn this case,for every
13、 point x of I,there exists an().fx 0 xf at .0()fx It is easy to see that the derivative of f atis equal0 xto the value of the derived function Therefore,the derivative is a new functionthat is,it is called the derived function of f on I,()fx defined on I,:fIR.dydxdfdx(),fx denoted by or Examples of
14、derivatives14,()1f xx If find.()fx SolutionBecause()11()()1()()xxxf xxf xx xxxxxxxxx xx 020()()()lim11lim()xxf xxf xfxxx xxx we haveFinish.Definition of derivatives15(1)()0c (where c is a constant)(2)()lnxxaaa (0a ,1a )(3)()xxee (4)1()()xxR (5)(sin)cosxx (6)(cos)sinxx (7)1(ln)xx (8)1(log)(0,1)lnaxaa
15、xa Since the proof for these formulae are all similar,we prove only some of them.001()limlimxxxxxxxxaaaaaxx Proof:To prove(2),by means of the definition of derivative,we have1xat Let ,ln(1)log(),lnatxxta then0 x 0t and as .(some useful formulae for derivatives)Definition of derivatives16Then,we have
16、 proved the formulae(2).ThereforeProof(continued)01()limxxxxaaax 0ln limln(1)xttaat lnxaa Similarly,to prove formulae(5),we notice that 0sin()sin(sin)limxxxxxx 02cossin22limxxxxx 0sin2lim cos22xxxxx cos x The other formulae can be proved by the same way.Do it by yourself.(some useful formulae for de
17、rivatives)Examples of derivatives17Because(0)(0)|fxfxxx 000(0)(0)|(0)limlimlim1xxxfxfxxfxxx ()|f xx Discuss the derivability of the function at x=0.(0)(0)ff So,.we have000(0)(0)|(0)limlimlim1xxxfxfxxfxxx Hence f is not derivable at0.x Finish.SolutionExamples of derivatives18000(2)(3)2)limxf xxf xxx
18、000(3)()1)limxf xxf xx Solution:000(3)()lim33xf xxf xx g00000(2)()()(3)limxf xxf xf xf xxx 00000(2)()(3)()lim(2)323xf xxf xf xxf xxx gg002()3()fxfx 03()fx ,05()fx .Suppose that the function f is derivable at x0,find the following limits:000(3)()1)limxf xxf xx 000(2)(3)2)limxf xxf xxx Examples of der
19、ivatives1911x 0 x While ,0()ln(1),0 xxf xxx ().fx Let,findSolution:0 x ()1fx While ,.01limln(1)1hhhx 0ln(1)ln(1)()limhxhxfxh Examples of derivatives20and(0)f 0ln1(0)ln(10)(0)lim1hhfh Solution(continued)0 x While ,we have1 0(0)0limhhh (0)1f So,.HenceFinish.1,0()1,01xfxxx .,0()ln(1),0 xxf xxx ().fx Le
20、t,findGeometric interpretation of derivative21If function f is derivable at point 0 x,then we know that the slope of tangent line of the curve()yf x at the point 00(,()xf x exists and the derivative is 0()fx.0000()()tanlim()xf xxf xfxx AB0 x0 xx x 00()()f xxf x Oxy()yf x The tangent line to y=f(x)at
21、(a,f(a)is the line through(a,f(a)whose slope is equal to f (a),the derivative of f at a.()()()yf afaxa Geometric interpretation of derivative22 Find equations of the tangent and the normal to the curveyx()()fxx 11(1)2yx12(1)yx And the equation of the normal is Finish.at the point(1,1).Solution:By th
22、e formula of derivative,we have11212x 12 x 1(1)2f So .Therefore the equation of the tangent at(1,1)isor23yx .1(1)2yxor.Geometric interpretation of derivative23O0 xx()yf x yNote:If a function f does not derivable at point 0 x,there are many kinds of possibilities.The first case is that 0()fx and 0()f
23、x are both exist,but they are not equal.Geometric interpretation of derivative24The second case is that 0()fx (or),then the inclination of the tangent line at 00(,()xf x is 2.O0 xx()yf x yOxy13yx For example the derivative of function 13()f xx at 0 x is infinite.Geometric interpretation of derivativ
24、e25The third case is that 0()fx and 0()fx do not exist.For example,let us consider the derivative of function 1()sinf xxx at point 0 x.xyO1()sinf xxx Theorem If the function 0:()fU xR is derivable at 0 x,then it must be continuous at 0 x.Relationship between derivability and continuity26exists and i
25、s finite,0000()()lim()xf xxf xfxx Proof:orhence000()()()()(0)f xxf xfxxxx ,000()()()()(0)f xxf xfxxxxx ,where0lim()0.xx By the definition of the derivative,we know that the limitRelationship between derivability and continuity27This implies the statement.000lim()()xf xxf x 000lim()()xf xxf x Proof(c
26、ontinued)000lim()lim()xxfxxxx 0 orFinish.ThusTheorem If the function 0:()fU xR is derivable at 0 x,then it must be continuous at 0 x.Relationship between derivability and continuity28Note Continuity of the function f at 0 xis only a necessary condition for its derivability at 0 x,but is not a suffic
27、ient condition.Oxy|yx Note The last theorem is valid for an interval.In other words,if f is derivable on an interval I,then f must be continuous on I.Relationship between derivability and continuity29Question:How can a function fail to be derivable?Oxy|at0yxx (a)A“corner”or“kink”xyO1sinat 0yxxx Oxy1
28、3at0yxx(b)A vertical tangent(c)A discontinuityOxy0 xRelationship between derivability and continuity301sin,0,()00,xxf xxx Discuss the continuity and the derivability of the function 0.x atSolution:Because001lim()limsinxxf xxx0(0),f1sin(0)(0)(0)(0)xffxfxxx 1sin,xf(x)iscontinuous at x=0.Since1sinx0,x and the limit of does not exist as f(x)is notderivable at x=0.Finish.ReviewThe definition of derivativeThe derivability of a function in/on a intervalThe relationship between derivability and continuity of a function31
