微积分大一基础知识经典讲解

1、 微积分大一基础知识经典讲解 Chapter1 Functions(函数) 1.Definition 1)Afunction f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. 2)The set A is called the domain(定义域) of the function. 3)The range(值域) of f is the set of all possible values of f(x) as x varies through

2、 out the domain. ?)(x(x)?gNote: f2?1xf(x)?,g(x)?x?1Example )xg(x)?f x?12.Basic Elementary Functions(基本初等函数) 1) constant functions f(x)=c 2) power functions a,a?)?x0f(x 3) exponential functions x,a?0,aaf(x)?1 domain: R range: ),?(04) logarithmic functions f(x)?logx,a?0,a?1 domain: range: R ),?(0a5) t

3、rigonometric functions f(x)=sinx f(x)=cosx f(x)=tanx f(x)=cotx f(x)=secx f(x)=cscx 6) inverse trigonometric functions domain range graph 1? x (x)=arcsinor fxsin ,1?1? ?,22 1?xcos for x)=arccosx ( 11,? 0, 1?xtan or x)=arctanxf(R ? )(?,22 1?xcot xxf()=arccot or R ? )0(, 3. Definition is defined ) , th

4、e composite function( and Given two functions fg复合函数g?fby )?x)(gf(?)(fgx 2 Note )h(xx)?f(g(?(f?gh)(find each function and its domain. Example If ,2?xxandg(x)?f(x)? g)g?ffc)?fda)f?gb)g?4 )?x?f(2)xSolutiona)(f?g)(x)?f(g(x?2?2?x 2x?2or(?,:domainx x?(x)?g(x)?b)(g?2f)(x)?g(f,?0x? 40,?domain:?02?x?4 x?f(x

5、)?x?(c)(f?f)(x)?f(fx)0,?domain: x?2?2g(2?x)?g?)g)(x?g(g(x)?)d(,?02?x? 2,2?domain:?0?2?x2?using constructed 数) is elementary function(初等函4.Definition An combinations ) and composition 除乘, division减(addition加, subtraction, multiplicationstarting with basic elementary functions. 2)9x)?cos?(F(xis an ele

6、mentary function. Example 2F(x)?f(xg(h(x)(x()?gx?9(x)?cosxfx)?h sinx?1e?x(xamplefx)?logE is an elementary function. a2x1)Polynomial(多项式) Functions nn?1?axxx?a?ax?Ra(Px)? where n is a nonnegative integer. 01n?n1a?0.?The degree of the polynomial is The leading coefficient(系数n) . nIn particular(特别地), a

7、?0.?constant function The leading coefficient 0a?0.?linear function The leading coefficient 1a?0.?quadratic(The leading coefficient 二次) function 2 3 a?0.?cubic(三次) function The leading coefficient 32)Rational(有理) Functions P(x) where P and Q are polynomials. .)?0is such that(fx)? Q(x,xx Q(x)3) Root

8、Functions 4.Piecewise Defined Functions(分段函数) 1?xifx?1? ?x)Examplef(?1?xifx?5. 6.Properties(性质) 1)Symmetry(对称性) even function: in its domain. x),?)?f(xf(?xsymmetric w.r.t.(with respect to关于) the y-axis. odd function: in its domain. x?x?f(),f(?x)?symmetric about the origin. 2) monotonicity(单调性) A fun

9、ction f is called increasing on interval(区间) I if Ixx)?xin)f(x?f( 2211Iinx?x?f(x)f(x)? if It is called decreasing on I2211) 有界性3) boundedness(xbelow?xample1f(x)e boundedE xabovee bounded?fExample2(x)? below andabovefromboundedsin)(Example3fx?x ) 周期性4) periodicity ( 4 Example f(x)=sinx Chapter 2 Limi

10、ts and Continuity 1.Definition We write L?x)limf(x?aand say “f(x) approaches(tends to趋向于) L as x tends to a ” if we can make the values of f(x) arbitrarily(任意地) close to L by taking x to be sufficiently(足够地) close to a(on either side of a) but not equal to a. Note means that in finding the limit of

11、f(x) as x tends to a, we never ax?consider x=a. In fact, f(x) need not even be defined when x=a. The only thing that matters is how f is defined near a. 2.Limit Laws Suppose that c is a constant and the limitsexist. Then )(xandlimglimf(x)x?ax?a )xlimg(f(x)?)f(x?g(x)?lim1)limx?ax?ax?a )(x?x)limg)g(x)

12、?limf(f2)lim(xx?ax?ax?af(x)lim)(xfx?a 0?(x)if3)lim?limg g(x)limg(x)a?x?axx?aNote From 2), we have )(x?climflimcf(x)x?ax?ann,nis a positive)f(xlim?f(x integer.lim x?ax?a3. 1) 2) 5 Note 4.One-Sided Limits 1)left-hand limit Definition We write L?x)limf(?x?aand say “f(x) tends to L as x tends to a from

13、left ” if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x less than a. 2)right-hand limit Definition We write L)?limf(x?x?aand say “f(x) tends to L as x tends to a from right ” if we can make the values of f(x) arbitrarily close to L by taking x

14、to be sufficiently close to a and x greater than a. 5.Theorem )xf(?x)?Llimlimf(x)?L?limf(?x?ax?ax?a |xlim|Example1 Findx?0Solution |x| limFindExample2 x0x?Solution 6.Infinitesimals(无穷小量) and infinities(无穷大量) limf(x)?0?We say f(x) is an infinitesimal as is 1)Definition x? ?,wherex?. some number or 22

15、x?limx0 Example1 is an infinitesimal as .?0xx?011x?.?0?lim is an infinitesimal as Example2 xx?x?limf(x)?0?limf(x)g(x)?0 and g(x) is bounded.2)Theorem ?x?x 6 Note 1 Example 0?limxsin x0x?3)Definition We say f(x) is an infinity as is ?limf(x)? ?,wherexx?some number or .?11? Example1 is an infinity as.

16、1x?lim? 1?xx?1?1x?22x?x?lim is an infinity asExample2 .?x?x4)Theorem 1f(x)?)lim?lim?0a )(xf?xx?1?lim?except possiblyat 0x)?0,f(x)?nearlimb)f( )xf(?x?x?421?23 1?24xx42xxxlim?limExample1 0? 141?3x?x?x?3 4x232?2 3?2n2n22nnlim?limExample2 ? 12313n?n?n?3 2n1?231x?2 3x?limlimxample3E ? 782x?7x8?x?x? 2xxa?

17、nifn?m? b?nnn?1a?ax?ax?01nn? Note m?limif?n0? 1m?m?bx?bx?b?x?0m?1m?ifn?m?wherea(i?0,?,n),b(j?0,?,m) are constants anda?0,b?0,m, n are 0ji0nonnegative integer. Exercises 2?bn?an2?3?a?(0),b?11.)lim(6) 1?2n?n?2?x1 )1(b1?bax?)2lim(?)1a(),? x?x 7 b?ax )?a?(2),b?3)lim(?2?2 1x?1?x2nn3?3nn1?25)?(?lim.1)2 ?lim2) 211n?n?41?4n5)(?25?n?n?11?1?41132n? n22lim)?3 14)lim(?)? 112223nnn?n?n?1? n331111 1?)?(5)lim?)n?()limn?1n6 )n(3n1?2?12?2?n?n?24x?3x ?)3.1lim 24?x2x?33x?h)(x? 2x3?lim2) hh0?213x?5x? 33)lim? 24x?3x?x?3030202032)2?(2x?3)?(3x? ?)l