1286编号微积分大一基础知识经典讲解.pdf

1 Chapter1 Functions(函数函数) 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(定义域定义域) of the function. 3)The range(值域值域) of f is the set of all possible values of f(x) as x varies through out the domain. )()(xgxf:Note 1)(, 1 1 )( 2 xxg x x xfExample)()(xgxf 2.Basic Elementary Functions(基本初等函数基本初等函数) 1) constant functions f(x)=c 2) power functions 0,)(axxf a 3) exponential functions domain: R range: 1, 0,)(aaaxf x ), 0( 4) logarithmic functions domain: range: R1, 0,log)(aaxxf a ), 0( 5) trigonometric functions f(x)=sinx f(x)=cosx f(x)=tanx f(x)=cotx f(x)=secx f(x)=cscx 6) inverse trigonometric functions domainrangegraph f(x)=arcsinx or x 1 sin 1, 1 2 , 2 f(x)=arccosx or x 1 cos 1, 1, 0 f(x)=arctanx or x 1 tan R ) 2 , 2 ( f(x)=arccotx or x 1 cot R ), 0( 3. Definition Given two functions f and g, the composite function(复合函数复合函数) is defined bygf ))(())((xgfxgf Note )))((())((xhgfxhgf 2 Example If find each function and its domain.,2)()(xxgandxxf ggdffcfgbgfa)))) ))(())(()xgfxgfaSolution)2(xf 4 22xx 2,(2:domainorxx xxgxfgxfgb2)())(())(() 4, 0 : 02 , 0 domain x x 4 )())(())(()xxxfxffxffc)0,:domain xxgxggxggd22)2())(())(() 2, 2 : 022 , 02 domain x x 4.Definition An elementary function(初等函数初等函数) is constructed using combinations (addition 加, subtraction 减, multiplication 乘, division 除) and composition starting with basic elementary functions. Example is an elementary function.)9(cos)( 2 xxF )))((()()(cos)(9)( 2 xhgfxFxxfxxgxxh is an elementary function. 2 sin 1 log)( x e xxf x a Example 1)Polynomial(多项式多项式) Functions where n is a nonnegative integer.RxaxaxaxaxP n n n n 01 1 1 )( The leading coefficient(系数) The degree of the polynomial is n. . 0 n a In particular(特别地), The leading coefficient constant function . 0 0 a The leading coefficient linear function . 0 1 a The leading coefficient quadratic(二次二次) function . 0 2 a The leading coefficient cubic(三次三次) function . 0 3 a 3 2)Rational(有理有理) Functions where P and Q are polynomials. . 0 )(such thatis, )( )( )(xQxx xQ xP xf 3) Root Functions 4.Piecewise Defined Functions(分段函数分段函数) 1 11 )( xifx xifx xfExample 5. 6.Properties(性质性质) 1)Symmetry(对称性) even function: in its domain.xxfxf),()( symmetric w.r.t.(with respect to 关于) the y-axis. odd function: in its domain.xxfxf),()( symmetric about the origin. 2) monotonicity(单调性) A function f is called increasing on interval(区间) I if Iinxxxfxf 2121 )()( It is called decreasing on I if Iinxxxfxf 2121 )()( 3) boundedness(有界性) belowbounded)( x exfExample1 abovebounded)( x exfExample2 belowandabovefromboundedsin)(xxfExample3 4) periodicity (周期性) Example f(x)=sinx 4 Chapter 2 Limits and Continuity 1.Definition We write Lxf ax )(lim and 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 f(x) as x tends to a, we never consider ax 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)(limand)(limxgxf axax )(lim)(lim)()(lim) 1xgxfxgxf axaxax )(lim)(lim)()(lim)2xgxfxgxf axaxax 0)(lim )(lim )(lim )( )( lim)3 xgif xg xf xg xf ax ax ax ax Note From 2), we have )(lim)(limxfcxcf axax integer.positiveais,)(lim)(limnxfxf n ax n ax 3. 1) 2) Note 4.One-Sided Limits 1)left-hand limit 5 Definition We write Lxf ax )(lim and say “f(x) tends to L as x tends to a from 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 Lxf ax )(lim and 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 to be sufficiently close to a and x greater than a. 5.Theorem )(lim)(lim)(limxfLxfLxf axaxax ||limFind 0 x x Example1 Solution x x x || limFind 0 Example2 Solution 6.Infinitesimals(无穷小量) and infinities(无穷大量) 1)Definition We say f(x) is an infinitesimal as is 0)(limxf x where,x some number or . Example1 is an infinitesimal as 22 0 0limxx x . 0 x Example2 is an infinitesimal as xx x 1 0 1 lim .x 2)Theorem and g(x) is bounded.0)(lim xf x 0)()(lim xgxf x Note Example 0 1 sinlim 0 x x x 6 3)Definition We say f(x) is an infinity as is some )(limxf x where,x number or . Example1 is an infinity as 1 1 1 1 lim 1 xx x .1x Example2 is an infinity as 22 limxx x .x 4)Theorem 0 )( 1 lim)(lim) xf xfa xx )( 1 limat possiblyexcept near0)(, 0)(lim) xf xfxfb xx 13 124 lim 4 23 x xx x Example1 4 42 1 3 124 lim x xxx x 0 13 322 lim 2 2 n nn n Example2 2 2 1 3 32 2 lim n nn n 3 2 xx x x 78 12 lim 2 3 Example3 2 3 78 1 2 lim xx x x Note mnif mnif mnif b 。