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 out the domain.

)()(xgxf :Note

1)(,11)(2xxgxxxfExample)()(xgxf

2.Basic Elementary Functions(基本初等函数) 1) constant functions f(x)=c 2) power functions

0,)(axxfa 3) exponential functions 1,0,)(aaaxfx domain: R range: ),0(

4) logarithmic functions 1,0,log)(aaxxfa domain: ),0( range: R

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 domain range graph

f(x)=arcsinx or x1sin ]1,1[ ]2,2[

f(x)=arccosx or x1cos ]1,1[ ],0[ f(x)=arctanx or x1tan R )2,2(

f(x)=arccotx or x1cot R ),0( 3. Definition Given two functions f and g, the composite function(复合函数) gf is defined by

))(())((xgfxgf Note )))((())((xhgfxhgf Example If ,2)()(xxgandxxf find each function and its domain. ggdffcfgbgfa)))) ))(())(()xgfxgfaSolution)2(xf

422xx

]2,(}2{:domainorxx

xxgxfgxfgb2)())(())(() ]4,0[:02,0domainxx





4)())(())(()xxxfxffxffc )[0, :domain

xxgxggxggd22)2())(())(() ]2,2[:022,02domainx

x

4.Definition An elementary function(初等函数) is constructed using combinations (addition加, subtraction减, multiplication乘, division除) and composition starting with basic elementary functions.

Example )9(cos)(2xxF is an elementary function.

)))((()()(cos)(9)(2xhgfxFxxfxxgxxh

2sin1log)(xexxfxa

Example is an elementary function.

1)Polynomial(多项式) Functions RxaxaxaxaxPnnnn0111)( where n is a nonnegative integer.

The leading coefficient(系数) .0naThe degree of the polynomial is n. In particular(特别地), The leading coefficient .00aconstant function

The leading coefficient .01alinear function The leading coefficient .02aquadratic(二次) function The leading coefficient .03acubic(三次) function 2)Rational(有理) Functions }.0)(such that is{,)()()(xQxxxQxPxf where P and Q are polynomials. 3) Root Functions 4.Piecewise Defined Functions(分段函数)

111)(xifxxifx

xfExample

5. 6.Properties(性质) 1)Symmetry(对称性)

even function: xxfxf),()( in its domain. symmetric w.r.t.(with respect to关于) the y-axis. odd function: xxfxf),()( in its domain. symmetric about the origin. 2) monotonicity(单调性)

A function f is called increasing on interval(区间) I if Iinxxxfxf2121)()(

It is called decreasing on I if Iinxxxfxf2121)()( 3) boundedness(有界性) below bounded)(xexfExample1

above bounded)(xexfExample2 below and above from boundedsin)(xxfExample3 4) periodicity (周期性) Example f(x)=sinx

Chapter 2 Limits and Continuity

1.Definition We write Lxfax)(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 axmeans that in finding the limit of f(x) as x tends to a, we never 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 limits)(limand)(limxgxfaxaxexist. Then )(lim)(lim)]()([lim)1xgxfxgxfaxaxax )(lim)(lim)]()([lim)2xgxfxgxfaxaxax

0)(lim)(lim)(lim)()(lim)3xgifxgxfxg

xf

axaxaxax Note From 2), we have )(lim)(limxfcxcfaxax

integer. positive a is,)](lim[)]([limnxfxfnaxnax 3. 1) 2) Note 4.One-Sided Limits 1)left-hand limit

Definition We write Lxfax)(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 Lxfax)(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)(limxfLxfLxfaxaxax

||limFind0xx Example1 Solution xxx||limFind0 Example2

Solution 6.Infinitesimals(无穷小量) and infinities(无穷大量)

1)Definition 0)(limxfxWe say f(x) is an infinitesimal as  where,x is

some number or . Example1 2200limxxx is an infinitesimal as.0x