微积分大一基础知识讲解

1、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.2.

2、Basic Elementary Functions(基本初等函数)1) constant functionsf(x)=c2) power functions3) exponential functions domain: R range: 4) logarithmic functions domain: range: R5) trigonometric functionsf(x)=sinx f(x)=cosx f(x)=tanx f(x)=cotx f(x)=secx f(x)=cscx6) inverse trigonometric functionsdomainrangegraphf(x

3、)=arcsinx or f(x)=arccosx or f(x)=arctanx or Rf(x)=arccotx or R3. DefinitionGiven two functions f and g, the composite function(复合函数) is defined byNote Example If find each function and its domain. 4.Definition An elementary function(初等函数) is constructed using combinations(addition加, subtraction减, m

4、ultiplication乘, division除) and compositionstarting with basic elementary functions.Example is an elementary function. is an elementary function.1)Polynomial(多项式) Functions where n is a nonnegative integer.The leading coefficient(系数) The degree of the polynomial is n.In particular(特别地),The leading co

5、efficient constant functionThe leading coefficient linear functionThe leading coefficient quadratic(二次) functionThe leading coefficient cubic(三次) function2)Rational(有理) Functions where P and Q are polynomials.3) Root Functions4.Piecewise Defined Functions(分段函数)5.6.Properties(性质)1)Symmetry(对称性)even f

6、unction: in its domain.symmetric w.r.t.(with respect to关于) the y-axis.odd function: in its domain.symmetric about the origin.2) monotonicity(单调性)A function f is called increasing on interval(区间) I if It is called decreasing on I if 3) boundedness(有界性)4) periodicity (周期性)Example f(x)=sinxChapter 2 Li

7、mits and Continuity1.Definition We write 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 ten

8、ds 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 LawsSuppose that c is a constant and the limitsexist. ThenNote From 2), we have 3. 1)2) Note4.One-Sided Limits1)left-hand limitDefinition We write and say

9、“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 limitDefinition We write and say “f(x) tends to L as x tends to a from right ”if we can make the values of f(x) arbitrarily cl

10、ose to L by taking x to be sufficiently close to a and x greater than a.5.TheoremSolutionSolution6.Infinitesimals(无穷小量) and infinities(无穷大量)1)Definition We say f(x) is an infinitesimal as is some number or Example1 is an infinitesimal asExample2 is an infinitesimal as2)Theorem and g(x) is bounded.Note Example 3)Defini