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1、Chapter1 Functions(函数)1.定义1)a function 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 of The function。3)The range(范围)of is The set of all possible values of f f(x)as x varies through out The domain。2.基本元素功能(默认初等函数)1。

2、)主机功能F (x)=C2)电源功能3) exponential functionsDomain: R range:4) logarithmic functions域：范围：r5) trigonometric functionsf(x)=sinx f(x)=cosx f(x)=tanx f(x)=cotx f(x)=secx f(x)=cscx6)inverse trigonometric functions多明Range图表F(x)=arcsinx orF(x)=arccosx orF(x)=arctanx orrF(x)=arccotx orr3.定义Given two functions。

3、 f and g，the composite function(复合函数)is defined by笔记(消歧义)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、multiplication乘、division除)and compositionstarting with basic elementary functions。Example is an elem。

4、entary function。Is an elementary function。1)Polynomial(多项式)FunctionsWhere n is a nonnegative integer。the leading coefficient the degree of the polynomial is n .In particular(特别是)，the leading coefficient constant functionthe leading coefficient linear functionThe leading coefficient quadratic(辅助)func。

5、tionThe leading coefficient cubic (3次)function2)Rational(合理)FunctionsWhere P and Q are polynomials。3)根功能4.Piecewise Defined Functions(函数打折)5.6.属性(属性)1)Symmetry(对称)Even function: in its domain。Symmetric w.r.t .(使用with respect to信息)the y-axis。Odd function: in its domain。Symmetric about the origin。2)单调。

6、a function f is called increasing on interval(间隔)I ifIt is called decreasing on I if3) boundedness(边界)4) periodicity(周期性)Example f(x)=sinxChapter 2 Limits and Continuity1.Definition We writeAnd say f (x) approaches (tends to趋势)L as x tends to a if we can make the values of(x)arbitrarily(任意)close to 。

7、l by taking x to be sufficiently(足够)close to a (on either side)note means that in finding the limit of f f(x)as x tends to a，we never consider x=a. in fact，f(x)need not even be defeed2.Limit Lawssuppose that c is a constant and the limits exist . thenNote From 2)，we have3.1)2)笔记(消歧义)4.One-Sided Limi。

8、ts1)left-hand limitDefinition We writeand say f(x)tends to l as x tends to a from left if we can make the values of(x)arbitrarily close to l by taking x to be sufficiently close to a and x less than a .2)right-hand limitDefinition We writeand say f(x)tends to l as x tends to a from right if we can m。

9、ake the values of(x)arbitrarily close to l by taking x to be sufficiently close to a and x greater than a .5.TheoremSolutionSolution6.Infinitesimals(无限)和infinities(无限)1)definition we say f(x)is an infinitesimal as is some number orExample1 is an infinitesimal asExample2 is an infinitesimal as2)Theorem and g(x) is bounded .笔记(消歧义)Example3)定义we say f(x)is an infinity as is some number orExample1 is an infinity asExample2 is an infinity as4)Theorem笔记(消歧义)m，n areNonnegative integer。Exercises。