微积分大一基础知识经典讲解

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.

⇔=)()(x g x f :Note 1)(,1

1)(2+=--=x x g x x x f E xample

)()(x g x f ≠⇒

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

2) power functions

0,)(≠=a x x f a

3) exponential functions

1,0,)(≠>=a a a x f x domain: R range: ),0(∞

4) logarithmic functions

1,0,log )(≠>=a a x x f a domain: ),0(∞ range: R

5) trigonometric functions

f (x )=sin x f (x )=cos x f (x )=tan x f (x )=cot x f (x )=sec x f (x )=csc x

Given two functions f and g , the composite function(复合函数) g f ο is defined by

))(())((x g f x g f =ο

Note )))((())((x h g f x h g f =οο

Example If ,2)()(x x g and x x f -== find each function and its domain.

g g d f

f c f

g b g

f a οοοο))))

))(())(()x g f x g f a =οSolution )2(x f -=422x x -=-=

]2,(}2{:domain -∞≤or x x

x x g x f g x f g b -===2)())(())(()ο

]4,0[:0

2,

0domain x x ⇒⎩⎨

⎧≥-≥ 4)())(())(()x x x f x f f x f f c ==

==ο )[0, :domain ∞

x x g x g g x g g d --=-==22)2())(())(()ο

]2,2[:022,

02-⇒⎩

⎨

⎧≥--≥-domain x 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+=x x F is an elementary function.

)))((()()(cos )(9)(2

x h g f x F x x f x

x g x x h ===+=

2

sin

1

log )(x e x x f x

a -+

=E xample is an elementary function.

1)Polynomial(多项式) Functions

R x a x a x a x a x P n n n n ∈++++=--0

111)(Λ where n is a nonnegative integer.

The leading coefficient(系数) ⇒≠.0n a The degree of the polynomial is n . In particular(特别地),

The leading coefficient ⇒≠.00a constant function The leading coefficient ⇒≠.01a linear function

The leading coefficient ⇒≠.02a quadratic(二次) function

The leading coefficient ⇒≠.03a cubic(三次) function 2)Rational(有理) Functions

}.0)(such that is {,)

()

()(≠=

x Q x x x Q x P x f where P and Q are polynomials.

3) Root Functions

4.Piecewise Defined Functions(分段函数)

⎩⎨

⎧>≤-=11

1)(x if x x if x x f Example 5.

6.Properties(性质) 1)Symmetry(对称性)

even function: x x f x f ∀=-),()( in its domain.

symmetric w.r.t.(with respect to 关于) the y -axis.

odd function: x x f x f ∀-=-),()( in its domain. symmetric about the origin.

2) monotonicity(单调性)

A function f is called increasing

on interval(区间) I if

I in x x x f x f 2121)()(<∀<

It is called decreasing on I if I in x x x f x f 2121)()(<∀> 3) boundedness(有界性)

below bounded )(x e x f =E xample1

above bounded )(x e x f -=E xamp le2

below and above from bounded sin )(x x f =Example3

4) periodicity (周期性)