微积分大一基础知识经典讲解
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Chapter1 Functions(函数)
1.Definition 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(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 6) inverse trigonometric functions
3. Definition
Given two functions f and g , the composite function(复合函数) g f is defined by
))(())((x g f x g f =
Note )))((())((x h g f x h g f =
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 ))))
))(())(()x g f x g f a = Solution )2(x f -=422x x -=-=
]2,(}2{:domain -∞≤or x x
x x g x f g x f g b -===2)())(())(()
]4,0[:0
2,
0domain x x ⇒⎩⎨
⎧≥-≥ 4)())(())(()x x x f x f f x f f c ==
== )[0, :domain ∞
x x g x g g x g g d --=-==22)2())(())(()
]2,2[:0
22,
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)( 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)