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微积分大一基础知识经典讲解

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2020-10-06 18:25
tags:高中数学微积分

高中数学导数为0可以得到-高中数学选修1-1b

2020年10月6日发(作者:计能)


Chapter1 Functions(函数)
tion 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.
Note: f(x)?g(x)?

x
2
?1
Examp lef(x)?,g(x)?x?1
?f(x)?g(x)

x?1
Elementary Functions(基本初等函数)
1) constant functions
f(x)=c
2) power functions
f(x)?x
a
,a?0

3) exponential functions
f(x)?a
x
,a?0,a?1
domain: R range:
(0,?)

4) logarithmic functions
f(x)?log
a
x,a?0,a?1
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
sin
?1
x

[?1,1]

22

[0,
?
]

f(x)=arccosx or
cos
?1
x

[?1,1]

f(x)=arctanx or
tan
?1
x

f(x)=arccotx or
cot
?1
x

3. Definition
Given two functions f and g, the composite function(复合函数)
f?g
is defined by
(f?g)(x)?f(g(x))

R
R
(?
??
22
,)



(0,
?
)

Note
(f?g?h)(x)?f(g(h(x)))


Example If
f(x)?xandg(x)?2?x,
find each function and its domain.
a)f?gb)g?fc)f?fd)g?g

2?x?
4
2?x

Solutiona)(f?g)(x)?f( g(x))
?f(2?x)
?
domain:{xx?2}or(??,2]

b)(g?f)(x)?g(f(x))?g(x)?2?x

?
x?0,
?domain:[0,4]

?
?
2 ?x?0
c)(f?f)(x)?f(f(x))?f(x)?x?
4
x

domain: [0,?)

d)(g?g)(x)?g(g(x))?g(2?x)?2?2?x

?
2?x?0,
?domain:[?2,2]

?
?
2?2?x?0
tion 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
F(x)?cos
2
(x?9)
is an elementary function.
h(x)?x?9g(x)?cosx
Examplef(x)?log
a
x?
f(x)?x
2
x
F(x)?f(g(h(x)))

e
sin
x
?1
2
is an elementary function.
1)Polynomial(多项式) Functions
P(x)?a
n
x
n
?a
n?1x
n?1
???a
1
x?a
0
x?R
where n is a nonnegative integer.
The leading coefficient(系数)
a
n
?0.?
The degree of the polynomial is n.
In particular(特别地),
The leading coefficient
a
0
?0.?
constant function
The leading coefficient
a
1
?0.?
linear function
The leading coefficient
a
2
?0.?
quadratic(二次) function
The leading coefficient
a
3
?0.?
cubic(三次) function


2)Rational(有理) Functions
f(x)?
P(x)
,
Q(x)
{xxis such that Q(x)?0.}
where P and Q are polynomials.
3) Root Functions
ise Defined Functions(分段函数)
?
1?xifx?1

Examplef(x)?
?
xifx?1
?
5.
ties(性质)
1)Symmetry(对称性)
even function:
f(?x)?f(x),?x
in its domain.
symmetric w.r.t.(with respect to关于) the y-axis.
odd function:
f(?x)??f(x),?x
in its domain.
symmetric about the origin.
2) monotonicity(单调性)
A function f is called increasing on interval(区间) I if
f(x
1
)?f(x
2
)?x
1
?x
2
inI

It is called decreasing on I if
f(x
1)?f(x
2
)?x
1
?x
2
inI

3) boundedness(有界性)
Example1f(x)?e
x
bounded below

Example2f(x)??e
x
bounded above

Example3f(x)?sinxbounded from above and below

4) periodicity (周期性)
Example f(x)=sinx
Chapter 2 Limits and Continuity
tion We write
limf(x)?L

x?a
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
x?a
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.
Laws
Suppose that c is a constant and the limits
limf(x)andlimg(x)
exist. Then
x?a x?a


1)lim[f(x)?g(x)]?limf(x)?limg(x)

x?ax?ax?a
2)lim[f(x)g(x)]?limf(x)?limg(x)
x?ax?ax?a
f(x)
f(x)
lim
x?a
3)lim?iflimg(x)?0

x?a
g(x)
x?a
limg(x)
x?a
Note From 2), we have
cf(x)?climf(x)

li m
x?ax?a
lim[f(x)]
n
?[limf(x)]
n,nis a positive integer.

x?ax?a
3.
1)
2)
Note
-Sided Limits
1)left-hand limit
Definition We write
lim
?
f(x)?L

x?a
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
lim
?
f(x)?L

x?a
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.
m
limf(x)?L?limf(x)?L?limf(x)

??
x?a
x?ax?a
Example1 Findlim|x|

x?0
Solution
Example2 Findlim
|x|

x?0
x
Solution
tesimals(无穷小量) and infinities(无穷大量)
1)Definition
limf(x)?0?
We say f(x) is an infinitesimal as
x??,where ?
is
x??
some number or
??.

Example1
limx
2
?0?x
2
is an infinitesimal as
x?0.

x?0


Example2
lim
11
?0?
is an infinitesimal as
x???.

x???
xx
x??x??
2)Theorem
limf(x)?0
and g(x) is bounded.
?limf(x)g(x)?0

Note
Example
limxsin
x?0
x??
1
?0

x
3)Definition
limf(x)????
We say f(x) is an infinity as
x??,where ?
is some
number or
??.

Example1
lim
?
x?1
x??
11
???
is an infinity as
x?1
?
.

x?1x?1
Example2
limx
2
???x
2
is an infinity as
x??.

4)Theorem
a)limf(x)????lim
x??x??
1
?0

f(x)
1
???

f(x)
b)limf(x)?0,f(x)?0near?except possiblyat ??lim
x??x??
421
?
2
?4
4x?2x?1
x
xx

Example1lim

?lim
x??
x??
1
3x
4
?1
3?
4
x
23
2??
2
2n
2
?2n?3n
n

?
2

Example2lim
?lim
n??
n??
1
3
3n
2
?1
3?
2
n
1
2?
3
3
2x?1
x

??

Example3lim
2

?lim
x??
8x?7x
x??
87
?
2
x
x
?
a
n
?
b
ifn?m
n
?
a
n< br>x
n
?a
n?1
x
n?1
?
?
?a
0
?
?
Note
lim
?
0ifn?m

?m?1
x??
bx
m
?b?
?
?b
0< br>?
mm?1
x
?ifn?m
?
?
?
32wherea
i
(i?0,
?
,n),b
j
(j?0,
?
,m) are constants anda
0
?0,b
0
?0,
m, n are
nonnegative integer.
Exercises


an
2
?bn?2
1.1)lim?3?a?(0),b?(6)

n??
2n?1
x
2
?1
2)lim(?ax?b)?1?a?( 1),b?(?1)

x??
x
3)lim
ax?b
x?1
x?1
?2?a?(2),b?(?2)

)lim
3n
2
2.1
?n3
n??
4n
2
?1
?
4
1?
1
3)lim
2
?< br>?
?
1
2
n
n??
?
4

1?
11
3
3
?
?
?
3
n
5)lim
n??
(
1
1?2
?
1
2?3
?
?
?
1
n(n?1)
)?1

3.1)l im
x
2
?3x?4
x?2
x
2
?4
??


3)l
3x
2
?5 x?1
x?
i
?
m
x
2
?3x?4
?3< br>
5)lim
x??
(1?
1
x
)(2?
1
x
2
)?2
< br>7)lim
3?x?1?x2
x?1
x
2
?1
??< br>4

3
9)lim
x?1
x?1
x?1
?
2
3

1 1)lim((x?2)(x?
1
x???
1)?x)?
2

4.1)lim
x
2
?3x
x?3
(x?3)
2
??

3)lim
x??
(5x
2
?2x?3)??

5 .
x
lim
???
(4x
2
?4x?5?2x)??1
2)lim
5
n
?(?2)
n
1
n??< br>5
n?1
?(?2)
n?1
?
5

4)l im
n??
(
1
n
2
?
3
n
2< br>?
?
?
2n?1
n
2
)?1

6)lim
1
n??
(n?1?n)n?
2

(x ?h)
3
?x
3
2)lim
h?0
h
?3x
2

(2x?3)
20
?(3x?2)
30
2
2 0
?3
30
4)lim
x??
(5x?1)
50
?
5
50
6)lim
2x
3
?3x?1
x??
4x
5
?2x?7
?0

8)l im
x?1
(
1
1?x
?
3
1?x
3)??1

10)lim
(1?x)(1?2x)(1 ?3x)?1
x?0
x
?6

2)lim
x
3
?2
x??
3x?4
??











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