【大一】Calculus (微積分)

1571

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2024.10.15 公開

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ページ1:

Definition
Functions
Calculus
X →
domain (定义域)
/ y range ()
Rational function: f(x) = P(x)
Q(x) 0
Q(x)
Combination of functions:
· (f⋅g-h) (x) = f(g(4x))
Inverse functions (x. 8 = 2)
→ inverse
条件: one-to-one
• graph: reflective about
y = x
Trigonometric functions:
ex F(x) = cos (x+2)- find f.g-
f(x)=212
82-805x
1%)=x+2
Sin cos - domain : R / munge: 19151
• arcsin - domain
MISI / range. I syst
tan
dranin X & nel range: R
• cot domain: X4 ne range: R
• arccos
arctan
domnin R
• sec
— domain
20+ range 18121
• Arccot
CSC
- domain
X#n
runge
. are sec
domain 120/21
· areese -
domain 12/21
range: sys, yo
dianin IIS range osyst
range
經
domain R
range o±yEx
Fange o≤ y ≤ x +
Nature Exponential / Logarithmic functions
尤拉数e
:
e=
lim (1+ 1 ) = 2 1 1
|
= 2.7182818284...
la x = log X
Limit
limit laws:
lim f(x) = lim
Sum/Difference/Product/Quotient Rule: L+M L·M
-g(x)=
-M
M
35
• Constant Multiple Rule:
lim (k f(x)) = k· L
11%
Power/Root Rule
I'm [f(x)] = ["
(AGN)
x+C
x-c
①左右极不相等(discontinuous)
dimit 不存在
③ 沒有界限
③Oscillating behavior (th211 )
h
Vertical line test :
intersects the curve
> check "function"
only
in
one
plane
(x)
Horizontal line
test:
#
check "one
10
one
intersects only in
one place
ex:
用图形判断
(x)
lim [a f(x) + b g(x)] = aL+6M
y = sin
y = cos

ページ2:

left-hand / right-hand limit:
lim f(x) = him f(x) = lim f(x) = L
1572
line foxy
x1ct
if Mo but Lo
doesn't
exit
=
x-c gix)
indeterminate form
if L-M=0
Squeeze (Sandwich) theorem:
特例
g(x) = f(x) = h(x)
lim sinx = 1
X40
x
,
lim g(x) = lim hex) = L
msin kok
->
B+α
lim
1-105%
x
0
=
lim (1+x)* = e
X-
one-sided limit
Step function
integer function
asymptotes
(R)
介
-C
f(x) = L
-101 sine 01
-101 cos 0 ≤ 101
#
9 0
0
• floor function/
(F)
greatest integer
• ceiling function FX7
function [X/X
tane
2
8
Sine
>
2
2
•horizontal asym
9-8
asymptote
lim fix) b
or
lim fexs - b
X-700
X--00
vertical
asymptote
lim f(x) = ±∞0-
or
lim f(x) = ± 00
xat
ex:
f(x) =
长陈法
x-3
2-4
oblique asymptote
分子次数大分母1次
++
function +$2 (+ 0)
Lager
remainder
as x 100
(x)-₤-1
三、
Continuty
continuous
definition: o fils is defined.
Olim ter exist
kinties-fees
• 10+ fas
removable
经重新定义断点极限值可使之连续
discontinuous
non-removable
Composition of
Continuous functions
f(x) g(x) of at c
I'm f (g(x)) = f ( Im
x+c
236
Intermediate value cheorem
→
(f+91(x)
continuous at c
(中间值、介值定理)
if f(x)在[a,b]区间连续
丁
RJ a b 1 ] Tz tz C
fus-k

ページ3:

limit exit
可微⇒连续 ⇒
limit exit
#不一定
ex continuous bt non differentiable :
f(x) = 1x1
连续
可微
ex limit exit.
bt discontinuous :

ページ4:

可微⇒连续
limit exit
#不一定
ex continuous bt differentiable:
f(x) = 1x1
non.
limit exit
连续
可微
ex limit exit
bt discontinuous :
条件:
•左右极相等(discontinuous )
limit exit
有界
continuous
• ₺ Oscillating behavior (211)
• f(c) is defined
①
lim f(x) exist
值
*15 (lim f(x) = f(x))
differentiable
无尖点 (cusp)
垂直切线

ページ5:

Differentiation:
tangent
line/Perivative at
a
point
slope lim
Derivative
f(x. + h) - fox.) = f (x.)
f'(x) = lim fexs - fras - ties = fr = y'
x a
differentiable continous
=
limit exit
一不一定
0
fixs
x
= Dx fos = D₁t
limit out
continous
可微
differentiable
Differentiation rules
小
(cusp)
垂直切线
• general power rute
n-1
x = nx
• constant rule c = 0
• constant multiple rule:
-c-f(x)
= c = f(x)
sum
1 / difference rule: f (f(x) = g(x) = f(x) = g(x)
• product rule :
(f.g)
=
fig + f.
(轮流似)
• quotient rule:
· chain rule (连鎮率): | (fog) (x) =
• 1x ex- ex
=
derivatives of Trigonometric functions
f
f
sin %
LOS X
COS X
sin
tan %
sec %
CSC %
-CsCx cot
Sec
sectan
tan
f'(g(x)) - g'(x) (13)
dy dy du
dx
du dx
Sin
Cos
sec
CSC
cot-CsC
derivative of Natural Exponential / Logarithmic functions
e* = e*
dx
dx
e
4(x)
=
e
4(x)
u (X)
cot
Sin xe làm sản (xt Sản X
AN-
AX
cxx-x (1-4)
4%
a'
a. In a
dx
u(x)
u(x)
a"
a
u'(x). In a
= him [cax.
sag
Lossing-0 = cosx

ページ6:

ln x
& In
4 =
x
u =
Implicit differentiation:
=
(隐微分)
to deal with implicit functions (#yalues).
求
·两边同时对微
连锁律
有
接
的移到 边
告
log x = 0 ( 11 - 21 x ) = 1 x x
I log u to the low) = ulna
ex
y = x
ex y' + y² - 5y-x²= -4
x
Ly
derivative of Inverse functions
+8+-+-+-+ (-4)
18848-81-3
2%
f(f(x))=x
→f(f(x)). f(x)=1
Logarithmic differentiation (24 82 182 57 12 ) :
ex
u
₤ loglut
x²+81-987-0
37 - 18 - 1 - 93 - 9x =
切…
数相×÷
f(x) 20x)
h()
ex Dif fun-x². x»
适用
两边
取处理,再微
fio es ex4x
=> fix = xnx
底数、指数皆变数 f(x) gum)
fx-
fax
derivatives of Inverse Trigonometric functions
馀
f
f'
1
sinx
cos
tan'
CSC
分
|
seck
cot
Indeterminate Form:
x√x-T
1+02
將其他型式钱威
罩
。
0.00
f(x)
L'Hôpital's rulet Art - bir firm
lim fix) g (x) = lim gix
x c
27c
or
live fix
(f(x))
f(x)
g'(x)
sing=
= X
隐彼
x
2x
-cosy y² = 1
· sin "x = J = Losg-1-x"

ページ7:

100 -0
lim (fix) - g(x)) = lim
Integral
antiderivative:
- Stix dx
taka
=
8(x)
Foxglas
F(x)+ C
34-10
不定积分,
ff(x) dx
area
ax"
文
×|-
ekx
akx
X
n+1
+ C
n = =
C
n+1
In x c
kx
e
+ C
k
|
a
k x
+ C
x = 0
COS
Sin
sin
+ C
3
+ C
Skf ex dx = k ffcx dx
(f(x) = g(x))
+01
dx = f f(x) dx 1 g(x)
sin
x + C
secx+C
sec² kx
tank x + c
tan x+C
1+x
CSC
½ coth + c
seckx tankx
sec k + c
+(x+人
tan
a
csck cot
esch + c
Riemann
sum
AX-
upper sum (Un)
Definite Integral
lim fla+k b-a) (b)
k = 1
AX
lower sum
(L)
U AL
Sn = lim | Un | = lim | Ln = A
rules
=
S. f (x) d x
Sfixdx
-order So flourds
• zero with interval: S^ fix+ dx
= 0
定积分
+ C
midpoint rule

ページ8:

Sa [kfox) = g(x)] dx = k fo fix dx ± lf" g(x)dx
• additivity: So fix) dx + S fix dx = S fix dx
total area
A = S fix dx - Si fex d x + S, fix) dx
(x軸上侧
+- 下
下侧
A = S [f(x) = g(x)] dx
=
Så [fixs-gox)] da + få [gras - fixi) da + So [fixs-gixi) da
odd
=>
Sa₁ f(x)dx =
。
a
fixi
even ⇒
Mean value theorem
Sa fix)dx = 2f f(x)dx
of definite integrals
f在区间连续,則其中有
average value
av (f) = ± √ fexs dx
Fundamental theorem of calculus
①F'(x) = £S" fit) dt = fix)
ffx dx = Fx Fib - Fras
b- a
fcc) = ba fi foxydy
y-fuo
f(c)(b- a) -
f(x)dx
- (running Chain rule backward)
ex √ √x²+x (2x + 1) dx
ex fx edx
Substitution method:(变数变換)指对分式 三角
0
Sfcus - u' dx = ffous du du
it
u dx du )
L integrals of Trigonometric functions
f
[ fix dx
tank x
|| In | seck | + c
seck
In | seck
+ tank*
C
csckx
In | csckx + cotx
c
cotx
In cos | + c
-In | csck | + c
CSC
Stan x dx = In |sec x +
secx+c
f sin x dx = fű du
cos x
In 1416
-In casx + c = In secol+C
= ln | sin | + c
f sec x dx = ln | sec x + x +
sec x=
secx (secx+ tan x)
secx+tan x
tanx C
Usecx+tan x, du secx tan x + secx)dx
"sec x + secx tan x dx = súdu = Inju₁ + c
sec x + can x
|| secx + canx + c
tan
sin
cos
sec
CFC
cot

ページ9:

t
=
Stous u' dx - S fous du
Techniques of Integration
x²+2x+2
d x n t s ce
dx
=√(x+1)² + 11
·tan" (x + 1) + C
4X+ 3
x²+2x+2
dx
2X+2
dx
x²+2x+2
24-x+2x+2
du = (x+2)dx
Sú du - In y + c = b
ex
S
3x-7x
3x+2
·dx徦分式长带分式
f (3x+21(x-3) dx + √ 1x+2 dx
3+2
30+2
- x² - 3x + √ z du
24=326+2
X = 3x + 2 n 3x+2 | + C
(相* 对反三角)
4X++
x+2x+2
= Sudu - S
= 2/nx+2x+1)
dx
tan (x+1)+C
Integration by
part
(分部积分)
2 product rate)
- Sudv +fvdu
=>
Judv=
-d(uv) = udv
+vdu
(uv) = u v + u v
→ definite int
⇒uv =
-fvdu
= uv-
= Suvdx + Suv dx » Suvdx =
Ludv-av-fvdu
=
Sx cos x dx
dv-cosxdx
v = sin x
令u=X
du = dx
=
x sin x
=
x sin x cos x + C
-S sin x dx
→>
Jlnx dx
u = lnx
= xlux -
dv - dx
v = x
xdx
x + c
uv
- Suvdx
不同形式
u
dv
3
V

ページ10:

2-2
8-424
5h-4-4
Im fi exists at every point in (1.3)
BRD)=-2
h)-3
* lim
1153+8)
31. lim (+) = -1 or 3 x A
32 in 1 - ) = -2
x (4-x)
36. lim
37.
(1)(√++)
4
43. lim (2 sin x-1)
44. lim sin²x
44. (sin x)²
Sin 00
46
làm tan
45. lim sec x
46. lim tan x
20
47. lim
42
1+x+ sin x
3 cos x
x +4
25-7-49 4+7
47.
49. lim √x + 4) cos (x + x)=
lim x++ him cos(x+5)
14-117
2-4
107
Which of the blowing namen do the fiction ys
+ him sinsh
I'm in teft
30. I'm 6x² (cotx) (csc >^) = lim ex cos x lim x
Sing
x+x60x
3.
= 1
sing cosso
33.50
9+0
breaking print
16
√√6 - V/5² + 116 + 6
15.
lim bith
16
43 2 14 2 lim sin x = lim I
M
ItH:
17. a
b
√20-11
V20-1)
17. (a)
(x+3)= |
(b) lim (x+3)(-1) = -|
20
(A)
18 lim
sin x
19. a
20-1
35. lim sin 11-cost
-sh-11
小
16. lim
√6+ √571+6
2√6
t-o
-Cost
lim sind -1
36
Jim
sisinh
Sinh
· him sindy wesy. I'm sinsy wisse sing lim(2012) (1271) (1976) (2017) (477) = ?
max 2015)g
41.
43 li tanh
sinty costly
line and sanjo -3 him that fund = 3
sin +8
Lime (1050) (1+2010)
(2 sin cos 8) (1+ cos 8)
+5. i 1-1053%
20
24-70
1-60539
3%
34 lim x-xx-x(1-5)
Sin 39
Sinx
(a)
Sin
不存在
lim
sin t = 0
lim sin x
2-5
夾桥
9. lim in 2x
11.
300x
2-1+sin
24.
12. lim
lim (x+x-1
00-15
8x-3
x--x--x
Sint
x²
lim
COSX (106x-1) cosse+!
lim Los x (cos`A-1) lim
20
(cos+1)
(+1)
46 lim
(b) lim x sin ½ = lim sint
t-o
x
19 (α) | 161 -1)
lim sing =
xno
x
lim 1-cosys
0
x
Limit exit
continous
(可微
X-0
21 (4) ₤1
(b)
22) 4-4-
live
of 1
tanx
+ sin x
= −1+
Sint
。
12. lim
14
45
cus lim sinx
= 0
2+7-5 sin r
x10
x
A
1
**
10 Lose
0
x
x
√3
14. lim √x²+1
x+-%x+1
36lim
lim
-X
-30
√x6+9
lim
1/15-
x178
=α
62. lim
31 lim
39
X-00
2x-xx+7
x/5+37+ √7
lim +
*-11-X-2
lim
+1
+
4
47 lim
*10
-
41. I'm
*+-8+
x+8
t
44 lim
x70x(x+1)
++
51. lim 11+ CSC 9 ) =
= ∞
b. x-0
d. x-l
107.
1-cosp
Sin = 1-1050
= 3
→
1- coso = 2 Sin²
-✓
110 ---
x
1634-301

ページ11:

ⓇD f(x) = (x-2) g(x) · find f'(L)
f'(z) = lim
lim
x10
h+
f12th)-foy
sin x
x
h
+
A
lim
· = lim g(z+h)=2
lim
x-10
tan
lim sinx
006-%
X
= 0
sin x
%
X
x
Sin A
B
tan
S
<
1
日
S
Sing
cos A
1712
12 sine 2 cose
由夾挤。
lim sing
|
= 0
lim sin
x15
X
= Xim
26-70
* lim sinx
(1) Find the horizontal asymptotes for f(x)=2x+√4x²-3x+5. (10%)
(2) Find the vertical asymptotes and the slant asymptotes for g(x)=
( lim f(x) = lim
37-5
2x-√4x-3x+5
lim
3 -
3
24
(L)
2+
-+/4
+
X-
x-
Let f(x)=[x]-√√x+[x].
(1) Determine if f(x) is continuous at x=0.5
(2) Determine if f(x) is continuous at x=2
-(1) lim f(x) = 1 - √o.s to flo.s) = f(x) continuous x=0.5
270.5
sin
lim tan x
x10
x
lim sin x
x10
= lim Sinx · lim cos x
sint
sint
=0=0
g(x) = x +1-
3
X-1
-3
as x-100
=
g(x) = x+1
y=x+14
(2) lim f(x) = 3 - √2+4= |
*12*
A
lim f(x) = 2 - √2+1=2-√3
26+2
· him fixes not exit => not continuous
[x] o
derivative of (a) y (x² + e²), (b) y = e²
x+ ex
x
J'= 1 + ex-e
(b) = e.ex
g' exe ex ex
x
- ex
(x-ex
++
Let f(x)=√x-5x (x50). (8%)
(1) Find the range of f(x)
(2) Find the inverse function(x) of f(x)
(3) Find the domain and range off'(x)
43581720)
公式解,
52√25-40
4 fix 2 x ≤0.720
-- fox) = 5-1
(3)
720
(a) =
J≤0
'lim sin (1+x) - Sin" ( I'm ( ) ) = sin" —
Xim
x105x-x
Sin 5x
50
=== lim sinsx
x (105x-1)
Sch
TC
6
=
sin³ 5x
3-1
1143
13. g(x) = 2 slope
x-2
at (3.3)
lim
3+h
13+1-2
h chris
18. f(x)=√x+1 (8-3)
kim
√18+1+1 - 3
h
|22
X-
y=
(1=0)
h-1
y=-1
+1
h+1
h
lim fcaths-(Fra)
fing
hoso

ページ12:

15
lim
12+ secx
√2+1
-√3
X-20
cos (π-tanx)
cos π
(可微>连续)→直接代
Dif ye
Los %
y= e' (sinx)
zy. y² = 2x + cos xy + (y + xy )
3-6
8185
-sec (tunx), g sec (TMMX) · Tan [Tanx) Sex*x
EXAMPLE 8
is positive.
Show that the slope of every line tangent to the curve y 1/(1 - 2x)³
y= (1-2x)3
→ y² = −3 (1-2×) 4.-6
-6(1-2x)
EXAMPLES that the pot (24) The
**
3x²+ 38-7' - 98 + 7x y' -
(2)412-483-94-43
378-7718-2)
1218-4818-4
=e
x =
x =
X
× 7+ y² 7x • kayə
ex flux
X
e
fbx 1 of hx
+
35 y=xex + ex-g₁ex + xC® 60 + 3*°*e*®
dx
· log"
u
40. g. x sec(x), y² = 2x seci 1+ * sec / tan -*
=
ulna
u
77. y=e* +54 g² = ext 2x +5 g" - ze³ + zx ex* zx
10.
28.
3-1
xy-
- cot
P191
· (xy) y+xz² == <sc² (xy)-(+377
Ang = xe²-2018 - e++ xie" g'
32(x+3)-(4-7)
5-5
-
2 (x²+ y²) (2x + Lg. y ) = 2(x-y) (1-y')
• Stan x dx = In |sec x | - c
u = cos %
du=-sinxdx
- sin x dx = - fu du
cos
-In11+C
- In cas x1 + C = In | sex x1 +6
• f sec x dx = ln | s
sec x=
sec X+tan X + 1
C
secx (secx+ tan x)
secx+tan x
In a
端点不可微
35. X³y² - 9 = 2xy + x zg.y = o © g₁ - Z
42. X² cosy - siny = 0 0
2% cos³g + x^2 cosy (sing). y - (cosyry"
45. y² - y² - x²
• Scot x dx = - In cscx + c
Fussin x
u = sec x + tan x, du - (secx tanx + sec'x)dx
secx+secx tan x
sec x + tan x
· dx = √ du = ln |
u + C
= In | sec x + tan x | + c
sin x
+C
CSCX C
In 1 sin x 1 + c
• fcsc x dx = -ln 1 c
CSC X =
In csc x + cot X | + C
CSC x (CSC x - cotx)
CSC x cotx
ệu CSCX cotx du - tcsr cotx sex do
=
u
C
= ln cscxcot x + c
+ C
CSC x cotx
- In
sex-cotx
+ C
6.
ju + √x ) dx du= = x + dx
25 | sin³ & Los 1 dx
u =
stu 4 = du = cos & dx
51 for x te six dx
√x
2 ] u du - 2 = u ³ + c
3 ) u³du - {ub + c • — sin ' * + C
-/mx + C
Je" du = e" + c·e"
14
JA = cos ( x ) d x
du - - x x d x
-26. f tan" 1 sec - x dx
-u-tan
du + sec & lx
56
I lu ve de
--fcos'n du
> f u' du = ± 4² + 6 + + m²(2)+(
du corx dx

ページ13:

19.
21
27.
4-5
0+x
80
-sine
1+ cos 29
(504x)
日
-co50
-5420-2
3
lim
In (ex-1)
lim
35985)(59)
=
lim
x40
Xn x
(→)
sing
KIM -4 cos LB
=
secx
39. lim Unxj
41.
- In (sin x)
lim
S
tax ) = lim lax-(x-1)
lime
7-1
X-1
48.
lim (ex-1)"
49.
-Sin cos
tano-o
lim
-α
(x-1)(lnx)
(ln x)+(6-1)1)
xcosx+sings
-six
1- Los & + sing
secto -|
-
Lim
-
+0" 20*
lim 25in10
tanto
2
51. lim
55. lim x
51. In f(x)= ln x²² - lux
lim. In f(x) = lim lax - Im☆ --|
Pe
53. In f(x) = In Un xxx - lalagy
I'm in fix) = live in (lax)
X
所 = e = 1
30 (x)--(5-x)
-√55x50
g'(x)--(1-x)*(-2x)
critical
point
x= -√
x
50
f'(x) =
2x (x-2)- x²
x²-4x
(2-2)
(1)
critical
point
X °
4
55. In f(x) = ln xxxx |
Si = e"