文档介绍：该【微积分CALCULUS知识点总结】是由【书犹药也】上传分享，文档一共【7】页，该文档可以免费在线阅读，需要了解更多关于【微积分CALCULUS知识点总结】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。学习必备欢迎下载
ADERIVATIVEFUNCTION

yf(xx)f(x)
f(x)=y=limlim
x0xx0x

a)Findy,
b)Findtheaveragerateofchangey,
x
y
c)Findthelimitlim.
x0x

Considerageneralfunctiony=f(x),afixedpointA(a,f(a))andavariablepoint
B(x,f(x)).Theslopeofchordf(AB=x)f(a).
xa
NowasBA,xaandtheslopeof
chordABslopeoftangentatA.
f(x)f(a)
So,limisf(a).
xa
xa
Thus,wecanknowthederivativeatx=a
istheslopeofthetangentatx=a.

f(x)f(x)
C(aconstant)0
xnnxn1
cosx
sinx
cosxsinx
1
tanxsec2x
cos2x
1
arcsinx
1-x2
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1
lnx
x
1
logxloge
axa
exex
axaxlna
u(x)v(x)u(x)v(x)
u(x)v(x)u(x)v(x)u(x)v(x)
u(x)uvuv
(v0)
v(x)v2

dydydu
Ifyf(u)whereuu(x)then.
dxdudx
f(x)eg(x)f(x)eg(x)g(x)
g(x)
f(x)lng(x)f(x)
g(x)
f(x)u(x)v(x)elnu(x)v(x)ev(x)lnu(x),
v(x)u(x)
f(x)ev(x)lnu(x)[v(x)lnu(x)]
u(x)
,ParametricfunctionandImplicitfunction
dy11
Inversefunction:,f(x),
dxdxdy[f1(x)]
.,yarcsinx,xsiny
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dydydt
Parametricfunction:,
dxdxdt
.,y(t),x(t)→t1(x),y[1(x)]
dydydtdydt(t)

dxdtdxdxdt(t)
Implicitfunction:F(x,y(x))0,F(x,f(x))0.
xacost
x2y2-a20,,t[0,2]
yasint
dyacost
y(x)cott
dxasint

d2yf(xx)f(x)
f(x)lim
dx2x0x
dydydtcsc2t1
y(x)[y(x)]
xdxdxdtasintasin3t
f(n1)(xx)f(n1)(x)
f(n)(x)lim
x0x

y=sinxycosxsin(x),ycos(x)sin(x2)
222

y(n)sin(xn)
2
BAPPLICATIONSOFDIFFERENTIALCALCULUS

a)IfSisanintervalofrealnumbersandf(x)isdefinedforallxinS,
then:
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f(x)isincreasingonSf(x)0forallxinS,and
f(x)isdecreasingonSf(x)0forallxinS.
b)Findthemonotoneinterval
Finddomainofthefunction,
Findf(x),andxwhichmakef(x)0,
Drawsigndiagram,findthemonotoneinterval.
,Horizontalinflection,Stationarypoint
CINTEGRAL

Wedefinetheuniquenumberbetweenallloweranduppersumsas
bf(x)dxandcall“itthedefiniteintegralf(xof)fromato”,b
a
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n1nba
.,f(x)xbf(x)dxf(x)xwherex.
iain
i0i1
n1
Wenotethatasn,f(x)xbf(x)dxand
ia
i0
n
f(x)xbf(x)dx
ia
i1
n
Wewritelimf(x)xbf(x)dx.
ia
ni1
Iff(x)0forallxon[a,b]then
bf(x)dxistheshadedarea.
a

b[f(x)]dxbf(x)dx
aa
bcf(x)dxcbf(x)dx,cisanyconstant
aa
bf(x)dxcf(x)dxcf(x)dx
aba
b[f(x)g(x)]dxbf(x)dxbg(x)dx
aaa
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bf(x)dxF(x)bF(b)F(a),whereF(x)f(x)dx
aa
af(x)dx0(f(x)odd),af(x)dx2af(x)dx(f(x)even)
aa0
Iff(x)0onaxbthenbf(x)dx0
a
Iff(x)g(x)onaxbthenbf(x)dxbg(x)dx
aa
Theaveragevalueofafunctiononaninterval[a,b]
1
fbf(x)dx
avebaa

IfF(x)f(x),thenf(x)dxF(x)C
11
Formulas:xndxxn1C,axdxaxC
n1lna
sinxdxcosxC,cosxdxsinxC,
tanxdxlncosxC,cotxdxlnsinxC
dxdx
arcsinxC(x21),arctanxC
1x21x2
USubstitution
f(g(x))g(x)dxsubstitutionu=g(x)f(u)du
IntegrationbyParts
udvuvvdu
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