APCALCULUS
Derivative
ADERIVATIVEFUNCTION1Thederivativefu
ctio
orsimplythederivativeisdefi
edas
fx＝y＝limylimfxxfx
xx0
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x
2Fi
dthederivativefu
ctio

aFi
dy
bFi
dtheaveragerateofcha
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cFi
dthelimitlimyxx0
3Geometricsig
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Co
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ctio
yfxafixedpoi
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t
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xa
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ge
tatA
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is
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4Rules
fxCaco
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x
si
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x
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x
1
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x
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11x2
1
fAPCALCULUS
l
x
logxaexax
uxvxuxvx
uxvx
Derivative
1x1logexaex
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uxvx
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5Thechai
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6I
versefu
ctio
Parametricfu
ctio
a
dImplicitfu
ctio

I
versefu
ctio
dy1fx1
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f1x
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y
2
fAPCALCULUS
Derivative
dy
Parametricfu
ctio


dy
dt

dxdxdt
ieytxt→t1xy1x
dydydtdydttdxdtdxdxdtt
Implicitfu
ctio
Fxyx0Fxfx0
xacost
x2y2a20
t02
yasi
t
yxdyacostcottdxasi
t
7Highderivative
fxd2ylimfxxfx
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tasi
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f
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1xxfx
1
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x
ysi
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2
2
2
y
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x
2
BAPPLICATIONSOFDIFFERENTIALCALCULUS
1Mo
oto
icity
aIfSisa
i
tervalofreal
umbersa
dfxisdefi
edforallxi
Sthe
：
3
fAPCALCULUS
Derivative
fxisi
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go
Sfx0forallxi
Sa
dfxisdecreasi
go
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S
bFi
dthemo
oto
ei
tervalFi
ddomai
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ctio
Fi
dfxa
dxwhichmakefx0
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Statio
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CINTEGRAL
1Theideaofdefi
itei
tegral
Wedefi
etheu
ique
umberbetwee
alllowera
duppersumsas
b
a
f
xdx
a
dcallit“thedefi
itei
tegralof
fxfromatob”
4
fAPCALCULUS
Derivative
ie

1

f

xi
x
b
a
f

xdx
f
xxi
i0
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where
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otethatas


1

f
xxi

b
a
f
xdx
a
d
i0


f
xxi

b
a
f
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Wewrite
lim



i1
f
xxi

b
a
f
xdx
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abthe

b
a
f
xdx
istheshadedarea
2Propertiesofdefi
itei
tegrals
b
a

f
xdx

ba
f
xdx
b
a
cf
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r