미적분-1.indd

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2 k k k= k= =4 =-3 = +5- = - 3 P t =t -t +3t- t=3

3 (NASA) NASA 44

0 (Galilei, G. ; 564~64) C m f() f()=-4.9 +C(m) mm3m f () mf () mf () m.. -9.8 3. 3 f {}=-4.

4 0 (Galilei, G. ; 564~64) C m f() f()=-4.9 +C(m) mm3m f () mf () mf () m f {}=-4.9@+3 f {}=-4.9@+ f {}=-4.9@+ f() f'() f() f() + + ( )'=( +)'=( +)'= + + F() f()f'()=f() F() f()

5 : (itegral) : f()d f() f() F() G() F'()=f()G'()=f() {G()-F()}'=G'()-F'()=f()-f()=0 0 C G()-F()=C G()=F()+C f() F() f() F()+C C : f()d : f()d=f()+c C : f()d=f()+c f() f() F'()=f() : f()d=f()+c C : d : d ()'= : d=+c (5 )'=0 : 0d=5 +C : 3d : 3 d : 4 d f() C : f()d= +3+C : f()d= +3 -+C 46

6 F() F'()=f() f() F() ;!; ;3!; ;4!; «± «0 { «± } ' =«: «d= «± +C + + =«=0 : d= 3 ± +C 0+ =+C 0 : «d= «± +C C + : d=: d= 3 ± +C= +C + : d= 3 ± +C= +C : d : d : d 4 m : μ «d : (μ )«d 47

7 f() g() F() G() F()=: f()dg()=: g()d F'()=f()G'()=g() k {kf()}'=kf'()=kf() : kf()d=kf() kf()=k: f()d : kf()d=k: f()d {F()+G()}'=F'()+G'()=f()+g() : { f()+g()}d=f()+g() F()+G()=: f()d+: g()d : { f()+g()}d=: f()d+: g()d 3 {F()-G()}'=F'()-G'()=f()-g() : { f()-g()}d=f()-g() F()-G()=: f()d-: g()d : { f()-g()}d=: f()d-: g()d : kf()d=k: f()d k : { f()+g()}d=: f()d+: g()d : { f()-g()}d=: f()d-: g()d 48

8 0 : (3 +-5)d C : (3 +-5)d=: 3 d+: d-: 5d =3: d+: d-5: d = C 3 = + -5+C + -5+C 5 : (3 +-)d : (6 --)d : (+5)(-3)d : 3 d-: 3 d f() f'()=3 -+ f(0)=3 f'()=3 -+ f()=: (3 -+)d= - ++C f(0)=3 C=3 f()f()= f()= f() f'()= - + f(0)= f'()=(-) f()=- 49

0 6 cm0.5 cm0.5 cm 3. a b a b 3 0 85. m M 3.

9 0 6 cm0.5 cm0.5 cm 3. a b a b m M cm0.0 cm m M M-m M-m 50

10 S m M m S M m M S m M = = S [0] (=) - 0{ } { } { 4} { } S«S«- S«= 0+ { } ++ { 4} S«= 4{ + ++(-) } (-)(-) S«= 4 6 S«= {- }{- } 6 S =@ - = =@ - = 5

11 T«T«T«= { } + { } + - T«=+ { 4} + { } T«= 4{ + ++(-) + } (+)(+) T«= T«= {+ }{+ } 6 S S«<S<T«limS«S limt«lims«= lim {- }{- }= 6 3 limt«= lim {+ }{+ }= 6 3 S= 3 =@ - = = lim S«lim T«lim S«lim T«= = S =# [0] S 5

12 0 r h (-) h r h h r r (-)r 4 V«r rk k h h r h r h (-)r h V«=p{ } +p{ 4 } ++p[ ] pr h = 4{ + ++(-) } r k r r =h h r = kr pr h (-)(-) = pr h = 4{- }{- } 6 V V= limv«pr h = lim 4{- }{- } 6 pr h = 4 3 V= pr h 4 3 a h h a 53

13 03 f()= = 4 f{}=@ 4 º=0 = = «=. º {=} k- f{ k } k. k 3. f( )(k=) D b-a D= - = 3 k= =f() [ab] f()æ0 =f() =a=b S [ab] a=º ««=b D b-a D= S«S«=f( )D+f( )D++f(«)D S«= f( )D k= =f{} º = a - = b 54

14 S«S S= lims«= lim f( )D k= =f()[ab] f()<0=f() =a=b S S«S«={-f( )}D+{-f( )D} ++{-f(«)}d S«=- f( )D k= S«S S= lims«=- lim f( )D k= =f()[ab] lim f( )D k= º =f{} =f() a b :Ab`f()d :Ab f()d f() a b a = - b = :Ab f()d [a b] f() ab =f() [ab] b-a :Ab f()d= lim f( )D {D= =a+kd} k= f() [ab] :Ab f()d=:ab f(t)dt=:ab f()d 55

0 :) d f()= f()[0] a=0b= b-a k D= = =a+kd= 4k f( )= = :) d= lim f( )D k= 4k 8 = lim = lim 4 k k= k= 8 (+) = lim 4 5=4 = 4 4 6 4 :) 3d :)

15 0 :) d f()= f()[0] a=0b= b-a k D= = =a+kd= 4k f( )= = :) d= lim f( )D k= 4k 8 = lim = lim 4 k k= k= 8 (+) = lim 4 5=4 = :) 3d :) (- )d :)3 d :) (- )d 0 f() - k k f{ } f{ } :) f()d k=0 k= =f{} - f{ } - k k=0 f{ k } k=0 f{ k } k= f{ k } k= =f{} :) f()d :) f()d f() 56

16 f(t)=t+ t t=0 f{t}=t+ t= S(). S() S'() S{} t. S() 3. S'() f() =f(t)[ab] f(t)æ0 [ab] =f{t} =f(t) tt=at= S{} S() a b t S()=:A/ f(t)dt ` DS() DS DS=S(+D)-S() 57

17 [+D] [+D] =f(t) D>0[+D] =f(t) M m =f{t} m D DS M D ` m M ûs a b +û t D<0[+D] =f(t) M m M D DS m D ` =f{t} M ûs m a +û b t D D DS m 4 M D D 0 DS lim m lim 4 lim M D 0 D 0 D D 0 =f(t)[ab] D 0 m f() M f() DS d f() lim 4 f() f() S() f() D 0 D d d S()=f() d d :A/ f(t)dt=f() d f() [ab] d :A/ f(t)dt=f() a<<b d 58

18 d :&/ (t +3t+)dt= +3+ d d d :%/ (3-8+9)d :_/# (3t-)(t+8)dt d d =f(t)[ab] S()=:A/ f(t)dt S'()=f() S() f() f() F() S()=:A/ f(t)dt=f()+c C ` S() =a S(a)=0 =a S(a)=F(a)+C=0 C=-F(a) S()=:A/ f(t)dt=f()-f(a) =b t :Ab f()d=f(b)-f(a) F(b)-F(a) [F()]bA [ab] f() F() :Ab f()d=[f()]ba=f(b)-f(a) 59

19 0 :! (4+3)d :_3! 3 d : (4+3)d= +3+C :! (4+3)d=[ +3]!=4-5=9 : 3 d= +C :_3! 3 d=[ ]3_!=3 -(-) = :!3 d :_3@ (3 -)d :_! (-6)d :) (4 +3 )d a=ba>b :Ab f()d :Aa f()d=0:ab f()d=-:ba f()d a>b f() F() :Ab f()d=-:ba f()d=-[f()]ab=[f()]ba [ab] 03 :@- 3 d :!0 (+4)d :@- 3 d=[ ]-@ =(-) - =-9 :!0 (+4)d=-:) (+4)d=-[+ ])=-(3-0)=

20 4 d (-4+)d :!- (3 +)d 04 :A/ f(t)dt= --3 f() a f()=- =a :Aa f(t)dt=0 (a+)(a-3)=0 a=- a=3 f()=-a=- a=3 5 :!/ f(t)dt=4-5+a f() a 6 [-] =f() g()=:_/! f(t)dt - =f{} -3 =f()=f'()=:)/ f(t)dt C B A 6

21 04 :) ( +)d :) d+:) d :) ( +)d=[;4!; + ])=;4%; :) d+:) d=[;4!; ])+[ ])=;4%; f()= g()=. :) f()d :) f()d. :) f()d-:) g()d :) { f()-g()}d 3. :) f()d :) f()d+:! f()d :) d= :) d= :) d=:) d 3 6

22 f()g() F()G() k : kf()d=k: f()d=kf()+c :Ab kf()d=[kf()]ba=kf(b)-kf(a)=k{f(b)-f(a)} :Ab kf()d=k:ab f()d : { f()+g()}d=: f()d+: g()d =F()+G()+C :Ab { f()+g()}d=[f()+g()]ba ={F(b)+G(b)}-{F(a)+G(a)} ={F(b)-F(a)}+{G(b)-G(a)} =:Ab f()d+:ab g()d :Ab { f()-g()}d=:ab f()d-:ab g()d f()g() [ab] :Ab kf()d=k:ab f()d k :Ab { f()+g()}d=:ab f()d+:ab g()d :Ab { f()-g()}d=:ab f()d-:ab g()d 63

23 0 :_! (6 +4+)d :) (+) d-:) (-) d :_! (6 +4+)d=6:_! d+4:_! d+:_! d =6[ ]_!+4[ ]_!+[]_! 3 =4+0+4=8 :) (+) d-:) (-) d=:) {(+) -(-) }d =:) 4d=4:) d =4[ ])= 8 :! ( -+3)d :_# (+3)(-)d :! fi d+:! (3-fi )d :! (-)(+)d-:! 3 d a b c f() F() : f()d=f()+c :Ac f()d+:cb f()d=[f()]ca+[f()]bc ={F(c)-F(a)}+{F(b)-F(c)} =F(b)-F(a) =:Ab f()d 64

24 abc f() :Ac f()d+:cb f()d=:ab f()d 5 :! ( -5)d+:@3 ( -5)d=:!3 ( -5)d=[ - ]3!= :) (3 -)d+:! (3 -)d :Ab f()d=-:ba f()d :_! (- )d-:# (- )d 0 :) - d f()= - -+ (0 ) f()= -- ( ) :) - d=:) (-+)d+:! (-)d = - =[- +])+[ -]! = + = 3 :_! - d :_3@ - d 65

f()[ab] b-a lim f( )D=:Ab f()d {D= =a+kd} k= 03 + +3 ++ lim + +3

a=0b= D= = =a+kd= + +3 ++ lim =:) f()d=:) d=[ ])= 3 3 3 4 + +3

25 f()[ab] b-a lim f( )D=:Ab f()d {D= =a+kd} k= lim lim = lim [{ } +{ } +{ } ++{ } ] k = lim { } k= b-a k f()= a=0b= D= = =a+kd= lim =:) f()d=:) d=[ ])= lim fi 3 lim [{+ } +{+ } +{+ } ++{+ } ] r (-) k r k r r k r 66

26 0 f() C : f()d=3 +7+C : f()d=- ++C 3 0 : fi d : (4+5)d 0 : (3 -+5)d : 3 d+: 3d :) d :) d 03 4 :)/ (t +4)dt :!/ (t-) dt 03 5 :) ( - +4)d 04 :! (-)( ++)d :_! ( -+)d+:_! ( +-)d :_! (3 +-4)d+:@3 (3 +-4)d 67

27 0 (3) =f() ( f()) + f() 0 = = S 0 [0] - 0 S«S S= lim S«= lim = k= =# 3 = - 3 f()= -+5 lim :@/ f(t)dt :_@ + d :_3! - d 04 :_0# ( -5)d-:#0 ( -5)d :) 3 d+:@0 3 d AB =BC =B=90 ABC AB B B B B«BC AC C C C B A B B C C C 04 3 C«lim - B C k= B- B- B C- C- C 68

28 0 f()= d d F()=: [ [ : f()d]]d d d F(0)=3 F() 0 f() f'() f() 6 =f'{} 0 - f() - 3 =f() =f'() lim f()=f()=3 =f'{} 03 :)3 f()d - 4 a +b+c=0 ab(a<b) a(b-a) :Ú (a +b+c)d= f() :Kk f()d=k lim 4:! f()d 04 69

미통기-3-06~07(052~071)

미통기-3-06~07(052~071) 06 F() f() F'()=f()F() f() : f()d f() f() f() f() F()f() F()+C : f()d=f()+c C F'()=f(): f()d=f()+c C d [: f()d]=f() d : k d=k+c k C : «d= + +C =0C + : k f()d=k: f()d k : { f() g()}d=: f()d : g()d =f()