IV 2 3

44

Leibiz, G. W. ; 646~76 Newto, I. ; 642~727 itegral calculus differetial calculus calculus 2 _ 45

0 02 (x+2)(x-2) (x+2) (a+b)(a-b)=a -b (a+b) =a +3a b+3ab +b 2 f(x)=x g(x)=x +2x f(x)=x«f(x)=x«( ) f'(x)=x«3 f(x)=x +x A(2) =f(x) (af(a)) -f(a)=f'(a)(x-a) 46

: f(x)dx 2x 2x x x x +x - (x )'=2x(x +)'=2x(x -)'=2x x x +x - 2x xf(x) f(x) d F'(x)=f(x) 2 F(x)=f(x) dx F(x)f(x) x x +x -2x 2x x (x )'=3x x 3x 47

4x x x -;2!; x -x x +3 F(x)f(x) G(x) f(x) F'(x)=f(x)G'(x)=f(x) {G(x)-F(x)}'=G'(x)-F'(x)=f(x)-f(x)=0 h(x) h'(x)=0 h(x)=c C 0 C G(x)-F(x)=C G(x)=F(x)+C f(x) F(x) f(x) F(x)+C (C) : (ite gral) : f(x)dx : f(x)dx=f(x)+c (C) C : f(x)dx=f(x)+c f(x) : f(x)dx f(x) F'(x)=f(x) : f(x)dx=f(x)+c (C ) 48

: 5dx : (5x +) dx (5x)'=5 : 5 dx=5x+c (xfi +x)'=5x + : (5x +) dx=xfi +x+c 5x+C xfi +x+c ( C ) : ( )dx=x +C : ( )dx=2x -3x+C : (-2)dx : (4x +2x)dx : 4x dx : (3x -)dx 2 f(x) ( C ) : f(x)dx=2x+c : f(x)dx=;3!;x +;2!;x +C F(x)G(x) F'(x)G'(x) F'(x)=G'(x) F(x)=G(x) 49

F(x)= 3 x + (=0234) F'(x) + 0 2 3 4 F(x) x ;2!;x ;3!;x ;4!;x F'(x) x x =x«(x)'={;2!;x } ' =x{;3!;x } ' =x C : dx :dx : dx=x+c: x dx=;2!;x +C: x dx=;3!;x +C =x { 3 x + } ' =x«+ Cx«: x dx= 3 x + +C + x«: x«dx= 3 x + +C (C ) + 50

: x dx= 4x 3+ +C=;4!;x +C 3+ : x dx : xfi dx f(x)g(x) F(x)G(x) F'(x)=f(x) G'(x)=g(x) : f(x)dx=f(x)+c : g(x)dx=g(x)+c C C []=kf(x) '=kf'(x) [2]=f(x)+g(x) '=f'(x)+g'(x) [3]=f(x)-g(x) '=f'(x)-g'(x) k{kf(x)}'=kf'(x)=kf(x) : kf(x)dx=kf(x)+c=kf(x)+kc =k: f(x)dx 2{F(x)+G(x)}'=F'(x)+G'(x)=f(x)+g(x) : { f(x)+g(x)}dx=f(x)+g(x)+c=: f(x)dx+: g(x)dx 3{F(x)-G(x)}'=F'(x)-G'(x)=f(x)-g(x) : { f(x)-g(x)}dx=f(x)-g(x)+c=: f(x)dx-: g(x)dx : k f(x)dx=k: f(x)dx (k) : { f(x)+g(x)}dx=: f(x)dx+: g(x)dx : { f(x)-g(x)}dx=: f(x)dx-: g(x)dx 5

: (x+)(x-)dx C : (x+)(x-)dx=: (x -)dx=: x dx-: dx : (x+)(x-)dx={;3!;x +C }-(x+c ) : (x+)(x-)dx=;3!;x -x+c -C C -C =C : (x+)(x-)dx=;3!;x -x+c ;3!;x -x+c : (x+)dx : (x +3x)dx : (6x -4x+3)dx : (x+) dx 2 f(x) f'(x)=3x +2x+ f(0)=4 f'(x)=3x +2x+ f(x)=: (3x +2x+)dx f(x)=3: x dx+2: x dx+: dx f(x)=x +x +x+c f(0)=4 C=4 f(x)=x +x +x+4 f(x)=x +x +x+4 f(x) f'(x)=2x+ f()=2 52

3 =f(x) (xf(x)) 2x-3 (2) f(x) =f(x) (xf(x)) f'(x) (xf(x)) f'(x) f'(x)=2x-3 f(x)=: (2x-3)dx=x -3x+C =f(x) (2) f()=-3+c=2 C=4 f(x)=x -3x+4 f(x)=x -3x+4 =f(x) (xf(x)) 4x -6x+ (0) f(x) : (x +6x)dx : (x +)(x -)dx 2 f(x) f'(x)=x +f(0)= d d f(x)=x 24 : f(x)dx : [ 24 f(x)]dx dx dx 53

. F'(x)=f(x) F(x)f(x) 2. F(x) f(x) : dx=f(x)+c C 3. f(x)g(x) : k f(x)dx= : f(x)dx (k) : { f(x)+g(x)} dx=: f(x)dx+ : { f(x)-g(x)} dx=: f(x)dx- 0 f(x) F(x)G(x) F()=5G()=3 F(x)-G(x) 02 : (5x +)(5x -)dx : (x+) dx-: (x-) dx : (t+) (t-) dt : ( ++)( -+)d 03 =f(x) (xf(x)) 3x -3 f(x) 54

04 f(x) F(x) f(x) F(x)=x f(x)-2x +x + f()=2 05 f(x) F(x) F(x)=f(x)+2x f(x) 06 =f(x) x DxD D=3x Dx+4(Dx) f(0)= f(x) 07 dv v m/s t C 2 =0.6 dt 0 æ 33 m/s 00 m 0.5 æ 200 m æ 400 m 55

t s(t) v(t) a(t) s'(t)=v(t) v'(t)=a(t) v(t)=: a(t)dt s(t)=: v(t)dt sº vº a(a) t v(t)s(t) v(t)=: a dt=at+c (C ) t=0v(0)=vº v(t)=at+vº s(t)=: v(t)dt=: (at+vº)dt=;2!;at +vº t+c t=0s(0)=sº s(t)=;2!;at +vºt+sº 00m 9.8 m/s 2 56

2 0 02 03 0 k k k= 0 k= (+) k= k= 2 k = 24 (+)(2+) k= 6 k 2 lim 3 k= {a } S«= a«= lim S«3 : (2x+)dx : (4x +3)dx F'(x)=f(x) : f(x)dx=f(x)+c C 57

(Kepler, J. ; 57~630) S S«T«S«S T«T«S«58

S«T«S S =x x= x [0, ] =x O x [0, ] x 2 3 (=) =x { } 2 { } 3 { } { } O 2 3 :: :: :: - ::::; S«x S«= { } 2 + { } 3 + { } ++ { } +2 +3 ++ = 2 (+)(2+) 6 S«= 4( +2 +3 ++ ) (+)(2+) S«= 2 6 S«=;6!;{+ }{2+ } S S= lim S«= lim ;6!;{+ }{2+ } S=;3!; ;3!; 59

=x x= x [0, ] =x O x =x x= x k =[ 333 (+) ] 2 k= 2 r h (-) :: h :: r h h x :: h h r 2r (-)r 3 32 r r h x:r= :h r x= (-) V«r V«=p { } h 2r +p { } h (-)r ++p[ ] h 3 32 pr h V«= 34{ +2 ++(-) } pr h (-)(2-) V«= 34 34 6 V«=;6!;pr h{- }{2- } V V= lim V«= lim ;6!;pr h{- }{2- } V=;3!;pr h ;3!;pr h 60

a (-) h h a a 2 =x x=x=2 x =x O 2 x r ;3$;pr r (CT) X 2 3 5mm 6

:Ab f(x)dx =x 0 S«lim S«AOB =x A O :: :: 2 3 :: B ::::; - x lim S«=x x= x =f(x)ab f(x)æ0 ab =f(x)x S =f(x) O xº = a f(x ) Dx x x x x x«= b x ab x xº(=a)x x x x«(=b) x x (k=2)dx b-a Dx= 3 62

S«S«=f(x )Dx+f(x )Dx++f(x )Dx S«= f(x )Dx k= S«S S= lim S«= lim f(x )Dx k= =f(x)ab lim k= f(x )Dx :Ab f(x)dx f(x) a b =f(x) a b :Ab`f(x)dx f(x)ab b-a :Ab f(x)dx= lim f(x )Dx {Dx= 3 x =a+kdx} k= :Ab f(x)dx x :Ab f(x)dx=:ab f(t)dt=:ab f(s)ds : f(x)dx : f(t)dt =f(x) ab :Ab f(x)dx x S x S O a S =f(x) b x S :Ab f(x)dx=s -S 63

:)3 x dx a=0b=3 3-0 3 3k Dx= 4= x =0+kDx= 2 f(x)=x :)3 x dx= lim f(x )Dx k= 3k :)3 x dx= { } 3 lim 2 k= 27 :)3 x dx= lim 2 k k= 27 (+)(2+) :)3 x dx= lim 2 3 6 27 2 :)3 x dx= 3=9 6 9 O =x x x x«x«x x 3 = 9 :!2 xdx :!2 x dx :) (x +)dx :) f(x)dx= lim f(x )Dx { Dx= x =kdx} k= - :) f(x)dx= lim f(x )Dx { Dx= x =kdx} k=0 64

[F(x)]bA f(t)=t [x] f(t)=t t S(x) 2 S(x) x S'(x) f(x) =t S(x) O x t =f(t)ab f(t)æ0 ab x =f(t)t=at=x t S(x) S(x)=:A/ f(t)dt O a S(x) =f(t) x b t xdxs(x)ds DS=S(x+Dx)-S(x) f(x) [ab] f(x) Dx>0 f(t)xx+dx M m m Dx DS M Dx O =f(t) M DS m a x x+dx b t DS m M Dx 65

2Dx<0 f(x)g(x) lim x a f(x)=a lim g(x)=b a x a x h(x) f(x) h(x) g(x) a=b lim h(x)=a x a 22 Dx Dx 0 DS m M Dx DS lim m lim lim M Dx 0 Dx 0 Dx Dx 0 f(t)ab lim M= lim m=f(x) Dx 0 lim Dx 0 Dx 0 DS =f(x) Dx DS d d lim = 24 S(x)= 24 :A/ f(t)dt Dx 0 Dx dx dx d 24 :A/ f(t)dt=f(x) dx f(t)ab d 24 :A/ f(t)dt=f(x) (a<x<b) dx d 2 dx :!/ (t +2t+3)dt=x +2x+3 d 2 :_/!(t +t)dt=x +x dx d d 24 :@/ (t -2t)dt 24 :#/ (t+) (t+2) dt dx dx 66

=f(t)ab S(x)=:A/ f(t)dt (a x b) S'(x)=f(x) S(x) f(x) f(x) F(x) S(x)=:A/ f(t)dt=f(x)+c (C ) S(a) a a S(a)=0 S(x) S(a)=:Aa f(t)dt=0 S(a)=F(a)+C=0 C=-F(a) C=-F(a) :A/ f(t)dt=f(x)-f(a) x=b t x :Ab f(x)dx=f(b)-f(a) F(b)-F(a) [F(x)] ba f(x)ab F(x)f(x) :Ab f(x)dx=[f(x)]ba=f(b)-f(a) :!3 (2x+)dx=[x +x]3!=2-2=0 :_!(6xfi -)dx=[xfl -x]_!=0-2=-2 :!2 3x(x+2)dx :_! (-3t +2)dt 67

a<b :Ab f(x)dx aæb :Ab f(x)dx a=b:aa f(x)dx=0 2a>b:Ab f(x)dx=-:ba f(x)dx a>b F'(x)=f(x) :Ab f(x)dx=-:ba f(x)dx=-[f(x)]ab=-{ F(a)-F(b)}=F(b)-F(a) ab :#3 x dx=0 :#0 (2x-)dx=-:)3 (2x-)dx=-[x -x]3)=-(6-0)=-6 :@2 (x +) dx : -)2 (4x +3x )dx x :!/ f(t)dt=x +x+a f(x) a x d d 24 :!/ f(t)dt= 24 (x +x+a) dx dx f(x)=2x+ x= :! f(t)dt=++a=a+2=0 a=-2 f(x)=2x+a=-2 x :!/ f(t)dt=x +ax+ f(x) a 68

ab f(x)g(x) F(x) G(x) k : kf(x)dx=k: f(x)dx=kf(x)+c :Ab kf(x)dx=[kf(x)]ba=kf(b)-kf(a)=k{f(b)-f(a)} :Ab kf(x)dx=k:ab f(x)dx 2 : { f(x)+g(x)}dx=: f(x)dx+: g(x)dx=f(x)+g(x)+c :Ab { f(x)+g(x)}dx=[f(x)+g(x)]ba :Ab { f(x)+g(x)}dx={ F(b)+G(b)}-{ F(a)+G(a)} :Ab { f(x)+g(x)}dx={ F(b)-F(a)}+{ G(b)-G(a)} :Ab { f(x)+g(x)}dx=:ab f(x)dx+:ab g(x)dx 2 2: { f(x)-g(x)}dx=: f(x)dx-: g(x)dx=f(x)-g(x)+c :Ab { f(x)-g(x)}dx=:ab f(x)dx-:ab g(x)dx 2 f(x)g(x)ab :Ab kf(x)dx=k:ab f(x)dx (k) :Ab { f(x)+g(x)}dx=:ab f(x)dx+:ab g(x)dx :Ab { f(x)-g(x)}dx=:ab f(x)dx-:ab g(x)dx 69

2 :) (x +3x)dx :!2 (x +x)dx+:!2 (2x -x)dx :) (x +3x)dx=:) x dx+3:) x dx=[;3!;x ])+3[;2!;x ]) :) (x +3x)dx ={;3!;-0}+3{;2!;-0}=;; 6 ;; :!2 (x +x)dx+:!2 (2x -x)dx=:!2 {(x +x)+(2x -x)} dx :!2 (x +x)dx+:!2 (2x -x)dx=3:!2 x dx=3[;3!;x ]2! :!2 (x +x)dx+:!2 (2x -x)dx=8-=7 ;; 6 ;; 7 :_2! (4x -x)dx :_! (x -x+)dx+:_! (x +x+)dx :) (x+) dx+:!0 (x-) dx :_! (x +x+)dx+:!- (x -x)dx f(x)abc f(x) =f(x) F(x) O a c b t :Ac f(x)dx+:cb f(x)dx=[f(x)]ca+[f(x)]bc :Ac f(x)dx+:cb f(x)dx={f(c)-f(a)}+{f(b)-f(c)} :Ac f(x)dx+:cb f(x)dx=f(b)-f(a) :Ac f(x)dx+:cb f(x)dx=:ab f(x)dx 70

abc f(x)abc :Ab f(x)dx=:ac f(x)dx+:cb f(x)dx 3 :)3 x- dx x-a -x-a (xæa) = -x+a (x<a) -x+ (0 x ) f(x)= -x- (<x 3) :)3 f(x)dx=:) (-x+)dx+:!3 (x-)dx = x- :)3 f(x)dx=[-;2!;x +x])+[;2!;x -x]3! O 3 x :)3 f(x)dx=;2!;+2=;2%; ;2%; :_0# x+2 dx :)2 x - dx f(x) ab =f(x) b-a Dx= 3 x =a+kdx lim k= f(x )Dx=:Ab f(x)dx O a = xº x x x«b = x«x 7

4 lim 4 ( +2 +3 ++ ) fi lim 4 ( +2 +3 ++ )= lim 4 k fi fi k= k ( +2 +3 ++ )= { } lim k= f(x)=x a=0b= b-a k Dx= 4 = x =a+kdx= xfi ( )=:) x dx=[ ])=;5!; ;5!; 5 [{ } 2 +{ } 3 +{ } lim ++{ } ] 3 lim 4{3 +6 +9 ++(3) } :) 0x dx :)2 (3x -x)dx 2 :)2 x -x dx :_! x«dx=0 ( ) :_! x«dx=2:) x«dx ( ) 72

. b-a 2. =f(x)ab lim f(x )Dx {Dx= 2 x =a+kdx} k= =f(x) a b d 3. f(t) [ab] 24:A/ f(t)dt= (a x b) dx 4. F'(x)=f(x):Ab f(x)dx=[ ]ba= - 0 :!2 3(x+)(x-)dx :_! x (x-)dx 02 f(x) f(x)=3x +2x+2:) f(x)dx 03 lim :!/ (t +)dt lim :@2 h (t +3t)dt x 2 x- h 0 h 73

04 :!/ (x-t)f(t)dt=2x -3x + f() 05 f(x)=:)/ (t-)(t-2)dt 06 f(a)=:)2 x-a dx ( 0 a 2) 07 f(x) - x f(-x)=-f(x) x f(x+2)=f(x)+2 :_2! f(x)dx= :!5 f(x)dx 08 fi +2fi +3fi ++fi lim 2 ( +2 +3 ++ )( +2 +3 ++ ) 74

Loretz, M. O. ; 880~ 962 0.5 B A O 0.5 Gii coefficiet = OAB Gii, C. ; 884~965 L(x) G :) {x-l(x)}dx G= 3 =2:) {x-l(x)}dx OAB 0.4 x<=0x== =L(x) L(0)=0L()= 2 L(x)=;6%;x +;6{; 75

3 0 02 t =x -6x +8x x =x -6x +8x O 2 4 x 2 3 =x+2 =x =x - =0 Pt s(t)=t +t t= t= =f(x) =g(x) x f(x)=g(x) P t s(t) v(t)=s'(t) a(t)=v'(t) 76

f(x)=-x+ A B =f(x) 2 :) f(x)dx A O A B 2 x 3 :!2 f(x)dx B - x =f(x)ab =f(x) xx=a x=b S ab f(x)æ0 =f(x) xæ0 x =x S=:Ab f(x)dx=:ab f(x) dx O a S b x 2 ab f(x) 0 =f(x) =-f(x) x<0 -x= x =-f(x) x -f(x)æ0 S=:Ab {-f(x)}dx=:ab f(x) dx O a S S =f(x) b x 77

x = -x (xæ0) -x (x<0) 3 ac f(x)æ0 cb f(x) 0 =f(x) S=:Ac f(x)dx+:cb {-f(x)}dx O a c b x S=:Ac f(x) dx+:cb f(x) dx S=:Ab f(x) dx x f(x)ab =f(x) xx=a x=b S S=:Ab f(x) dx =x -x x x=2 x A B =x -x 0 0 2 æ0 S =x -x S=:)2 x -x dx A O B x S=:) (-x +x)dx+:!2 (x -x)dx S=[-;3!;x +;2!;x ])+[;3!;x -;2!;x ]2! O 2 x S=;6!;+;6%;= =x - x =x -x x =x -3x+2 xx=3 =x -x xx=2 78

=f(x)=g(x) ab =f(x) =g(x) x=ax=b S abf(x)æg(x)æ0 =f(x) S S=:Ab f(x)dx-:ab g(x)dx S =g(x) S=:Ab { f(x)- g(x)}dx O a b x 2 ab f(x)æg(x) f(x) g(x) =f(x)=g(x) k =f(x)+k 0 g(x)+k f(x)+k S S =g(x)+k =f(x) O a S b x S=:Ab { f(x)+k}dx-:ab { g(x)+k}dx =g(x) S=:Ab { f(x)-g(x)}dx 3 ac f(x)æg(x) cb f(x) g(x) 2 S S=:Ac { f(x)-g(x)}dx+:cb { g(x)-f(x)} dx =g(x) S=:Ac f(x)-g(x) dx+:cb f(x)-g(x) dx S=:Ab f(x)-g(x) dx O S =f(x) S a c b x ab =f(x)=g(x)x=ax=b S S=:Ab f(x)-g(x) dx 79

2 =x =x x x x =x x(x-)=0 x=0 x= S =x =x S=:) (x-x )dx=[;2!;x -;3!;x ]) S={;2!;-;3!;}-(0-0) O x S=;6!; ;6!; =-x +2 =x =x -3x =x =x - x x=-2 x=2 =x - -2 O 2 x 2 =x =x =f(x) x A B 39 =f(x) :_2@{ f(x)-2}dx B A -2 - O 2 x 80

P t x=s(t) v(t) 2 s(t) v(t) :!t v(t)dt s() s(t) s'(t)=v(t) :!t v(t)dt=[s(t)]t!=s(t)-s() s(t)=s()+:!t v(t)dt Pt v(t) t=a s(a)px=s(t) dx v(t)= 2=s'(t) dt :At v(t)dt=[s(t)]ta=s(t)-s(a) t Ps(t) s(t)=s(a)+:at v(t)dt t=a t=b P s(b)-s(a)=:ab v(t)dt 8

Pt v(t) t=a P s(a) t Ps(t) s(t)=s(a)+:at v(t)dt t=at=b P s(b)-s(a)=:ab v(t)dt P t v(t)=3t +2t t=2 P t= t=2 P t P s(t) P s(0)=0 t=2 P s(2)=s(0)+:)2 v(t)dt s(2)=:)2 (3t +2t)dt s(2)=[t +t ]2)=2 s(2)-s()=:!2 (3t +2t)dt s(2)-s()=[t +t ]2!=0 2 0 P t v(t)=4t-t t=3 P t= t=3 P 82

Pt=at=b s v(t)>0 P t=a t=b s P s(a) s s(b) x s=s(b)-s(a)=:ab v(t)dt 2v(t)<0 P t=a t=b s s(b) s P s(a) x s=s(a)-s(b) s=:ba v(t)dt=-:ab v(t)dt 2 t=a t=b s s=:ab v(t) dt Pt v(t) Pt=a t=b :Ab v(t) dt 2 P t v(t)=-t+2 t= t=3 P t= t=3 P :!3 -t+2 dt=:!2 (-t+2)dt+:@3 (t-2)dt :!3 -t+2 dt=[-;2!;t +2t]2!+[;2!;t -2t]3@ 2 = v(t) :!3 -t+2 dt=;2!;+;2!;= O 2 3 t 83

P t v(t)=-t -t+2 t=0 t=2 P m 9.8 m/s t v(t)=9.8-9.8tm/s t= t=0 t=2 P t v(t)=t - t=2 P t=t=3 P t=t=3 P x= P t v(t) ( 0 t 8) v 2 O v(t) 2 3 4 6 7 8 t -2 Pt=3 Pt=t=4 3 Pt=0t=8 9 P 84

. f(x)ab =f(x) xx=ax=b S S=:Ab dx 2. f(x)g(x) ab =f(x)=g(x) x=ax=b S S=:Ab dx 3. Pt v(t) t=a Ps(a) t P s(a)+:at dt t=a t=b P s(b)-s(a)=:ab dt 4. Pt v(t) Pt=a t=b :Ab dt 0 x =x -3x+2 =x -x -2x x 02 =x + x -2 x x 03 =x () 85

04 P t v(t) v(t)=t -2t P P 05 f(x) x «± -4x «-3 f(x)= lim x «+ =f(x) x x 06 =x -x x A =x -x x=a x x B AB a ( a>) O =x -x B A x x=a 07 0 m/s t v(t)=0-;2!;t 86

_ 4_4 2 3 87

0 f(x) F(x)G(x) F(x)+G(x)=x +2x+3 F(0)= G(x) f(x) F(x) F'(x)=f(x) 02 :(x+) (x-) dx x :3dx+:3dx x+ x+ 03 f'(x)=x +x+a f(x) x= 0 =f(x) x= f'()=0 04 -x +a (x<) f(x)= x -x- (xæ) a =f(x) x= lim f(x)=f() x :)2 f(x)dx 05 f(x) f(x)=x +x+:) f'(x)dx d 3 :!/ f(t)dt=f(x) dx :!/ f(t)dt=xf(x)+2x -x + 88

06 =x -4x+a A B :2 a A O =x -4x+a B x =x -4x+a x=2 :)2 (x -4x+a)dx=0 07 PQ t 3t 2t-3 P Q 3 PQ Q t P xº+:at v(t)dt 08 f(x)=-x + x A(-0)B(0) k k A k =f(x) f(x)=-x + C :: k A B - O x A k AC k = x+ k C ABC S lim S =:)a (bx-x )dx ab k= 09 xº x f(x)(n) f(x)=k(x-xº) ( k ) f(x)(n) x=a x=b W(J) W=:Ab f(x)dx 0 cm 20 cm 50(N) 40 cm 50 cm J=N_m 89

x m x t/m m x F F=()_() x m 00 m50 m x m x+dx m 00_Dx m x_(00_dx)() x Dx 00 m 50 m :)5 0 00x dx=[50x ]5)0 =25000() 50 m 29 m 20 m 90

635 (CavalieriFB; 598647) 0 02 2 0 02 03 3 0 02 9