(004~011)적통-Ⅰ-01

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1 0 f() F'()=f() F() f(), : f() F'()=f(): f()=f()+c C 4 d d : [ f()]=f()+c C [: f()]=f() «`` n C : n = n+ +C n+- : =ln +C n+ f() g() : kf()=k: f() k : { f()+g()}=: f()+: g() : { f()-g()}=: f()-: g() 5 C : sin=-cos+c : sec =tan+c : sectan=sec+c : cos=sin+c : cosec =-cot+c : coseccot=-cosec+c 6 C : e =e +C e : a a = +C a>0, a+ ln a 7 : f() g(t) =g(t) : f()=: f(g(t))g'(t)dt 8 f()g() : f()g'()=f()g()-: f~'()g() 4

2 : f '()=f()+c C d [: f()]=f() 0 f()=: ( ++4) f()-f() f()=: (ln+e + ) h 0 f(+h)-f(-h) h e e+ e+ 4e+ e - e + f()=: (ln+e + ) f'()=ln+e + h 0 = h 0 f(+h)-f(-h) h { f(+h)-f()}-{ f(-h)-f()} h f(+h)-f() = h 0 + h =f~'()+f~'() =f~'() h 0 f~'()=(e+)=e+ f(-h)-f() -h 0 f() g()=: f(), d { f()+g()}= f() f()=e, g()=+ g() : F()= f() F() F() e - - -e e e e e 5

3 f()` F'()=f() F() f() f() f() 05, y f(+y)=f()+f(y)+ f() f '(0)=- f(-) f() F() F()=f()-4 y=f() (, 4) f() f() F()=f()-4 f()=f()+f '()- f '()= f()=: =6 +C C y=f()(, 4) f()=6+c=4 C=- f()=6 - f()=6 -= 04 f() F() F()=(-)f()+ - f(0)= f() 06 f() f()>0, y f(+y)=4f()f(y) f '(0)= f(ln)

4 f '() f() (>0) f '()= g f()=f(-)= k+ (<0) f() =0 k y=f() + f '() ( <) f '()=[ - ( >) y=f() =- f()=f(-) f(0)=0 f()>0 08 f() y=f '() f() 4 0 f(0) y O y=f '() f '() f() ( -+C (<-) f()=: f '()={ ;!; +C (-<<) ª -+C (>) C C C y=f() --0 f()= f() +C =-;!;+C C =C -;$; f()= f();!;+c =-+C C =C +;$; -0 y=f() y f '()=-<0 f '()=>0 f() =- C _ O y=f() =- y=f() y f()=f(-) y=f() 09 f(-)= f() y=f '() y _ O _ y=f '() f() f()= f() f(0)=0 C =0 f()= f()=;!;>0 7

5 4 «` n n : n n n+- : n = n+ +C C n+ n=- : =ln +C C 0 - : C ;!; -ln +C -ln +C ;!; +ln +C +ln +C -ln +C >0 f() F() F()=f()--ln f()=- f(e ) e e e e e - e + e e f() f '()= ' f()=;4&; f(') +ln +ln +ln 4+ln 4+ln «n f()=f()+f '()-- f '()= + f ()=: { + } f ()=ln - +C C f()=- ln -+C=- C=0 f()=ln - f(e )=ln e - =- e e (4, 8) y=f() (, f()) 'ƒ+ f(0) -;$; - -;@; -;!; 0 8

6 5 sincos sec e a (a>0a+) f'()=-sin, f(0)=- f() f{ p } 4 f() k p -p - - cos (>0) f '()=[ k (<0) f(p)=p, f (-)=0 p p ;@;p ;@;p-;!; ;@;p- ;@;p-;#; ;@;p- f()=: (-sin )=+cos +C C f(0)=cos 0+C=- C=- f()=+cos - p f{ }=;@;p+;!;-=;@;p-;#; f() f()=ln : 4 f()= n= f(n) 5 f() f()=: sin 4 sin 0 f() p =a f{ 55555} a 4 0 ;4!; ;!; ;!; ;@; ;$; 9

7 6 7 f() f()=: (-sin )sin f(p)=0f{ p } : f(a+b), : f(g())g'(), f'() : f() ;5!; ;4!; ;5@; ;!; ;5#; f()=: e y=f(){0, ;#;} f(') e ;!;e ' + ;!;e ' +;#; ;!;e + ;!;e +;#; e +;#; dt ln =t = f() f '()= f()=: e =: e t ;!;dt f(e)= e f()=;!;e t +C=;!;e +C C f(0)=;#; f(0)=;!;e 0 +C=;#; f() 8 C= f()=;!;e + f(' )=;!;e + f(e )=;%; >0 y=f() 6 f() f '() + f '()= y=f() f() ln ln ln 5 0

8 7 0 f() h 0 f(+h)-f(-h) =e cos h f(0)=- f(p) e -e p - -e p -;#; >0 f() F() f()=f()+ e, F()=e f() e -;!;(e p +) -;!;(e p +) -;!;(e p +) e -e e -e e e -e e f()=f()+ e f()+f '()=f()+e + e f '()=e + e f '()=(+)e f()=: f '()=: (+)e f()=(+)e -: e yy 0<<p f() f '()= cos f() p f()=(+)e +C C yy = f()=f()+e=e ( F()=e) f()=e+c=e C=0 f()=(+)e f()=e 9 f() f'()=ln, f(e )=e y=f()(a, a) a e f() g()=sin+;!; =a n= a n- p p- p- p 6p p- p- p- e e e e

9 0 n f() [a, b] [a, b] n º(=a),,, y, k, y, n-, n (=b) b-a D= 5 n n n k= f( k )D k =a+kd y y=f() O º ««= a = b f() a b :Ab f() :Ab f()= n n k= f( k )D ab f() F() :Ab f()=[f()]ba=f(b)-f(a) :Aa f()=0 :Ab f()=-:ba f() 4 f(), g() [a, b] :Ab kf()=k:ab f() k :Ab {f()+g()}=:ab f()+:ab g() :Ab {f()-g()}=:ab f()-:ab g() f() a, b, c :Ab f()=:ac f()+:cb f() 5 [a, b] f() [a, b] =g(t) g'(t) [a, b] a=g(a), b=g(b) :Ab f()=:ú f(g(t))g'(t)dt 6 f() g() f '(), g'() [a, b] :Ab f()g'()=[f()g()]ba-:ab f '()g()

10 f() F() :Ab f()=f(b)-f(a) 0 a, b :) p ~(sin~+) =a+bp ab f()=e -e - :) {6 +af()}=f'() a e ;e!; e- e e+ f()=e -e - :) {6 +af()}=:) {6 +a(e -e - )} =[ +a(e +e - )]) =+a(e+e - -) f'()=e +e - f'()=e+e - :) {6 +af()}~=f'() +a(e+e -)=e+e a(e+e -)=e+e - a= 0 :! {'+ ' } ln~ ln~+;!; ln~+;#; 0 n a«=:n n+ {e +(-e)«psinp}~ n= e a«e- e - e - (e-) (e-) ln~+;%; ln~+;&;

11 :Ab f()=-:ba f() 04 :) ;4 ; (+sin )sin (+cos )cos : sin+cos ;4 ; sin+cos -' '- +' (-') ('-) :Ab {f() g()}=:ab f() :Ab g() :Ab f()=:ac f()+:cb f() ' (>0) f()=g e +a ( 0) :_A f()-:a f() a e ;e!;-;4#; ;e!;-;4!; ;e!;-;!; ;4#;-;e!; ;4!;-;e!; f() =0 f()= f()=f(0) (e +a)= '=+a a=- :_A f()-:a f()=:_a f()+:@a f() 05 y=e y y=f() :) f() e e-+4ln e-+ln e-+4ln e-5+ln e-5+4ln y y=f() O ln _ :_A f()-:a f()=:_aa f() :_A f()-:a f()=:!~- f() :_A f()-:a f()=-:_! f() :_A f()-:a f()=-[:_0! (e -)+:) '] :_A f()-:a f()=-[[e -]0_!+[;4#; ;$; ])] :_A f()-:a f()=-[-{;e!;+}+;4#;] :_A f()-:a f()=;e!;-;4#; 4

12 g(a)=ag(b)=b 07 ln :!e (ln) e ;!;ln ln ;#;ln :Ab ~f(g())g'()=: a b ~f(t)dt (g()=t ln ;%;ln a f(a)=:!a f(a ) 'ƒln f(a) 8f(a) f(a) 6f(a) 0f(a) ln~ dt ln~=t = =t=0, =at=ln~a f(a)=:!a 'ƒln~ =:) ln a 'tdt f(a)=[;@;t ;#; ]) ln a =;@;(lna) ;#; y=f() :) ;!; 'ß f(+) y y=f() f(a )=;@;(ln~a ) ;#; =;@;(4~ln~a) ;#; O 4 f(a )=4 ;#; [;@;(ln~a) ;#; ]=8f(a) 06 :) ~(e -) e e- e- ;!;(e-) ;!;(e-) ;!;e+ 5

13 09 : ::!:/ " " + -' -' - +' + ' ' 4 :Ab f()g'()=[f()g()]ba-:ab f '()g() f() f()=f()f '() f(a)=0, : a 4a : a a =k (a>0, 0<k<) k k k k 4 k { f()} f() k 0 :! ' " - -:! ' " - ' ' p ' -p - -p ' p ' - - p 4 4 f(a)=0 f()=f()f '() f(a)=f(a)f '(a)=0 f(4a)=f(a)f '(a)=0 {f()} {f()} f()f '() :A a =[- ]Aa -:A a [- ] f()f '() =0+:A a =:A a f() dt =t = =at=a=at=4a f() f(t) :A a =: 4 a ;!;dt=: 4 a f (t) 5 dt=k a a t ;!;t 6

14 :) ; ; ~(+cos~)sin~~ ;!; ;4#; ;4%; ;#; p n a«=: cos~(n) ; ; 0 ; 0; { +a k } k= a k e ln :!e ~ 555 ~ e -;e#; -;e@; -;e!; :) p ~~e ~=pe p -:) p ~e ~ :) p ~e ~sin~~=:) p ~e ~cos~~ :) p ~e ~sin~~ +;e!; +;e@; =:) p ~e ~sin~~-:) p ~e ~cos~~ 7

15 5 f() f()=f(+a) q+na f() n 5 f() 0 < f()=ln(+) f()=f(+) :$5 f() :Pq f()=: p+na b-c b+c :Ab f()=: f(+c)=:a+c f(-c) a-c ;!;~ln~-;4!; ;!;~ln~-;4!; ;#;~ln~-;4!; ;#;~ln~-;4!; ;%;~ln~-;4!; f()=f(-) :_aa ~f()=:)a ~f() f()=-f(-) :_aa ~f()=0 f(p-)=f(p+) p+a p+a : f()=:p ~f() p-a f() f()=f(+) 6 f()=e :_! f'()(-sinp) e (e-) e 4(e-) 4e :! ;#; f()=7, :! ;$; f()= 0 : f() :! ;#; f()=7 =t :@ f(t)dt=4 :! ;$; f()= =t :#4 f(t)dt= 7 f() :! `f{;!;+;#;} f(-)=f(+) :) { f()+f(-)}~=8 :_! f()=0 :@ f()+:#4 f()=:@4 f()=7 :@)0)! f()=: ~ f()=:! f() =:! f()+:@ f() =:#4 f()+5:@4 f() =+5_7=88 8

16 6 [a, b] f(), g() 8 >0 f()=~ln~+k : ;e!; e f() <: ;e!; e f() k e f() g() :Ab f() :Ab g() :Ab f() :Ab f() f() :) f() -e <k<;e!; -e <k<e <k<e -e <k<;e!; <k<e f(0)=, f~'(0)= 0<a<b< f~'(a) f~'(b) (0, ) f~"()=e 5 e- e- e- 7 9 e- e- 0<< f~"()=e f~'()=e +C C f~'(0)= C=0 f~'()=e f~'()=e f()=e +D D f(0)= D=0 f~()=e f() f~'()=e, f()=e << f~'()æf~'()=e 9 f() f() e a, b a<b e :Ab f() :Ab e :Ab f() :Ab e :Ab { f()} :Ab e :!/ f~'(t)dtæ:!/ edt f()-f()æe(-), f()=e f()æe :) f()=:) f()+:! f() :) f()æ[e ])+[;E; ]!=;%;e- 9

17 0 f() d :A/ f(t)dt=f() d :? +a f(t)dt=f(+a)-f() F'()=f() d d :A/ f(t)dt= {F()-F(a)}=F'()=f() d :? +a d f(t)dt= {F(+a)-F()}=F'(+a)-F'()=f(+a)-f() f() 0 :A +a f(t)dt=f(a) :A/ f(t)dt=f(a) -a F'()=f() :A +a F(+a)-F(a) f(t)dt= =F'(a)=f(a) 0 0 a F()-F(a) :A/ f(t)dt= =F'(a)=f(a) a -a a -a f() [a, b] n b-a b-a f {a+ k} =:A b f() n k= n n f {a+ k} =:) b-a f(+a) f {a+ k} =(b-a):) f((b-a)+a) n k f { } =:) f() n k= n n n p p f { k} =:) p f() n k= n n n p p f {a+ k} =:A a+p f() n k= n n =:) p f(a+) =p:) f(a+p) 0

18 : f(t)dt a :A/ f(t)dt 0 >- f() :_/! f(t)dt=(+)f()-(+) (>-) f() ;#; ln {ln-;!;} d :A/ f(t)dt=f(), :Aa f(t)dt=0 ln {ln+;!;} f() :!/ f(t)dt= -a +a f() a :!/ f(t)dt= -a +a = 0=-a+a a= :!/ f(t)dt= - + f()= -4+ f()=7-+=6 6 0 f() :!/ f(t)dt= +a +b- 0 >0 f() :!/ f(t)dt= +~ln~- (>0) :E e dt e -e +5 e +e +5 e (e -e )+5 (e +e )+5 (e -e )+5 f(t) t f'(0)=6f() ab 4 5 6

19 04 æ0 f() f()=:)/ e (t -7t+)dt f() : f(t)dt a e e -e e -e -e +e -e +e e -e :A/ ~f(t)dt :A/ ~f(t)dt=:a/ f(t)dt d :A/ ~f(t)dt=:a/ f(t)dt+~f() [0, ; ;] f() f{;4 ;} :) ; ; f(t)dt= cos :)/ f(t)dt=sin:? ; ; f(t)dt { 0 ; ;} 05 f() f()=:)/ cos 0 'p f~'()æ0 0 'p f~"()æ0 :) 'p`f()<;!;p t dt ;5!; ;4!; ;!; ;!; -sin~:)/ f(t)dt+cos~_f()=-cos~: / f(t)dt-sin~_f() ; =;4 ; ' - :) ;4 ; ' ' ;4 ; ' f(t)dt+ f{;4 ;}=- : f(t)dt- ; ; ' " f{;4 ;}= :) ;4 ; ' ; ; f(t)dt+ : f(t)dt ;4 ; ' "f{;4 ;}= :) ; ; f(t)dt ' ' "f{;4 ;}= _= () f{;4 ;} ' f{;4 ;}= _ =;!; '

20 06 f() f(0)=;!; f()=e +:)/ (+t)f'(t)dt f() e e -;!; e +;!; e -;!; e +;!; 4e -;!; g() : f(t)dt h() g() :H?Æ~ f(t)dt f() F() g() :H?Æ~ f(t)dt=f(g())-f(h()) d g() :H?Æ~ f(t)dt=f'(g())g'()-f'(h())h'() f() t >0 f() :!/ (-t)f(t)dt= (ln+a)+ (>0) f(e) a e e 4e 5e 6e 7e >- f() :)/ (-t)f(t)dt=ln(+)- (>-) f(0)=0 -< < f( )<f( ) >- f()= :) f(t)=4t f() 4 5 dy :) f(t)=4t t=y t= =0y=0=y=t y dy :) f(t)=:) t f(y) t t :) f(t)= :) t yf(y)dy=4t t :) t yf(y)dy=4t t tf(t)_(t)'=6t f(t)=4t t= f()=4 09 f()=:? e t dt f~'() e e- e e+ e - e +

21 0 sin f()=: t dt [0, p] f() cos a, b ;ba; ;!; ;@; ;$; ;%; 4 b : f(t)dt a a, b :Ab f()=k~ k f() f()=;; 7 ;; -:! f(t)dt+[:! f(t)dt] 0:! f() f()=;; 7 ;; -:! f(t)dt+[:! f(t)dt] y=f() y y=f() g() g()=: (-t)f(t)dt - g"() g"() g() O _ :! f(t)dt=a a f()=;; 7 ;; -a+a a=:! f() a=:! {;; 7 ;; -a+a } a=[;7$; -a +a ]! a={;; 7 ;;-4a+a }-{;7$;-a+a } a=4-a+a a -4a+4=0 a= 0:! f()=0a=0 0 f() f()=e -:) f(t)dt :) f() e ;!;(e -) ;!;(e +) e - e e + 4

22 >0 f() f()=ln~-:!e f(t) dt f() e - -;!; -;4!; -;e!; - e 5 :A/ f(t)dt a -a F'(t)=f(t) :A/ f(t)dt=[f(t)]/a=f()-f(a) f()-f(a) f'(a)= = a -a h 0 f(a+h)-f(a) h 0 : + ~ {e t- sin;4t;p+t -} ~ dt - e 4 6 ;!;e + ;!;e + 4 f() f(0)=0 f(t) f '()=( +)e +:! ~ dt t f() e 0 e- e e - e f(t)=e t- sin;4t;p+t - f(t) F(t) 0 : + ~ {e t- sin;4t;p+t -} ~ dt - + = 0 : f(t)dt - + = 0 [F(t)] - = = 0 0 F(+)-F(-) F(+)-F()-F(-)+F() F(+)-F() F(-)-F() = =F'() =f() =(+ -) =6 5

23 5 :!/ tsin; ;tdt - ;!; ;!; ;4!; ;5!; 6 n pk p f{a+ } =:A a+p f() n k= n n =:)p f(a+) =p:) f(a+p) 6 >0 f() f'()= - (+)' f()=- :$ (t-)f(t)dt ' 4 4' 8 y=f() y y=f() O [0, ] f() g() n k k- k [g{ }-g{ }] n k= n n n :) g() :) g() :) f() :) f() :) { f()-g()}~ 7 0 <; ; f()=:)/ sec~t~dt / : ;4 ; f(t)dt ;4 ; -;4 ; ln~ ln' ln('+) ln('+) n k k- k [g{ }-g{ }] k= n n n =[g{ }-g(0)] +[g{ }-g{ }] +[g{ }-g{ }] n n n n n n n n n n- n +y+[g{ }-g{ }] n n n n- =g()-[g(0)+g{ }+g{ }+y+g{ }] n n n n n- k =- ~g{ } k=0 n n n n- k k- k k [g{ }-g{ }] = [- n n ~g{ } ] k= n n n k=0 n n =-:) g() =:) f() 6

24 8 n n [e +;n#; +e +;n^; +e +;n(; +y+e +;; n ;; ] e ;!;(e -e) ;!;(e +e) ;@;(e -e) ;@;(e +e) e -e 0 +y =(æ0, yæ0) n A(, 0) P, P, P, y, P n-, P n (0, ) OP k (k=,,, y, n) +y= Q k B(, 0) OQ k, BQ k n b a, b { } n n k= a O y Q«Q«P«P Q P«(0,`) P Q B O A(,`0) +y= p p ~ p 9 f() f'() f'(-)=-f'() :) f'()=-;!; f(0)=0 :) f()=-; 0; n k k f'{ -} n k= n n

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()