高等数学

Transcription

1 (CIP) /. :,00.8 ISBN Ⅰ. Ⅱ. Ⅲ. : Ⅳ.O3 CIP (00) 4775 : 80 :603 : : : E mail:dutp@dutp.cn URL:htp:// :85mm 60mm :9.5 :443 :~ : : : ISBN :34.00

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3 ,,. ( ),,.,,.,.,,,.,,.,, ( ), ;,,. 00.,,,,,,,,.,,,,,,.,,,,,,

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6

7 (Gompertz) (Role) (Lagrange) (Cauchy)

8 ( ) ( ) ( )

9 ( ) ( )

10 y =f(,y ) y =f(y,y )

11 .,,.,.,.,,.....,,,,,... y,d, D, y, y, y=f(),y,d( D f ),f f(). 0, f(0) y0, f() =0, y0=f(0) y = 0 =y0. D,, W W f, W = {y y=f(), D}. y, :y=g() y=f() y=φ ( ) y=y().,,,. :,,.,,. y=f() y=g(), D, D f()=g(), y=f() y=g().,y= ( R) y=sin +cos ( R), R, R

12 ; f()=+ g()= -, - Df=R, Dg={, R},, {, R},. :, y= u=t.. : (),., S r S=πr, {r r>0}. (),,,,,., : 0; ; 3 ; 4 arcsin arccos,.. ()y= ; ()y= 槡 9- ; (3)y=ln(-3); (4)y=arcsin(+); (5)y=arccos ( - 5 ) + 槡 5-. (), --6 0, (-3)(+) 0, 3 -, :{ 3 -}, D=(-,-) (-,3) (3,+ ). (), 9-0, (3-)(3+) 0, :{ -3 3}, D=[-3,3]. (3), -3>0, > 3. { }. 3 : >, D= 3, + (4), +, - +, - 0. :{ - 0}, D=[-,0]. 烄 - (5), 烅 5, { -4 6 烆 -5<< 5, -4 <5. ( 5- >0

13 3 ) :{ -4 <5}, D=[-4,5). () y=f() [a, b], y=f(+). () y=f(+) [a,b], y=f(). (), y=f(+), a + b, a- b-. y=f(+) :{ a- b-}, D= [a-,b-]. () y=f(+) [a,b], a b, a+ + b+. y=f() :{ a+ b+}, D=[a+,b+]. 3. y=f() f() y,f(),f(). 3 f()= -3, f(),f(a),f(+),f( ),f[f()-]. f()=() -3, f()= -3=-, f(a)=a -3, f(+)=(+) -3= +4-, f( )=( ) -3= -3= + -3, f[f()-]=[f()-] -3=[ -3-] -3= f(+)= --, f(). t=+, =t-, f(+)= --, 4. f(t)=(t-) -(t-)-=t -3t+, f()= -3+. ( ). () : Oy,,.,, G( ) t.,,,. () :,., 005,.

14 4 t y ,,. (3) ( ):,., y=a+b, y=a +b+c.,., :,,,. 5,, 0, 0.8, 0 30,.99, 30, y t 烄 0.8t, 0 t 0 y=烅.99, 0<t 30, 烆 (t-30), t> 30 [0,+ ), 3. 3 :(),. (). (3) :, 6;,, 7. 6 y= =, 0 {-, <0. ( -, + ), 4.

15 5 烄 sin 7 y=烅 0, (-,+ ), 烆 0 = (4), 6 7,, 8,. 8,, []. [ 4 ] =0,[ 槡 3]=,[-3.6]=-4,[]=., y=[],, (-,+ ), 6. (5),,. 烄 -, <0 9 f()= 烅 0, =0, 烆 +, >0 6 (-,+ ), =0. <0, - ; >0, =0, 0,,f(-)= -3,f()=5, f(0)= f()= +, 0 { e -, >0,( ) f(), f(-3),f(), f(0);(). () (-,+ ), =0. -3 (-,0], f(-3)=-3+=-; (0,+ ), f()=e -; 0 (-,0], f(0)=0+=; () 8.

16 6 7 8 y=y() F(,y)=0, y=y() F(,y)=0., e +y= y. : y y=f(),..,e +y= y= -e,,,e y +y=. 3 y=y() t =g (t) { y=h(t ), y=y() =g(t) { y=h(t )., y=槡 - ( [-,]) =cost { y=sin t ( 0 t π).... f() I (I f() D, D ), M, I, f() M, f() I., f() I. f() I : y=f() I y=-m y=m.( 9 ) :(),,.,y= (,+ ), M=

17 7 9, (,+ ),. y= ( 0,), 0,y=, M, (0,), M. () f() I, M.,y=sin (-,+ ), M =, (-,+ ), sin (, M).. f() I (I f() D, D ),, I, <, f()<f(), I,I f() ;, I, <, f()>f(), I,I f().,., I,,,,.( 0 ),,y= [0,+ ), (-,0] ; (-,+ ), y=., y= (-,+ ). 3. f() D, D f(-)=f(), f() ; D f(-)=-f(),

18 8 0 f().,., y,.( ),y=,y=cos ;y= 3,y=, y=sin ;y=ln ;y=0.. ()f()= sin ; ()f()=ln(+ 槡 + ); (3)f()=(+) 槡 - +. () D={ 0},, D, f()= sin. f(-)= sin (-) - =-sin - =sin =f (), () D=(-,+ ),, D, f(-)=ln(-+ 槡 +(-) ) =ln ( -+ 槡 + )(+ 槡 + +槡 + )

19 9 =ln +槡 + f()=ln(+ 槡 + ). =ln(+ 槡 + ) - =-ln(+ 槡 + )=-f(), (3) D=(-,],, f()=(+) 槡 f() D, T, D (±T) D, f(±t)=f(), f(),t f().,,, ( )., T f(), T,3T,,nT, f()., T(>0).,.,y=sin,y=cos π;y=tan,y=cot π...3 Q p : Q=50- p 5, p, Q, Q p., Q, p, : p=50-5q p Q.,,,,,,

20 0.. y=f() I, W ={y y=f(), I}, W y, y=f() y, y,, y=f(), =f - (y).,,y, =f - (y) y=f - ().,. y=.,,, y= (0,+ ) y= 槡,>0( 3 ); y=e (-,+ ) y=ln,>0 ( 4 ) ,,... y=c (C ) (-,+ ), {C}. (-,+ ),y C.. (0,C).( 5 ). y= α (α ) 5 α, α,,y=,y=,y= 3,y= 3,y= 3 (-,+ ),y= [ 0,+ ),y= - (-,0) (0, + ),y= - ( 0,+ ), α, α (0,+ ) ; (,). : α>0 y= α (0,+ ) ; α<0 y= α (0,+ ).( 6 )

21 6 3. y=a (a,a>0 ) (-,+ ), (0,+ ). (0,). a>,y=a (-,+ ). 0<a<,y=a (-,+ ) ( 7 ). :() y= a y= y. a =a -, y=a () e=.7888 y=e. (3) : y= α,, α; y=a,, a. 4. y=loga (a,a>0 a ) 7 (0,+ ), (-,+ ); y (,0); a>,y=loga (-,+ ) ; 0<a<,y=loga (-,+ ) ( 8 ). :() e=.7888 y=loge, y=ln.

22 () y=loga y=a, y=. 5. : y=sin, y= cos, y=tan, y=cot, y=sec, y=csc.,., sin π, sin90.sin. 8 : () y=sin = π 80 (-,+ ), [-,]; π ; ; ; - π +kπ, ( π +k ) π,k Z, π +kπ, ( 3π +k ) π,k Z ; y=- y=, sin ( 9 ). () y=cos 9 (-,+ ), [-,]; π ; ; ; ( 0+kπ,π+k π ),k Z, π+kπ,π+k ( π ),k Z ; y y=- y=, cos ( 0 ). (3) y=tan 0

23 { } 3 π +kπ,k Z, (-,+ ); π ; ; ; - π +kπ, π +k π,k Z ( ). (4) y=cot { kπ,k Z }, (-,+ ); π ; ; ; ( 0+kπ,π+k π ),k Z ( ). (4) y=sec= cos (5) y=csc= sin,. 6.,, sin π 6 =, sin 5π 6 =,, = π 6 +kπ,= 5π 6 +kπ (k Z).,.,,.,

24 4 [ ] [- π,π ],,, y= [ ] y=sin, - π,π,, y=sin arcsin, - π,π y=arcsin,.y= arcsin,,, y ( ). arcsin ( [- π,π ]), arcsin =π 6.,,. () y=arcsin [ ] [-,], - π,π ; ; ;. ( 3 ) () y=arccos [-,], [0,π]; ; ;.( 4 ) (3) y=arctan 3 4 (-,+ ), - π,π ; ; ;. ( 5 ) (4) y=arccot (-,+ ), (0,π); ; ;.( 6 )

25 , E v, E= mv, v t, v=gt, E t, E= mg t. E v t, t,v, E..3 y u y=f(u), D f, u u=φ ( ), D g, W g, W g D f, D g,y u y=f [ φ ( ) ], y=f(u) u=φ ( ),,y=f(u),u=φ (,u.,,,., ()y=arcsinu u=sin y=arcsin(sin)( R), y= arcsinu D f=[-,],u=sin D g=r, W g=[-,], W g D f ; ()y=arcsinu u=+, y=arcsinu D f=[-,], u=+ D g=r, W g=[,+ ), W g D f ; (3)y=arcsinu u=-, y=arcsinu D f=[-,], u=- D g=r, W g=(-,], W g D f,, u= - D g [- 槡 3,-] [, 槡 3], y=arcsin(- ), [- 槡 3,-] [, 槡 3]., y=e sin 槡 y= e u,u=sinv,v= 槡.,,..,

26 6.. ()y=sin ; ()y=sin ; (3)y=arctan 槡 - ; (4)y=ln(tane +3sin ). () :, y=sinu,, u=. y=sin :y=sinu,u=. () :, y=u,, u=sinv, v=. y=sin : y=u,u=sinv,v=. (3) :, y=arctanu,, u= 槡 v,, v=-. y=arctan 槡 - :y=arctanu,u= 槡 v,v=-. (4) :, y=lnu,, u=tanv,, v=e w, w= +3sin. y=ln(tane +3sin ) :y=lnu,u=tanv,v=e w,w= +3sin. :,, ( )..,.,., P()=an n +an- n- + +a+a0, P () Q() ( P (), Q() ),y=cos 槡 + +cos -cos. y= ,f()= +, 0,, { e -, >0.,., y= =, 0 {-, <0,, y= = 槡. f(),g(),f()>0, y=[f()] g (),,

27 7,,. y=[f()] g ()=e g ()lnf(). N=a log a N lnn m =mlnn..,..,,.,.... n yn=f(n), n,,3,, y,y,,yn, f(),f(),,f(n),., f(),f(),,f(n), {yn} {f(n)}., n yn. (),, 3,, n,, yn= n( n=,, ), { n }. (),,4 3,3 4,, + ( -) n- n,, yn=+ ( -) n- n ( n=,, ), + ( -) n-. n { } { } (3), 3,4 3,5 4,,n+ n,, yn= n+ n ( n=,, ), n+ n. (4),4,6,8,,n,, yn=n(n=,, ), { n }. (5) 0,,0,,0,,, yn =+ (- ) n (n =,, ), + { (- ) n }. (6) 0,,0,, 0, 3,, yn = (- ) n + (n=,, ), n (- ) n +. n { },,,. n, yn<yn+, y<y< <yn<yn+<, {yn}, n, yn>yn+, y>y> >yn>yn+>,

28 8 {yn}..,, (4), () (3), (),(5),(6). M, yn <M n, {yn} ;, {yn}. {yn},, n ( n, n ), {yn}..., n,. (),yn n, n,yn 0; (),yn n, n,yn ; (3),yn n, n,yn ; (4),yn n, n,yn ; (5),yn n 0, n, ; (6),yn n, n, 0., (4) (5), n,yn ; (),(),(3),(6), n,yn,...5 {yn}, n yn A, n {yn} A, {yn} A, {yn} A, lim n yn=a yn A(n ).,, lim. n yn, ()lim =0; n n ()lim + ( -) n- n n = ; n+ (3)lim n n = ; (4)lim n, { n} ; n (5)lim + (- ) n, + { (- ) n } ; n (- ) (6)lim n + =0. n n,,,.,., : () n, n () yn A, yn A

29 9 (),,,yn-a yn A, yn-a,yn A., ε(ε ). ε,yn-a <ε yn A. ε,ε,yn A.. (), yn= { n+ n }.,yn= { n+ n },,yn- <ε yn ε=0.5, yn- = n <0.5, n>. N=, n>n=, yn- <ε, y3,y4,y5, yn- <ε; ε =0.05, yn- = n <0.05, n>0. N =0, n> N=0,yn- <ε, 0 y,y,y3, yn- <ε; ε3=0.005, yn- = n <0.005, n>00. N3=00. n> N3=00,yn- <ε3, 00 y0,y0,y03, yn- <ε3;, ε, yn- = n <ε, n> ε. ε N, n>n,yn- <ε, N yn+,yn+,yn+3, yn- <ε. ( ε-n ) :.5, ε( ), N, n>n yn-a <ε, A yn, yn A, lim n yn=a yn A(n ). :() ε,, yn-a <ε yn A. ()N, ε, n.n>n yn-a <ε, yn-a <ε N., ε N.,lim yn=a : { yn}, N+ n A ε, {yn} y,y,,yn ( 7 ). ε-n yn= n+ (-) n-. n

30 0 7 yn- = n+ (-) n- - = n n, ε, yn- <ε, n <ε n> ε., N, n>n, ε yn= n+ (-) n-. n.. yn- <ε, n yn=f(n), yn=f(n) yn=f(n), n, n=,,3,. y=f(),,,, :() ;().. : >0, + ( ); <0 ( ), - ( );3 0 0, ( ).,, f()..6, f() A, A f(). lim f ()=A f() A( )., ( ε X ):.6 ε( ), X, >X, f()-a <ε, A f(), lim f ()=A f() A( ).,lim f ()=A : y=a-ε y=a+ε,

31 X, < -X >X, y=f() ( 8 ). 8, ε X lim f()=a, lim n yn=a, ε, X. 3 lim =0. -0 = = ε, -0 <ε, <ε, >. X= ε ε, >X -0 <ε, lim =0. f(), + f() - f()..7 >0, f() A, A f() +..7 lim f ()=A f() A( + ). + ε( ), X, >X, f()-a <ε, A f() +, lim f ()=A f() A( + ). +.8 <0, f() A, A f() -..8 lim f ()=A f() A( - ). - ε( ), X, <-X, f()-a <ε, A f() -, lim f ()=A f() A( - ). -, +, - f(), :. f() lim f () lim f () - +, lim f ()=A lim f ()=A lim f ()=A. - +, lim f () lim f (),, - +

32 , lim f()., y=arctan, lim arctan. lim - y=e, lim e. arctan=- π, lim arctan= π, lim - y=,. + e =0, lim e, + lim - =0, lim + =0, lim =0. : 0 0, 0 - ( 0 ); 0 0, 0 + ( 0 );3 0 0, 0( 0 ). () 0, f()=+ g()= - - ( 9 ),, =, =.,, f()=+, g()= - -,. 9, :.9 f() =0 ( 0 ), 0, f() A, A f() 0,

33 3 lim 0 f()=a f() A( 0)., :lim (+)=,lim - - =.,, ( ε-δ ).. 0, δ, 0 δ (0-δ,0+δ) 0 δ, 0,δ, U(0,δ), U(0,δ)={ 0-δ<<0+δ}={ -0 30, U (0,δ), 0 δ, U 0 (0,δ), U 0 (0,δ)=(0-δ,0) (0,0+δ) ={ 0< -0 <δ}. <δ} f() 0 ( 0 ), ε( ), δ, 0< -0 <δ, f()-a <ε, A f() 0, lim 0 f()=a f() A( 0). : ε f() A,δ 0 ;ε,δ ε. 0< -0 0, 0 f() f() 0, f() 0 f().,lim 0 f()=a : y=a-ε y=a+ε, δ, (0-δ,0) (0,0+δ), y=f() ( 3 ). ε-δ, lim 0 f()=a 3 : ε, δ, f()-a <ε δ. 4 lim (+)=. (+)- = -. ε, (+)- <ε, - <ε. δ=ε, 0< - <δ,. (+)- <ε lim (+)=.

34 4 5 lim (3-)=4. (3-)-4 = 3-6 =3 -. ε, (3-)-4 <ε, - < ε 3. δ= ε 3, 0< - <δ,. (3-)-4 <ε lim (3-)=4. () ,, 0 0, 0 -, 0 0, + 0, f() ;, ,..0 f() =0 ( 0 ), 0 -, f() A, A f() 0, lim f()=a f() A( 0 - ) f() 0 ( 0 ). ε( ), δ, -δ<-0 <0, f()-a <ε, A f() 0,. lim f()=a f() A( 0 - ). - 0 f() =0 ( 0 ), 0 +, f() A, A f() 0, lim f()=a f() A( 0 + ) f() 0 ( 0 ). ε( ), δ, 0<-0 <δ, f()-a <ε, A f() 0, lim f()=a f() A( 0 + ). + 0 lim f() lim f() ,, :., f() 0 lim f()=a lim f()=a lim f()=a. 0-0,, lim f()=a

35 5 6 f()= -, <0 { +, 0 0. f(), =0, f() ( 3 ): lim -f ()=lim ( - ) =,lim +f ()=lim ( + ) =, lim -f ()=lim +f ()=, 0 0 0, f() lim 0 f()=. 7 f()= 0. f() : f()= =, >0 {-, <0. f() =0, f() ( 33 ): lim -f ()=lim (-)=-,lim +f ()=lim =, lim -f () lim +f (), 0 0 0, f()., : 3 33 () (, 0 ).,, ; (),., ( ),.,. (3) lim,,,...3 0,, +, -, 0 +, 0 -. ( ) lim 0 f(),. ( ) lim 0 f()=a, δ>0, f() 0

36 6 (0-δ,0) (0,0+δ). -0 lim 0 f()=a, ε=, δ>0, 0< <δ, f()-a <, f()-a < f()-a + A <+ A, M=+ A, f() <M, f() 0 (0-δ,0) (0,0+δ)., f()=, lim f()=, f()= = ( (0,) (,)), f()= (-,+ ). 3( ) lim 0 f()=a A>0( A<0), δ>0, 0< -0 <δ, f()>0( f()<0). A>0, lim 0 f()=a, ε= A, δ>0, 0< -0 <δ, f()-a < A, 0<A <f ()< 3A, f()>0. A<0., lim 0 f()=a, 0., f()=, lim f()= >0, f()= = ( (0,) (,), { 0< - <}) f()>0. lim 0 f()=a, δ>0, 0< -0 ( f() 0), A 0( A 0).. <δ f() 0, f()=, = ( (0,) (,), { 0< - <}) f()>0, lim f ()=>0. : 0< -0 <δ, f()>0( f()<0), f() A> 0( A<0)., f()=, =0 ( (-,0) (0,), { 0< - <}) f()>0, lim 0 f ()= , 0,. α, β,γ., -4, +4 ; 0,,sin ;

37 7,, ; n, n. :(),. 0-0,0-00,.0. (),,.. lim 0 f()=a lim [f()-a]=0, α()=f()-a, 0..3 lim 0 f()=a :f() A α lim 0 f()=a f()=a+α, α 0. : :.. lim sin. lim =0,, ; sin, sin, lim sin=0 ( 34 ). lim sin lim =0, 0, ; sin 0, sin

38 8, lim 0 sin =0 ( 35 ). 35, 0,,, , 0,, 0 +,,, 槡,, +, 槡 :, 0 ; 槡 0 ; 0 ; + 0. lim 0 + =0 槡,lim 0 + =,lim 0 + =,lim =.,,,. 0. 0,..3 α β, α 0, lim β α =0, β α, β =o(α); lim β α =, β α ; lim β α =C (C 0 ), β α ;, C=, β α, β α, α~β., 0 +,, =o(); 槡

39 9 ; ; +. :, α β, lim β α. lim β α 0 ( ), 0 0, ,,.,,,, ; 0,, 0, ; π, tan, π, tan ; 0 +, ln, 0 +,ln ; :().. 0 0,0 00,. ; (),,. (3),,. lim =,lim =,lim tan=. 0 (4),, ;,. lim ln=+,lim ln= (5),., n, yn=[+(-) n ]n,. π.3.3, :.4, f(), f() ;, f(), f() 0 f()., ( )., 0, ; ;

40 30 π, tan ; tan f(),g(), limf()=a,limg()=b, ()lim[f()±g()]=limf()±limg()=a±b; ()lim[f() g()]=limf() limg()=a B; lim[kf()]=klimf()=ka(k ); lim[f()] n =[limf()] n =A n (n ); (3)lim f () g() =limf () limg() =A B ( limg ()=B 0)., ( )., () (). :, lim,, 0,,. 0 (),. limf()=a,limg()=b, f()=a+α, g()=b+β, 0 α β. f() g()=(a+α)(b+β ) =AB+(Aβ+Bα+αβ ), 0,Aβ, Bα,αβ, A β+bα+αβ,, lim(f() g())=a B=limf() limg(). lim ( 3 +-). =(lim ) 3 +lim -lim = 3 +-=7. 3 lim lim. lim ( -)=(lim ) -lim = -=3 0, (3 ++)=3(lim ) +lim +lim =3 ++=5. lim ( 3 ++ ) = lim ( - ) =5 3 =5.

41 3, P()=a0 n +a n- + +an, lim P() 0, lim P()=P(0); 0 P () Q(), lim P() 0 Q(), 0 Q(0) 0, P(), lim 0 Q() =P (0) Q(0)., : f() D, 0 D, lim 0 f(), lim 0 f()=f(0) lim +-3. lim ( +-3)= + -3=0,, lim (4-)=4 -=3 0,,, lim 4- = 4 - =0 3 =0, 4- lim +-3 =. f() :() lim 0 g(), lim 0 g()=0,. 4- () :lim +-3 =3 0 =. (3) f() lim, lim g()=0, 0 g() 0 lim f() 0,,. 0 + lim - =,lim 槡 - = 5,lim 0 sin = 3,lim =. - e 4 lim lim ( +-3)= + -3=0, lim ( -)= -=0,,, 0 0,,.,

42 3 =, -,, -( ),.. (+)(-) =lim (+3)(-) =lim + +3 =. 5 槡 - lim , 0 槡 - =lim 4 ( 槡 +)( 槡 -) =lim 4槡 + = 4. 6 lim 槡 ,.,,,. =lim (+ 槡 +) 0 (+ 槡 -)(+ 槡 +) (+ 槡 +) =lim =lim (+ 0 0 槡 +)=. 7 槡 lim ,,,. (3+ 槡 4-)(3+ 槡 4+) =lim - (+)(3+ 槡 4+) 3+3 =lim -(+)(3+ 槡 4+) =lim 3 -槡 =3. 8 槡 -- lim 槡 ,,,. (- 槡 -)(- 槡 +)(+ 槡 +) =lim (+ 槡 -)(+ 槡 +)(- 槡 +) (-)(+ 槡 +) =lim (-)(- 槡 +) =lim (+ 槡 +) (- 槡 +) =

43 33 f(), lim 0 g() 0 0, f() g(),,,. 9 lim , ( ),,,,,., ( ), =lim - 3 = =. 9,,, 0 lim =lim = =0. 3 : lim 3 +- =.,, : 烄 0, m<n a0 m +a m- + +am a0 lim = b0 n +b n- 烅, m=n. + +bn b0 烆, m>n a0 0,b0 0;m n... ()lim ; 5 ()lim ; (-5)(3 (3)lim +) ( -)(+6) 3; n (4)lim - n n -3n+ ; (),

44 lim -- =. (), lim =0. (3),,,., 5, 9; 5,, (-5)(3 +) lim = 9 ( -)(+6) 3. (4),,, 槡 4 lim n - lim n n -3n+ =.,,, 9,,. 槡 4+ =lim + + =. :,,,. lim 槡 , 0. 3 lim n+ - n 3 n +4.. lim n qn =0( q <), 3 n+,. =lim n n+ ( 3 ) - ( ) ( 3 ) n+ n+ =0. : 9 3,,. 4 lim n n n n n. n,,,

45 35 n,. ++ +n n (n+) =lim =lim = n n n n. :,. 5 lim lim - = 3,lim =, ,. ( ). +- =lim (-)(++ ) =lim (-)(+) (-)(++ ) =lim -(+) ++ =-. 6 lim =lim (-)(+) =. 7 lim (+ 槡 - 槡 -3). + lim 槡 +=,lim 槡 -3=, ,.,,,,,,. (+ 槡 - 槡 -3)(+ 槡 + 槡 -3) =lim + (+ 槡 + 槡 -3) 5 5 =lim + 槡 ++ 槡 -3 =lim 槡 + + 槡 =0. + 槡 ,, - 0 : () 0 0,., ;,. (),.,, :

46 36 a0 m +a m- + +am lim = b0 n +b n- 烅 + +bn a0 0,b0 0,m n. 烄 0, m<n a0 b0, m=n. 烆, m>n (3) 0 0., - 0 ( 0 0, ), lim 0 φ ( )=a,lim f(u)=a, 0 u a φ () a, f[ φ ()] 0, lim 0 f[ φ ()]=lim u a f(u)=a.. a 0,.,,,., lim u a f(u)=f(a), lim 0 f[ φ ()]=f(a)=f(lim 0 φ ( )) f[ φ ()],. 8 - lim 槡 -4. y= 槡 - -4, : y= 槡 u,u= u= , - lim -4 =lim + = 4. y= 槡 u u 4, 槡 4 =.,. - = lim 槡 -4 = lim 槡 + = 槡 4 =. 9 lim + 5. ( ) y= + 5, : y=u 5,u=+. u= +, lim + =. y=u 5 u

47 37, 5 =.,. 5 =lim + 0 lim e. = lim (+ )5 = 5 =. y=e, :y=e u,u=. u= 0, lim =0. y=e u u 0, e 0 =.,. =e lim ( ) =e 0 =. lim 0 e. y=e, :y=e u,u=. u= 0, lim 0 =. y=e u u ( lim e u = u + +,lim e u =0, lim e u ). u - u 0 -, -,e 0, lim e =0, 0-0 +, +,e +, lim e =+, lim e lim 0 sin. y=sin, : y=sinu,u=. u= 0, lim 0 =. y=sinu u ( 36 ). 36 0,,sin.

48 , : sin ()lim 0 = ; ()lim + =e. Ⅰ( ) f(),g(),h() 0 (0-δ,0) (0,0+δ) : g() f() h() lim 0 g()=lim 0 h()=a, lim 0 f()=a.,. lim 0 sin=0. < π, 0 sin. lim =0, Ⅰ lim sin= Ⅱ( ). {n} n n+ {n} ; {n} n n+ {n}..,,, Ⅱ :,. yn=- n : 0,, 3,3 4,.,yn, yn<., Ⅱ,lim n yn. lim - n n =..5.. sin Ⅰ.lim 0 =

49 39 sin (-) - =-sin - =sin,,sin,. O, 37. AOB=(0 < < π ), OB A D, AOB < AOB < AOD, AOB = OA BC= sin AOB = 37 AOD = AO AD= tan sin<<tan, sin< < tan. sin < sin < cos, cos< sin <. sin lim cos=lim =, Ⅰ lim =. : () 0 0 ; () sin. 3 tan lim 0. tan lim 0 =lim 0 4 sink lim 0 ( k 0). lim 0 sink =lim 0 sin ( cos ) =lim 0 cos lim sin 0 =. sink k k. sint t=k, =k lim t 0 t =k =k.

50 40 5 sina lim 0 sinb ( a 0,b 0)., 4. 6 lim 0 -cos. -cos lim =lim 0 sina sina lim sina lim 0 sinb =lim 0 = = a 0sinb sinb b. lim 0 sin 0 熿 = sin 燄 lim 0 燀 燅 7 lim sin( -4) -. = lim 0 熿 sin 燄 燀 燅 =. sin( lim -4) - =lim (+)sin( -4) (+)(-) =4. 8 lim 0 arcsin. t=arcsin, =sint, 0,t 0. lim 0 arcsin =lim sint t 0 t =. sin 9 lim 0 sin. sin lim 0 sin =lim sin 0 sin lim 0 =0.. n Ⅱ.lim + n n =e(e,.78), 3. 3 n ( + n n ) , n,+ n, n. n,+ n, e n

51 n n ( n ). lim + n n n= + =e. 4,, lim + =e : (), + ( ) ; (),,. 0 lim ( ). 0 + t=, 0, t, lim - t lim ( + ) =lim + 0 t t. t=-,,t 0. lim - ( ) =lim t 0 + ( ). ( ) =lim + ( ) ( + ). lim lim + 3 lim [ ] =e. ( + t) - t = [ lim ( + ) t ] - =e -. =e. 烄 烌 lim =lim + + =lim + 烆 烎 4 lim lim 5 lim lim t 0 - t =lim ( ) =lim - 3 [ ( ) ] - ( + ). - - ( + ) lim - (- )(-) =lim ( + = ) lim + ( ).5.3 ( + ) - =e -. [ ] 6 lim =e-6 =e. =e- =e -4. e,,

52 4. 0 :sin~;tan~;arcsin~;arctan~;ln(+)~;e -~ ;-cos~ ;n 槡 + -~ n. :,. : 0,sin ~,.,,sin( -)~ -;,sin ~. 6 lim 0 tan7 tan9. tan7 lim 0 tan9 =lim 7 09 = lim 0 -cos tan5. -cos lim 0 tan5 =lim 0 5 = 5. 8 lim + arctan. lim arctan=0 π + = , :,..5 u u u, u-u ( ), Δu, Δu=u-u.,,. y=f() 0, 0 Δ, y Δy=f(0+Δ)-f(0),,,...6 y=f() 0, Δ, Δy

53 43 f() 0. lim Δ 0 Δy=0, y=f() 0 Δ, Δy ( 38 ). y=3-0. Δy =f(0+δ)-f(0) =[3(0+Δ) -]-(3 0-) =60Δ+3(Δ) lim Δy=lim (60Δ+3(Δ) )=0 Δ 0 Δ 0 y= , =0+Δ, Δ 0, 0, lim Δy=0 Δ 0.7 lim [ f()-f(0 )] =0, 0 lim 0 f()=f(0). y=f() 0, 0, f(), f() 0 f(0), f() 0. lim 0 f()=f(0) lim f()=f(0), f() 0 ; + 0 lim f()=f(0), f() y=f() 0, f() 0 f(0), 0 f().. lim sin. 0 lim sin=sin0= lim +5. f()= +5 =,..8 (a,b). lim +5 = +5 =4 9. y=f() (a,b), f()

54 44 y=f() (a,b), lim +f ()=f(a),lim -f ()=f(b), a b f() [a,b].,, f() g() 0, f()+g(), f()-g(), f() g(), f () g() ( g() 0) u=φ ( ) 0,y=f(u) u0, u0=φ ( 0), y=f [ φ ( ) ],.,. 4 lim 槡 槡, [- 槡 5, 槡 5], [- 槡 5, 槡 5], lim 槡 5 lim 4 e +cos(4-) 槡 =槡 5- =. +cos(4-) e 槡 -3, [ 0,9) (9,+ ), 4 [0,9), e +cos(4-) 4 +cos(4-4) lim 4 槡 =e 槡 4-3 =- (e 4 +). 6 f()=+ g()= - -. g() =,,f()=g(). :,. lim f ()=lim (+)=,lim g ()=lim - - =, :f()=, lim f ()=f()=, g(), lim g () g() 39 40,f() =, g() =.

55 烄 -, <0 7 f()= 烅 0, =0, f() =0. 烆 +, >0 f() =0, lim +f ()=lim (+)=, lim -f ()=lim (-)= , =0, f() =0 ( 4 ). 8 f()= +, {, =, f() =. f() =, f()=, lim f()=lim (+)=, lim f()= f(), f() = ( 4 ). 9 f()= ln, { -, <, f() =. f() =, lim +f ()=lim ln=0,lim -f ()=lim (-)=0, + - lim f ()=0 f()=ln=0, lim f ()=f()=0, f() = ( 43 )., f(), () ;() ;(3) g() =. 7 f(). 8 f().,. 9 f() =,,, f() =..9 y=f() 0, 0 f()., f() 0, 0

56 46 f() : () 0,f() ; ()lim 0 f() ; (3) lim 0 f(), lim 0 f() f(0) , 0 f() :lim f() lim f() lim f(), 0 f()., 0, f() 0. :lim f() lim f() lim f() lim f(), 0 f() f()= =0. f()= =0, f()= =0. lim 0 =, =0 f() ( 44 ). f()= sin =0. sin f() =0, lim 0 =. 烄 sin f()= 烅 0, 烆 =0 lim f sin ()=lim 0 0 =, f(0)=, lim f ()=f(0)=, f() =0 0 ( 45 ) f()= +, {, =, f() =, f()=,

57 47 f() =. +, f()= <0 { +b, 0, =0, b. lim -f ()=lim ( +)=,lim +f ()=lim (+b)=b f() =0, lim 0 f(), lim -f ()=lim +f (), b= ( ) y=f() [a,b], f()..0( ) f() [a,b],., 46,f() [a,b]. m, M..( ) f() [a, b],m M f() [a,b]. m M c( m<c< M), ξ (a,b), f( ξ )=c. 46 ( ) f() [a,b], f(a) f(b)<0, ξ (a,b), f( ξ )= =0 (,). f()= 5-5-, [,], f()= 5-5 -=-5<0,f()= 5-5 -=>0, f() f()=(-5) (-)<0,, (,) ξ, f( ξ )=0(<ξ<) =0 (,). 4 3 =. f()= 3 -, [0,], f(0)=-<0,f()=>0, f(0) f()<0,, (0,) ξ, f( ξ )=0. 3 =.

58 ( ),,.,., P, Q P,, Q=Q(P).,..,,, ;,,.,., P = P(Q) P Q,. : ()Q= a-p b ( a>0,b>0),,. P=0, b Q= a b, ; P=a, Q=0, a,. ()Q= a P+c -b (a>0,b>0,c>0) (3)Q= a-p b (a>0,b>0) (4)Q= a- 槡 P b (a>0,b>0) (6)Q=ae -bp (a>0,b>0) (5)Q= 槡 a-p b (a>0,b>0),..7.,,.,.,.

59 49 K=K(P).,,, ;,,.,. : ()K=-d+cP (c>0,d>0) K 0 P d c, d c,. ()K= ap-b cp+d ( a>0,b>0,c>0,d>0)..7.3, C, C=C(). : C0 C(). C0,,, ; C(),, C()= C0+ C() =0,, C(0)=C0. C, C ()= C () A ( ) :C()=3000+0, ( ). =0, C(0)=3000( ), ; =00, C(00)=5000( ), C (00)=50( ); =000, C(000)=3000( ), C (000)=3( );,., C()=3000+0,, 0,,..,,..7.4, R P R()=P R()., P,, P(),

60 50 R()=P(). A 30, R()= 30, R(0)=0;R(00)=3000;R(000)= A 烄 30, P()= 烅 7, 5000< 0000 烆 4, >0000 : 烄 30, R()= 烅 (-5000), 5000< 0000 烆 (-0000), >0000 烄 30, =烅 , 5000< 0000 烆 , >0000 R(0)=0( ); R(00)=3000( ); R(000)=30000( ); R(7000)=04000( ); R(000)=333000( )..7.5, L() R() C(), L()=R()-C(). 4, : L(0)=R(0)-C(0)=-3000( ); L(00)=R(00)-C(00)=-000( ); L(000)=R(000)-C(000)=7000( ).,.,. L()=0 R()=C() 30= =300.,, A (, ) 300,,,.,,,..7.6 Gompertz (Gompertz) : y=ka bt lg(a)<0,0 <b <, 44,,

61 5.,, ;, ; k,. 47. () : y,d, D, y, y, y=f().,y,d( D f ),f f(). 0, f() y0, f() =0, y0=f(0) y = 0 =y0. D,, W W f, W ={y y=f(), D}. () :. (3) : ( ). (4) :,. (5) : y=y() F(,y)=0, y= y() F(,y)=0. (6) : y=y() t =g (t) { y=h(t ), y= y() =g (t) { y=h(t ).. :,,, () : f() I (I f() D, D ), M, I, f() M, f() I., f() I. () : f() I (I f() D, D ),, I, <, f( )<f( ), I

62 5,I f() ;, I, <, f()>f(), I,I f().,. (3) : f() D, D f(-)=f(), f() ; D f(-)= -f(), f().,. (4) : f() D, T, D (±T) D, f(±t)=f(), f(),t f(). 3. y=f() I, W ={y y=f(), I}, W y, y=f() y, y,, y=f(), =f - (y).,,y, =f - (y) y=f - (). ; y=. 4. y u y=f(u), D f, u u=g(), D g, W g, W g D f, D g,y u y=f[g()], y=f(u) u=φ ( ),,y=f(u),u=φ (,u.,., ;, ( ). 5. () (,, ) :y=c :y= α :y=a :y=loga (C ) (α ) (a>0,a,a ) (a>0,a,a ) :y=sin,y=cos,y=tan,y=cot,y=sec,y=csc :y=arcsin,y=arccos,y=arctan,y=arccot () :,.. ()

63 53 :, A. ( 4): 4 lim f()=a lim n n=a ε>0 N>0, n>n n-a <ε lim f ()=A ε>0 X>0, >X f()-a <ε lim + f ()=A ε>0 X>0, >X f()-a <ε lim - f ()=A ε>0 X>0, <-X f()-a <ε lim 0 f()=a lim f()=a + 0 lim f()=a - 0 ε>0 δ>0, 0< -0 <δ f()-a <ε ε>0 δ>0, 0<-0<δ f()-a <ε ε>0 δ>0, δ<0-<0 f()-a <ε () :, 0,. α, β,γ. (3) :,,. (4) : α β, α 0, lim β α =0, β α, β =o(α); lim β α =, β α ; lim β α =C (C 0 ), β α ;, C=, β α, β α, α~β.. () ( ) lim 0 f(),. ( ) lim f()=a, δ>0, f() 0 0 (0-δ,0) (0,0+δ). 3( ) lim 0 f()=a, A>0( A<0), δ>0, 0< -0 <δ, f()>0( f()<0). lim f()=a δ>0, 0< -0 <δ f() 0( 0 f() 0), A 0( A 0). ()

64 54 f(),g(), limf()=a,limg()=b, Ⅰ.lim[f()±g()]=limf()±limg()=A±B; Ⅱ.lim[f() g()]=limf() limg()=a B; lim[kf()]=klimf()=ka(k ); lim[f()] n =[limf()] n =A n (n ); Ⅲ.lim f () g() =limf () limg() =A B ( limg ()=B 0)., ( )., Ⅰ Ⅱ. lim 0 φ ( )=a,lim u a f (u)=a, 0 φ () a, f[ φ ()] 0, lim 0 f[ φ ()]=lim u a f(u)=a.,,,., lim u a f(u)=f(a), lim 0 f[ φ ()]=f(a)=f(lim 0 φ ( )) f[ φ ()],. 3.. :lim 0 f()=a f()=a+α, α 0. :,. 3. () : lim f()=f(0), 0 f().( 0 ). (). (3) ;( ) lim f ()=A lim f ()=A lim f ()=A. - + lim f()=a lim f()=a lim f()=a (4). (5),,. (6) 0 0,, - 0 :

65 55 0 0,., ;,.,.,, : a0 m +a m- + +am lim = b0 n +b n- 烅 + +bn a0 0,b0 0,m n. 烄 0, m<n a0 b0, m=n. 烆, m>n , - 0 ( 0 0, ),. (7) sin :lim 0 = :a. 0 0 ; b. sin. sin lim 0 = lim sin =0, : sin lim 0 = sin,lim =lim sin=0, lim 0 sin =lim 0 :lim + sin =0,lim sin sin =lim =. =e. :a., + ;b.,,. ). (8) (, : 0 :sin~;tan~;arcsin~;arctan~;ln(+)~;

66 56 e -~;-cos~ ;n 槡 + -~ n.. () :lim 0 f()=f(0) lim Δ 0 Δy=0 ( ). () : y=f() (a,b), f() (a,b) ; y=f() (a,b), lim a +f ()=f(a),lim b -f () =f(b), f() [a,b]... f() 0, 0 f() : 0,f() ; lim 0 f() ; 3 lim 0 f(), lim 0 f() f(0). :lim f() lim f() (, ) :lim f() lim f() ( ) : 4. : y=f() [a,b], f() : f() [a,b],. : f() [a,b],m M f() [a,b]. m M c( m<c<m), ξ (a,b), f( ξ )=c. : f() [a,b], f(a) f(b)<0, ξ (a,b), f( ξ )=0.. Q=Q(P).. K=K(P). 3. C()=C0+C(). 4. R()=P.

67 57 5. L()=R()-C().. : ()y= +-8 ; A ()y= 槡 -6; (3)y= ; (4)y= 槡 +3+ 槡 + - ; (5)y=ln(3+) ; (6)y=arcsin - 3 ; 槡 (7)y= 9- ln(-) ; ( 8)y= / 槡 + -ln ; (9)y= (-4)ln - ; ( 0)y= ln 槡 - +. 烄槡. f()= -, 烅, : 烆 -, << ()f() ;()f(0),f( ) f(),f(, 3 ). 烄 sin 3. f()=, 0 烅, :()f() ;()f(-π),f(0),f(), 烆 0, =0 f( π ). 4. f()=e,f[ φ ()]=-, φ () 0, φ (). 5. ()y=ln y= ln ; ()y= y= 槡 ; (3)y= y=sin +cos ; 6. : ()y= ; (4)y= y=eln. ()y=cos+ sin ; (3)y=sin+cos; (4)y=ln + - ; (5)y= ( e -e - ); (6)y= e- - e : ()y= 槡 3 + ; ()y=sin 3 ;

68 58 槡 (3)y=e ; (4)y=arcsin(ln); (5)f()=arctan ; (6)y= sin ; (7)y=e sin ; (8)y=lnsin 槡 ; (9)y=cose - ; (0)y=e tan ; ()y= 槡 ln(+ 槡 ); ()y=tan 5 槡 ln(arcsin ). 8.,,,. ()n=a n ( a <); ()n= n- n+ ; (3)n=-(-) n ; (4)n=(-) n- n. 9.,,,,. ()lim e - ; (3)lim sin; 0 - (5)lim - ; ()lim arctan; (4)lim sin; ( 6)lim tan. 0. : n+ ()lim =0; ()lim n n n 3n+ = 3 ; + (3)lim = ; (4)lim (3-)=.,.() f()= <0 { +, 0, lim -f (),lim +f (), lim f () 烄 cos, 0 () f()= +, 0< 烅, lim 3 f (),lim f ()., 0 > 烆.,, π ()ln,( 0 + ); ()ln,( + ); (3) /,( 0 + ); (4) /,( 0 - ); (5) + -4,( ); (6)sin-,( π ); (7) n,( n ); (8) + (-) n,(n ). n 3. : ()lim ( 3-4+ ); ()lim ;

69 59 (3)lim ; ( - 4)lim 槡 + ; 槡 (5)lim 槡 8; (6)lim 5 槡 +4+3 (7)lim + 7 ; ( 8)lim cos. 4. : ()lim cos ; ; ()lim ( -3)sin ; 槡 3 - 槡 3 arctane (3)lim ; (4)lim ; + 槡 + (5)lim -- ; 5. : ()lim - -- ; ( e 6)lim 0. ( -4+4 )lim 3-8 ; (5)lim n (n -)(n -) 3n 槡 (7)lim ; ; (6)lim + 槡 4+ ; ( (-5) n + n 8)lim (-5) n+ + n+ n ( 3 ) ; (9)lim -4 - ; (0)lim ()lim ; ()lim (3)lim ; (4)lim +6+5 槡 ; ( +h ) (5)lim - ; (6)lim ; h 0 h 0 - 槡 - - 槡 + (7)lim ; 槡 5-4- 槡 (8)lim ; 槡 + -3 (9)lim ; 槡 + -槡 - (0)lim ; 4 槡 -- 槡 0 - 槡 - - ()lim +3- ; ( 3-3 )lim 4 +- ; (3)lim ; ( (3+) 0 (-) 0 4)lim ; (+3) 30 (3)lim ( 槡 +- 槡 -); (4)lim (n- 槡 - 槡 n+ ). + n 6. : ()lim sin π ; ( sin )lim ππ- ;

70 60 sin5 (3)lim 0 sin7 ; ( 4)lim ; 0 + 槡 -cos (5)lim 0 -cos sin ; ( 6)lim 0 arctan. 7. : ()lim + lim (-) / ; 0 (3)lim + +5 ; ( 4)lim ; (5)lim - n ; ( 6)lim - 3+ ; n 5 (7)lim (9)lim ()lim ; ( ) ( n+ ) 5 + ( ) ; -4 ( + ) ( - ) + 8. : arcsin ()lim 0 sin3 ; ln(+) (3)lim 0 arctan4 ; -cos3 (5)lim 0 -cos4 ; ( 8)lim ; ( 0)lim 3 n 槡 ; ( 3- ) 5-3 ( + ) ; n+3 ( n+ ) n+ ; ( )lim (+cos) 3sec. π lim 0 tan3 e - ; ( e 3-4)lim 0 sin 3 () ; ( 6)lim (e / -). 9. : ()lim + n ( n ) ; ( )lim n n n n n n (3)lim n ( n(n+ ) ; ( sin(-) 4)lim +3-0 ; sin(-) (5)lim - ; ( 槡 -3-6)lim 0 sin ; -cos (7)lim ; 槡 +sin - (8)lim 0槡 arctan ; ln(+ (9)lim ) 0 sec-cos ; ( tan-sin 0)lim 0 sin 3. +a+b 0. lim - =3, a b.. lim (+5) -k/ =e -0, k. 0 烄. y= -, <0 烅槡 - =0. 烆, 0 ;

71 6 烄 3. f()= sin, 0 烅 =0. 烆 0, =0, 4. f()= { -, > =. 烄 e, <0 0, =0 5. f()= 烅, =0 槡 + -, >0 烆 烄 f()= -, 烅, a,f() =. 烆 a, = 烄 7. f()= 烅 sin, <0 b, =0. a,b,f() =0. sin +a, >0 烆 烄 a+b, 0 8. a,b, f()= 烅 sinb 烆, >0 =0. 9. f(0), f()=arctan =0. 30., : ()f()= lg(+), lim f (); 8 ()f()= , lim f (); 0 (3)f()=lnarcsin, lim f (). 烄, <0 - 槡 - 3. f()= 烅, =0, f(). ln(+), 烆 >0, 3. f()= e <0 { k+, 0 ( -,+ ), k =0 (3,4). 34. e += C :C=C()=a+b 槡 3. =9, 3 ; =6, 68.

72 6 a b. 36. R, : =0,,4, R R=0,6,8. R. 37., 40, 00, ; 00, 0, 8 ;. R ,, : 50, 30 ; 50,, R. 39. P : R. =8000-8P 40. C, 00, 0. P, =50-P., L. 4. :Q =0000-4P, Q ( ),P ( / ). (). (). (3). B. f() [-,], y=f( ).. f() [0,], y=f(e -). +, 3. f()= 0 < { <,, f(). 4. f( 槡 +)=+ 槡, f()=. 5. f()=,g()=, f[g()]=,g[f()]= lim (+) =. 3 7.lim nsin π n n =. sin 8.lim + 槡 =. +sin 9.lim =.

73 63 0.lim - + =.. 0,-cos asin, a=. 烄 sin, <0. f()= 烅 k, =0, =0, k=. ln(+), >0 烆. y= 槡 A.(-,) C.[-,-) (-,] 槡. y= - ln(3-). A.{ <3} C.{ <3 } B.[-,] D.[-,-) (-,) (,] B.{ } D. 3. f(+) [0,], y=f(). A.[0,] B.[-,0] C.[,] D.[0,] 4. f(-)= 3 -, f()=. A B C D lim f()=a( ), f() 0. 0 A., f(0)=a C. 6.. A.lim 0 e - C.lim sin 7.lim 0 =. B., f(0) D., B.lim e 0 D.lim - A.0 B. C.- D. 8.lim =. +e A.0 B. C., D. 9.lim cos =. A.cos B.π C.0 D.cosπ

74 ,. A.e B. sin C.cos D.sin. y=e -,. A. 0 + B. 0 - C. + D. -.,- + 槡 7-3. A. C. 3.. sin A.lim 0 =0 B. D. B.lim sin = C.lim 0 sin = D.lim sin = sin 4.lim 0 sin =. A. B. C. D.0 5.,. A.lim =e B.lim (-) =e 0 C.lim - =-e D.lim - =e - 6.lim ( ) ( - k ) =e, k=. A.- B. C.- D. 7. f()= -, 0< { -, < 3 =. A.f() = C.lim +f () 8. y= A.= C.=,=- B.lim -f () D.lim f () B.=- D. 9. f()=ln(9- ), f(). A.(-,-3) C.[-3,3] B.(3,+ ) D.(-3,3) - 槡 ,f()=, f() =0, f(0)=.

75 65 A. B. C.- D.- 烄, 0. f()= 烅槡 +4- =0, k. 烆 k, =0 A.0 B. 4 C. D.4

76 ,.,,,.,..... y=f(), M(0,y0). M (0+Δ,y0+Δy), M M MM. M M, MM, M M, MM MT M. MT α, MM β, y=f() M(0,y0) Δy tanα=lim tanβ=lim β α Δ 0 Δ =lim f(0+δ)-f(0). Δ 0 Δ.,s [0,t], s t s=s(t), t0 [0,t]. t0 t0+δt, [t0,t0+δt] Δs=s(t0+Δt)-s(t0), v= Δs Δt =s (t0+δt)-s(t0). Δt Δt, [t 0,t 0+Δt] ;, v t 0. Δt,. Δt 0, v,

77 67 t0, Δs v(t0)=lim Δt 0 Δt =lim s(t0+δt)-s(t0) Δt 0 Δt,,,,..... y=f() 0, 0 Δ( 0), f() Δ 0, Δy=f(0+Δ)-f(0) Δy lim Δ 0 Δ =lim f( 0+Δ ) -f ( 0 ) Δ 0 Δ, f() 0 ( f() 0 ), y=f() 0. =0+Δ, f (0), y =0, dy d = 0, df d = 0 Δy f (0)=lim Δ 0 Δ =lim f(0+δ)-f(0) Δ 0 Δ f()-f(0) f (0)=lim 0-0 y= =. Δy=(+Δ) - = Δ+(Δ) Δy Δ =+Δ Δy f ()=lim Δ 0 Δ =lim (+Δ)=. Δ 0. y=f() (a,b), y=f() (a,b). y=f() (a,b), (a,b), f (), f (), f(),,..3 f (), y, dy d, df d y=f() 0 f ( 0 ) y=f() ( 0,y0)

78 68. f (0), y=f() (0,y0) y-y0=f (0)(-0) :f (0)=0,, y=y0; f (0)=,, =0. y= (,)., (,)..4 k=f ()= y-=(-) -y-=0 y=f() 0,. Δy.3 y=f(), lim Δ 0 - Δ, f() 0, f Δy - (0); lim Δ 0 + Δ, f() 0, f + (0). y=f() 0 y=f() 0. y=f() [a,b], f() (a,b), f - (b) f + (a). 3 f()= sin, < 0 { ln(+), 0 =0. f() =0. f(0+δ)-f(0) sinδ-ln f - (0)= lim = lim Δ 0 - Δ Δ 0 - Δ = f(0+δ)-f(0) ln(+δ)-ln f + (0)= lim = lim Δ 0 + Δ Δ 0 + Δ = lim ln(+δ) =lne= Δ f - (0)=f + (0)= y=f() 0, 0. y=f() 0,

79 69 Δy lim Δ 0 Δ =f (0) lim Δy=lim Δy Δy Δ 0 Δ 0 Δ Δ=lim Δ 0 Δ lim Δ=f (0) 0=0 Δ 0 y=f() 0. 4 f()= =0,. ( ) f()= =0, f(0)=0 lim -f ()=lim lim 0 +f ()=lim 0 + =lim 0 - (-)=0 =lim 0 +=0 lim 0 f()=0=f(0) f()= =0 ; ( ) Δy=f(0+Δ)-f(0)= 0+Δ -0= Δ f - (0)= lim Δ 0 - Δy Δ = lim Δ 0 - f + (0)= lim Δ 0 + Δy Δ = lim Δ 0 + f()= =0. f - (0) f + (0) Δ Δ = lim Δ 0 - -Δ Δ =- Δ Δ = lim Δ Δ 0 + Δ = 烄 5 f()= sin, 0 烅 =0. 烆 0, =0 : f() =0, f(0)=0 f() =0. : lim f ()=lim sin 0 0 =0 lim 0 f ()=f(0)=0 f (0)=lim 0 f() =0 f()-f(0) -0 =lim 0 sin =lim 0 sin.,.,.,,.

80 70..6., : () Δ Δy=f(+Δ)-f(); () Δy Δ =f (+Δ)-f() ; Δ (3) Δ 0, Δy Δ, f(+δ)-f() y =f ()=lim. Δ 0 Δ 6 y=c(c ). Δy=f(+Δ)-f()=c-c=0 Δy Δ =0 Δy y =lim Δ 0 Δ =0 7 y=. Δy=f(+Δ)-f()=(+Δ)-=Δ Δy Δ = Δy y =lim Δ 0 Δ = 8 y= 槡. Δy=f(+Δ)-f()= 槡 +Δ - 槡 Δy Δ = 槡 +Δ - 槡 Δ Δy y =lim Δ 0 Δ =lim 槡 +Δ - 槡 (+Δ 槡 - 槡 )(+Δ 槡 + 槡 ) =lim Δ 0 Δ Δ 0 Δ(+Δ 槡 + 槡 ) +Δ- =lim Δ 0 Δ(+Δ 槡 + 槡 ) = 槡. 9 y=loga(a>0,a ). ( ) = ( loga +Δ Δ ) Δ 0 ( loga +Δ Δ ) + Δ Δ Δ 0 ( ) Δy=f(+Δ)-f()=loga(+Δ)-loga=loga + Δ Δy Δ = Δ loga +Δ Δy y =lim Δ 0 Δ =lim = logalim = logae= lna.

81 7 (loga) = lna., a=e, (ln) =. 0 y=sin. Δy=f(+Δ)-f()=sin(+Δ)-sin=cos+ ( Δ ) sinδ Δ cos+ Δy Δ = sinδ Δ Δy y =lim Δ 0 Δ =lim 熿 Δ 0 cos+ ( Δ sin ) Δ 燄 =cos 燀 Δ 燅 =cos. ()(c) =0(c ) ()( α ) =α α- (α R) (3)(a ) =a lna(a>0,a ) (4)(e ) =e (5)(loga) = lna ( a>0,a ) (6)(ln) = (7)(sin) =cos (8)(cos) =-sin (9)(tan) = cos =sec (0)(cot) =- sin =-csc ()(sec) =sec tan ()(csc) =-csc cot (3)(arcsin) = 槡 - (4)(arccos) =- 槡 - (5)(arctan) = + (sin) =cos (cos) =-sin

82 7 (6)(arccot) =- + y=tan = π 4. y = cos k=y π = = 4 cos π = 4 = π 4, y=tan π 4 = y-= ( - π 4 ), -y+- π =0.... u(),v(), u()±v(), u() v(), u () v() (v() 0) : (u±v) =u ±v., u=c, (cv) =cv. (uv) =u v+uv : (uvw) =u vw+uv w+uvw. u u v-uv 3 ( v ) = (v 0), u=c, c c v =-. v v v,,. f()=u()v(), Δy =f(+δ)-f()=u(+δ)v(+δ)-u()v() =[u(+δ)v(+δ)-u()v(+δ)]+[u()v(+δ)-u()v()]

83 73 =[u(+δ)-u()]v(+δ)+u()[v(+δ)-v()] =Δu v(+δ)+u() Δv Δy =Δu v(+δ)+u() Δv = Δ v(+δ)+u() Δv Δu Δ Δ Δ [ ] Δy y =lim Δ 0 Δ =lim Δ v(+δ)+u() Δv Δu Δ 0 Δ =u ()v()+u()v (). y= cos+lnπ, y. f()= - 4 f()= y =(5 ) +(3-3 ) -( ) +(4cos) +(lnπ) = ln-4sin. ( + ) 4 ( - )( + 4 ) 4 = 8 -, f (). f ()=( -8 ) -( ) = f ()=-8-=0. 3 y=e cos, y. y =(e ) cos+e (cos) =e cos+e (-sin) =e (cos-sin). 4 y= sin ln, y. y =() sin ln+ (sin) ln+ sin (ln) =sinln+cosln+sin. 5 y=tan, y. y=tan= sin cos, y =(tan) = sin (sin) cos-sin (cos) = cos cos 6 y= --, y. 槡 y= -- 槡 = y = = cos +sin cos = cos =sec. 3-7 y= - +, y.

84 74 y = ( -) (+)-(-)(+) (+) = - (+)-(-) (+) =- (+). 烄, 8 f()= 0 烅烆 e, >0. f(), =0 f(),,. <0, >0, f ()=( ) = f ()=(e ) =e +e =e (+) f(0+δ)-f(0) (Δ) f - (0)= lim = lim Δ 0 - Δ Δ 0 - Δ =0 Δ f(0+δ)-f(0) Δ e f + (0)= lim = lim Δ 0 + Δ Δ 0 + Δ = f() =0,f (0).,.. 4 f - (0) f + (0), <0 f ()= { e (+), >0 y=f(u),u=φ ( ), y=f [ φ ( ) ], dy d =f (u) φ () y =y u u : y=f(u),u=φ ( t),t=s(), y=f { φ[ s(t )] } 9 y=(+) 3, y. y =y u u t t y=(+) 3 y=u 3,u=+. y=u 3,u=+, y =y u u =(u 3 ) u (+) =3u =6(+) 0 y= 槡 a + (a ), y. y= 槡 a + y= 槡 u,u=a +. y= 槡 u,u=a +, y =y u u =( 槡 u) u (a + ) = = 槡 u 槡 a +.

85 75 y=sin (3-), y. y=sin (3-) y=u,u=sinv,v=3-. y=u,u=sinv,v=3-, y =y u u v v =(u ) u (sinv) v (3-) =u cosv (-) =-sin(3-).,,. y= tan, y. y = tan ln tan tan = ln sec tan ln =- cos. 3 y=ln - 槡 +, y., y= [ ln (- )-ln(+ )] y = {[ ln (- ) - ] [ ln(+ )] } = [ ] (- ) - ( + ) - + = = y= arcsin(ln), y. y =() arcsin(ln)+ [ arcsin(ln ) ] =arcsin(ln)+ (ln) 槡 -ln =arcsin(ln)+ 槡 -ln. 5 y= n, y. - y =n n- n n- n- ( -)- = - - ( -) ( -) =- nn- ( +) ( -) n+...3, : y y=f(), ; y F(,y)=0,.,e y +y-e =,sin(+y)=lny.,,

86 76. :,, y, y, y. 6 e y +y-e = y. y, e y.., y, e y y +y+y =0 y =- y. +e y 7 sin(+y)=lny y, y. y, sin(+y) lny.., y, 8. ()y=arcsin; cos(+y) (+y) =lny+ (lny) cos(+y) (+y )=lny+ y y y = ylny-ycos (+y) ycos(+y)-. ()y=arctan. () y=arcsin =siny,y - π,π. =siny,, =cosy y y = cosy = 槡 -sin y = 槡 - (arccos) =- 槡 - () y=arctan =tany,y - π,π. =tany,, =sec y y. y = sec y = +tan y = +

87 77 (arccot) = y+lny= M(,)., y, y+y + y y =0 M(,) y =- y y+, M(,) y = =- y= + =- y-=- ( -) +y-3=0...4 y=f(), lny=lnf(),, y... 0 y= sin.,,.,., y= sin,, lny=sin ln sin y =cos ln+ y y =y ( cos ln+ sin ) y= ( +5) ( +) (-) 槡 3 +. =sin cos ln+ sin,.,.,, lny=ln(+5)+ln( +)-ln(-)- 3 ln (+)

88 78 y y = (+) [ ] y =y (+ ) [ ] = ( +5) ( +) (-) 槡 (+ ) y= y y.,, lny=y ln y y =yln+y ln+y y = ( ln+)y -yln.3, v(t) s=f(t) t, v(t)=f (t)., v(t) t a(t), a(t)=v (t)=[f (t)]., a(t) f(t) t, f(t) t, f (t). f(t) t, a(t)=f (t)., y=f() f (), f (), f () f(), f (),y, d y d,d f d., f () f(), f (),,(n-) f (n-) () f() n, f (n) (),y (n), dn y d n,dn f d n. y=f() n, n.. f (k) (),k 4. y=f() 0 0,

89 79 f (0),f (0),f (4) (0),,f (n) (0).,,,. y= , y,y (4),y (5). y = 3-0+=6-0+ y =6-0= y = 3 = 3 = y (4) =0,y (5) =0., n f()=a0 n +a n- + +an-+an, f()= ln, f (). f (n) ()=a0n,f (n+) ()=0 f ()=ln+ f ()=ln+3 f ()= f ()= 3 n : ()y=sin; ()y=a. ( ) ( + π ) ( ) ( + π ) ()y =cos=sin + π y =cos+ π y =cos+ π, ()y =a lna y =a lna lna=a (lna) y =a lna (lna) =a (lna) 3, ( ) π =cos+ =cos+ π y (n) =sin ( + nπ ) y (n) =a (lna) n. 4 y=e sin y -y +y=0. ( ) =sin +π =sin +3π y =e sin+e cos y =e sin+e cos+e cos-e sin=e cos y -y +y=e cos-(e sin+e cos)+e sin=0

90 ,., Δ, Δy., y=f(), Δy=f(+Δ)-f()., Δ, Δy.., S, S=. Δ, ΔS=(+Δ) - =Δ+(Δ). : Δ Δ,. (Δ), Δ 0, Δ., Δ, Δ ΔS,, Δ. Δ S., S=, S =( ) =., S S.. y=f(), Δy lim Δ 0 Δ =f ().3, Δy Δ =f ()+α, α Δ 0, Δy=f ()Δ+α Δ Δ 0,α Δ Δ, f ()Δ y=f()..4 y=f() 0, f (0)Δ y=f() 0, dy, dy=f (0) Δ, y=f() 0. y=f(),, dy=f () Δ y=, d=dy=() Δ=Δ

91 8 d Δ, dy=f () d f ()= dy d, dy d,,. y= 3 =,Δ=0.0. Δy=(+0.0) 3-3 =8.06-8=0.060 dy=y () Δ=3 0.0=0..4. dy Δy y=f(), 3. M0(0,y0), M0 M0T, α, tanα=f (0) 0 Δ, M(0+Δ,y0+Δy), 3 M0N=Δ,NM=Δy 3 NT=M0N tanα=f (0)Δ=dy, y=f() dy M0(0,y0). TM Δy dy, Δ..4.3 dy=f () d,, d.,..d(c)=0.d( α )=α α- d 3.d(a )=a lnad 4.d(e )=e d (c ) 5.d(loga)= lna d 6.d(ln)= d (α ) (a>0,a ) (a>0,a )

92 8 7.d(sin)=cosd 8.d(cos)=-sind 9.d(tan)= cos d=sec d 0.d(cot)=- sin d=-csc d.d(sec)=sec tand.d(csc)=-csc cotd 3.d(arcsin)= 槡 - d (-<<) 4.d(arccos)=- 槡 - d (-<<) 5.d(arctan)= + d 6.d(arccot)=- + d 7.d(u±v)=du±dv 8.d(u v)=vdu+udv 9.d(cu)=c du (c ) 0.d u =v du-u dv (v 0) v v ()y=e 3 cos; () ()y= sin. y =3e 3 cos-e 3 sin=e 3 (3cos-sin) dy=y d=e 3 (3cos-sin)d () y = cos-sin.4.4 dy=y d= cos-sin d, y=f(u) u,. u, dy=f (u) du. u, u=φ ( ), du=φ () d, u y=f[ φ ()]

93 83 dy=y d=f (u) φ () d=f (u)[ φ () d]=f (u) du, y=f(u), u, dy=f (u) du,. 3 y=arctan, dy. dy= + d = + - d=- d + 4 y= 槡 +sin, dy. dy= 槡 +sin d (+sin )= sin cos sin d= 槡 +sin 槡 +sin d 5 3 +y 3 =e y dy., d( 3 +y 3 )=d(e y ) d( 3 )+d(y 3 )=e y d(y) 3 d+3y dy=e y (yd+dy) (3y -e y )dy=(ye y -3 )d dy= yey -3 d 3y -e y.4.5 y=f() 0,, Δ,. Δy dy=f (0) Δ Δy=f(0+Δ)-f(0) f(0+δ)-f(0) f (0) Δ f(0+δ) f(0)+f (0) Δ. 6 0, 6.. r V=f(r)= 4 3 πr3 ΔV, dv, r=5,

94 84 dr=- 6. dv=f (r) dr=4πr dr=4π 5 (- 6 ) ΔV dv y= 槡 y ( :0 ), ( :0 ). =00.05, 0=00,Δ=0.05, Δ 0,. f(00.05)=f(0+δ) f(0)+f (0) Δ =( 槡 00)+ ( 槡 ) =00 Δ = 槡 = =50.005(0 ) 8 arctan0.98. f()=arctan,0=,δ=-0.0, f(0+δ) f(0)+f (0) Δ=arctan0+ Δ + 0 arctan0.98 arctan+ + (-0.0) = π : Δ,ln(+Δ) Δ. f()=ln,0=, Δ,, ln(+δ)=f(0+δ) f(0)+f (0) Δ =ln+ Δ=Δ ln(+),.,, :. f() 0 sin,tan e +,(+) α +α

95 Δy f (0)=lim Δ 0 Δ =lim f(0+δ)-f(0). Δ 0 Δ 85 y=f() (a,b), f() (a,b). f(). Δy f ()=lim Δ 0 Δ =lim f(+δ)-f(). Δ 0 Δ f() 0 dy=f (0) d.dy Δ, Δ,dy Δy. f() dy=f () d. 3. f (0) y=f() (0,y0). dy y=f() (0,y0) Δ. Δy y=f() Δ. 4. f () f(), f ().(n-) f (n-) () f() n, f (n) (). 5. y=f() 0, y=f() 0 ;,y=f() 0,. 6. u, dy=f (u) du.. ( ). y=f[ φ ()] (u±v) =u ±v (u v) =u v+u v (cu) =c u u v u -u v ( v ) = (v 0) v dy d =f (u) φ ()

96 86 3. F(,y)=0 y,,, y. 4.,,, y.. y=f() (0,y0) y-y0=f (0)(-0). Δ, Δy dy=f (0) Δ f(0+δ) f(0)+f (0) Δ.. f (0), : ()lim Δ 0 f(0-δ)-f(0) Δ A ; ()lim h 0 f(0-h)-f(0+3h). h. f()= 槡 -, f (). 3. s=t 3 +, t=3. 4. =0 ()y= 槡 3 ; ()f()= 烄 +4, 0 烅烆 e, <0. 5. : () y=5, y,y (-),y (5); () y= 3,y=,y= 0.6,y= a b,y=( m ) n, y ; 槡 3 (3) y=lg,y= -,y=e a, y. 6. f()= + 3 =. 7. : ()y= 槡 - + +arctane; ()y= 3-3 +tan;

97 (3)y=(+ 槡 )- 槡 ; ( 4)y= - 槡 + ; + 槡 (5)y=e sin; (6)y= ln-sec; (7)y= ln cos ; ( 8)y= - + ; 槡 (9)y= π + ; (0)y= e - 槡 +log3 cota 烄 +, 8. f()= <0 烅烆 e -, 0, f (). 9. : ()y=(-) 3 ; (3)y=arcsin( 3 ); ()y= 3 槡 - ; (4)y=3 槡 cos ; (5)y=lnln(ln [ )] ; (6)y=ln 槡 -sin +sin ; 槡 (7)y= + +槡 - ; (8)y=( +e - ) 3 ; 槡 + -槡 - (9)y=sin n sin(n); (0)y=arctan - ; ()y=(+)3-4 槡 ; ()y=log3cot 槡 +; (3)y=e - 槡 sin cos3; (4)y=ln ( a ). 0. y=(+)- 槡,.. dy d : () +y 3-3y =4; ()+arccos y =sine ; (3)e y -yln+arccot 槡 =0; (4)sin( +y )-ay+b=0; (5)e +y -ye +y=; (6)e y -ye =0.. +y -y=4 y. 3. dy d : 5 (-) ()y= 3 槡 + 3 ; ( 槡 + (4-) )y= (-) ; (3)y= cos ln 槡 +; 槡 (4)y=3 ; (5)y=(cos) ln ; (6)y=(- ) cos3 ; (7)y= e - ; (8)y = y. 4. : ()y=ln(- ), y. ()y=sin 3, y ;

98 88 (3)y= sin, y (0); (4)y= e 槡 +lnsin+arctan, y ; (5)y=ln(-), y (n) ; 5. : ()y= +ln -ln ; (6)y= - +, y (n). y=ln 槡 - 3 +arctane ; (3)y=e - cos 3 ; ( 4)y=(e -e - ) ()ad=d; ()bd=d; (3) 槡 d=d ; (4) d=d ; (5) d=d; (6) + 槡 - d=d ; (7)sind=d; (8)cosad=d; (9)e -3 d=d; (0)sec tand=d. 7. : () 槡 ; ()e.0 ; (3)sin3 ; (4)arctan , 0cm, 0.cm.. B f(). y=f() =a, f(a)=0,lim a (-a) =, f (a). A.0 B. C. D.. y=f() = y=5-0,. A.f ()=5,f()=0 B.f ()= 5, f()=0 C.f =5,f =0 D.f ()= 5, f()=0 3. f() 0,. A.f() 0 B.f() 0 C.f() 0 D.f() 0, 4. f()= e <0 { a-b, 0 =0, a,b. A.a=,b= B.a=,b=-

99 89 C.a=-,b= D.a=-,b=- 烄, 5. f()= 3 3 烅, f() =. 烆, > A. B., C., D. 6., f(-)=-f(), f (-0)=-k, f (0)=. A.-k B.k C.- k D. k 7. f ()=, d d f (e - )=. A.- e - B.-e - C.-e -3 D.-e f =, f ()=. A. B y=f ( - ), dy=. C. A.- f ()d B.- f d D. - C. f ()d D.- f ( - ) d 0. f()=a n +a n (n ), f (n) (0)=. A.0 B.an C.an D.a n,. f()= {, >, =. A. C.. f() 0, ( A. f() 0 C. f() 0 B. f ()= D. ). B. f() 0 D.lim 0 f()=a, A f(0). s= b a ( at+e -at ),a b, t= a. f(). f() f(0)=0, lim 0 =. f()-f() f()=sin, lim - =. 3. f()=cos 3, f (0)=.

100 90 f( 3-Δ ) -f( 3) 4. y=f(), lim =-, y=f() (3,f(3)) Δ 0 3Δ. 5. f()=a0 n +a n- + +an-+an, F()=f() =0 F (0)=. 6. y= ( + ), dy d =. 7. y=sin, d y d =. 8. y=f() 0, Δ, Δy. 9.d = ln d. 0. f(t)=lim t ( +3t ), df (t) t==. 烄. f()=, - 烅 =- 烆槡 +, >-. y= 3 槡 3+y+=0 3. y=cos( )sin, y. 4. y=ln( 3 +cos 4 )+π +π π, y. 5. y= 槡 3 +e + - +, y. 6. y=f() e y +sin(y )=+y, dy.

101 3 3,.,, ; ; ;, Role 3. y=f() : () a, [ b] ; () (a,b) ; (3)f(a)=f(b). (a,b) ξ, f ( ξ )=0(a<ξ<b). () f() a, [ b] M m. M=m, a, [ b] f(), f ()=0, (a,b), (a,b) ξ, f ( ξ )=0, M=m. M m, M>m, (3) M m (a,b), M ξ (a,b), : : M=f( ξ ) f(), [ a, b ] (3.) f ( ξ )=0. (3.), ξ +Δ [ a, b], f( ξ +Δ)-f( ξ ) 0. Δy Δ =f ( ξ +Δ)-f( ξ ) 0 Δ Δy Δ =f ( ξ +Δ)-f( ξ ) 0 Δ () : f ( ξ )= lim Δ 0 + Δy Δ 0 (Δ>0) (Δ<0)

102 9 f ( ξ )=0.( ) f ( ξ )= lim Δ 0 - Δy Δ 0 : AB,, AB. 3. :, ξ (a,b), f ( ξ )=0. ξ (a,b).. f()= 槡 - 0, [ ],, ξ. f()= 槡 - (-, ], -, ( ], [ 0, ]. f ()= ( 槡 - = ) 槡 - - 槡 - = 4-3 槡 - 3 f()= 槡 - (-,), (0,), f(0)= f()=0, f()= 槡 - 0, [ ]. f ()= 4-3 槡 - = 0, = 4 3 (0,), ξ = 4 3 (0,), f ( ξ )= c=0 (c ).. a b, a<b, f()= 3 ++c a, [ b]. (a,b) ξ f ( ξ )=3ξ +=0, ξ.,, 3 ++c= Lagrange 3. y=f() : () a, [ b] ; () (a,b). (a,b) ξ, f ( ξ )= f (b)-f(a) b-a ( a<ξ<b). f ( ξ )= f (b)-f(a) b-a d d f ()- f (b)-f(a) [ b-a ( -a ) ] =ξ=0,

103 3 93 F()=f()- f (b)-f(a) b-a ( -a) (),(),F() a, [ b], (a,b), F(a)=F(b)=f(a).,, (a,b) ξ, F ( ξ )=f ( ξ )- f (b)-f(a) b-a =0, f ( ξ )= f (b)-f(a) b-a ( a<ξ<b). : 3 AB f (b)-f(a) b-a, f ( ξ ), : AB, ξ, AB. :, f(b)=f(a). : (a,b) ξ, f ( ξ )=0.,. 3 3 y=ln, [ ], ξ. y=ln (0,+ ), f() (0,+ ),, [ ]. y =+ln, f() (0,+ ), (,). y=ln, [ ]. f()=0,f()=ln f ()=+ln= f ()-f() - = 4 e ξ= 4 e (,), f ( ξ )= f ()-f() <ln (+)(>0). f()=(+)ln(+), f(), (-,

104 94 + ), 0,+ [ ). f ()=+ln(+) f() (0,+ ), f() 0, [ ], ξ (0<ξ<), : f(0)=0 f()> f()-f(0)=f ( ξ )(-0)(0<ξ<) f ( ξ )=+ln(+ξ ) >( ξ >0) ln(+)> + ( >0) ( ),. f() (a,b), f ()=0, (a,b), f() (a,b). (a,b),, <, f() [, ], f()-f()=f ( ξ )(-) (a<<ξ<<b) f() (a,b).( ) f ( ξ )=0, f()=f() f() g() (a,b), f () g (), (a, b), (a,b) f() g(), f()-g()=c(c ).( ) ,, ;, f() (a,b), () (a,b) f ()>0, f() (a,b) ;

105 3 95 () (a,b) f ()<0, f() (a,b)., (a,b), <, f() (a,b), f() [, ], : f()-f()=f ( ξ )(-)(<ξ<), () f ()>0, f ( ξ )>0, ->0, f()> f(),,, f() (a,b).().( ) : 3.3 (a,b) f ()>0( f ()<0) (a,b) f () 0( f () 0),. : 5 f()= 3 +. (-,+ ),f ()=3 0, =0 f ()=0, f() (-,+ ). : f(),, 3 4,f() (a, ) (,3), (,) (3,b). f ()=0 f (). 3 4 f() : () f() ; () f(), f ()=0, f () ( ); (3) f () ( ); (4), 3. 6 f()= 3-3. (-,+ ). f ()=3-3=3(+)(-) f ()=0, =-,=,,, (-, -),(-,),(,+ ), 3 : 3 (-,-) - (-,) (,+ ) f () f() 3 f()= 3-3 (-,-) (,+ ) ; (-, ). 7 f()= f()=3 3 - ( -,+ ) f ()= = 槡 3 槡

106 96 f ()=0 =8. =0,f (). =0 =8 (-,0), (0,8) (8,+ ), 3 : 3 (-,0) 0 (0,8) 8 (8,+ ) f () f() 3 f()=3 3 - (-,0) (8,+ ) ; (0,8). 6 y=(-)ln. y=(-)ln (0,+ ) y =0 =. y = - +ln=-+ln (0,+ ). 0<<,y <0, y=(-)ln (0,) ; >,y >0, y=(-)ln (,+ ) Cauchy 3.4 f() g() : () a, [ b] ; () (a,b), g () 0(a<<b). (a,b) ξ, f ( ξ ) g ( ξ ) =f (b)-f(a) g(b)-g(a) ( a<ξ<b).,,. ( 3 5 ):, =g () { y=f( ) ( a<<b) Oy y=f(), A(g(a),f(a)) B(g(b),f(b)), AB f (b)-f(a) g(b)-g(a), dy d =f ()d g ()d =f () g () 3 5 (g(),f()), f ( ξ ) g ( ξ ) C (g( ξ ),f( ξ ))

107 3 97. : AB, C, AB., g()=, =g ()= { y=f( ) ( a b), y=f()(a b),, Ⅰ f() g() : ()lim 0 f()=0,lim 0 g()=0; ()f() g() 0 ( 0 ), g () 0; f () (3)lim 0 g () =A ( ). f() lim 0 g() =lim f () 0 g () =A ( ).,. : () lim 0 f()=,lim 0 g()=,,, Ⅱ. Ⅰ Ⅱ 0 0 ( 0 ). 3, lim f () g () 0 0, f () g (),,. -tan lim , tan 0, 3 0 Ⅰ : -tan -sec lim =lim = lim tan 0 ( ) =- 3. 槡 - lim 3 槡 , Ⅰ :

108 =lim π 3 -arctan lim =lim 0 0, Ⅰ : - + =lim e lim +e cos. 0 0, Ⅰ : = 3 4. =lim + + =. e =lim -e - 0 sin ( 0 0, Ⅰ) e +e - =lim 0 cos =. 5 ln(+) lim (n> ). 0 n 0 0, Ⅰ : 6 e lim -- 0 sin. + =lim =lim 0 n n- 0 n n- (+) = 0 0, Ⅰ : e =lim -- ( ) 0 7 ln lim + ( n>0). n e - =lim 0 =lim e 0 =., Ⅱ : =lim =lim =0 + n n- + n n

109 lim 0 + lncot ln., Ⅱ : =lim e 9 lim cot csc =-lim 0 + sincos =-lim sin lim cos = , Ⅱ : e =lim + =lim e + = :, lim f () g () ( ),. lim f () g(),. 0 lim +sin 3-sin., lim (+sin) (3-sin) =lim +cos 3-cos,. : sec lim tan. + +sin lim 3-sin =lim sin 3- = +0 sin 3-0 = 3. π sec lim tan =lim sectan sec =lim tan sec π π π sec =lim sectan =lim sec tan = π π,, :. sec lim tan =lim cos cossin =lim sin = π π 0 0, 0 -, 0 0, π

110 ,. lim 0 +sin ln. 0,, 0 0 ln =lim 0 + csc 3 lim π + -arctan. 0 =lim + 4 lim + 0 =lim 0 + -csc cot =-lim sin 0 + tan=0. π -arctan =lim + - e. [ ] =lim + + = -, 0 0. (+) =lim -e ( ) =lim 0 5 lim 0 sin - -. (+)-e -e =lim 0 =-.. +-e =lim 0 ( 0 0 ) =lim -sin 0 sin =lim -sin -sincos =lim =lim 0 -sin 4 3 =lim 0 6 lim - - ln. -. -cos =lim 0 4sin 4 = 3. lim - - ln =lim ln-+ (-)ln 0 0

111 =lim =lim - +ln -+ln ( 0 0 ) =lim - ++ln =-. 7 lim 0 + (sin). 0 0, (f()) g ()= e g ()lnf() 0, 0 0, : lnsin sin cos =lim +elnsin =e lim 0 + =e lim =e - lim 0 + sin cos =e 0 =. 0 y=(sin), :lny=lnsin,, lnsin lim lny =lim lnsin=lim sin cos =lim lim () -.. =lim 9 lim (cot) ln =lim lncot e 0 + ln =e lim 0 + =-lim sin cos=0 0 + lnlim 0 +y=0 lim =, 0 +y=e0 lim (sin) = 0 + -=e lim ln -=e lim ln e lncot ln =e lim cot (-csc ) =e -. =e lim sincos =e -.

112 ,. 3. f() 0, ( 0), ()f()<f(0), f(0) f(), 0 f() ; ()f()>f(0), f(0) f(), 0 f().,. : (), ( 3 6 f() f(5) ). (), 3 6 f() (a,b) f() f(4) f() f(5). (3) f() 0 :f(0) 0,,. 3 6 y=f(), f ()=0( 4 ), f () ( 5 )., f ()=0 ( 3), f () ( 6). f () ( ) f (). ( ) f(), f (0)=0 f (0).( ) 3 6, ;, ( ). 3.6( Ⅰ) f() 0, 0. () 0 f ()>0, 0 f ()<0, 0 f() ; () 0 f ()<0, 0 f ()>0, 0 f() ; (3) 0 f (), 0 f().( ) , :

113 3 03 f() ; f (); 3 ( ); 4 Ⅰ ; 5,. f()=(-) (+) 3. (-,+ ) f ()=(-)(+) 3 +3(-) (+) =(-)(+) (5-) f ()=0, =-,= 5, 3=. f ().,,3 (-,-), -, 5, 5,,(, + ). 3 3 : 3 3 (-,-) - -, 5 5 ( 5, ) (,+ ) f () f() 3 3 f() = 5 f( 5 ) = , = f()=0. f()= 3 槡. (-,+ ) f ()= 3-3 = 3槡 3 f ()=0,, =0 f (), =0 (-,0),(0,+ ) <0,f ()<0; >0,f ()>0. =0 f(), f(0)=0. f(), Ⅱ. 3.7( Ⅱ) f() 0, f (0)= 0,f (0) 0, () f (0)>0,0 f() ; () f (0)<0,0 f().( ) 3 f()= 3-6. R f ()=6 -=6(-),

114 04 f ()=0 =0 =. f ()=- f ()=-<0,f () =>0. =0 f(), f(0)=0;= f(), f()=-8. : Ⅱ, f (0)=0,f (0) 0, Ⅱ f() a, [ b], 0 [ a, b ], [ a, b ], f() f(0)(f() f(0)), f(0) f() [ a, b] ( ), 0 f() a, [ b] ( )..,,. :., a, [ b]. :, ( ). ( ),,. : () f() f (); (), a, [ b] ( ) ; (3),,. 4 f()=(-5) 槡 3 -, [ 3]. f ()= 槡 ( -5) - 3 =5 (-) 3槡 3 f ()=0, =, =0 f (), -, [ 3], : f(-)=-6,f(0)=0,f()=-3 槡 3 4,f(3)=- 槡 3 9 :f() -, [ 3] f(0)=0, f(-)=-6. 5 f()= , [ ]. f ()=3 +6+3=3(+) 0, f() 0, [ ], f() 0, [ ] f()=8, f(0)=. :() 5 : f() a, [ b] ( ), f() a, [ b] f(b)( f(a)), f(a)( f(b)). (),, f(), ( ; ),.

115 3 05,. : 6 a,,,, 3 7, (0<< a), s=4(a-), : V=4(a-) ( ) V =4(a-) -8(a-)=4(a-)(a-3) V =0 = a 3, =a( ) V ( a 3 ) =8 =-8a<0, (3-a)= a = a 3, ( 0,a),. a 3,, 6 7 a , 6, 3,, h, F =0 V=π h=08, F=6 π +3 πh=π F =4π =槡 π = 3 3 槡 π F =4π >0 (>0) = 3 3. 槡 π,. = 3 3, 槡 π

116 ,, q, R(q)=8q-q, C(q)=q+3, : L(q)=R(q)-C(q)=8q-q -q-3=6q-q -3 L (q)=6-q L =0, q=3, L =-<0, q=3., q=3, 3, L(3)=6( ).. 9 C(q)=q 3-6q +5q+000,. C(q)= C (q)-000 =q -6q+5( ) q C (q)=q-6 C (q)=0 q=3. C (q)=>0 q=3,, q=3, 3, C(3)= , 000, 4. ( ). () ( ); (). (),,. :F=4., 000, :F= , : F=F+F= ( >0) () :

117 3 07 F =0 :=000( ). F = F = >0 =000,,, 000,, , ( ),,,, 3 8,,,,,,.,. 3 9 f()=. f()=- ( 3 0),, [a,b] f ()+f() f + ( ) ( 3 ), f() [a,b] ; [a,b], f + f()+f() ( 3 ), f() [a,b]. 3 9, f() (a,b), f(), (, ). f(), (, ).

118 08 3 3, y=e, y=sin (0,π).,,. 3 9 f(),,,, f (), f(), f (). f (),, f ()>0, f(). 3 0 f() f (), f ()<0, f() f() [a,b], (a,b), (a, b), ()f ()>0, f() (a,b) ; ()f ()<0, f() (a,b).( ) y= 3. y =3,y =6, (-,0),y <0,, (0,+ ),y >0,. f()=e -. f ()=e - -e -,f ()=(-)e -, f ()=0 =. >,f ()>0, f() (,+ ) ; <,f ()<0, f() (-,).,,,,. 3.4 y=f() (a,b), (0,f(0)),,,,.,, y =f() f (), f (), :,,,.,,, f ().

119 y=+(+) 3. y =6(+),y =4(+), y =0, =-, (-,- ) - (-, + ) y y -,-, -, + 4 y= 9 3 槡 +., - (, ). y = 3-3 +,y = y =0, =, =0 y, 3 5 f (). 3 5 (-,0) 0 (0,) (,+ ) y y :,,. y= 4 =0,y =0, y= 4, =0,.,y= =0,y, (0,0). : (0,f(0)) y=f(), : ()f(0) f(), ()f (0) y=f() P, y=f(), ( 3 3 )., :.. y=f() lim + f ()=c lim - f ()=c, y=c y=c y=f(). : lim + f ()=c lim - f ()=c,

120 0 ; lim + f ()=c lim - f ()=c c =c, ; lim + f ()= c lim f ()=c c c, - y=c y=c.. y=f() =a, lim a +f ()= 3 3 lim -f ()=, =a y=f(). a : lim +f ()= lim -f ()=, a a =a; lim +f ()= lim -f ()=, a a =a. 3. f() :lim [f()-(a+b)]=0, y=a+b f(). : [ ] f() lim -a-b b, : b lim =0. [ ] f() lim -a-b =lim [f()-a-b] =lim [f()-a-b] lim =0 [ ] =lim f() - b a -lim [ ] f() lim - a =0 f() a=lim. a, lim [f()-(a+b)]=0, b=lim [f()-a]. 5 y=e -. lim e - =0, y=0 y=e -, lim e - =+, + - y=0, y=e -,. 6 y= - (-). - lim (-) =0, y=0 ; =, - lim (-) =, = y=

121 3 - (-). 7 y= 3-. () lim 3- =, ; () =3, lim 3 3- =, =3 y= 3- ; (3) f() a=lim =lim 3- =lim 3- =-, [ ] b=lim [f()-a]=lim 3- - (- ) =lim 3- =-6 y=--6 y= ,,,,. :.,, ( )...,,. y= 3, (-,+ ),,, (0,+ ), y=cos (-,+ ), π, [0,π]. 3. y y,. 4., y= () (-,+ ),, =0 y=, = y=0.. ()y = 3 - = (-), y =0, =0,=. y =36-4=(3-), y =0, 3=0, 4= 3.

122 , (-,0) 0 0, 3 3 ( 3, ) (,+ ) y y y (0,) 3, 7 (3). (4) ( 3 4). 9 f()=. 槡 π e- (,0) () (-,+ ),, y, [0,+ ) 3 4 (-,+ ). =0 f()= 槡 π. ()f ()=-, f ()=0, =0. 槡 π e- f ()= 槡 π e- ( -). f ()=0, =,=-. (3) lim f()=0, y=0 f() (-,-) - (-,0) 0 (0,) (,+ ) y y y 槡 π e 槡 π e (4) (0,+ ), (-,0) ( 3 5).,, y, y,,,,,, :.,.

123 3 3 0 y= e +. () y=f()= e + -, <-, f()<0, <-, ; > -, f()>0, > -,. () y = e (+), y =0, =0, y = e ( +) (+), =-, y, y =0. 3 =0,=-,, 3 8: 3 5 (3). 3 8 (-,-) - (-,0) 0 (0,+ ) y y y (0,) lim - f()=, =- y=f(). e lim - +, -, e 0, +, lim =0 - +,y=0. (4) ( 3 6). e ,,...

124 4 : C q :C=C(q), q., : q0, ( ). C=C(q) q0, q q0 q0+δq, C ΔC=C(q0+Δq)-C(q0). Δq Δq q0, :ΔC C (q0)δq., Δq=, ΔC C (q0), : ΔC=C(q0+Δq)-C(q0) C (q0). : q0, ( ) C (q0).,, C (q0). : C=C(q) q0 C (q0) C q, C=C(q),, : q, ( Δq=), =C(q+)-C(q)=C(q+Δq)-C(q)=ΔC(q). q,δq= q,, : =C(q+)-C(q)=ΔC(q) dc(q)=c (q)δq=c (q) q.. R q, R=R(q),, : q, ( Δq=), : =R(q+)-R(q)=R(q+Δq)-R(q)=ΔR(q) R (q) q. 3. L q, L=L(q),, : q, ( Δq=), : =L(q+)-L(q)=L(q+Δq)-L(q)=ΔL(q) L (q) q.,

125 3 5 L(q)=R(q)-C(q) :L (q)=r (q)-c (q),.. C(q)= q ( ), :() ; () 000, () C (q)=0.008q; ()q=000 C (000) 000 =9 ( );q=000 C (000)= =6( ). : 000, 6. p=50-0.3q, C(q)=00q+ 800( ). :(). (),,. () R(q)=pq=(50-0.3q)q=50q-0.3q L(q)=R(q)-C(q)=-0.3q +50q-800 R (q)=50-0.6q,c (q)=00,l (q)=r (q)-c (q)=50-0.6q. () L (q)=r (q)-c (q)=50-0.6q=0, q=50. L (q) q=50 = -0.6<0, q=50 L(q),,, L(50)=6950( ).,,,, ( Δ ), y ( Δy y )., : %, 0.5%,., :y= 0,y 00 44, Δ=,Δy=44, Δ = 0 =0%, Δy y = =44%, =0 =, 0%,y 44%,. Δy /y Δ/ =44% 0% =. ( 0,), 0 %,y.%, =0 =, y=.

126 y=f(), Δy/y lim Δ 0 Δ/ =lim Δy Δ 0 Δ y =f () y y=f(), E, E=f () y. y=f() %, f() E%. =0, E(0)=f (0) 0 f(0), f() 0,. =0, %,f() E(0)%. 3 y=00e 3 E =. y =300e 3,E=f () f() =3,E == 3=6. =, %, 6%( ).. () : Q p, Q=Q(p), p p : EQ p =Q (p) Q(p) p Q=Q(p) p, Q (p), EQ p. () : S=S(p) p, p : ES p =S (p) S(p) p (3) : R(p) R p, : ER p =R (p) R(p) p =Q (p)(+eq p ) pq(p) p =+EQ p R(p)=Q(p)p, [ ] R (p)=q(p)+q (p)p=q(p)+q (p) p Q(p ) =Q(p)(+EQ p ). EQ p : EQ p >,, ; E Qp =,,

127 3 7 ; 3 EQ p <,,. 4 Q=e -p 5, : () ; () p=3,p=5,p=6. ()Q =- 5 e-p 5,EQ p =Q (p) p Q(p) =-p 5 ; ()EQ p (3)=-0.6,EQ p (5)= -,EQ p (6)= -., EQ p (3) =0.6<, p=3,, p=3, %, 0.6%. ; EQ p (5) =, p=5,. ;. EQ p (6) =.> p=6, %,.%.. : ( EQ p <0) [ ] R (p)=q(p)+q (p)p=q(p)+q (p) p Q(p ) =Q(p)(+EQ p )=Q(p)(- EQ p ) EQ p <,.,R >0,R p., ;, ; EQ p >,.,R <0,R p., ;, ; 3 EQ p =,.,R =0,R. 5 Q=f(p)=- p. () ; () p=6 ; (3) p=6, %,, (4)p, () Q (p)=-, EQ p =- p -p/ = p p-4 ()EQ p (6)=- 3. %, 0.33%( ). (3)E Qp (6) = 3 <,, %,. R

128 8, R. ER p =+EQ p ER p (6)=+EQ p (6)= 3 6 %, 0.67%( ). (4) R=p Q(p)=p- p, R =-p. R =0, p=, R = - <0, p=,,,, R()=7.. :()f() a, [ b] ; () (a,b). (3)f(a)=f(b). : (a,b) ξ, f ( ξ )=0.. :()f() a, [ b] ; () (a,b). : (a,b) ξ, f ( ξ )= f (b)-f(a) b-a. 3. :()f() g() a, [ b] ; () (a,b) g () 0 : (a,b) ξ, f ( ξ ) g ( ξ ) =f (b)-f(a) g(b)-g(a). :, ;,. lim f () g() 0 0, limf () g () =A ( ), lim f () g() =limf () g () =A ( ); lim f () g () 0 0, f () g (),, :

129 3 9 lim f () g() =limf () g () =limf () g () = :() ;() (, +, -, 0, 0 +, - 0 ). () (a,b) f ()>0, f() (a,b) ; () (a,b) f ()<0, f() (a,b).. () : 0 f(), f (0)=0 f (0). () : 0, f (), 0 ; f (), 0 ; f (), 0. (3) : f() 0, f (0)=0, f (0)>0, 0 ; f (0)<0, 0 ; f (0)=0,. 3. () : a, [ b] ; ;3,. () : f() a, [ b],. f(),. 4. () : (a,b) f ()>0, f() (a,b) ( ); (a,b) f ()<0, f() (a,b) ( ).. () : : 0 y=f(), f (0)=0 f (0) : f() 0, f () 0, 0 ; f () 0, () lim f () lim f (), ( + -, lim f ()=c,lim f ()=c c c, + - :y=c,y=c.

130 0 =a. () lim a +f ()= lim a -f ()=, y=f() f() (3) lim =a, y=f(), b=lim (f()-a), y=a+b. 6. : () ; () ; (3) ; (4) ; (5) ; (6) ; (7). () : C (q) q ; R (q) q ; L (q) q, :L (q)=r (q)-c (q); () : EQ p = p dq Qdp p, % EQ p %; ER p = p dr Rdp =+EQ p p, % ER p %; (3) EQ p <-( EQ p >),, ; 0>EQ p >-( EQ p <),, ; EQ p =-( EQ p =),,. A. ξ. ()f()= --3, [-,. 5 ]; ()f()= +, [-, ]; + <5 (3)f()= { 5, [ 0, 5 ]; (4)f()= 槡 6-,0, [ 6 ].. ξ.

131 3 ()f()= 3,[ -, ]; ()f()=ln,, [ e ]; (3)f()= 3-5+-, [-, ]; (4)f()=e +,0, [ ]. 3. :arctan+arccot= π, (-,+ ). 4. : ()y=e -; ()y= 3-3 ; (3)f()= ; 槡 (5)f()= +0 ; (7)y=(+) (-) 4 ; (9)y= 槡 -ln(+); 5. : ()lim - ln-+ ; lnsin (3)lim 0 - cot ; (4)f()= -ln (+) ; ( 6)y=arctan-; (8)f()= -ln; (0)y= 槡 ( lncos )lim 0 ; ( tan-sin 4)lim 0 sin ; e (5)lim -e - 烄 ; ln + 烌 (6)lim ; 0 + 烆 arccot 烎 ln - π (7)lim π + tan ; ( )lim ; (9)lim 0 ( - ln(+ ) ; ( 0)lim ( - - e - e ); ()lim ln; 0 + (3)lim (-)tan π ; (5)lim (tan) sin ; : ()y= (-) 3 ; (3)y=3 3 - ; (5)y= + ; ()lim ( e - ); ( 4)lim (+sin) ; 0 (6)lim () +. + ()y=e - ++; (4)y= + ; ( 6)y=(+) 槡 3 ; (7)y=arcsin-; (8)y=-ln(4) ; (9)f()=arctan- ln (+ ); (0)f()=e - ; ()y= 3-6 ; ()y=sin.

132 . 7. f()=asin+ 3 sin3 = π, a, 3 8. f()= +p+q =4 4, p,q. 9. Ⅱ, : ()y=(-3) (-); ()y= ; (3)y=sin+cos,0, [ π ]; (4)y=+e : ()y= ,0, [ 5 ]; ()y= +3 -, [, 5 ]; [ ] [ ] (3)y= +, -, ; (4)y= 槡 3 ln, 8,.. a,,. A, a, b, 3. 0 /, 40, 50, 5 0, 4. q=84-p( q,p ), 5 q +q, q=50 00 ( ), : () ;() (3) , 00, 3. ( ), ( ). : () F() ; (), F() F() 6. : ()y=+ 槡 3 5 ; ()y= + ; (3)y= (- ) ; ( 4)y=ln(+ ); (5)y=e - ; (6)y= a,b, (,3) y=a 3 +b. 8. : ()y=+ ; y=+ln;

133 3 3 (3)y= ln - ; ( 4)y=(+)e : ()y= (+) ; y= 3-3 +; (3)y=e - ; (4)y=-ln(+). 0. :C(q)=3+ 槡 q,r(q)= 5q q+, q,.. :q=800-0p,c(q)=5000+0q,, q=50 q=400.. : ()y=k α ; ()y=e k ; (3)y=4- 槡 ; (4)y=0 槡 q p :q=600 ( ) 4. p, p=3 4. p(q)=0e -q, 6, q,p, : (). () p=5,. (3) ( = ),. 5. y p y=p(75-5p),.8,. 6. q=800-0p,c(q)=5000+0q, : (),,. ().(3) q=400,. B. f()= 5 3,. A.=0 C.=0 B.=0 D.=0. f() a, [ b] ξ (a,b) f(), ξ f(). A. C. B. D.

134 4. 3. f() (a,b) f() (a,b) A. B. C. D. 4. f() f () a, [ b] :f (a)=0,f ()>0, (a,b), f() (a,b). A. C. B. D. 5. f() 0 f (0)=0,f (0) 0, 0 y=f(). A. C. 6. f()= ln. 槡 A. C. B. D. B. D. 7.. ()lim 槡 + sin (3)lim 0 sin ()lim (4)lim π + -arctan A.(),() B.(),(3) C.(),(3) D.(),(4) 8. 0 y=f(),. A.f (0)=0 C.f (0)=0 f (0) B.f (0) D. 9. Q=000e -0.p,. A. B. C. 0 D A.f()=ln,, [ e] B.f()= 槡 3 -, [-, ] C.f()=tan,0, [ π ]. f()= -.. f()=-(-) 3. e -e 3.lim (-) =. D.f()=, [-, ]

135 f()=asin+3sin 3 =π, a=. 5. y=+ln, [ e] ξ. 6. p+q-50=0, EQ p =. 7. =a y=f(), y=f() =a. 8. f()=+ 5, (+). 9. f()=a- 槡 3 -b. 0. f()=arctan( -).. lim 0 cos(sin)-cos 4.. f()= 槡 3 (- ). 3. f()=ln( 3 +) ,, 60, 30., , 00, 0%, 640..

136 4. :.,.,,,., () F() f(),, f(), F(), F() f()., F()=sin, F ()=(sin) =cos=f() cos sin. f()=cos,, cos, (sin) =cos, sin. sin cos. 4. f() I, F(),, F ()=f() df()=f()d F() f() I., ( ) = (-,+ ), (-,+ ) ( +C) = (C ), +C.C,,. ()

137 4 7, F() f(), F ()=f(), F()+C(C ) f() ; G() f(), G ()=f(), G()=F()+C f().,, ; F()+C. : f() I,.,.. 4. f() f(), f ()d F() f(),, f ()d = F()+C,,f(),f()d,C.,, C. cosd =sin+c d = +C. f()= α. α+ α+ d. =α, α+ α+ α, α d = α+ α+ +C. f()= =0. >0, (ln) =, <0, d =ln+c

138 8 [ ln(- ) ] = - ( -) = - ( -)=,, d =ln (-)+C d =ln +C ( 0). 4.. f() C, C,.,y =F()+C, f(). C,, f().,., y = F() +C y=f() y. 4., C,., =0 y =y0, y0 =F(0)+C C =y0-f(0),. 3 (,3). y = F().,y = F ()=k =, y = F()= d = +C =,y =3 4 3= +C C = y = ()( f ()d) =f() d f ()d =f()d; () F ()d = F()+C df ()= F()+C.

139 4 9, ;,. k, kf ()d =k f ()d.. (k f ()d) =k( f ()d) =kf(). k f ()d kf(),. 3, [ f ()±g( )d ] = f ()d± g ()d.. 3, 4..4 [ ()±f()± f ±fn( )d ] = f ()d± f ()d± ± fn ()d., : () 0d =C ; () kd =k +C (k ); (3) α d = α+ α+ +C (α,α ); (4) d =ln +C ; (5) a d = a lna +C (a >0,a ); (6) e d =e +C; (7) sind =-cos+c ; (8) cosd =sin+c ; (9) sec d = cos d =tan+c ;

140 30 (0) csc d = sin d =-cot+c ; () tansecd =sec+c ; () cotcscd =-csc+c ; (3) 槡 - d =arcsin+c =-arccos+c ; (4) + d =arctan+c =-arccot+c.,,. 4..5,, cos d. -+5cos 5 ( +) 槡 d. d = d- d+ 5cosd = d- d+5 cosd =ln sin+C =ln - +5sin+C. ( +) 槡 d = ( 5 + )d = 5 d+ d = C = C ( 3 - ) a 3 3 d. a ( 3-3 ) 3 d = ( a -3a a )d 4 =a 3 d-3a4 3 3 d+3a 4 3 d- d 槡 - d. 槡 =a a a C.

141 槡 - d = 槡 ( ) d =4 d+3 d- d 5-8 ( - 槡 )(+ 槡 ) d. = C. ( - 槡 )(+ 槡 ) d = - 槡 - d = - d d d. =ln - -+C =ln - 槡 -+C. + d = +- + d = ( - + ) d =-arctan+c. 4 + d = d = d+ + d = ( -)d+ + d = arctan+c. (+ ) d. (+ ) d = - d =- + -arctan+c. tan d. tan d = ( sec -)d = sec d- d =tan-+c. 3 sin d. sin d = -cos d = d- cosd = - sin+c. 4 cos cos-sin d.

142 3 cos cos-sin d = cos -sin cos-sin d 5 sin cos d. sin cos d = sin +cos = ( cos-sin)(cos+sin) d cos-sin = ( cos+sin)d =sin-cos+c. sin cos d = cos + sin d = ( sec +csc )d =tan-cot+c. 4.,.,,, ( arcsin) 槡 - d. (arcsin) :(arcsin) 槡 -, (arcsin) arcsin, 槡 - (arcsin) =f(arcsin) arcsin, 槡 - 槡 - = (arcsin), (arcsin) 槡 - (arcsin) (arcsin) ( arcsin) 槡 - d = ( arcsin) d(arcsin) = u du ( arcsin =u)

143 4 33 = 3 u3 +C ( ) = 3 ( arcsin) 3 +C ( u =arcsin),. (arcsin) (arcsin). arcsin φ ( ), f( φ ()) φ (). f( φ ()) φ (),, : u =φ ( ), f (u)du = F(u)+C, f ( φ ()) φ ()d = f ( φ ())dφ ( )= F( φ ())+C,. φ (),. cos d. d =d - =-d cos d =- cos d e d +e. =sinu+c =sin +C. e d =de =- cosudu =u e d +e = +e de = +u du( e =u) 3 槡 3+4 d. d =d, =arctanu+c =arctane +C. 槡 3+4 d = 槡 3+4 d = 槡 d(4 ) = 槡 d(3+4 )= 8 u du

144 34 = 8 3 u3 +C = ( 3+4 ) 3 +C 4 ( 3-4) 7 d. 3d =d(3-4), d = 3 d (3-4) ( 3-4) 7 d = 3 ( 3-4) 7 d(3-4)= 3 u7 du = 3 u8 8 +C = ( 3-4) 8 4 +C., u, : ( 3-4) 7 d = 3 ( 3-4) 7 d(3-4)= 3 (3-4) 8 +C 8 = ( 3-4) 8 4 +C. 5 3 d. e- 3 d =-3 3 d - 3 e- e- 3 =-3e- +C. 6 sin (-5)d. sin (-5)d = sin (-5)d = sin (-5)d(-5) =- cos (-5)+C. 7 tand. tand = sin cos d =- dcos =-ln cos +C. cos 8 tan3 d. cotd =ln sin +C. tan3 d = tan tand = ( sec -)tand = ( sec tan-tan)d = sec tand- tand = tandtan+ln cos = tan +ln cos +C.

145 cos d. cos d = +cos d = d+ cosd = d+ cosd () = +sin 4 +C. 0 secd. secd = sec (tan+sec) tan+sec = tan+sec d (tan+sec) =ln tan+sec +C. d = sectan+sec tan+sec d cscd =ln csc-cot +C. d. a + d = a + a + a = a arctan a +C. 6+ d. d = a + ( d a ) a 6+ d = 4 + d = 4 arctan 4 +C. 3 a 槡 - d. 槡 a - d = a 槡 4 槡 9-6 d. 槡 9-6 d = 槡 3 - (4) 槡 - d = - d a =arcsin a +C. a a = 4 arcsin4 3 +C. 5 a - d. d = 4 3 槡 - (4) d4

146 36 a - d = (a-)(a+) d = a a- + a+ d [ ] = a a- d+ a+ d = [-ln a- +ln a+ ] +C = a+ ln a a a- +C. d = -a ln -a a +a +C. 4..,,. - 槡 d. 槡, 槡 =t,. 槡 =t, =t,d =tdt, ( -t dt ) - 槡 d = -t tdt=- -t- -t dt=- dt- =- [ dt+ ] -t d (-t ) =-t+ln [ -t ] +C =- [ 槡 +ln - 槡 ] +C.,. : f ()d, =φ ( t), f ()d = f ( φ (t)) φ (t)dt φ (t), φ (t), φ (t) 0, =φ ( t) t=φ - (), f ( φ (t)) φ (t)d = F(t)+C f ()d = F( φ - ())+C. 6 d + 槡. t= 槡, =t,d =tdt, d + 槡 = t +t dt= t+- +t dt= dt- dt +t

147 d 槡 -3. =t-ln +t +C = 槡 -ln + 槡 +C. t= 槡 -3, =t +3,d =tdt, d 槡 - 3 = ( t = (-3) d 槡 - 3 槡. +3)tdt = t ( t +3)dt= 3 t3 +6t+C +6槡 -3+C. t= 6 槡, =t 6,d =6t 5 dt, d 槡 - 槡 3 = 6t 5 dt=6 t 3 -t t3 t- dt=6 ( t 3 -)+ t- dt 9 槡 e -d. =6 t +t++ dt=t 3 +3t +6t+6lnt- +C t- = 槡 +3 3 槡 +6 6 槡 +6ln 6 槡 - +C. t= 槡 e -, =ln(+t ),d = tdt +t, 槡 e -d = t dt = +t t +- +t dt= dt- dt +t =t-arctant+c =槡 e --arctan 槡 e -+C. 0 槡 a - d. =asint, d=acostdt,t=arcsin a, t [- π,π ], 槡 a - =a槡 -sin t=acost ( 4 ) 槡 a - d =a cos tdt = a ( +cost)dt = a ( t+ sin t ) +C 4 = a ( t+sintcost)+c = a arcsin a +a a a 槡 - a +C

148 38 = a arcsin a + a 槡 - +C. d. (- ) 3 =sint, d =costdt,t=arcsin,t - π,π, 槡 - = 槡 -sin t=cost ( 4 3) d = (- ) 3 sin t cos 3 t costdt = sin t cos t dt 4 3 槡 +a d. = tan tdt = ( sec t-)dt=tant-t+c = 槡 - -arcsin+c. =atant, d =asec tdt,t=arctan a, t (- π,π ), 槡 +a ( 4 4) -lna). =a槡 tan t+=asect 槡 +a d = asec t asect dt= sectdt =ln sect+tant +C =ln 槡 a +a 3 槡 -a d. + a +C =ln 槡 +a + +C ( C =C =asect, d =asect tantdt,t=arccos a, t [ 0, π ], 4 4 ( 4 5) 槡 -a =a槡 sec t-=atant

149 4 39 槡 -a d = asect tant atant dt = sectdt =ln sect+tant +C =ln 槡 a -a + a +C =ln 槡 -a + +C ( C =C -lna). 4 5,. : 槡 n a +b(a >0), 槡 n a +b =t; a 槡 - (a >0), =asint; 槡 +a (a >0), =atant; 槡 -a (a >0), =asect;,., : () tand =-ln cos +C ; () cotd =ln sin +C ; (3) secd =ln sec+tan +C ; (4) cscd =ln csc-cot +C ; (5) d = +a ln a - a -a +C ; (6) d = -a ln -a a +a +C ; (7) d = a + a arctan a +C ; (8) d a 槡 - =arcsin a +C ; (9) d 槡 ±a =ln +槡 ±a +C. 4.3,,

150 40, d, e- lnd,,.. u =u(),v =v(), : udv =uv- vdu.,, uv, vdu. udv vdu. udv vdu. u =u(),v =v(),,, d(uv)=vdu+udv, udv =d(uv)-vdu. udv =uv- vdu. : uv d =uv- vu d.. e- d e -,. u =,dv =e - d =d(-e - ), du =d,v =-e -. e- d = d (-e - )=-e - + e- d =-e - -e - +C., u=e -,dv=d=d, du=-e- d,v=, e- d = e- d = e - + e - d,,,.,u dv,, :

151 4 4 () dv v; () vdu udv. cosd. u =,dv =cosd =d sin, cosd = d (sin)= ( sin- sind ) = [ sin- sind ( ) ] 3 sind. = sin+ 4 cos+c. u =,dv =sind =d(-cos), sind = d(-cos)=- cos+ cosd =- cos+ cosd., sind,,,. u =,dv =cosd =dsin,, cosd = dsin =sin- sind =sin+cos+c sind =- cos+(sin+cos+c) =- cos+sin+cos+c ( C =C)..., u,v. 3, : n e a d, n sinad, n cosad, n, n u, dv. 4 3 lnd. ln, ln u, 4 3 lnd = lnd 4

152 4 = 4 4 ln- 4 4 dln = 4 4 ln- 4 4 d = 4 4 ln C. 5 arctand., arctan u, arctand =arctan- darctan =arctan- + d =arctan- + d(+ ) =arctan- ln (+ )+C. 4 5, : m lnd, m arcsind, m arctand, m -, ln,arcsin,arctan u, m d dv. 6 e cosd. cos u, e cosd = cosde =e cos- e dcos =e cos+ e sind =e cos+ sinde =e cos+e sin- e dsin =e cos+e sin- e cosd,,. e cosd,, e cosd =e cos+e sin+c

153 e cosd = (cos+sin)+c e : u =e,dv =cosd. 6, : ( C = C ) 4 43 cosbd, ea ea sinbd., e a u, cosb,sinb u.,,. 7 e 槡 d., t= 槡, =t,d =tdt,, 8 3 e d. e 槡 d = tet dt= tdet =(te t - dt) et =(te t -e t )+C =e 槡 (e 槡 - )+C., e,, u=,dv= e d =d e, 3 e d = d e = e - e d = e - d e = e - d e = e - e +C , F()( C(q), R(q) ) F ()( C (q), R (q) )., F () F(). F()= F ()d, C. 4.4., Q p Q =Q(p)., p =0,

154 44, Q0, Q0 = Q(p)p=0. Q (p), Q(p) (4.), C Q(p)p=0 = Q0. Q(p)= Q (p)dp (4.) Q p, 800( p =0,Q =800). ( ) Q (p)=-500ln ( ) Q p. (4.) [ ] Q(p)= Q (p)dp = -500ln p =-500ln p ln +C p =500 ( ) +C dp p,,p=0,q =800, C =300. Q p 4.4. p Q(p)=500 ( ) +300 q C (q), C0, q, C C(0)=C0. C(q)= C (q)dq (4.) q C (q)=e 0.q, C0 =90,. (4.) C(q)= C (q)dq= e0.q dq = 0. e0.q d(0.q) =0e 0.q +C C0 =90, q=0,c(0)=90 C = C(q)=0e 0.q +80 q R (q), q

155 4 45 R(q)= R (q)dq (4.3), C R(0)=0 (, 0 0). 3 R (q)=3q-q ( : / ), q ( : ),, q (4.3) R(q)= R (q)dq= ( 3q-q )dq =6q - 3 q3 +C,, R(0)=0 C =0. R(q)=6q - 3 q3 R(q)= R (q) q =6q- 3 q R (q)=6-3 q R (q)=0, q=4, R (q)=- 3 <0, q=4,,. 4, L(q) R(q) C(q), : L(q)= R(q)-C(q) L (q)= R (q)-c (q) q R (q), C (q), q L(q)= L (q)dq= [ R (q)-c (q )dq ] (4.4). : f() I, F(), F ()=f() df()=f()d I, F() f() I.

156 46. : f() f(), f ()d. F ()=f(), f ()d = F()+C.. ()( f ()d) =f() d f ()d =f()d; () F ()d = F()+C df ()= F()+C... kf ()d =k f ()d. 3.. [ ()±f()± f ±fn( )d ] = ()d± ()d± f f ± ()d. fn. :,,.. : f (u)du = F(u)+C, u =φ ( ), f ( φ ()) φ ()d = f ( φ ())dφ ( )= F( φ ())+C 3. :,. : 槡 n a +b(a >0), 槡 n a +b =t; a 槡 - (a >0), =asint; 槡 +a (a >0), =atant; 槡 -a (a >0), =asect; 4. : () : u =u(),v =v(), () : udv =uv- vdu. n e a d, n sinad, n cosad(n ), u= n, dv. m lnd, m arcsind, m arctand(m - ), dv = m d, v = m+ m +, u.

157 4 47, m =0, dv =d. 3 ea cosbd, ea sinbd,, u v., u =e a, u =e a.. Q(p)= Q (p)dp., C Q(p) p=0 = Q0(Q0 ).. C(q)= C (q)dq., C C(0)=C0(C0 ). 3. R(q)= R (q)dq., C R(0)=0 (, 0 0). 4. L(q)= L (q)dq= [ R (q)-c (q )dq. ]. : A () ( -3 )d; () ( 3- +4e -sin)d; ( 槡 - ) (3) d; (4) 槡 槡 槡 d; (5) 3 槡 d ; (6) 槡 (-3)d; (7) - e- e 槡 - d ; (8) d; (9) ( -) 槡 d ; (0) -9 槡 +3 d ; () e - e - d ; () ( -3 ) d; (3) ( +) 3 d; (4) cos d ;

158 48 (5) cot d; (6) sec (sec-tan)d.. 3, (,5),. 3. : () ( -6) 7 d; () 3 d; (3) arcsin 槡 - d ; (4) e d; 3+e (5) 槡 cos 槡 d; (6) d ; (7) 槡 - d ; (8) 槡 4 -d; (9) 槡 +ln d ; (0) d; 6+9 () d; () 6-9 槡 6-9 d ; (3) +4+5 d ; (4) +-6 d ; (5) cos sin d ; (6) cos d; (7) sin3 d; (8) sin4 cosd. 4. : () 槡 + d; () + 槡 3 d ; (3) 槡 (- 4 槡 ) d ; (4) 槡 4- d ; (5) 槡 (+ 3 ) d ; (6) 槡 - d ; (7) 槡 - d ; (8) 槡 +4 d. 5. : () sind ; () ln ( +)d; (3) arctand ; (4) e - d; (5) ln d; (6) arcsind ; (7) ln d; (8) 3 (ln) d; (9) e sind; (0) lnln d.

159 Q p, 000( p =0, Q =000), Q (p)=-40,. 7. q C (q)=e 0.5q, 6, C(q),. 8. R (q)=3-0.q,q, 0 0, R(q), 9. q( :kg) C (q)=000-0q+q ( : /kg), 9000, kg 3400, : (), ; () B. f ()d f(). A. B. C. D.. f() ln, f ()=. A. B.- C.ln D.e 3. f() a, f(). A. a lna +C B.a ln a +C C. a ln a +C+C D.a ln a+c+c 4. (a,b), f ()=g (),. A.f()=g() [ ] [ ] C. f ()d = g ()d 5. sin4. B.f()=g()+C D. f ()d = g ()d A.cos4 C.- 4 cos4 B. 4 cos4 D.- 4 sin4 6. f ()d = F()+C, e- f(e - )d =. A.F(e )+C B.-F(e - )+C C.F(e - )+C D. F (e - ) +C 7. f() sin, f ()d =.

160 50 A.sin- cos+c C.sin-cos+C B.sin+ cos+c D.sin+cos+C 8. esinθ sinθcosθdθ=. A.e sinθ +C C.e sinθ cosθ+c B.e sinθ sinθ+c D.e sinθ (sinθ-)+c 9. F() f()=0.4-6, f(0)=60( : / ), F()=. A C d d e- =.. d =. 3 + B D f ()d =sin+c, f ()=. 4. f ()d = F()+C, f (5+)d =. 5. f ( )= ( >0), f()=. 6. f()= cos, f ()d =. 7. f ()d =arcsin+c, f()=.. : () +sin d ; () 槡 + d ; (3) arccot 槡 + d ; (4) 3 -+ d.. : () sec3 d; () ( arcsin) d; (3) e 槡 - d; (4) ln (- 槡 )d. 3. f() sin, f ()d.

161 5 5.,. ;,, y=f()( 0), =a,=b(a<b) y= 0( ) ABCD. 5. () n. a=0<<< <n-<n=b, a, [ b] n 0, [ ], [, ], [ n-, n ], i-, [ i ],i=,,,n. Δ, Δi=i-i-,i=,,,n, Δ=ma { } Δ i. i n 5,, n ( 5 ). i ΔSi,i=,,,n. (). i-, [ i ](i=,,,n) ξ i,, f( ξ i) f( ξ i)δi. ( 5 ) (3) n. ΔSi f( ξ i)δi,i=,,,n.

162 5 n n f( ξ i)δi, i= ( 5 ), n n S = ΔSi f( ξ i)δi i= i= (4). a, [ b], n, 5 n Δi,, f( ξ i)δi i= S. n, n, S. a, [ b], Δi,, :. n Δ 0 i= S =lim, : f( ξ i)δi =, v t v =v(t), t=a t=b, a, [ b]. tn-,t n () n. ( 5 3) a, [ b] n t0,t [ ],t,t [ 3 ],, [ ], [ ti-,t i ],i=,,,n Δti =ti -ti-,i=,,,n Δt, i Δt= ma { } Δt i. i n Δsi,i=,,,n. 5 3 (). ti-,t [ i ](i=,,,n) λi, v(λi), v(λi)δti, Δsi v(λi)δti,i=,,,n. (3) n. n n v(λi)δti i= [ a, b]

163 5 53 n n s= Δsi v(λi)δti i= i= (4). a, [ b], n, Δti, n,n, v(λi)δti i=. n,, s. [ a, b], Δti,, s,. n Δt 0 i= s=lim v(λi)δti, :..,,.,, f() a, [ b], a, [ b] n a =0 < < < <n- <n =b [ i-, i ] (i=,,,n) Δi =i-i- (i=,,,n) Δ = ma { } i i n Δ i-, [ i ] ξ i, n f( ξ i)δi i= Δ 0,, a, [ b], ξ i, f() a, [ b], f() a, [ b], b a f ()d, n Δ 0 b f ()d =lim f( a ξ i)δi i= f(),f()d,,a,b,a, [ ] b.

164 54, b a f ()d, f() a, [ b ]., b a f ()d = b a f (t)dt, b a f ()d, a<b,,,, :, b a f ()d =- a b f ()d, : 5. a a f ()d =0 f() a, [ b], f() a, [ b]. ;. 5. f() a, [ b], f() a, [ b].,,. a, [ b], f(),.., y =f() 0, =a, =b(a<b), S y =f() a, [ b] S = b a f ()d, a, [ b], f(), S = b a f ()d = b ad =b-a, a, [ b], ( 5 4).,. f() 0, y =f(), =a, =b ( 5 5)., b a f ()d. S =- b a f ()d

165 5 55 f() a, [ b], 5 6, b f ()d, a. S = c a f ()d- d c f ()d+ b d f ()d. a, [ b], f()>0,f ()>0, : f(a)(b-a)< b a f ()d <f(b)(b-a). a, [ b], f()>0,f ()>0, y=f(), abca = b a f ()d, f(a)(b-a)< b a f ()d <f(b)(b-a). - 槡 - d = π. y= 槡 -, [-, ] ( 5 8),, y = 槡 - -, [ ] ; π. 槡 - d = π ,,., b a [ f ()±g( ]., )d = b a f ()d± b a g ()d b kf()d =k a b f ()d a (k )

166 56 3 f()=, 4( ) b d =b-a a a,b,c, b a f ()d = c a f ()d+ b c f ()d c [ a, b ], 5 9, ABCD = ACEF +BDEF, b a f ()d = c a f ()d+ b c f ()d. c a, b c f ()d = a c f ()d+ b a f ()d 5 9 c b,. 5( ) b a f ()d = b c f ()d- a c f ()d = c a f ()d+ b c f ()d f() g() a, [ b] :f() g (), b a f ()d b a g ()d. [ a, b],g()-f() 0, b a g ()d- b a f ()d = b a, b a f ()d b a g ()d. : [ g ()-f( )d ] =lim [ Δ 0 g ( ξ i)-f( ξ i)δi ] 0 () d 0 3 d; () 0 e d e d. () 0, [ ], 3, 0 d 3 d. 0 (), [ ], e e, 6( ) e d e d f() a, [ b] M m, 5. 7( ) m(b-a) b a f ()d M(b-a) f() a, [ b], a, [ b]

167 5 57 ξ, b f ()d =f( a ξ )(b-a) ( 5 0): a, [ b],f( ξ ) abba abdc. f( ξ ). f( ξ )= b-a b f ()d a f( ξ ) f() a, [ b], f() a, [ b] f() a, [ b],,.,.,., :, f() a, [ b], [ a, b ], f ()d., a f ()d,, a, [ b], a Φ(), Φ()= a f ()d.,,, t, Φ()= a f ()d = a f (t)dt Φ() : Φ(), 5, Φ(),, Φ(), Φ(). 5.3 f() a, [ b], Φ()= f (t)dt, [ a, b] a 5

168 58 f() a, [ b],, lim Δ 0 Φ ()= ( a f (t)dt) =f() Φ(+Δ)-Φ() =f(), [ a, b ]. Δ +Δ [ a, b ], Φ(), Φ(+Δ)-Φ()= +Δ f (t)dt- f (t)dt a a = a f (t)dt+ +Δ = +Δ f (t)dt f (t)dt- a f (t)dt Φ(+Δ)-Φ()=f( ξ )Δ, ( ξ +Δ ) f( ξ )= Φ (+Δ)-Φ() Δ Δ 0, +Δ, ξ, f(), lim Δ 0 Φ(+Δ)-Φ() =lim Δ f ( Δ 0 ξ )=lim f ( ξ ξ )=f() Φ ()=f()., : f(), f(). 4.,. Φ ()= f (t)d t =f() : a , f() a, [ b],f() f() a, [ b] b f ()d = F(b)-F(a)= F() b a a F() f(), Φ()= f (t)dt f() a,, C: F()= Φ()+C F()= f (t)dt+c a

169 5 59, =a, F(a)= a a f (t)dt+c =C F()= a f (t)dt+f(a), a f (t)dt= F()-F(a). =b, b a f (t)dt= F(b)-F(a) b f ()d = F(b)-F(a) a (Newton) (Leibniz).,.,. Φ() : ()Φ()= t +t dt ; ()Φ()= dt; +t 3 () Φ ()= ( a f (t)dt) =f() t +t dt= +. Φ ()= d d () Φ ()= ( f (t)d t ) =f() a (3)Φ()= 槡 +t dt. Φ ()= d d dt= d +t 3 d ( - dt)=-. +t (3) 槡 +t dt,, u=, Φ ()= d d 0 d. 槡 +t dt= d du u 槡 +t dt du d = 槡 +u ( ) = 槡 + 0, [ ], 3 3, d = 槡 槡 - d = 8 3-0= 8 3. arcsin, 槡 - 槡 3 0 槡 3 d =arcsin 0 槡 - =arcsin 槡 3 -arcsin0= π 3.

170 60 π 3 4 sin cosd. - π 3 π 3 π 3-3sin cosd =3 sin d(sin)= sin3 π - π 3 3 = 3 3 槡 槡 3 [ (- )] = d. π 3 - π 3 槡 3 4.,. - = -, 0 { -, <, - d = 0 0(-)d+ (-)d 6 - d. = -, - d = - ( - ) =-. = + =.. >0,. [-, ],. [-, ], , 4,. 9 + 槡 d.,,, 槡, u = 槡. : u = 槡, =u, d =udu,

171 5 6 + 槡 d = u +u du :, [ 9 ], ; u= 槡 u,, u, [ 3 ], 9 + 槡 d = 3 u +u du = 3 u +u du = 3 (+u)- +u du = 3 - du + u 3 =[ u-ln +u ] =4 (-ln5+ln3).,.,,,,., b a f ()d, : 5.5 f() a, [ b], =φ ( t), : () φ (t) α, [ β ] ; () φ (α)=a, φ ( β )=b; (3) φ (t) α, [ β ] φ (t), 槡 3 槡 4- d. b a f ()d = β α f ( φ (t)) φ (t)dt. =sint, d =costdt,: 槡 3;t: π 6 π 3. 槡 d. + π 3 槡 4- (cost) d = π π 6 4sin t dt= 3 cot π tdt 6 π 3 = (csc π t-)dt= (-cott-t) 6 = 槡 π 槡 3- π 6 槡 3 =- 3 + 槡 3- π 6 u =+, du =d,: 3,u: 0. 3 π 6 d = sincosd. 0 u du = 0 ln u = ln5 u =sin,du =cosd,:0 π 6, u:0. π 3 π 6

172 6 π 6 = u = udu 0sincosd 0 0 = 8 π 6 π 6 π 6 = sind(sin)= 0sincosd sin 0 π 6 π 6 0 = 8 = 0sincosd sind =- cos 0 4 = 0 8 ( ),,,. 5 f() -a, [ a], : a -a f ()d =0. f(), f(-)=-f(). a -a f ()d = 0 -a f ()d+ a 0 f ()d 0 -a f ()d, =-t,d =-dt,:-a 0,t:a a f ()d = 0 a f (-t)(-dt)= 0 a f (t)dt=- a 0 f (t)dt=- a 0 f ()d, f(), a -a f ()d = 0 -a f ()d+ a 0 f ()d π 6 =- a 0 f ()d+ a 0 f ()d =0 a -a f ()d = a 0 f ()d u(),v() a, [ b], a b, (uv) =u v+uv uv = (uv) -u v b auv d = b a(uv) d- b avu d =uv b a - b a vu d.

173 e - d. u =,dv =e - d, du =d,v =-e -, 0e - d = -e e - d =- 0 e -e- =- 0 e. π 7 cosd. 0 u =,dv =cosd, du =d,v =sin, π 0 π π cosd =sin - sind = 0 π π 0 +cos = π lnd. u =ln,dv =d, du = d,v =, 9 0arcsind. lnd = ln - d =ln- 4 =ln u =arcsin,dv =d, du = 槡 - d,v =, 0arcsind =arcsin 0-0 = π + 槡 - d 槡 - 0 = π ,,,, :,, b>a, f() a, [ b], lim b + b f ()d, a f() a,+ [ ), + f ()d, a

174 64 + f ()d = lim a b + b f ()d. a + f ()d. lim b + b f ()d, a a + f ()d. a, f() - (, b] b f ()d = lim - a - b f ()d. a f() (-,+ ) + f ()d = - c f ()d+ - + c f ()d c,, ;,. + e - d. 0 b>0, lim b + b e - d = lim (-e - ) b = lim (-e -b 0 b + +)= 0 b + + e - d. 0 + e - d =. 0. b>0, lim b + b e - d = lim 0 b + b d(-e - )= lim (-e - b + 0 b + 0 b e - d) cosd. 0 b>0,, + cosd a <, lim a - a = lim (-be -b +-e -b )= b + + e - d =. 0 lim b + b cosd = lim sin b = lim sinb 0 b + 0 b + (-3) d. d (-3) = lim a - a d(-3) (-3)

175 - =- lim a - =- lim a - (-3) d = ( - 3 ) a (-- ) = a- 3,, : + f ()d = F() + a a. 5 + d d = d d 0 + =arctan +arctan - 0 = lim - (0-arctan)+ lim + (arctan-0)= π f() (a,b], a +,f(), ε(ε<b-a), f() a+ε, [ b], lim + b f ()d, f() (a,b] ε 0 a+ε, b a f ()d, b a f ()d =lim ε 0 + b a+ε f ()d b a f ()d. lim ε 0 + b a+ε f ()d, b a f ()d., a., b b-ε f ()d =lim a ε 0 + f ()d a b a f ()d = c a f ()d+ b c f ()d (b ) (c a <c<b), ;,. 6 3 槡 - d.

176 66 lim 槡 - =,. + lim ε ε 7 0 槡 - d. d =lim 槡 - 3 ε ε 3 =lim + 槡 - ε 0 +ε (-) - d(-) =lim ε 0 + (- 槡 ε)= d =. 槡 - lim 槡 - =,. - -ε lim ε d. -ε d =lim arcsin 槡 - ε =lim [ arcsin(-ε)- 0] = π ε 0 + 槡 - d = π. lim =,. 0 - d = 0 d+ - d 0 d. - 0 d =lim : : - ε ε ( ) d =lim - ε 0 + =lim -+ ε 0 + ε =+ d = - ε - =--=- =0,,,.,,,.

177 5 67 f() a, [ b],. 5.6,. : (). n, n S = ΔSi i= ()., (3) (4) S =lim ΔSi f( ξ i)δi,i=,,,n. n n S = ΔSi f( ξ i)δi i= i= n Δ 0 i= f( ξ i)δi = b a f ()d, f( ξ i)δi f()d, ds., Q () a, [ b], ; (),+d [ ] ΔQ dq =f()d d, Q. :,, [ a, b ]( ); a, [ b],+d [ ], Q ΔQ, Q dq =f()d; 3 f()d, a, [ b], Q = b f ()d. a,,+d [ ],,,, y=f(),y=g()(f() g ()) =a,= b(a <b) ( 5 )., [ a, b ], a, [ b],+d [ ] f()- g() d, ds = [ f()-g( )d. ] f()-g( [ )d ], a, [ b], S = b [ f ()-g( )d. ] a

178 68 g()=0,, S = b a f ()d. =φ ( y), =ψ ( y)( φ (y) ψ ( y)) y=c,y=d(c<d) ( 5 3). y, c, [ d ], [ c, d] y,y+d [ y] φ (y)-ψ ( dy, ds = [ φ ( y)-ψ ( y)dy. ] [ φ (y)-ψ ( y)dy ], c, [ d], S = d [ φ( y)-ψ ( y)dy. ] c 5 y = y =4., 5 4. { y = (-,4) (,4). y = 4, -, [ ],,4, 8 ( 3 ) =3 3. S = -(4- )d = = a + y b =(a,b>0) S = πab., 5 5, S S S y = b 槡 a a - (0 a) =0 y =0., 0, [ a ],,y = b 槡 a S =4S =4 a b 槡 0 a a - a - d( =asint) = a 4b π π 0a cos +cost tdt=4ab 0 dt =4ab t +sint 4 π 0 =4ab π 4 = πab. y =0,

179 , y =ln (e, ). k= =e = e, y- = e ( -e), y = e. 5 5 S : [, e] S., S, 0, [ ] S 0, [ ], e 0, S = 0( e -0 ) d = e 0 = e, [ e],y = e y =ln, S = e ( e -ln ) d = e ( e -ln+ ) 5 6 = e - e - S =S +S = e -. y, 0, [ ],,e y ey, = ( e- e ) - (-0)= e -. S = 0(e y -ey)dy = e y - e y,,.,,, y =f() =a, =b(a<b) ( 5 7),.., a, [ b ], a, [ b],+d [ ] f() d, 0

180 70 dv = π[ f( )] d. V = b π [ f ( )] d. a =φ ( y) y =c,y =d(c<d) y y ( 5 8) V = d π [ φ ( y) ] dy. c O P(h,r), =h ( 5 9),., OP y = r h, V = h π r 0 h[ ] d = π r h h d 0 = π r 3 h 3 h 0 = 3 πr h. 5 y = y = 槡 V V,V y = 槡, =,V y =, =, V =V -V = 0 π ( 槡 ) d- 0 π ( ) d = π [( 槡 ) - ( ) ] d 0 = π [ - 4 ] d 0 ( ) = π = 3 0 π.

181 π 0( 槡 - ) d V,.. 6 y=, y= y= y y. y, V = π y dy = π - y = 3 π. 7 C (), C0, R (),. C() R(), C()= 0C (t)dt+c0. R()= 0R (t)dt , /,. () , 00 ; (). () ΔR = R(300)-R(00) = 300 R ()d- 00 R ()d 0 0 = 300 (0-0.0)d 00 = (0-0.0 ) 300 = () L() R() C(), L()= R()-C() 00 = [ ] 0(0-0.0t)dt-

182 7 = /, 0- /. () (), () L()= R()-C(). L(),, L ()= R ()-C ()= (0-)- =0-4. L ()=0, =5. L ()=-4 < 0, =5,. 5. (), 7, L(7)-L(5)= 7 L ()d = 5 7 5(0-4)d = (0- ) 7 5 = f() a, [ b], b a f ()d. b f ()d, f() a, [ b ], a,. b a f ()d = b a f (t)dt. : f() a, [ b], f() a, [ b] ; :() f() a, [ b], f() a, [ b] ; () f() a, [ b], f() [ a, b]. 3. b a f ()d S, y = f(), = a, =b.

183 5 73 (), b a f ()d ; (), b a f ()d ; (3), b a f ()d.. b [ f ()±g( )d ] = a b f ()d± a b g ()d. a. b akf()d =k b a f ()d (k ). 3. f()=, b d =b-a a 4. : a,b,c, b a f ()d = c a f ()d+ b c f ()d 5. : f() g() a, [ b] :f() g(), b a f ()d b a g ()d 6. : f() a, [ b] M m, m(b-a) b a f ()d M(b-a) 7. : f() a, [ b], a, [ b] ξ, b f ()d =f( a ξ )(b-a). : f() a, [ b], Φ()= f (t)dt, [ a, b] a f() a, [ b], Φ ()= ( a f (t)dt) =f(). : f() a, [ b],f() f() [ a, b], b f ()d = F() b = F (b)-f(a) a a 3. : f() a, [ b], =φ ( t), : () φ (t) α, [ β ] ; () φ (α)=a, φ ( β )=b; (3) φ (t) α, [ ] β φ (t),

184 74 b a f ()d = β α f ( φ (t)) φ (t)dt :,,. 4. : u(),v() a, [ b],. b b uv d =uv - a a b vu d a. f(), + f ()d a + f ()d = lim a b + b f ()d a lim b + b f ()d, ;,. a, + b f ()d = lim - a - b f ()d a f ()d = - c f ()d+ - + c f ()d c,, ;,.. b f ()d, c [ a, b] lim a +f ()= lim -f ()=, c c b a f ()d, c. () a. b a f ()d =lim ε 0 + b a+ε f ()d () b. b f ()d =lim a ε 0 (3) a a <c<b. + b-ε a f ()d b a f ()d = c a f ()d+ b c f ()d..,, a, [ ] b ( ); Q dq =f()d;

185 5 75 [ a, b], Q = b f ()d. a. () y=f(),y=g() =a,=b(a<b) S = b f ()-g()d a () =φ ( y),=ψ ( y) y=c,y=d(c<d) 3. S = d φ( y)-ψ ( y)dy c () y =f() =a, =b(a<b) V = b π [ f ( )] d a () =φ ( y) y =c,y =d(c<d) y y 4.. V = d π [ φ ( y) ] dy c C()= 0C (t)dt+c0 R()= 0R (t)dt A. f() a, [ b],0 [ a, b ], 0 f ()d, a. : () d; () b a d. 3. : () d; () 0 槡 4- d; 0 π (3) (4) - sind; π cosd. π -π

186 76 4. : () π sind >0; () 0 π 0cosd >0; (3) d = 0 ; ( 4) a 槡 a - d = πa , : () d 0 d; 0 (3) e lnd e ln d; 6. : π () π 4 (4) π 0d sind; 0 π 4 0sind cosd. 0 ()F()= 槡 +t dt; ()F()= 0 -t e 3t dt; (3)F()= 槡 +t dt ; (4)F()= costdt. 7. : () 3 d; () 0 a 0( +)d; (3) + d; (4) 4 槡 (+ 槡 )d; 槡 3 (5) d; 槡 (6) 0 槡 - d ; 槡 3a (7) d; (8) 0 a + 0 槡 4- d ; (9) d ; (0) 3 0+ d ; π 6 () 0tan θdθ; () π cos d; 0 (3) 3 f ()d, f()= +, 0 { 3, >. 8. : () π sin + π π 3 d ; () - d (+5) 3; π (3) 0sinφcos 3 φdφ ; ( 4) π (-sin 3 θ)dθ; 0 π 槡 (5) π 6cos udu; (6) 槡 - d; 0 dy; (8) 槡 - d; 槡 (7) 槡 8-y - 槡 (9) a 0 槡 a - 槡 槡 3 d; (0) d ; 槡 +

187 5 77 () d ; () - 槡 d ; + 槡 (3) 槡 d ; (4) 3 4 槡 -- a d ; 0 槡 3a - (5) te -t dt; (6) 0 e d ; 槡 +ln (7) 0 π d - ++ ; ( 8) coscosd; - π π (9) 槡 cos-cos 3 d; (0) - π π 槡 +cos d : () π ( )d; () 5cos - 4 d; - π (arcsin) (3) - 槡 - d ; (4) 3 3 sin d ; (5) 4 - d; (6) 0 π -π 4 sind. 烄 +, 0 0. f()= 烅, f (-)d. +e, 0 <0 烆. : () e - d; 0 π (3) π 3 () cosd; 0 ); (4) π 0sin d( 6 cos d ; (5) 9 ln 槡 d ; π 3 槡 3 (6) arctand; 0 π (7) 0e cosd; (8) 4 lnd; (9) π (sin) d; 0 (0) e sin(ln)d; () e ln d; () arcsind. e 0. : () + e ln d ; () + d; (3) + +- d ; (4) d ; (5) + d; (6) d ; 槡

188 78 (7) 0 槡 d ; (8) ln 0 d ; (9) e d; (0) d. 3. ()3y = 3 =y ; ()y =sin,y =cos, =0 = π ; (3)y = 3 +y =4 ; (4)y =3,y = y = ()y =e,y =e - =,. ()y = y =,. (3)y = =,. (4)y =,y = y =3, y. (5)y =ln, =0,y =0 y =, y. (6) =y +y =, y. 5. V = 4 3 πr3. 6..( : (-a) +y =r y, 0<r<a) 7. t( : ) Q (t)=0+0t-3t, /, () 00 ; () q C (q)= 50 q+30, 900, 0. C ()=+0.5( / ), R ()=6-( / ), () 5 ; () B. f() a, [ b] f() a, [ b]. A. B. C. D.. f() a, [ b] f() a, [ b]. A. B. C. D.

189 f() a, [ b], y=f(), =a,=b,y=0. A. b a f ()d B.- b a f ()d C. b a f ()d D. b a f ()d 4. f()d =k 0 =. 0f()d, k A.4 B.3 C. D. 5. d d b aarctand =. A.arctan B.arctanb-arctana C.0 D f () a, [ b], f (b)=a,f (a)=b, b f ()f ()d =. a A.a-b B. ( a-b) C.a -b D. ( a -b ) π 7. M = - π 4)d,. π sin cos d,n = π e -e ( - ) +cos d,q = π ( 3 e - - π A.N < Q < M B.M < Q < N C.N < M < Q D.Q < M < N 8. f()= 0(t-)e t dt, f(). A. -e B. e- C. -e D. e- 烄 9. f()= 0, <0 烅,k >0, 烆 ke -k, 0 + ()d = ( - ). A. B.- C. D. 0. b a0d =. A.0 B.a-b C.b-a D.. a f (t)dt. A.f () C.f() B.f () D.f() 0 sintdt. lim =. 0 tdt 0 A.0 B.- C. D. 3.. A. e d B. 0 - 槡 - d

190 80 C d D. e 0. ln d 4.. A. + d B 槡 - d C. + 槡 d D. + d 5.. A. 0 e - d B. - 0 槡 d C. d D. 0 ln 0 d 6.. A. + ln(+)d B. 0 4 槡 - d C. d D. 0 (-) -3 + d 7. y =, =-, =,y =0. A. d B. - 0 d C. 4 槡 ydy D. 0 4 槡 ydy 0. 0e - d =.. b a(-a)(-b)d =. 3. f()= {, 0 f ()d =., 0, < 4. y y e t dt+ 0 sintdt=, dy 0 d =. 5. a 03 d =8, a =. 6. f(), b a f ()d = + b c f ()d 7. + d 3 -- = 槡 (-). 9. y = -, = 0, =,y = y =ln, =,y =,,.

191 5 8. y = e = a(a > 0) 7, a =.. : π π π () =0; () = cosd. - sind π -cosd π 0., : () a, [ b], f() >0,f () >0,f () <0, (b-a) f (a)+f(b) < b f ()d < (b-a)f(b) a () a, [ b], f() >0,f () <0,f () >0, (b-a)f(b) < b f ()d < ( b-a) f (a)+f(b) a (3) a, [ b], f() >0,f () <0,f () <0, (b-a) f (a)+f(b) < b f ()d < (b-a)f(a) a 3. f()= +槡 f ()d, 0 f ()d. 4. f(0)=,f()=3,f ()=6, 0f ()d. 5. F()= 0te -t dt f (t-n)e n dt=sin, f(). 7. f() a, [ b], φ ()= a(-t) f(t)dt, ( a,b). : φ ()= a(-t)f(t)dt. 8. φ ()= 0t(t-)dt. 9. : m (-) n d = 0 0 n (-) m d. 0. f()= e -t dt, f()d. 0. f(3+)=e, 0 f ()d.

192 6,,,,,., y, S=y., y, S, S y., - (Cobb Douglas) Q=cK α L β, c,α, β, L>0,K>0,Q,Q K L.,, O, O, ( ) y ( ) z ( ),.,,,, 90 y, z.,, Oyz. O, 6.,. y Oy ; y z Oyz ; z Oz.,. 6,y,z, Ⅰ. Ⅰ,Ⅱ,Ⅲ,Ⅳ Oy, z ; Ⅴ,Ⅵ,Ⅶ,Ⅷ Oy, Ⅰ Ⅴ,Ⅱ Ⅵ, Ⅲ Ⅶ,Ⅵ Ⅷ Oy., M,. M, y z P,Q,R.,y,z P,Q,R

193 6 83. M,y,z, (,y,z)., (,y,z), P, y y Q, z z R, P,Q,R y z, M (,y,z). 6.,, M (,y,z).,. : O(0,0,0). 6 (,0,0),y (0,y,0),z (0,0,z). Oy (,y,0),oyz (0,y,z),Oz (,0,z). 6.. M(,y),M(,y) MM = (-) 槡 +(y-y), M(,y,z),M(,y,z) MM = (-) +(y-y) 槡 +(z-z) M(,y,z) O(0,0,0) MO = 槡 +y +z. M(-,,0) M(,0,3). A(,0,0), AM = AM : (,0,0). 槡 (+) +(0-) +(0-0) = (-) +(0-0) 槡 +(0-3) = -4+3,=,.,.. f(,y)=0, F(,y,z)=0. 6., S F(,y,z)=0 :

194 84 () S (,y,z) F(,y,z)=0; () S (,y,z) F(,y,z)=0, F(, y,z)=0 S, S F(,y,z)=0. P0(,-3,0),. P(,y,z), PP0 =r. 槡 (-) +(y+3) +z =, P0(,-3,0), (-) +(y+3) +z =. 3 P(,-,) P(,0,) P. P(,y,z), PP = PP. 槡 (-) +(y+) +(z-) = (-) +(y-0) 槡 +(z-) +y-z+=0 P.,P PP., z=0 Oy,, z=k (0,0,k) Oy. z. Oyz z=y z, : z= +y O, z +y -z = z +y =a F(,y,z)=0,y,z,. F(,y,z)=0 (,y,z ),

195 6 85,. 6..4,., Oy, P (, y), P (,y). E,,, : D={(,y) (,y) E} C={(,y) >y } Oy y = : C={(,y) +y <} Oy, P0(0,y0) Oy,δ. P0(0,y0) δ P(,y), P0 δ, U(P0,δ), U(P0,δ)={P PP0 <δ} U(P0,δ)={(,y) 槡 (-0) +(y-y0) <δ},u(p0,δ) Oy P0(0,y0),δ>0 P(,y).., Oy..,,,.,,,...,., {(,y) +y }, {(,y) +y>0}, {(,y) +y 0}.

196 ,,. : V r h V=πr h., r,h {(r,h) r>0,h>0} (r,h),v. V r h., Q K( ) L( ) Q=40K 3 L 3, K,L(K>0,L>0), Q. Q K L. 6. D Oy, D (,y), f, z, z,y, z=f(,y),(,y) D,y,z,D.,y 0,y0, z z0 z=f(,y), =0,y=y0, z0=f(0,y0), f(,y) ( 0,y 0 ).. {z z=f(,y),(,y) D} z= 槡 - -y., +y. - -y 0 D={(,y) +y } D,, 6 8. z= 槡 ln (+y)., +y>0 { > 0

197 6 87 D={(,y) +y>0,>0}, D +y>0,>0 (,y), {(,y,z) z=f(,y),(,y) D} z=f(,y)., z= - 槡 -y, ,. P(,y) P0(0,y0), (,y) (0,y0). P(, y) P0(0,y0) 0, ρ= PP0 = 槡 (-0) +(y-y0) z=f(,y) P0(0,y0) ( P0(0,y0) ), P(,y) P0(0,y0), f(,y) A, f(,y) (,y) (0,y0) A, : lim f(,y)=a lim f (,y)=a 0 (,y) ( 0,y ) 0 y y 0 : (,y) (0,y0),f(,y) A, (,y) (0,y0),f(,y) A., (,y) (0,y0), 0. (,y), (y ) ( 0,y0),, (,y) ( 0,y0) f(,y).

198 88 (,y) (0,y0),, (,y) (0,y0), (,y) (0,y0),. 3 siny lim (,y) (,0). siny y = y. lim y =0 siny, lim (,y) (,0) (,y) (,0) =0. 4 f(,y)= y +y, (,y) (0,0). (,y) (0,0), y=, f(,y) y= = y +y = y= (,y) y= (0,0),f(,y). y=, f(,y) y= = y +y = y= 5 (,y) y= (0,0),f(,y) 5. (,y) (0,0),f(,y), (,y) (0, 0),f(,y).,. 5 lim ( +y). y 0 y 0,,y 0 lim ( +y)=. y z=f(,y) (0,y0), z=f(,y) (0,y0). lim 0 y y 0 f(,y)=f(0,y0), 5 z= +y (0,0). z=f(,y) D, z=f(,y) D. D D.

199 6 89 f(,y) (0,y0), z=f(,y) (0,y0), (0,y0)., z= y - y =,.,. f(,y) D, f(,y) D ,.,,,,. z=f(,y) (0,y0), y=y0, Δ, (0,y0) (0 +Δ,y0) ( 6 ), Δz=f(0+Δ,y0)-f(0,y0) z=f(,y) (0,y0)., z=f(,y) (0,y0) y Δz y=f(0,y0+δy)-f(0,y0) z=f(,y) (0,y0), Δz lim Δ 0 Δ =lim f(0+δ,y0)-f(0,y0) Δ 0 Δ, z=f(,y) (0,y0), f (0,y0) f (0,y0), z,z (0,y0). ( 0,y ) 0, z=f(,y) (0,y0) y : Δz y lim Δy 0 Δy =lim f(0,y0+δy)-f(0,y0) Δy 0 Δy f (0,y0) y f y (0,y0), z,z y (0,y0). y ( 0, y ) 0 z=f(,y) D (,y) ( y),,y, f(,y) D ( y),. f (,y)( f y (,y)) f (,y), z, z f (,y) y, z y, z y.

200 90, f(,y), y,, y=f(). f(,y) y,, y.. f(,y)= 3-3y +y 4. y,, f (,y)=3-3y f y (,y)=-6y+4y 3 f(,y)=yln( +y ), f (,), y f (,0)., y, y,, f =y +y ( +y ) y = +y f y =ln ( +y )+y +y ( +y ) y=ln( +y )+ y +y f (,)= y +y (,) = y f (,0)= ln( +y )+ y +y (,0) =0 3 z= y (>0, ), : z y + z ln y =z. 4 (0,0). f(,0)=0,f(0,y)=0 z =yy- z y =y ln z y + z ln y = y yy- + ln y ln=z 烄 y f(,y)= +y, (,y) 0 烅烆 0, (,y)= 0 f(0+δ,0)-f(0,0) 0 f (0,0)=lim =lim Δ 0 Δ Δ 0 Δ =0 f(0,0+δy)-f(0,0) 0 f y (0,0)=lim =lim Δy 0 Δy Δy 0 Δy =0

201 6 9, f(,y) (0,0). 4, (0,0), (0,0) ,. A,B,,y C. C=C(,y) C C, (,y), A,. C y C y, (,y), B,. 5 Ⅰ Ⅱ, C(,y)= +0y+3y ,y Ⅰ Ⅱ ( : ), Ⅰ 300 Ⅱ 50,. Ⅰ Ⅱ y C (,y)=+0y C y (,y)=0+6y Ⅰ 300 Ⅱ 50 C (300,50)= =300 C y (300,50)= =6300, Ⅱ 50, 300 Ⅰ 300 ; Ⅰ 300, 50 Ⅱ p, m, Q=Q(p,m) Q p ; Q m. E p = p Q Q p, m, p %, Q

202 9., Em, Em= m Q Q p p, Q %, Q. - Q=f(L,K)=λL α K β L Q K β Q L =λαlα λl α K =α ; β K Q Q K =λ βl α L β λl α L = β β z=f(,y) z z y, (,y)., z=f(,y)., : z z z z = z = (f (,y)) =f (,y) z y = y z y = y z = y y f y (,y) f y (,y). = (f (,y)) y=f y (,y) = (f y (,y)) =f y (,y) y = (f y (,y)) y=f yy (,y),. 6 z= 3 +y 3-3 y 3. z =3-6y 3, z y =3y -9 y z z z = z = ( 3-6y 3 )=6-6y 3 z y = y = y ( 3-6y 3 )=-8y z y = = y ( 3y -9 y )=-8y

203 z = z y y y = y ( 3y -9 y )=6y-8 y. 7 z=arctan y. z = -y + y =- y +y z y = + y = +y ( y +y ) ( +y ) z = - y =- -y +y ( +y ) = y ( +y ) z y = y - z y = 6 93 =- +y -y y ( +y ) = y - ( +y ) = +y - ( +y ) = y - ( +y ) z = y y = - y +y ( +y ) = -y ( +y ). z y = z y z,,,,,,. 6. z=f(,y) z z y, D, y D, z y = z y., ,,. z=f(,y) (0,y0),,y Δ,Δy, Δz=f(0+Δ,y0+Δy)-f(0,y0) z=f(,y) (0,y0)., Δz, Δ,Δy Δz., y, S=y, +Δ,, y y+δy.

204 94 6 3, ΔS =(+Δ)(y+Δy)-y =yδ+δy+δδy, ΔS, yδ+δy Δ Δy, ΔΔy ρ = Δ 槡 +Δy.,, yδ+δy ΔS z=f(,y) (0,y0), (0, y0) Δ,Δy, z=f(,y) Δz=f(0+Δ,y0+Δ)-f(0,y0) Δz=AΔ+BΔy=o( ρ ),A B Δ Δy, y,o( ρ )( ρ = Δ 槡 +Δy ) ρ 0 ρ, z=f(,y) (0,y0), AΔ+BΔy z= f(,y) (0,y0),,dz (0,y 0 ) df(0,y0) dz =df (0 (0,y0)=AΔ+BΔy,y ) 0 z=f(,y) (,y), Δ, Δy, dz Δz, Δz dz z=f(,y) D, D., z=f(,y) (0,y0) dz A B 6.( ) z=f(,y) (0,y0), () z=f(,y) (0,y0) ; () z=f(,y) (0,y0), A=f (0,y0),B=f y (0,y0), z=f(,y) (0,y0), z=f(,y) (0,y0) : dz =f (0,y0)Δ+f y (0 (0,y0)Δy,y ) 0, Δ,Δy d,dy z=f(,y) dz =f (0,y0)d=f y (0 (0,y0)dy,y ) 0 z=f(,y) D, z=f(,y) D (,y) dz=f (,y)d+f y (,y)dy : y=f(,y),,

205 6 95, z=f(,y), f (,y),f y (,y), z,. 烄 y 4 f(,y)= +y, (,y) 0 烅 (0,0) 烆 0, (,y)= 0, (0,0). 6.3( ) z=f(,y) (0,y0) f (0,y0),f y (0,y0), (0,y0), z=f(,y) (0,y0)., : {. 8 z= y + 3 +y 4. z =y +3, z y =y +4y 3 dz=(y +3 )d+(y +4y 3 )dy. 9 z=e sin(+y). z =e sin(+y)+e cos(+y), z cos(+y) y =e dz= z d+ z y dy=e [sin(+y)+cos(+y)]d+e cos(+y)dy 0 C A B,y : C= -0.5y+y. A 00 05,B 50 5, ΔC dc=cδ+c yδy,=00,δ=5,y=50,δy=, =(-0.5y)Δ+(y-0.5)Δy ΔC ( ) 5+( ) =975 A 00 05,B z=f(u,v), u=φ ( ),v=ψ ( ),

206 96 z=f( φ (), ψ ())., φ () ψ () z. z, z. 6.4 u=φ ( ),v=ψ ( ), z=f(u,v) ( φ (), ψ ()), z=f( φ (), ψ ()),. dz d = z du u d + z dv v d z=u v,u=cos,v=sin, dz d. z u =uv, z v =u du d =-sin, dv d =cos dz d = z du u d + z dv v d =-sin cos+cos 3 z=arctan(y), y=e, dz d.,. : z = y + y, z y = + y,dy d =e dz d = y + e = ( +)e + y + y + e.,. 6.5( ) u=φ (,y),v=ψ (,y) (,y),z= f(u,v) (u,v), (u,v)=( φ (,y), ψ (,y)), z=f( φ (,y), ψ (,y)) (,y), z = z u u + z v v, z y = z u u y + z v v y 3 z=( +y ) y, z, z y. u= +y,v=y, z=u v. z = z u u + z v v =vuv- ()+u v lnu y = y( +y ) y- +y( +y ) y ln( +y ) [ ] =( +y ) y y +yln( +y +y ) z y = z u u y + z v v vy =vuv- (y)+u v lnu

207 6 97 =y ( +y ) y- +( +y ) y ln( +y ) [ ] =( +y ) y y +ln( +y +y ),,,,, z u,v, u,y,,y, u,v. :, ;,, ;,,. :,,,. 4 z=e u cosv,u=y,v=+y, z, z y. z = z u u + z v v =eu cosv y-e u sinv =e y [ycos(+y)-sin(+y)] z y = z u u y + z v v y =eu cosv -e u sinv =e y[cos(+y)-sin(+y)] 5 z=u lnv,u= y, v=-y, z, z y., z u =ulnv, z v =u v, u = y u y =- y, v =, v y =- z = z u u + z v =ulnv v y +u = ln(-y)+ y y (-y) z y = z u u y + z v v y =ulnv - y + u [ ] =- ln(-y)+ y 3 y (-y ) 6 z=f( -y,y) f, z, z y. v v ( -) u= -y,v=y z=f(u,v)., :

208 98 z = z u u + z v v = z u +y z v z y = z z u y + z v v y =-y z u + z v 7 u=f(,y,z)=e +y +z,z= siny, u, u y. u = f + f z +y +z z =e +ze +y +z siny =(+ sin y)e +y +z u y = f y + f z +y +z z y =ye +ze +y +z cosy =(y+ 4 sinycosy)e +y +z 8 z=y+u,u=φ (,y)( φ (,y),y ), z,z y z z u =y+ u =y+ φ z =(z ) = y+ φ φ ( ) = z y= y y+ φ =+ φ y F(,y)=0 y,, F(,y)=0 y dy d,,. F(,y), F y 0, F(,y) y =f(), y=f() F(,y)=0, F(,f())=0,f(),., F y 0,. F + F dy y d =0 F - dy d = F y =- F (,y) F y (,y), F(,y,z)=0 z=z(,y), F(,y,z)

209 6 99, F z(,y,z) 0, z (,y,z) z (,y,z) =-F F z(,y,z) y =-F y F z(,y,z) 9 ye +arctany=y y=y(), y (0). F(,y)=ye +arctany-y, F (,y)=ye,f y (,y)=e - y +y (,y) y ()=- F ye y(+y )e F y (,y) =- =. e - y y -(+y )e +y =0, y=0. y (0)= y (+y )e y -(+y )e =0. =0 y=0 0 e -y -z+e z =0 z,y, z, z y. F(,y,z)=e -y -z+e z, F =-ye -y,f y=-e -y,f z=-+e z z =-F F z = ye-y e z -, z y =F y F z = e-y e z , z=f(,y) M0(0,y0), M0(0,y0) M(,y) f(,y)<(0,y0)( f(,y)>f(0,y0)) f(0,y0) z=f(,y) ( ), M0(0,y0) z= f(,y) ( )... z=f(,y)= +y, (0,0),f(0,0)=0. (0,0) (0,0). z=f(,y)= 槡 - -y, (0,0),f(0,0)=. (0,0) (0,0) f(0,0)=>f(,y),(0,0) (,y),,.

210 ( ) z=f(,y) (0,y0), (0,y0), f (,y)=0, f y (,y)=0 z=f(,y) (0,y0),, (0,y0), (0,y0) (,y) f(,y)<f(0,y0)., (,y0) (0,y0), f(,y0)<f(0,y0). f(,y0) =0,, f (0,y0)=0 f y (0,y0)=0. z=f(,y), ( ). :,.., (0,0) z=y, (0,0),. z(0,0)=0, (0,0).,., z= 槡 +y (0,0), (0,0), 0, 6 5.,., ( ) z=f(,y) (0,y0), (0,y0), f (0,y0)=0,f y (0,y0)=0, A=f (0,y0),B=f y (0,y0),C=f yy (0,y0) () B -AC<0, z=f(,y) (0,y0), A<0,f(0,y0) f(,y) ; A>0,f(0,y0) f(,y). () B -AC>0, f(0,y0). (3) B -AC=0, f(0,y0). f(,y)= 3 +y 3-3y. 烄 f (,y)=3( -y)=0 烅烆 f y (,y)=3(y -)= 0 ( 0,0) (,).

211 6 0 f (,y)=6,f y(,y)=-3,f yy (,y)=6y. (0,0), (0,0). (,), A=0,B=-3,C=0 B -AC=(-3) -0=9>0 A=6,B=-3,C=6 B -AC=(-3) -6 6=-7<0 (,). A=6>0, (,), f(,)=-. f(,y)=y -4 y+3 4. (0,0). 烄 f (,y)=-8y+ 3 =0 烅烆 f y (,y)=y-4 =0 f (,y)=-8y+36,f y (,y)=-8,f yy (,y)=. (0,0), A=0,B=0,C= B -AC=0-0 =0 6.7 (0,0) f(,y). y= f(,y) =- 4, y=4 f(,y)=3 4, (0,0), f(,y)>0 f(,y)<0, f(0,0)= , f(,y) D, f(,y) D, f(,y) D, D., : f(,y) D,, ( ) f(,y) D ( )., f(,y) D,.,,,.,,. 3 8m 3,,

212 0 m, y m, 8 y m.,=y=. A= ( y+y 8 y + 8 ) = y+ 8 y +8 y 烄 A = y- 8 =0 烅 A y= ( - 8 y ) =0 烆 ( >0,y>0),, D:D= {(,y) >0, y>0}. D (,), =y=,a, m, m, m,. 4 A B, p=,p=8( : ), C( : ) C(,y)= +y+y,, : : (,4). R(,y)=+8y L(,y)=R(,y)-C(,y) =+8y- -y-y (>0,y>0) 烄 L =-4-y=0 烅烆 L y=8--4y= 0, D:D={(,y) >0,y>0}, =,y=4, L(,4)= ,,,.,. :,y,z, V=yz,,y,z (y+yz+z)=.,,,., φ (,y)=0, z=f(,y), z=f(,y).

213 6 03 z=f(,y)( ) z=f(,y). y=y() φ (,y)=0, z=f(,y), z=f(,y()) z=f(,y), φ (,y), φ y (,y) 0.,, z.,,. dz d =f (,y)+f y (,y) dy d =0 dy (,y) d =-φ φ y(,y) f (,y) φ (,y) =f y (,y) φ y(,y) f (,y) φ (,y) =f y (,y) φ y(,y) =-λ λ. (,y) 烄 f (,y)+λφ (,y)=0 烅 f y (,y)+λφ y(,y)=0 烆 φ (,y)= 0 (6.) F(,y,λ)=f(,y)+λφ (,y) (6.),y,λ. (6.) (,y,λ) F(,y,λ)., z=f(,y) φ (,y)=0 F(,y,λ). (6.) F(,y,λ),λ,. : () ( ) F(,y,λ)=f(,y)+λφ (,y) (), 烄 F =f (,y)+λφ (,y)=0 烅 F y=f y (,y)+λφ ( y,y)=0 烆 F λ =φ (,y)= 0 (,y,λ), (,y). (3) (,y).,.,,,.. 5 S( : ),y( : )

214 04 S= y 0+y,. 5, L, L= 5 S--y= y 0+y --y +y=5.., F(,y,λ)= y 0+y --y+λ (+y-5) 烄 00 (5+) -+λ=0 烅 00 (0+y) -+λ=0 (5+) =(0+y) +y-5=0, =5,y=0.,. 5 0,. 6 y( ), C(,y)=8 - y+y ( ), +y=4.. +y=4, =5,y=7. F(,y)=8 -y+y +λ(+y-4) 烆 +y- 5=0 烄 F =6-y+λ=0 烅 F y=-+4y+λ=0 烆 +y- 4=0 (5,7), C(5,7)= = ,,.,. :, P(,y) P 0 ( 0,y0),f(,y)

215 6 05 A,.., A.. ()z= 槡 +y; ()z= 槡 +y + ; 槡 -y (3)z=ln(--y); (4)z= 槡 +y 槡 -y ; 槡 (5)z= 4-y ln(- -y ) ; ( 6)z=arccos +y 6 + 槡 +y -4.. =, 槡 3 y=-, z=ey +ln(-y). 3. f(,y)= -y+y, f(+δ,y)-f(,y),f(,y+δy)-f(,y). 4. z=f(u,v)=u v, f y, y,f(+y,-y). 5. f +y, y = -y, f(,y). 6. -y ()lim 0 +y ; y (3) lim (,y) (0,0) lim (,y) (,0) ln(+e y ) ; 槡 +y y ; (4)lim +y. 0 槡 y+ - y +(-y) 7.. y 0 () f(,)= y + -y, f (,0),f y (,0); () f(,y)=sin(+y), f (π,0),f y (π,0). 8.. ()z= 3 y-y 3 ; ()z= y -y ; (3)z=arctan y ; ( 4)z= 槡 3 -y ; (5)z= 槡 ln(y ); (6)z=e sin cosy; (7)z=sin(y)+cos (y); (8)z=(+y) (+y ) ; (9)z=ln y ; ( 0)u=sin(y+e z ).

216 06 9. z=e - ( + y ), : z +y z y =z. 0. : z=ln(e +e y ).. ()z= 3 +3 y+4y 3 ; ()z= y ; (3)z=arctan +y -y. z z - y z ( y ) =0.. z=ln(+ +y ), (0,-). 3.. ()z=y+ y ; z=ln 槡 +y ; (3)z=e +y ; (4)z=(y). 4.. () z=uv,u=e,v=sin, dz d ; () z=e -y,=sint,y=t 3, dz dt ; (3) z=arcsin(u,v),u=3,v=4 3, dz dt ; (4) z=arctany, y=e, dz d ; (5) z=u +v, u=+y,v=-y, z, z y. (6) z=u lnv, u= y, v=3-y, z, z y. (7) z=arctan(u,v), u=-y,v= y, z, z y ; (8) z=e uv, u=ln 槡 +y,v=arctan y, z, z y. 5., f. () z=f(+ 槡 y), z, z y ; () z=f( +y, y ), z, z y. (3) z=f( y,e y ), z, z y. 6.. ()siny+e -y =0; ()+y+z- 槡 y z=0.

217 z=y+f(u), u= y, F(u). : z +y z y =z+y. 8.. ()z= -y+y +9-6y+0; ()z=4(-y)- -y ; (3)z=e (+y +y); (4)z=-( +y ) q q C(q,q)=50q+00q+q +qq+q =0000 () C(q,q) q q ; () q=3,q=6,. 0. R,.. k,,.. y( : ), 34, y C(,y)=6 +0y -y A B,,y( : ), ( : ) L(,y)=- +8y-3y -, 000kg, 3000kg.,, 4., p p, q q, q=48-0.4p,q=0-0.p C=35+40(q+q), 5.., R( : ) ( : ) ( : ) R= (),, ; ().5,,.

218 08 B. (,-,) Oyz.., M(4,,9),N(-,,). 3. f(+y,-y)=y, f(,y). 4. f(,y)= 槡 - +槡 y z=f( +y ), f, z =, z y =. 6. z = - y,z y (,). (,) z (,) = 7. f(+y,-y)= -y, f (,y)+f y (,y)=. 8. z = siny, z =, z y =, z =. y 9. z= y (3,). 0. z=ln(y), dz=.. z=f(,y) +yz+z =, z =.. (,0) f(,y)= -+y +9.. (,3,-),. A.(+) +(y+3) +(z-) = 槡 4 B.(-) +(y-3) +(z+) =4 C.(+) +(y+3)+(z-) =4 D.(-) +(y-3) +(z+) = 槡 4. z= y + y, z -, 3 =., z y = A. 4 3 B C. 3 D.0 3. f(,y)= -y f(y,+y)=. A. --y C.+y- y B. y --y D.(+y) -y 4. f(,y)= y +y, f y, =. A. y +y B. +y y C. y + D

219 f(,y)= 槡 ln( -y ). A. -y B. -y>0 C. -y>e D. -y> 6. f(,y)=f()f(y) (0,y0),, f (0,y0). A.lim h 0 f(0+h)-f(0) h C.lim h 0 f(0+h)-f(0) h 7.,. B. f (0+h)-f(0) h D.lim h 0 f(0+h,y0+h)-f(0,y0) h A.f(,y) (0,y0), f(,y) (0,y0) B.f(,y) (0, 0 ), f(,y) (0,y0) C. f(,y) (0,y0), f(,y) (0,y0) D. f(,y) (0,y0), f(,y) (0,y0) 8. z= y, z y (e,)=. A.e B. e C. D.0 9. z=f(u), u=3-y, z y =. A.9f B.4f C.-6yf D.-yf 0. (0,y0) f (0,y0)=0,f y (0,y0),. A.(0,y0) f(,y) C.(0,y0) f(,y). z=f(,y), ( A. z=f(,y) C. z, z y. z= ln (+ +y ), dz (,) =. B.(0,y0) f(,y). D.(0,y0) f(,y). ), z y = z y. B. z=f(,y) D.z y,z y A. 3 ( d+dy) B.d+dy C. (d+dy) 槡 3 D. ( d+dy). z=arctan y +3 槡 ln(-y ).. f(,y), f u + f v =,

220 0 g + g y. g(,y)=f ( y, ( -y ), 3. +y +3z +y-z-9=0, z z (,-,). 4. f(,y)= -3y +y 4 5. f(,y,z)=ln+lny+3lnz, +y +z =5r Ⅰ, f(,y,z), a,b,c, 5 abc 3 7 a+b+c 5 6.,. =0-5P+3P y,y=0+3p-p y, P,P y, c= -y+y +37.5,. 7.,,Q. Q= α, α, β β, α+β=. p p, :,.

221 7 7.,., ,.., Oy D, D z, z=f(,y), f(,y) 0 D ( 7 ).. V.,, =., (,y) D, f(,y),. 5,,,. 7 () D n Δσ,Δσ,,Δσn,, z, Ω n ΔΩ,ΔΩ,,ΔΩn( Δσi ΔΩi, Δσi i,,δωi i, ). n V = ΔΩi i= () f(,y),,.,

222 , ΔΩi f( ξ i, η i)δσi ( ( ξ i, η i) Δσi) (, ΔΩi ) (3) n V = f( ξ i, η i)δσi i= (4) V, n,., :.,. n λ,. n λ 0 i= V =lim f( ξ i, η i)δσi. Oy D, (,y) ρ (,y), ρ (,y) 0, ρ (,y) D, M. D n Δσ,Δσ,,Δσn, λi Δσi,Δσi i. 7. λ= ma { λ i }, i n ρ (,y),, i ρ ( ξ i, η i)δσi ( ( ξ i, η i) Δσi ) n M ρ ( ξ, i η i)δσi i= n λ 0 i= M =lim ρ ( ξ i, η i)δσi 7,.,,,. 3. f(,y) D, D Δσ,Δσ,,Δσn.,Δσi i,,λi. λ= ma {λi}( ( i n ξ i, η i) Δσi) f( ξ i, η i)δσi (i=,,n)

223 7 3 n λ 0 i= lim D f(,y)dσ. n f( ξ i, η i)δσi i= f( ξ i, η i)δσi, f(,y) D, n λ 0 i= D f(,y)dσ=lim f( ξ i, η i)δσi :f(,y),f(,y)dσ,dσ, n,y,d, f( ξ i, η i)δσi. i= () f(,y) D, f(,y) D. :,. () D f(,y)dσ dσ Δσi. D, D,,,, dσ ddy( ddy ), D f(,y)dσ (3) f(,y) 0, f(,y), D α, β. ( ), D [α f(,y)+β g(,y)]dσ =α D f(,y)dσ+β D g(,y)dσ ( ) ( )( ). D D,D, 3 f(,y)dσ= f(,y)d+ f(,y)dσ D D D D,f(,y),σ D,

224 4 σ= D dσ= D dσ :. 4 D,f(,y) φ (,y),, D f(,y)dσ D φ (,y)dσ - f(,y) f (,y) f (,y), D f(,y)dσ D f(,y) dσ 5 ( ) M m f(,y) D,σ D, m σ f(,y)dσ M σ D 6 ( ) f(,y) D, σ D, D ( ξ, η ), f(,y)dσ=f( ξ, η ) σ D ( +4y +9)dσ,D +y 4. D f(,y)= +4y +9 D 烄 f = =0 烅 f 烆 y =8y = 0 (0,0), f(0,0)=9, f(,y)= +4(4- )+9=5-3 (- ) 3 f(,y) 5 fma =5,fmin =9 36π=9 4π I 5 4π=00π f(,y) D, D f(,y)dσ D,

225 7 5 z =f(,y)..d D = {(,y)a b,c y d}, f(,y)dσ= D b d a d f (,y)dy. c : [a,b] 0, yoz =0,, [c,d], z = f(0,y). 7 4., A(0)= d c f (0,y)dy., [a,b] yoz 7 4 A()= d c f (,y)dy,, V = b = aa()d a b [ d f (,y)d ] c y d, f(,y)dσ= D a b [ d f (,y)d y] d. c y, ( )., f(,y) y, y c d ; ( ) [a,b]. f(,y)dσ= f(,y)ddy D = D b d a d f (,y)dy c,, y f(,y)dσ= f(,y)ddy D = D c d [ b f (,y)d ] a = d cdy b a f (,y)d dy D e y dσ, D =0, = y =0,y =.( 7 5) D = {(,y)0,0 y }, e ydσ= D 0 [ e y d ] 0 y d = 0 y ( ) e y= [ ] y= 0 d

226 6 = (e -)d = (e -) = =e- 0 =0 - D ( 3 - y 4 ) dσ, D =- = y =-,y =.( 7 6) D D = {(,y)-,- y }, ( y 4 ) dσ= - [ - - ( 3 - y 4 ) d y] d ( - ) ( - ) [ ] [ ] = dy+ 0 d = - =4 - =8 = =0 =8. y= 3 y d y= 0 3 d =8 d :. : f(,y)ddy f() g(y), D f(,y)=f() g(y), D = {(,y)a b,c y d},, f() g(y)ddy = D [ b f ()d ] a [ d g (y)d y] c f() g(y)ddy = D b ad d f ()g(y)dy c = b a f ()d d c g (y)dy = [ b f ()d ] a [ ] d g (y)d y c 3 D y ddy, D =,= y=0,y=3.

227 7 7 D = {(,y),0 y 3}. y ddy = D d 3 0 y dy = 3 ( 3 ) y =. 4 D e +y ddy, D =0, = y=0,y=. D = {(,y)0,0 y }. e +yddy = D 0e d e y dy = ( e ) 0 0.D. ( ) e y 0 = (e-). D y D, y ( X )( 7 7). 7 7 D ( Y )( 7 8). 7 8 D y, ( 7 9). D ( 7 0), D,.

228 () D y. D = {(,y)a b,y() y y ()}, D f(,y)dσ= b a d y () y () f (,y)dy. : [a,b] 0, yoz =0,, [y(0),y(0)], z =f(0,y). 7., A(0)= y ( ) 0 y ( 0 ) f (0,y)dy., [a,b] z=f(,y) yoz A()= y () y () f (,y)dy 7, V = b a () D. A()d = a b y () [ ] f (,y)d y d, y () f(,y)dσ= D a b y () [ f (,y)d y () ] y d = b d a y () f (,y)dy. y () D = {(,y)c y d,(y) (y)}, () (3) D. f(,y)dσ= D c d ( y) [ f (,y)d (y) ] dy = d dy c ( y) f (,y)d. (y)

229 7 9 D = {(,y)a b,y() y y ()}, D = {(,y)c y d,(y) (y)}, D f(,y)dσ= b a d y () y () f (,y)dy = d dy c (y) f (,y)d. (y) :,, D, y, y. D :, y =a,=b, y=g(),y=h(), y., y,,,.,,., D, y. : D ; ; 3. 5 D yddy, D =,y = y =. D y ( 7 (a)), D [,], [,], D, y, y = y =. () D = D yddy = d ydy = = {(,y), y }, y= ( ) ( 3 4 ) - y ( 3 -)d = y=d = = 9 8 ( -)d D ( 7 (b)), D y [,], [,] y, D y,, =y =. () D = {(,y) y,y },

230 0 7 yddy = D dy yd = y ( ) y = =y dy = y (4-y )dy = (4y-y 3 )dy = y - y4 4 = ( +y)ddy, D y = y =, y, D y ( 7 3(a)), y, D = {(,y) 0, y 槡 }, ( +y)ddy = D 槡 0d ( +y)dy y= 槡 d y= ( ) = y+ y 0 = d = = D ( 7 3(b)),, D = {(,y) 0 y,y 槡 y }, ( +y)ddy = D 槡 y 0dy ( +y)d y = ( y ) = 槡 y dy =y 0

231 7 = 4-3 0( 3 y3 3 y6 -y ) dy = 8-5 y5 y7 - ( 4 ) y 4 = D yddy, D =0,y =0 +y =. D y,. () D y ( 7 4), y,. () D = { } (,y) 0,0 y 槡 - D yddy = 0 = 0 = 0 = 槡 d - ydy 0 ( ) y 0 槡 - (- ) 3 ( 3 ) -5 5 d 0 = 5, 7 4 () D,, y. (). D yddy = 0 槡 d -y y dy,,,., D, f(,y). 8 D (-y)ddy, D y =,-y+3=0 +y-3=0. 0

232 D y,. () D,, y, D = { (,y) y 3, ( y-3) 3- } y, D (-y)ddy = 3 dy 3-y ( y-3) (-y)d = 3 ( -y) =3-y dy = ( y-3) = (y -4y+3)dy = 9 4 ( 3 y3 -y +3 y ) 3 =-3 () D y, y,. D y D D : - 0,y y =, y =+3, D = {(,y) - 0, y +3}; 0,y y =, y =3-, D = {(,y) 0, y 3-},,,, y. 9 D yddy, D =y y =-. D y,. () D, 7 5(a),, y, D = {(,y) - y,y y+ }, yddy = D dy - +y yd y = - y =+y =y dy = (y 3 +4y +4y-y 5 )dy = 45-8 () D y, 7 5(b), y,, = D D D, D = {(,y) 0,- 槡 y 槡 },D = {(,y) 4,- y 槡 },,,,, y,. 0 d 0 siny dy.

233 d 0 siny dy, D y, siny dy, siny,,,. D y, y,, D = {(,y) 0, y },, D,, y, 7 6 D = {(,y) 0 y,0 y }. 0d siny dy = siny ddy D = dy 0 y 0siny d = 0 ysiny dy 7 6 = ( -cos) D f(,y)ddy,.,,. :, D; D,. D : ( ) ; ( ). D,,. d 0 f (,y)dy+ 0 槡 d - f(,y)dy., =0 =, = =,. =0 =,y 0, 0 y =0 y = ; = =,y 0 y = 槡 -, y=0 y= 槡 -. D, 7 7.,,, y 0

234 4, : 0d 0 f (,y)dy+ = dy 0 + 槡 y -y f(,y)d y 0dy y 0 f (,y)d+ dy 槡 d - f(,y)dy 0 f (,y)d. 槡 - y, y y =0 y=, y = y=,. y=0 y=, 0 y/, 0 y =0 =y/ ; y = y =, = 槡 -y =y/, = 槡 -y =y/. D, 7 8.,, y,, : y 0dy = d 0 y 0 f (,y)d+ dy + f (,y)dy 槡 -y f(,y)d R. +y = R, +z = R. 7 8, Ⅰ ( 7 9((a)) V, V =8V. Ⅰ, D( 7 9(b)), 7 9 { } D = (,y) 0 R,0 y 槡 R - z = 槡 R -.,

235 7 5 V = 槡 R - ddy = D R d 0 R 0 槡 - 槡 R - dy = R (R - )d = 0 3 R3 V =8V = 6 3 R3 7...,,., +y -y y y, f(,y)dσ. D. r θ., Oy ( ), r, θ, (r,θ).,... r θ. y, :θ=f(r),f(r,θ)=0,{r=r(t),θ=θ(t)}.. f(,y)dσ : D D, ; +y, r,θ. f(,y)dσ. D O D D. :r=, :θ=, D n ( 7 0). Δσi ri ri+δri θi θi+δθi,, Δri riδθi, Δσi Δri riδθi =riδriδθi, dσ=rdrdθ : P (,y),

236 6 (r,θ), :=rcosθ,y=rsinθ; f(,y) f(rcosθ,rsinθ). f(,y)dσ D D f(,y)dσ= D f(rcosθ,rsinθ)rdrdθ. 7 0 D f(,y)ddy = D f(rcosθ,rsinθ)rdrdθ.,,,y rcosθ,rsinθ, dσ rdrdθ, ddy rdrdθ.,. D,. () D r, r θ : O D, 7, D = {(r,θ)α θ β, r(θ) r r(θ)}, D f(,y)ddy = β α 7 dθ r (θ) r (θ) f (rcosθ,rsinθ)rdr; O D, 7, D = {(r,θ)α θ β, 0 r r(θ)}, f(,y)ddy = D β dθ α r( θ) f (rcosθ,rsinθ)rdr; 0 3 O D, 7 3, D = {(r,θ)0 θ π,0 r r(θ)},