Mathematics | Mathematical analysis » Dancs István - Analízis

     

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 (1) z1 z2 = r1 r2 (cos(α1 + α2 ) + i(sin(α1 + α2 )), 3 3  z n = r n (cos nα + i sin nα) n  r1 z1 = (cos(α1 − α2 ) + i sin(α1 − α2 )) (z2 6= 0). z2 r2 (2) (3)   Y"ZB 1 ?-&',2 424.?4'%/C%2%2M%? )*,'-2;4  z1 z2 = r1 (cos α1 + i sin α1 )r2 (cos α2 + i sin α2 ) = r1

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  9  95%/A%?;%22M4*)(/%. )*%&),' 5?+ 43)*-<%(&27++ (&+(.% < 4 )*R/% E%>5/ 4   (1 + i)  (1 + i) π/4 (1 + i)100   2  () >5 /A E% / √ √ 2 2 cos α = 1/ √ 1 + i = 2 (cos(π/4) + i sin(π/4)) .

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x| 5 1, sin π/2 = 1, sin π = 0, sin(x + π) = − sin x. 7    (8)  cos  sin 2 2π       9  V Y" > 5 5 > 5 1 2 K 2 2 + + ZB % ? , )-&-.%/A%? % %)%? .-?;%&/%?)% ' ? %?;(/%) ? %&;4 6' 5 2 G# 8 W# 2 # R + + + + 34?<%*(& 4* ))*%/CP//;(?A C-?-) 4 , Q ,) )*%& ? A Y0E%ZB>5/ 1D)4. E,&4>%V+?245?,2KU4)-&,34? 5 U D M=: 2 + + < )-&,34? )4. E,&-) . %; 4 )*%&%E% 4* ;,?A-? 5 2 # + ? %. 4* ?%. ;,?A4Y 46)*%&%E% YGZB CP//;(?A%.% 5 5 > 5 = 1 35*-?AQ+,)+# 4@2)-&-. DU 2 # J 2 +R + )*-&,)4 4 & (? . %? &% . ) *,'6. 4 () AZ ; = 5 M 2 # 1 2 D5 + 7 + ++ 33 4 CP//;(?A )*,'6. * 43)*.-?;%&/%? 4 ' 4 '%/%?/%>%++ 4 )-&-. 46DUA )*-&4+4B  cos(x + 2π) = cos x   sin(x + 2π) = sin x.  cos x sin x cos2 sin2 cos2  4U-2 2 cos x = ∞ X x2n (−1)n (2n)! n=0 ! · ∞ X x2n (−1)n (2n)! n=0 ! = ∞ X cn , n=0 X x2j x2k x2j+2k

(−1)k = (−1)j+k (2j)! (2k)! j+k=n (2j)!(2k!) j+k=n   (−1)n x2n X (2n)! (−1)n x2n X 2n = = . (2n)! j+k=n (2j)!(2k!) (2n)! k=0 2k X cn = (−1)j @*+ '4=> C-2A+4+=6.< >% %2M33 4 CP//;(?A+ 5) 5)*,'-2=6. %>>5/ 4 )*5?+5/B sin2 ! ∞ X 2n+1 n x (−1) (2n + 1)! n=0 ∞ 2n+1 X n x (−1) (2n + 1)! n=0 4U-2 <4 46DUA )*-&4+ >%V?QD5K=4 )%&5?+B 2 sin x =  dn−1 2j+1 · 2k+1 ! = ∞ X dn = n=0 dn−1 = X = (−1)j+k ∞ X dn−1 . n=1 x2j+2k+2 (2j + 1)!(2k + 1)! j+k=n−1  n−1  (−1)n−1 x2n X (2n)! (−1)n−1 x2n X 2n = = . (2n)! (2j + (2n)! 1)!(2k + 1)! 2k + 1 j+k=n−1 k=0 x x = (−1) (−1)k (2j + 1)! (2k + 1)! j+k=n−1 X j 1U-/A*%2MM ),'-2,)-.3%C%=%*()(U% 3%2,+6?.(+ 4*-?-)),/-+# 1 35?-'5,25) +(+%23M2 .4E=6< U4 U4 2 2n 2n = (1 + 1) 0 = (1 − 1)2n  2n  X 2n = , 0 5 n, k k=0   2n X k 2n 1 5 n. = (−1) , k k=0 $ %
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* &  @*%.%+ R))*%4>;4 52 %+;% .5;-?;4 4E=6< U-/A '5?>%? )*,'&4B () 1 %)()+(?B )*,'-2,),?4. 6+-2)K C-&'62,=,34? %*%.%+ V/A%2%'3% ;(;% 4*+ .4E=6< U-/A 15n 2 2n =2  n  X 2n cos2 n=1 2 2k k=0 2n  n−1  X 2n =2 . 2k + 1 k=0 sin2 (−1)n x2n 2n−1 2 , (2n)! (−1)n−1 x2n 2n−1 = 2 , (2n)! cn = () Q/A `*,'-2=6. 5 %*%. 6+,? 4 2 cos x = 1 + dn−1 ∞ X (−1)n x2n (2n)! n=1 1+ ∞ X (−1)n x2n n=1 (2n)! 2 2n−1 + 4)-?2K4? )*,'-2=6. 5 4 1+ ∞ X (−1)n x2n n=1 (2n)! cos2 x + sin2 x 2 2n−1 1+ − (2n)! n=1 (2n)! n=1 ∞ X (−1)n x2n .5C%=%*()+B n=1 ∞ X (−1)n−1 x2n ∞ X (−1)n−1 x2n 22n−1 . 22n−1 = (2n)! cos2 x − sin2 x 2 sin x = ∞ X (−1)n−1 x2n n=1 # 2 () .5C%=%*()+B 2n−1 2 2n−1 (2n)! =1+2 2 2n−1 − ∞ X (−1)n x2n ∞ X (−1)n x2n n=1 =1+ (2n)!

n=1 (2n)! 22n−1 = 1. 22n−1 ∞ X (−1)n (2x)2n (2n)! = cos 2x. Y8+(+Z%2 @)*%&33%?5?+B5) U4)-?2K ),'-2,)-. )*%&%E%2?%.< '5?+ 4* %2MM3%?# @2M)R& 5) 4 35?-'5,25) n=1 sin (x + y) = ∞ X (−1) n=0 ∞ X 2n+1 n (x + y) (2n + 1)! 2n+1 X 2n + 1 1 = (−1) xk y 2n+1−k (2n + 1)! k n=0 k=0 = ∞ X n=0 n (−1) n 2n+1 X xk y 2n+1−k . k! (2n + 1 − k)! k=0 $ %
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n=0 n=0 k=0 k=0 ∞ X n X n 9-)+ Q&=6. C%2 4 () 4 )*-&4+,+# $6>=6.< U-/A sin x cn = n X cos y sin x sin y = n (−1)k n X cos x sin y = P∞ n=0 dn , 4U-2 4U-2 k=0 n X x2k y 2n−2k+1 x2k y 2n−2k+1 dn = (−1) · (−1)n−k = (−1)n . (2k)! (2n − 2k + 1)! (2k)! (2n − 2k + 1)! k=0 k=0 k = U 2 2 K + + 4 2K4? '%/A.45C%',)=%*()(;% ))*%;%+;%%4)-? 34?4* .4E 6.< ;4 5. 4*-?-)),/ 5/4-2,)4 5)B -/A  n=0 cn , X x2k+1 y 2n−2k x2k+1 y 2n−2k · (−1)n−k = (−1)n . (2k + 1)! (2n − k)! (2k + 1)! (2n − 2k)! 9,)&()*&M2 ;52,/-)< U-/A k=0 P∞ cos x sin y. # sin (x + y) cos (x + y) = = ∞ X n=0 ∞ X (−1) 2n n (x + y) (−1)n (2n)! 2n X  2n  1 X 2n k 2n−k (−1) = x y (2n)! k n=0 k=0 n xk y 2n−k . k! (2n − k)! k=0 5 L(*P. 2 M # 1 + 43% ) )*6'', '%??A 5?>%X: +4/-. R))*%/(&%B 3%? n=0 ∞ X sin (x + y) = sin x cos y+ n > 0, k 7/A3-?+)6.C%2 4E,&-) 5?>%X: ()E,&4+24?

k 2n X n n−1 X X xk y 2n−k x2k y 2n−2k x2k+1 y 2n−2k−1 = + . k! (2k)! (2k + (2n − k)! (2n − 2k)! 1)! (2n − 2k − 1)! k=0 k=0 k=0 5 5 U U 1+4/C%?7 )*6''4 2 +5 C-&'62(&4 +'(/ + %)% (? ) / 4*< 4 %' ( .)*P?. 4&&4 4 '%/%/A%*()&%< -/A 4 V > 5 = = U (.(+ ?4 % ? ,2 6#
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C-?()+-) 2,+?5)-&4 $6>=>56.< )*6' cos x cos y = (−1)n = − (−1)n−1 P1∞ n=1 cos x  cos y n=0 cn , n n X X y 2n−2k x2k y 2n−2k x2k (−1)n−k = (−1)n . cn = (−1) (2k)! (2n − 2k)! (2k)! (2n − 2k)! k=0 k=0 P P∞ sin x sin y = ∞ n=0 dn = n=1 dn−1 , k 9,)&()*&M2 ;52,/-)< U-/A dn−1 = = 4U-2 n−1 X k=0 n−1 X x2k+1 y 2(n−1−k)+1 n−1−k (−1) · (−1) (2k + 1)! (2 (n − 1 − k) + 1)! k (−1)k k=0 n−1 y 2n−2k−1 x2k+1 · (−1)n−1−k (2k + 1)! (2n − 2k − 1)! n−1 X x2k+1 y 2n−2k−1 . (2k + 1)! (2n − 2k − 1)! 5 = 2 K 2 D 2 2 K R4*))%;% ++ + + + + 4 () 4 &% .4E( 6 ) C-&'6 , 4 -) . C % %*()(;% 34? ;% ;4 + .4E=6< U-/A V YF.RZB*%2Q+1() 2%U%()+)(/%)<CP//;(?A%.% > 5 = U 2 K 2 K 2 M 2 2 K : + + % ? , )-&-. 4.-* % % %.< () Q / 4)-? '%/ ;, A 5 5 5 > = 2 # 2 + + + 7 ++ + % 3? * QE6)E-5)-&-.?, )*-.,)-) '-) +-?C-/A &%E&- 6.,K<()6+4&4*+

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

?(2.P25 +4/= 45 '-?-+-?5+,),U-* 4 .%2 < U-/A 4*4 4'% 4)U4+KB 24,+)*45?6) )-&4.%*%>M2%/%%)%+ ++%)3%?;4? 4* %2#M331 .-)*'-?5?>6))-&4 -++4.?4<U%U2A4%++ D(2)*%&: 4< 5' ;%22A34M22%/.2%-5)2;%334-& = (−1) k=0 cn sin dn−1  cos (x + y) = cos x cos y − sin x sin y. cos ! ! (x + y) (an )0 n (−1) an n=0 s2n − s2n−2 = −a2n−1 + a2n < 0 s2n+1 − s2n−1 = a2n − a2n+1 > 0 s2n − s2n−1 = a2n > 0 an 0 P∞      s2n 1 sin x2n+1 x2n+3 > (2n + 1)! (2n + 3)! x2 < (2n + 2)(2n + 3), 0<x< √ 6 n=0 $ %
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'34? % 3 ? * Q E 6) 4 5 = > 5 5 2 2 2 M R R + ++ + + + )-&< () 4 3 * -?AQ ,) % % (? '-? 4. )*%& ? 4 )-&-. E,&-) () E,&4 4 ? . *% Q ))*%/% >U = U U U 5. 4 U24?+4,&(&4+(<4*44 # 9 R*(%)C%??, + + + () CP//;(?A -? 4 6. % , < -/A 4 < .4-& M 2 M 2 R;%+.%* %/A%? + %?)(/%.B YG#"SZ () U 5 1)%''.-)*55<?%(& K 2 M 2 2 2 ++ + + + 6)*CP//;(?A&%;-?4 .-* %/A%? % ?)(/3%?4* % (3% % (;%?%';, -* .4 A 5 5 5 2 # 1 2 M 2 2 + + + 4* C%??, %)% (3%? ) ',) . %/A%? % ?)(//% ) %* U@%*2+A%4 +.(< +>%%/A%? 5# 2)*%&:+ U44)?,%/A%?2=6.2M+4*2%?)(/% 2 + %UU%*2M+2D%(?)(/% 4 4 .34? ?(*? YF YS 5 U YF 5 U # 1 2 + Z () Z 3 * -?AQ ,), -* * Z / 4*, ), -*/ >5 2 M 2 M2 + %2M)*R& 5) =%/A%P. '%/< U-/A ?A52;,?;42K4? # 1 R*3P<42)MYG#E-?"SZ%/A%? % ?)(/(3 E% 5 2 2 + + + + ( % % )*%& ? 4? ;4? A U 5 2 2 # @ + )*%&+< 4'%? ?%' P&%)?%.4?%;%*P?.# @)*%&5?+ 4 '4< #()1;%/AP. 5 2 2 +

4'K>-? %/A(&< 4'%+%2'A&%:%? '%/U)*,'% * %? A A > > D 2 + ++ 4'%/,&-*-U4+,&-6.)*,'&4B < % %*+ ?4/A-? 6&;4+ '53%?>5)/ (?4/A )<() # @ + + .()(&>M%332M>'4()=>;%++E-?% +R-)4334? 4* (& (.( /A(3.(? 4 )*,'5 5 = 5 5 2 K 2 2 K 2 # + +R + + + + &P <)*,'Q /(EE% 33 ' %/A E-? -)),/ / . ) *,',.4* (& (.( YSZB 1 YG#"SZ 424++5 %/A%?2M+2%?)(/%.3M2B () U4 <() Q/A 9: : 
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0 π 0<π<4 π 1 − cos x x x x3 − < < 2 4! x 2 0 < |x| < √ 1− sin x x2 < < 1, 6 x 6   1 − cos x x x3 x = lim − = lim = 0, lim x0 x0 x0 2 x 2 4!   sin x x2 = lim 1 − = lim 1 = 1. lim x0 x x0 x0 6 YN%*(&ZB+ %33%? >5 > 5 U 1 YG#"S4*Z 45?24+%&;4 2 M 2 2 2 2 ++5 ',)+ + . %/A%? % ?)(/ 34 4 4 E-* Q < 4 < ; U 2 # @ 2 2 2 M2 + + 6 '34? * C% 4)*?, 44 %/A%? % 3 # 1 () (&+(.% 4* 4>>QD5K) C-&'62,.3; K2=R??%B sin x > 0 sin π2 = 1 cos π sin π x − x3 /3! 0<x< 2 π 2 π cos 2 + sin 2 = 1 √ 6 cos π = cos(π/2 + π/2) = cos π/2 cos π/2 − sin π/2 sin π/2 = − sin2 π/2 = −1, sin π = sin(π/2 + π/2) = cos π/2 sin π/2 + sin π/2 cos π/2 = 0. $ %
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x cos π + cos x sin π = − sin x, cos(x + π) = cos x cos π − sin x sin π = − cos x. YWZB 1* 4>>5D5K) C-&'62,.3K2B cos(x + 2π) = cos x cos π − sin x sin π = cos x, sin(x + 2π) = cos x sin π + sin x cos π = sin x. $ %
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'%/ % ;% 5 > 5 > # @ M M 2 + + + + ++ + ++ 7+ .-& 4 4& 4' 4 4 '%/ % -))*, 4 () <() 4 %/A)(/?A E-? 34?B5 5>M+4&+()4'&4%)M 7+U-))*B  t # t t (t + h) (s(t + h) − s(t)) s(t) s(t + h) s(t) h s(t + h) − s(t) . h 5 > 5 U M K2 M2 14.-&E&-3U2(4'4)*%'3% M 2 M R+2 + + + + + + 4& 4'?4/A),/, <C % / B 4. ) *,'(& (.%& )%?CP// % 4 5 5 > >= U = U = D M 2 2# ++ + 4 .-4 E) 52%*%? 4 '-?+< U6.4 4 /A-&)6 &%*2%U%%+MP.<2%/5-/A)%33<4-& .4E+ =6%2%?4+5<  4)%3%))(/% 5U>-/AME-?+34? 5 5 + + 4 ?4 ?A (& (.% 4 4' 4* U >5 >>5 5 > 5 U U5 2 2 # @ M D + + + + + + 4& 6?. 4 ?6 , -* * E% / % / 6 ,)6?. )*%& ? '%/ % % < ) *%? )4. ;4 5 2 .% 4 ;%??  h t h h U'44++,&(&;%*%++(.%=P+<4' 5 5 = 2 2 2 2# 1= 2 2 2 + + + + + + E%&)*% ?%' ( %* . C% ( % ?P % % ?C% %*% 3%? 4'-) %XE-?, C-/4 3%<() ?(U,?A

42.42'4*,),+ 5)'%&=P. '%/# s(t + h) − s(t) h0 h lim     '&( ) -(. $'( & ,& )-( ") *#0  7  5 7 02  f   2     x ∈ R   2 2   f 2 x    5        2   (    2   )   9  h 7     3     f (x + h) − f (x) , h  5 3  h 6= 0    0 2 2    2  f     (x + h) "F" "F0 >5 U > 5 5# U = 1 D5 2 2 2 R + ++ + %&%? 4 ,?A4 -) ',) C-&',34? ) )*-.,)-) Q & ? 4 % % 4* % ( ) A = 5 U > 2 R ;%*%+ P. 3%<4-& 4 P ?)(/ ,?A4 -)B Y8#"Z 1* < '-)+ 4 4-? .R&?A%*%+(3M2 ;%U%+M<4U-2 4 (&+%2'%%++#  # y 7 y (y 6= x) y f (y) − f (x) . y−x x  (x + h) f  6 '&( ) -(. $'( & ,& )-" ") *#  & ,(/" ! &  )! + )0  7  5 7 02  f   2     x   2 2

  f 2 x       2    h 7  f (x + h) − f (x) h h = 0  2  f (x + h) − f (x) h0 h lim        2     2  f     x       2   2          f 0 (x), d f (x), dx df (x) , dx Dx f (x), Y8#0Z Df (x).  3  5  2       1      f 0 (x)               2      9   2   7 f 2 Y8# 5 >5 U > > " 2 D5 2 2 2 : ++ + 4 4 Z 4 4 %&%? 4 ,?A4 -))4 / -*6?.< 4.-& %??%. (& % % ')*%& %? 4* U4*%2A%?.% 5 U U5 U U 2 2 + + + + + + 4 4 ,&(& (.( < ) *%?4 E-? -)4?4.-& 4& 4?6 , -*< 44* ;%?? U%*# /A R))%C-/242;4 4 >%V?QD5K .(+ 424=4B x  # h  x x y = x+h f (x + h) − f (x) f (y) − f (x) = lim . yx h0 h y−x f 0 (x) = lim 1%*(&>%+V4?QD5KE-?)%&+ 54?+ 4CP//;(?A +%2'+=%*;%< CP//;(?A(&+%4*2'%()E-?5

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> 5 > = 124*2?552A4%?>%'%/4 K> D K # R R + ,) ' &%.6&*Q % ?Q ?4. '-? 6?. 4 & %? ))*% .Q ,? 6. C-/ ; ; ; V 5 5 > 5 > D K D5 K 2 M 2 K 2 2 + 4* E-? %/A >? 6.D54*K) 2(?E()6.E%>5%/)4 <R;%?+%*6 MB (1E()% 4 2(+%(+5%()% R.?A%*%&?A%%+(3%?<+(3%?<?Q() 4+< 54.-& + . 4* E-? %/A .R& ;4/A 4'5 C-&',25)4? .P2R?R)%? )*%'2(2%+%)B f (x), n f (x), n . Dxn f (x) = Dx (Dxn−1 f (x)) Dx0 (x) = f (x). #  f f 0 (n−1) x  x . f n (x) = (f (n−1) )0 (x)  d dx n . d f (x) = dx  d dx (n−1) U U >>5 5 > 9 2 2 2 2 2 2 R + %/ .% %' Q %?P?.< -/A -/A 4* % / % ? %;%*()%.% () '%/, 4 E,)-.4 ))* >5 > 5 5 U U5)4?/34? U 2 2 + + + + 4 5?>46.D5K%&? 4;2,4E6?42K >?%;%*P.< %#)*14< %4>5'4/, 4*,2+ %CP//;(?A 4' %)*%& ?( 45?'%/+ 4 V > 5 = 5 = 2 M M 2 D K +'%/,%)*P?. + . %& * < % %) % ?Q ,34?< ; 2 4E->,))*%&5?+ 4 5)

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g 0(0) = 0 lim    *   1 >%V?QD5K )*%&5?+B 1   9   7   x 7 (sin 2x + x2 ) 2      x = 0   (sin 2h + h2 ) − (sin 0 + 02 ) sin 2h + h2 = lim = h0 h0 h  h  sin 2h sin 2h = lim 2 + h = 2 · lim + lim h = 2 · 1 + 0 = 2. h0 h0 h0 2h 2h lim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x 7 |x| R R+ 2 0<x |y| − |x| y−x d |x| = lim = lim = 1, yx yx dx y−x y−x x y x<0 y # x d |y| − |x| (−y) − (−x) |x| = lim = lim = −1. yx y − x yx dx y−x  x=0  Y4'5 24*Z >(&%&+5;%2,'2U%()4+K<()5 +4&4+-',?A CP//;(?A >%&5;,2+.P=424*R?3RM)(/% '54++ ?%' 4-?-) 4 43)-27+(& +(. CP//;(??A% UU44 CP//;(?A<4*4 4 CP//;(?A< 2%):.Q+;% 4* U42'4&4# x 7 |x|, x ∈ R {0}  s(x) = sgn  1 −1  0<x x<0 R {0} $ %
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+%U,+ 4 =-33-2>425 >%&5;,2+ #  9            4 x = 0  2 1    # −2(0+h)−(−2·0) = −2h = −2 h h 0.5(0+h)−05·0 0.5h = h = 0.5 h  %!   
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 %#% 0.5 −2  5 5 U 5 9/R,&&3%4*(&4?5?++5.M=/(?R&R# /1-&34? 2 2 2 M : + ++ + + + + C-/ 4 . -* 4 4 4 /%-'% & 4 ' 4*< -/A ' (& )%?%. %/A4* ;% 5 5 5 > 5 = U 2 2 K2 R + + . & %)% (3%? % % ' /%-'% & 4 '%//-? , )-.3 ) '%& P.< -/A 5?+M '%&4*-?34? U M2%/%) 4*%(&+5?4+=K2()55)E-?'%&++U-,2 QU+7,)-+ ++5) )6/,&&4<%(& M# O 2 R + + + (&+4.(EE%? . ??A%? '%/)*%&.%)* % 5 D M M 2 R + + )4. 4 . .%* .3%? 4* (& ? C-/4 ',?4. ;% V > > 5 = => 5# 2 2 + + + 4* %/4. '%/&4/4 ,)4< % ? , , )4 6 ,? C-/ 6. '4 E-? -)4? /4*? = 1 2 2 2 2 2 2 K>U R R + + + + + E&-3 ( '4)*%' ( % %)'%/. *% Q ()%.

*3%?<4&4 *-.4 , ' 4* 4 3%? ;4'%/C-/4 4 5 5 U 5 5 =K2 5 2 2 M 2 2 2 + + +  ?P?. 4 ' 4)' < -/A 4* (& ? ? %/A 4? %/A%?%) (& )P?.< 4'% . ) ' 6 ;4 A A A  U 5 5 =K2 5 # 1 2 2 M K +.%*()2M5+E-?;%)P.4* +34? CP//;(?A/&,C +-)Q+,),;4 4 .R;%E-?+%*+34?<4'% /.-*R&3(U4+6?.B %* 4 (&2A?4?)% ) '6 E-?=,?4.4* .(EE%? 2E&A%?% 3,2 = 5 2 ;%+++U-*<',)()5. 7/A (E*%2U%+=P.<'%U-/A+)*()E-? 2 2 2 2 R + + 4 /R&3(;%E-? 4 4 ' %?(& % % '3%? . *% 4* ;4 A ;4? = 5 2 M 2 2 R + %*%. 4 )*% . 4??, -334? '%/. *% Q . 4  5 5 Y8# M 2 2 2 G # R R ++  + + + ++ + + .%&%)%1*(&+ (&)%'? 2(2+<'%+=P.?( (+C(*%2%%33'K>;4?-? 45) 4%''-?Q %>-++.(44E-?+< U-/A4 / %*%&3(?2 5) U4?/),3&4Z72A-6. 4*+ 4 C-?E-?+++-)K2+<4)*%& (?A+< U5?-/A+ U-/A4* ;4/A E-?# + =-33&4;4/A 342&4%/A4&,?+ %)U%+ 4    f (x + h, f (x + h)) (x, f (x)) (x, f (x)) (x + h,

f (x + h)) h>0 h<0  & '&( ) -(. 02  7 2    f x   1      9    23 2 :; 2  (x, f (x))  7 f 2 (x, f (x))    (x, f (x))     3 :;   2     f 0 (x)    2  $ %
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  "S" 5 U = @*%. 6+K,?(&5',&?+M %/A%?%)%/A%? 2',++-)<()+(?A.R??A%? 2 + %/*4.+?4 +%2%5?+(++U#%+@=P*+ 4%/A)%& 4* (&5?+M :C-/4< >%C-? C% ) Q & 4 6. 4 5 = K)* 34?C-&/ 2 R ++(+% 3%? ) & /*Q+ P.#   !"#  $ ,()! &* & )&!& 0 7   3 1  f 2     a        2 2    7 f         (a, f (a))   95?++6>=6.<4* 5&,?A+4?/%?): %/A%?%)%/A%?2%+(?%.4*,2+42,?-)424.= 4 () '5;%2 4* %/A%?%)?%. ,+ %2 '%??5% 4* E-?+-?<%(&+ 1 .(+ 6+K335 %/A%?2M)(/%+ 5;-?;4

%/A',)3K2 4 5%)5<() 2 Q+-++6.# 4U-/A4? , 5 U U 5 1 M 2 2 D 2 2 2 2 2 M + + + + * (& ? C-/4 ' ,;4 .4E )4 34? )*%' ( % %)%? % %) 4* 4 C% C -/,)< -/A 4* (& ? A  5 5 5 U 5 U 2 2 2 # 9 R + +  + + 4?6. %/A%?%)< E-?>34?-24+ E-?-*,)+-) '%/C-/4 '6 4 /2'&3(4*,),%U- )P.)(/ -) -;,33 2%)* +%/A%/A,&/A47-= A2=C-/4 + 4 )5'4'%65+2,4)A+.R#;%14*+.%*C%(&2;M?%>+%()%++V?Q/-? %*2-'&4<4' D5K34?;%*%+P?. 3%# y − f (a) = f 0 (a) · (x − a).   f 0 (a) y = f 0 (a)x + b , (a, f (a)) f (a) = f 0 (a)a + b . b y − f (a) = f 0 (a)(x − a),    '&( ) -(. $ (#,*.  ,(# ,& )*.0 1              9 2 g g(h) = 0, h0 h lim 52 3  g(h) = 0, 5        9 3  2           9  3   g h h :   ;       2         7 5 3 h g o(h) 2    h 2    9     3 7         

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(-"-(. !!& 0   5   7  a ∈ R     2 2     f   2         3  3          2    2  9  2 52  a  α f (a + h) − f (a) = αh + o(h),   7    α   2 f 2  a    f 0 (a)      f (a + h) − f (a) = f 0 (a) · h + o(h).     1* ,2 Q+,))%'2(2%+%)%? Q/A C-/42'4U4+KB 1U-/A4? %>>5/ 5) +%++P.< 4 5) '%/C-/42'4*U4+K 4 ,2 Q+,)#

.P2RU?3)(/&% ;,24)*+,))42 41 >5 U > U 2 D5 2 K + + ++ * CP//;(?A % %? %&%? 4 ,?A4 -),?4. %* 4& ,)4 A ;% ;4 +4* =%2%?+5< U-/A 4     3  2        f (y) − f (x) f a 7    f 2      a     h = y−x  x f 0 (x) f (x + h) − f (x) f (x + h) − f (x) − hf 0 (x) − f 0 (x) = h h >U U 22%%.(E%*() + 7/A 5)2 '-? + +< 4 2,2?6K= 42 ,U;4-*2 +-)4&++,)4;4?6'%2 ,2 U%++-< +4'4&+5<()4?6 .(E%*()+4'%/4 ?62 ,>KU-*+R&++4&),' 4 6?.< -/A 4 >5 2 # @ =%2%?+5< U4-/A*E% / 4* %2MM >%V?QD5K34? 3%;%%+%++ .5) -&>K +%&'5?-2K/5,;42 E-?+-)4? 4*+,34??6 h h h f (x + h) − f (x) − hf 0 (x) = o(h). 5 > 5 > 5 1 2 2 2 2 R : + + + + + + + ( % %/A)*%& 3 * -?AQ ,)4 % % ?(&% 4*(& 4 6?. ?%. .P ? ?%;% < '%& &%? . P ; Y 5 = >= > 5 U > 5 D5 K 2 K 2 # @ +-) =+%%&2 %5*'()( + + + + + C-?4

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( 0 x h 7 ∆f f (x) ∆x x = 8    2   ∆f f (x) ∆x x f (x+h)−f (x) f (x) h x f (x) x 6= 0 f (x) 6= f (x+h)−f (x) f (x) (x+h)−x x  x f (x + h) − f (x) . f (x) h U > 5 U 2 2 K + 44.-&4* 4-??42()'%/ 5) U4+,&-C%2+(U+4%+2=%6. +4%2=&%%)P24+2?Q;%<>5 ()%&%?4* D54UCP//;(?A % %&

, 4 4* ; ,?A4>-) U4+,&(&+(.(+B A%?< # x 6= 0 = = f (x) 6= 0  ∆f f (x) lim h0 ∆x x f x x f (x + h) − f (x) = h x x 0 f (x + h) − f (x) = = lim f (x).  f (x) h0 h f (x) = lim h0 f (x) 1* %2MM '%//-?>-2,)-. %&%>'(?A(&% +,'4)*.->;4 %/A 7= %2?%;%*()+ ;%%+P?. 3%B  6 '&( ) -(. 02  3   7 5 1 f 2  x 6= 0   2 2      f (x) 6= 0    f 7 2      8    x  2    (4.3)       2   h = 0  2 7      3  3      f 2 x  2        9          x 0 f (x). f (x)
      >5 U > 5 > @4233%? D5 2 2 2 K + + + .42626)+ 4 E-? 34? 4 %&%? , ,?A4 -) . ) *,'Q ,),&4 )*/ , &%? )*%& < %/A 5 > U 5 M 2 2 K R + + + + 4>5 .Q%&%?+6?D5,25#)*,'QO-+2+,)4.(EE%? ? ?%? %&% 4*< -/A 4 3%;%*% 4?4 Q * ) ,&/A4 . ?A;%.%  ?(;%?

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      1* CP//;(?A >5 %&%?D54U,?A4>-)4 4 U%2A%?  (αf ) x f (x + h) − f (x) αf (x + h) − αf (x) =α h h 5 5 2 2 'K>4*-??4 -?Q&U24+.4EK<4'=6<53M2U-/A '45=;-33%2 %/ACP//;(?A)*,')-&-),?4.4 ' %)*%4 '4++%)*),')-&-)4 U 5 > > 5 2 # @ 2 2 2 22 % %? ' 4 34 4 ' %)*% 4 A 5
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>%V?QD5K=,+ U4)?,2=6.# O(;% 4* 42?4. 2%.(E%*() >5 %&%?D54U,?A4>-),+<%/A)%&: ,+424.Q+,))42 4>K>5< U-/A h=0       1  28  αf 0 (x) (αf )0(x) (f + g) f (x + h) − f (x) g(x + h) − g(x) (f (x + h) + g(x + h)) − (f (x) + g(x)) = + , h h h > >K U 5 2R))*%/2>42UU44+,&(& 2 R R + + + (.% 4* ))*%4 4? . 4 ,&(& (.% ? %.4* ))*%/%<4-??4 .4E =4'4'6.<5534UM2-/A><'%V?Q54;=D%-33> 5 U 2 2 2 5K )*%&54?+8#G+#,&(&,2 Q++,)(.%)%/Q#4* +)(/(;%2<4'%<()2A )%&Q/A5?4+ 34 - 4 ?4. ) ;4? 4+,&(&+(%< () # -2 4M2 () CP//;(?A%. C-2A+-?-)4 4* E-?+34?< () () @4U*%.3  0 0 f (x) + g (x)     28  0 (f + g) (x) f (y) − f (x) = (y − x)φ(y) φ ψ g(y) − g(x) = (y − x)ψ(y), x f 0 (x) = φ(x) g 0(x) = ψ(x) (f + g)(y) − (f + g)(x) = f (y) + g(y) − f (x) − g(x) = = (f (y) − f (x)) + (g(y) − g(x)) = = (y − x)φ(y) + (y −

x)ψ(y) = (y − x)(φ(x) + ψ(x)). 5 95E-?;+%34?<%*(& 2 C-2A+-?-)+ 4CP//;(?A%. 2 2 R + + ))*%/% ) C-?-)< %*(& 4 CP//;(?A CA 8#G# ,2 Q+,))*%&5?+ 4 R))%/CP//;(?A >%&5;,2U4+K<() A+-?-) 4 φ+ψ x (f + g)0(x) = φ(x) + ψ(x) = f 0 (x) + g 0 (x). $ %
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+ + + + + C-?-) 4* % %? , Q ,)Z . 4 ' 4*;44 4 ,&(& (.C-&', ) 6 4 -?),/4 A A 5 5 > > 5 2 2 2 2 2 2 < 4' '%/%/A%* . 4 34 4 4*%'544>K>5 4 ' %)*% V 5 > 5 5 2 D K # ++ 2(.+%*< UM#-/A 425='4-33+ %)*(;% ?Q 2<4'% )*%& ?2A )%&5?+ 8#G# ,2<4' Q+,) )%/Q4 +%)(/(;% () # 4@U*%.3 -2 4M2 () CP//;(?A%. C-2A+-?-)4 4* E-?+34?< () () h0 f (x + h) x     f (x) f (x)g 0 (x) + f 0 (x)g(x) (f g)0(x) 28  f (y) − f (x) = (y − x)φ(y) φ f ψ g(y) − g(x) = (y − x)ψ(y), f 0 (x) = φ(x) x (f g)(y) − (f g)(x) = = = = = g 0(x) = ψ(x) f (y)g(y) + f (x)g(x) = f (y)g(y) − f (x)g(y) + f (x)g(y) − f (x)g(y) + f (x)g(x) = g(y)(f (y) − f (x)) + f (x)(g(y) − g(x)) = (y − x)g(y)φ(y) + (y − x)f (x)ψ(y) = (y − x)(g(y)φ(y) + f (x)ψ(y)). 5 U 5 5 U 5 1> 2 K 2 2 2 2 + + ++ + %& , 4 ),/' 4 4 CP//;(?A ) C-?-)4* % %?<()' % C-?-)CP//;(?A%. ; A A ; A 5 2 R + + + CP//;(?A

)*-&4 4 () ))*%/% ) C-?-)< %*(& 4* A 5?+ 4 )*-&4+CP//;(?A >%&5;,2U4+K<() C-2A+-?-) 4 U%2A%?<%*(&+ 4 8#G# ,2 Q+,))%& @2M)*R& 3%2,+=6.< U-/A 4* 2%.(E%*() >%&5;,2U4+K 4 U%2A%? () g x (g(y)φ(y) + f (x)ψ(y)) x (f g)0 (x) = g(x)φ(x) + f (x)ψ(x) = f 0 (x)g(x) + f (x)g 0(x). 5    1/g x  0 1 g 0(x) (x) = − 2 . g g (x) 5 Y8# N 95?%'?6 2 2 # 2 ++ + + + 4 4 C-?-)4* ?( , Q ,)Z() <%*(& ;%2 42>4%&< U5;44,2U4+K%2),/' A 5 5# 5 > 5 5 D @ 2 M 2 2 D 2 +++ + ++ ( //( . ) ??%. 4)*%' % 4& ,),;4 ' ? %?% ( //(. ) '% % C%2Q&U4+=6. 4* >5 %&%?D54U,?A4>-),+<() '%/%?/%>%++ 4 42,335 ,+424.Q+,)B g h 1/g 1 h   1 1 − g(x + h) g(x)  = x g(x) 6= 0 g(x + h) h g(x) − g(x + h) g(x) − g(x + h) 1 = . hg(x + h)g(x) h g(x + h)g(x) $ %
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( −g 0 (x)/g 2 (x) 0 (1/g) (x)  0  0 1 1 f 0 (x) = f (x) (x) + f (x) (x) g g g  g 0 (x) f 0 (x)g(x) − f (x)g 0 (x) f 0 (x) − f (x) 2 = . g(x) g (x) g 2(x) 5  7 2 1 8#G# ,2 Q+,) )%/Q+)(/(;%2 35*-?AQ+6?.<4'%2A )*%&5?+ = () +-?-)M24* E-?+34?< 4U-2 4 CP//;(?A C-# @2A*%.3 =%2R2(4))%2E%.4E>5/=6<4* U-/A E-?+34?<() f (y) − f (x) = (y − x)φ(y) φ g (f (x)) = ψ(f (x)) x h=g◦f 0 g(f (y)) − g(f (x)) = (f

(y) − f (x))ψ(f (y)), ψ f (x) f 0 (x) = φ(x) () h(y) − h(x) = g(f (y)) − g(f (x)) = (f (y) − f (x))ψ(f (y)) = = (y − x)φ(y)ψ(f (y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/A 4* ;4? A >5 U > U D5 M 2 K# R + + + + CP//;(?A %&%? 4 ,?A4 -)4 4* E-? 34? 4 . ;% %* .(EE%? 4 4Q 4 f x g x (g ◦ f )0 (x) = φ(x)ψ(f (x)) = f 0 (x)g 0(f (x)).   7 2    x f 0 (x) 6= 0  Ux x y ∈ (x−δ, x+ δ) f x  f  f  f δ Ux  (x − δ, x + δ) h) ∈ Vf (x)  g◦f x f (y)−f (x) y−x f (x) (φ(y)ψ(f (y))) Ux x Ux h y ∈ Ux f (x) f (x) + h = f (y) Vf (x) (f (x)+ f −1 f (x)



f −1 f (y) − f −1 f (x) f −1 f (x) + h − f −1 f (x) = = h h y−x 1 = = f (y)−f (x) . f (y) − f (x) U > > 5 2 2 2 M M 2 + + ++ + ++ ++ + -/A 4* , 4 4 .Q ,) '%/%?/% % 4 4* % * %.3%? '-? 4. ' 4 % ;=O%)P%/AP.2#

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x + · · · + (x + h)xn−2 + xn−1 h n−1 = (x + h) + (x + h)n−2 x + · · · + (x + h)xn−2 + xn−1 . > U5 5 > 5 >5 1=&%?-332 2 2 2 2 2 2 R + + 4 34?C-?-)() 4?6 4 , ?A-) . &?A%*% (3%? '%/%/A%* . 434 4 %42 A > 5 >5 U > 5 = > D54UU,?A4 2 2 D5 2 2 + + ++ -))4 < %*(& 4 ' %)*.(? .4E%&%? , ,?A4 -) '%/%/A%* . 4 -33U 5 5 2 2 2 2 # ++ ++ + + % %? C% % % % %)Q () (& (.(;% <4' A ; A YGZB )'%&+ +&5/-?-'%+&5.6) 4*-?-)),/ 424E=,? h   nxn−1 h=0  ) sin( h2 ) 2 cos( 2x+h sin(x + h) − sin x 2 = lim = h0 h0 h h lim = lim cos( sin( h ) 2x + h ) · lim h 2 = cos x · 1. h0 2 2 5 5 2 +-2+)(.%K2(+E()3%?4)* + + ++ ? 6)*CP//;(?A3%;%% ()(?( ',&3%3 * -?AQ < U14*6+,&(& 5 U 2 2 # + + + () 4 .-)*>%&?5;6),2+=CP//;(?A C- UA4-?-)),/, 4)*?, 6. U 1 + CP//;(?A ,?4. 4 '%/ ,&-*,), -* 4 () 4 .R*;%+%++ CP//;(?A >%&5;,2,)5 )43,2A,;42 .4E=6< U-/A h0 cos cos x = sin

cos0 x = < U4 < 4*-?-)),/-+ sin h 1 h
π −x 2 h0 d sin(π/2 − x) = sin0 (π/2 − x) · (π/2 − x)0 = dx = cos(π/2 − x) · (−1) = − sin x. Y8*,),;4 ZB 12 4>K>5.< U-/A R))*%CP//() 424E=,?<4 +R&+ >%&5;,2,),&4;-?4+.-*K )43,2A 42.42'4  tan = sin / cos d tan x = dx 0  sin x cos x  sin0 x cos x − sin x cos0 x = = cos2 x cos x cos x − sin x(− sin x) cos2 x + sin2 x 1 = = = , 2 2 cos x cos x cos2 x C%YF2ZB+(;%`*,'E%&)%<2;4U4-/A>5 %&%?D54U,?A4 <4'>5+-)4*25'%)(+B x 6= (2n + 1) π2 cos x 6= 0  C%2+(+%2 35*+-)Q+# ex+h − ex eh − 1 eh − 1 = lim ex = lim ex · lim = ex · 1 = ex . h0 h0 h0 h0 h h h U U 5 1() U443%;%*% 2 2 2 D5 2 2 +,&(&+(.% + + + + '%/ 4 ,&-*,),?, C% 4)*?, 6. 4* %XE-?%? , ) CP//;(?A C-?-)),/, A U # 2 2 +()(?( 3% ,+-++ < 4+,&(&+(.%+ lim h limh0 e h−1 = 1 $ %
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( log ! y = ln x x = ey 1 1 d 1 d ln x = ln(ey ) = d y = y = . dx dx e x e dx YN3,ZB2A+1U4)*?, 5 > 5 5 2 K 2 +,)-.24+'%&'()*% + + 35*-?AQ2= ,.<;4 %)%? 4* ? %& , , ),&4;-?4 .-* C-&', ) )*4 ;%&*CP//;(?A ; 5 U 5 K K 2 + + + ? 4* < -/A 4 )* 34? C-&/ CP//;(?A%.?% ? () C;4? ;%&*P. A # %2M)*R& ?-)4.%/A%? > 5 = # @ 2 + 4*4 .-& 4 CP//;(?A %& , ,?4. 4* ; 5)'%&%+(3%?<4* 5?;%&CP//;(?A >%&5;,2,)5 )43,2A4 )%&5?+   ! y = arcsin x 1C-2A+4+U4+KB −π/2 < y < π/2 U4 # %/A%? ! = p )*,'-.&4 x = sin y 1 d arcsin x = . dx cos y y=arcsin x cos y = 1 1 − sin2 y y=arcsinx |x| < 1 y = arctan x 4*4 sin x = tan y p 1 − sin2 y = =p < %*(&+

4 .5%'%2+ )-& 4 R;%+%*M.(EE%? 1 1 − sin2 (arcsin x) =√ 1 , 1 − x2 # @.-&< U4)-?2K4? =,&;4 %2<'5?+ 4* %2M33B 1 d 2 arctan x = 0 |y=arctan x = cos y|y=arctan x = dt tan y 1 1 1 . = = = 2 2 1 + tan y y=arctan x 1 + tan (arctan x) 1 + x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x 8 7 2 3    5   1 0 < x   2 2    f

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( (xn )0 = nxn−1 = αxα−1 = ex 1 x cos x − sin x 1 cos2 x 1 − 2 sin x (αf )0(x) = αf 0(x) (f + g)0(x) = f 0 (x) + g 0 (x) d α x dx x 0 (f g)0(x)  0 f (x) g = f 0 (x)g(x) + f (x)g 0(x) ln0 x = = (f −1 )0 (f (x)) = (g ◦ f )0 (x) = (e ) f 0 (x)g(x) − f (x)g 0 (x) g 2 (x) 1 0 f (x) sin0 x = cos0 x = tan0 x = g 0(f (x))f 0 (x) cot0 x = 5 > 5 = 1 M 2 2 D5 2 2 2 R + + + + + + + . .%* ,3 , *4 34 .( 4? )E% , ) CP//;(?A %& , ,&4 %' ( .%* % P?.< 4'% ;% A ; > > 5 2?%' 2 2 M 2 M # O 2 R + + + + C-& 6 ? 4. % < % / %? C-? -)4. % )*?%. 4 . .%* E-? 34? 4.(EE%? A%. &4*5+.,334? ;% U >= > 5 = U 2 2 + + + 4 % /C-? -)433< -/A 6

6.B 4* CP//;(?A %& , 44* < 4?%' ; > > 5 = 5 2 # @ 2 2 + + '%/C-& Q 4* CP//;(?A 4* CP//;(?A %& , 4 * E%&)*% % % / ;4B ; ; > > 5 => U 5# 2 2 M 2 6/A4?4* 4 +6 ,)< % %33%? 4 &,?A34? .% '4 % ) )-&34? 4)*?, ?  1/(1 + x2 ) arctan x arctan x 1/(1 + x2 ) d arctan x dx = d arcsin x dx = d arccos x dx = 1 (x ∈ R) 1 + x2 1 √ , x ∈ (−1, 1) 1 − x2 1 √ , x ∈ (−1, 1) 1 − x2 = U > 5 24 '%/C% 2 + .4ED2%)-2M24++(34?'%/ %/A%*P.< -/A4* CP//;(?A %& , ; =.P,+2R1U?34++,3R,&-*52.,*4<%(& > >K 2 M2 D 2 +6.+ 6D+)4+-4*2)K'%/6)-&,;4 + ++ 3%?< U%%+%M# 4* CP//;(?A )4. , 4? 34? +-2)K )-& 5) 5/4*-2+%?4.;% > 5 5 U 5 5 1;(?A.R*;%>%&+5%;++,2,CP//;(?A 2 2 D5 2 2 + + %& , , ) )*43, 4 () 4 4 < %XE-?%? , ) () /4& '6)CP// ; A ;,?A U = 5 U 5 2 M 2 2 R : + + + + ),;4 .4E 4 6. 4. .%* C-&'6 , .4 <4'% %. / %? 4)*?-)4. 4&6 ? )*%& ;% A

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g(b) = f (b) + f 0 (b)(x − b) + M(x − b)2 = f (x). g(x) = f (x) g 0 (ξ) = 0. g0 4>K>5.<4'%2A3M2 4 424E= ,? 4*-??42 .4E=6< U-/A 4*4 ))*%C-/242;4 4 %>>5/5%.%+B ;4? -2A4? )*,'<4'%2A&% g 0(y) = f 0 (y) + f 00 (y)(x − y) − f 0 (y) − 2M(x − y) = f 00 (y)(x − y) − 2M(x − y), g 0 (ξ) = 0 f

(ξ)(x − ξ) − 2M(x − ξ) = 0,  Y8#"SZ f 00 (ξ) . M= 2 00 Y8#"NZ 5 /-?>-2U44+'%?% 5 5 12%')*%& 2 2 2 + + + + + + * %2M33:%?;(/&% % %&'()*% %)%? %/A 4 . 4 ' 4) < ? %&;4 6 '&4 ) (& % = U > > = > 5 2 # 1 2 2 R + ++ + + 4 6.;? 4 *%&% '(?A '4&4 -334 . &?A%*% (3%? ;44 %/A ,2 Q+,)34? 5) &R/*Q+=P.B b<ξ<x f (x) = f (b) + f 0 (b)(x − b) + f 00 (ξ) (x − b)2 . 2  (c, b) c < b b    !"#  $#,& )*. % , , &  !#0 7 2      02  (b, c)  f   2 ξ ∈ (b, x)    [b, c]  9 1 3  2     x ∈ f 00 (ξ) (x − b)2 . 2 CP//;(?A %2)MC-.7 Y &%?>:< 25?%,&5)Z $4A112-*& %&%.R*%>'(?A 2Q+()%+4.5%2E-?(/Q++M34??%# '-?%2)M>UC-4+7=6<E-25U5?-'4* >= U D + + + ) *%? '-) ',& ?%' )4. 4* 6 6.< -/A4*+ 54)< U U 2 2 2 M2 2

K 2 R + + + + + + CP//;(?A?%. 4 . *% Q ()( % (&()% -)* ?6 , -* 4& < 4?%' ;4 ;% ;4 U-/A 4* %2+(&() 4 ?(/A%+(;%2 4&,?A-)# ,))6?. '-)+ 4242'4*,).(?+ %/A E(2>,+# f (x) = f (b) + f 0 (b)(x − b) + x 7 f (b) + f 0 (b)(x − b)  b (x − b) f    *   f ! (x − b) 7  5 8  2 2        9   sin 2    0         9 7 2  2
5 > 5 = 1 2 M 2 + + ) ? 6)* CP//;(?A % ) .( %& , ; 25?%,&5) .R*%2Q+() 4 ?62 ,34? 4 42,335 2%)4B# 0.1 sin x = sin 0 + sin0 0(x − 0) + sin0 y = cos y () sin00 y = − sin y,     %*(&+ 4 1 00 sin ξ 2 sin ξ(x − 0)2 = x − x, 2 2 $ %
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4U-2 0<ξ<x # 1* x = 0.1 U%2A%? 4 U534 (&+(.% ?%' ?4/A-33<'5?+ 1 sin ξ , 0.12 5 2 200 +%U,+ 4

.R*%2Q+() U53,= 4 .5)%33 '5?+ R+ %*&%># 1* ,2+42,?-) %)%+ ;5)/,24+4 %2M++ ;%%))P?. 3% ?(U,?A %2?%;%*()+#   & '&( ) -(. $ &*,& ). % , , &  !#0 02  n7 2         f n4 b f 0 (b) f 00 (b) f (n) (b) . (x − b) + (x − b)2 + · · · + (x − b)n Tn (b, x) = f (b) + 1! 2! n!       f 7 2 b    2 2 b   5 3 2 b    2 n 4 2 4    3  4 5 8 7  9    7 n 4    f (x) − Tn (b, x) 1  %>5. $4A2-& E-25?-' U%2A%++ )*-.,)-) )*KU4)?,24+ '(/B %>&%?>: ;4/A * 25?-'+3%;4/A 2 # 1 R +  E-?+34? ;%++ U%2A%++ 2%U%+B E-?+ .R&P25< %>E-?C-.7+?,$24A2(2;-M&<E-E-? . *% Q () 5 2 # U 5 > 5# 5 = U 2 2 2 $ 2 R + + + + 4)*?-)< 4%/ACP//;(?A E? -''4 6 6?. *% Q %? , ? C-/ 6.< -/A 4 4A & 5?-''4>5 %&%? =K2 5 U 2E-)-.)*-& 2<%2(D5//(, 2 2 2 2 2 2 K 2 2 2 M + + +

++ + 4CP//;(?A%. ,?-)C% ( %# %1.'%R*%2%Q+()E-?4EE&-X '+,4'4&4 4 4.>(%)*?%. 4'%/C% %?%'%? U 5 2 K + + + + , 4 -)),/, 4/'(& <%*(& '%/2%EM< U-/A %&&% )-.C(2% %2M,2 Q+,)++42,2+4# 95 4 2%/%/A)*%&:33 C-&',++,&/A42=6. D)4# 5  n b  n 4  2  7     : ;    4   n b b b # n !      !"#  $ &* ,& ) . % , , &  !# 0   3   02  n 7 2 f (n + 1) 4      [b, c] (b < c)  9  1   93 2    2    x ∈ (b, c) f (x) = f (b) + ξ   4 (b, x) f 0 (b) f 00 (b) (x − b) + (x − b)2 + · · · + 1! 2! f (n) (b) f (n+1) (ξ) (x − b)n + + (x − b)n+1 = n! (n + 1)! n X f (i) (b) f (n+1) (ξ) (x − b)i + = (x − b)n+1 . i! (n + 1)! i=0 4?/)72A-*-++4? C-?+-) 4 4 %)%+<4'5.-& 4 )*,' ?62 4<%.-& 4 $4A2-& R*%2Q+()B # b  f

00 (0) 2 f (n) (0) n f 0 (0) x+ x +···+ x . f (0) + 1! 2! n! $ %
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  b<c x ! b b  M f (x) = n X f (i) (b) i=0 i! (x − b)i + M(x − b)n+1 . g n . X f (i) (y) (x − y)i + M(x − y)n+1. g(y) = i! i=0 ξ ∈ (b, x) g g(x) − g(b) d = g 0(ξ) = g(y) . x−b dy y=ξ g g(b) = f (x) M g(x) = f (x) g 0 (ξ) = d g(y) =0 dy y=ξ    5  d dy  f (y) + f 0 (y) (x − y) + · · · 1! f (n) (y) (x − y)n + n! @)*%&5?+ 4 CP//;(?A >%&5;,2+= 4B    7 9  = f (n+1) (y) (x − y)n , n! g f (n+1) (y) (x − y)n − (n + 1)M(x − y)n , n! () Q/A 4 Y8#0"Z )*%&5?+ 4 4 .R;%+%*M %/A%?2%+?%. +%)* %2%/%+ g 0 (y) = M f (n+1) (ξ) (x − ξ)n − (n + 1)M(x − ξ)n = 0, n! $ %
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"IG 4'%2A3M2 4>K>5.< U-/A Y8#00Z # 2+ >5 %&%?D5,2,)5 C-&'624 5/4*-2,)4 ;4? U,+&4< 4'5 %/A)%&: 4'59;%-)2 4+ +',&(+%2+D)4.3% 5)4 2C%,++2U64)*?, )*,'-2,) )-&-4+B M= f (n+1) (ξ) , (n + 1)!     f 0 (y) f (n) n f (y) + (x − y) + · · · + (x − y) = 1! (n − 1)!   00 f (y) f 0 (y) 0 + (x − y) − = f (y) + 1! 0!  (3)  f (y) f (2) (y) 2 + (x − y) − (x − y) + · · · + 2! 1!  (n)  f (y) f (n−1) (y) n−1 n−2 + (x − y) − (x − y) + (n − 1)! (n − 2)!  (n+1)  f (y) f (n) (y) n n−1 (x − y) − (x − y) = + n! (n − 1)! d dy = U > 5 U 5 1%2)*-?()&%;( 2 2 2 2 2 R + + + + % 4 4 E ,?)*,'6?.< -/A4 %& , , )6 ,??A%& ))*%/4?< -/A 4' 4* ; A U 5 > 5 = >5 U M 2 2 2 M 2 R + + + + % A%?+Q)*%&%E% 4*2<%(&%/A+.D)44/ 4*6%& ;+,-2)K,34?<4 4+4/. '4&4 4/34?+R',).R))*%4% A%?>,)?,)%2 ;% .%*> '%/<4 = 5 5 2 2 M 2 +

&%E% ?%/4 % % % E-* Q 33 4* ; ; .5%)5# f (n+1) (y) (x − y)n . = n!      *    9   x 7 2 2  5 8  9 53    3  2 e n 4   5 8  2     9 9     e      9 4  >5 >= 5 5 5 1 2 2 2 2 R R + + + * % . . *% Q () .%;() CP//;(?A?( 6 6. . ??A%? . ) *,'? <' % 4 '4/4)433 ; >: > 5 U U5 2 2 2 2 # 1= 2 D + + + + &%?)*%? 4 %& ;CP//;(?A , 4. ?4/A-? 3-?A6 4. % % ?%. % % ? %)% 3%? )*%&%? )()%. ;4/A6?.< 5 > > 5 = 5 > 2 2 M 2 2 1 + + + ++ ' ? %? %& , 4 ) < %*(& 4* % ) C% % .(& ()&% 4 4 )*B * ; ;, >5 5 = CP//;(?A % . $4A2-& 4EE&-X ',D5K 4 n   ex n ex 1* %2MM +(+%2 )%&5?+ ;4? -2A4? 1+ ex = 1 + ex x2 x3 x xn + + +···+ . 1! 2! 3! n! ξ ∈ (a, x) < U-/A x2 eξ x xn + +···+ + xn+1 . 1! 2! n! (n + 1)! $ %
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e x=1 1 eξ 1 1 + +···+ + , 1! 2! n! (n + 1)! 4U-2 4 ;424'52A%?<4 () .R*R++ 2(;M<),'# 1 e=1+ ξ 0 1 U534+4/ 3%D)2()(U%* +6>=6.< U-/A eξ (n + 1)! 0<e<4 <() Q/A 4* n=9 %)%+3%? .4E=6B e 4 eξ < < < 4/3, 628, 800 < 0.000001, 10! 10! 10! 4* ),' (&+(.(+ +%U,+ %/A'52 5-'->?,2 5)%33 U53,;42 4>= 4 '%/ 4* e R))*%/#   1+ 1 1 1 + +···+ 1! 2! 9! #! "  
   
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# 5 = @%??%. 2 2 2 + + + + + + + 33%?4)%/Q 4* 42E+)(/(;% -?+34?2 %/A4 >%&-25A;4?,2,+)(.+R%2*(E(& (. ( % ( ?%. %/A , 4 , ?-)Q ,), 3 * -?AQ 6.4+,&(&3%<()+( 5 U > U 2 2 R + + + + ,&/A4 6 ?.<4'% . 4 (.-?A %)*. * 4 4 A U # + .%15)*,'Q ,), -* U >

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f (b) − f (a) b−a f x 7 x f  b f 0 (ξ) ξ  a g f (b) − f (a) g(b) − g(a) U-/A>-))4 2A4? 5.4*'%/# 1 +.%*MK+34?C-&/ 4?24+%&;4 .7# 216'34?<4'% (+%2 )*%&KU5?+,?A45/%?>-)2M<4() 4+ ,2 Q+=4<U,?A4

;,24)* 4 .R2A;%&%4)* ;4? 2-%/A%* 5'%/C-/4 5 5 2 K 2 2 2 R + + + + ' 4*,)4-?34? 3 * -?A-) -)),// (?A% <'%& 4 .P ?3)(/ %)% %/ ;4 ?62 4 5) 2%U%+<%*(&+ '-?>=6. 5 4* ,2 Q+,)+ U,?A4>-) '%?+%)%?B ξ (a, b) f 0 (ξ)/g 0(ξ) g(b) − g(a)  $ %
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f (b) − f (a) · g (ξ) = g(b) − g(a) · f 0 (ξ). 1* ,2+42,?-)Q+-++ .R*(E(&+(.+(+%2+ 46DUA C(2% R*(E(&+(.+(+%2?% 5) )*-.,) ?%;%*?5# 5 5 5 1 2 2 # O R R R + + + + + + + + 3 * -?AQ ,) E-? -)4? . 4 . *(E(& (. ( % / 4*, ), %/AP. 4 . ;% ;% V > 5 # M 2 .%* .(EE%? % ? , +

CP//;(?A+ Y8#08Z =6. 3%24,+8#?50<8U>-/A%&5;;4?,2,),3-2AK24?4>K>5< U-/A)*,'< 4'%2A&% # 4 6/A4?5) %* +1%*2=+%)PC-/2<4.-&     h

h(t) = f (b) − f (a) · g(t) − g(b) − g(a) · f (t). # h0 (ξ) = 0 ξ ∈ (a, b) + E%>5+ ;(/E-? ,+&%?7>%*;%),'E-?# +1-)4?CP//;(?A 4 +(+%2 ,4*2 Q+5,)4?+%&;4# 92 -)6'.( / 3%2,++==6.<,34?U-/A4*-?-)2(+%(&5.+(%4 Q+ ;;%)*,?+ C%+62< 2U54'4=>)*%?5-?),/

h0 (ξ) = f (b) − f (a) · g 0 (ξ) − g(b) − g(a) · f 0 (ξ) = 0, ξ  h

f (b) − f (a) · g(b) − g(b) − g(a) · f (b)

− f (b) − f (a) · g(a) + g(b) − g(a) · f (a) =



= f (b) − f (a) g(b) − g(a) − g(b) − g(a) f (b) − f (a) = 0. h(b) − h(a) = 5 5 > = >K 2 2 2 2 2 + + + ? %&;4 6 '' ? %?E-? 44 . 4 ' 4) % ??% 44 CP//;(?A , 4 ? <4.-& 4* E-? 5 5 5 >K 2 2 2 # 2 + + 4*+=?453%&;44?<()6'-? 3%C%2P;%)*. 5)%334* 5?;4/A

?4/A-33< 'P2 ?4'4*5 ?4. ?%' Y,4*-?-) 4? (&4+(.<:4-& 5??5+'%&;46',2 4+6'-. 2 2 + + Z;(/E-? %*(& %&;4 6 '-?3% U 5 5 > 5 # O 2 2 M 2 2 + + ++ + + )*( ) (& (. % % < 4 4 '4X ' 6', (;% %/A %? %& , ?4. A ;4/A A ; 2?6 ,?4. %2 2%??5# 5 U U 1 2 2 2 2 2 R + + ++ + + + + + * , 4 , ?-)Q . *(E(& (. ( % %/A . 4)*?4 4*< -/A %??%. 4 )%/Q )(/(;% ,&/A4 U4+,&(&5 4*+(.4 '%/ 4+=6.24/A4-& U%2AU*%4++,&-,),?4. U< -/A -2A4?%/A U4+(.-?A 'K>)*%&(+# 1 U4+,&(&+(.% 5)*,'Q +U,),?, # h # (a, b) h ξ   0 ξ h (ξ)    f (x) g(x) 2 +R&+"P?.Z '5;4?<4'% + &% %/A E-? 34? A > 5 > Y U 2 2 K M 2 + + + ? 4 )*,' , ?4. ' ? 4 ?%;%* ?%. 4* (& (.% 4/A 4 4 ,&(& (.%Z ?6 4 ; 5 > 5 > U 0 2 2 K M 2 # + + + Z ' ? 4 )*,' , ?4. ' ? 4 ?%;%* ?%. 4 4 ,&(& (.% % % ? ;(/ @/A)*%&:%? )K2;4B 4

;4/A U4+,&-*4+24? <?%'(&+%2'%U%+MU,?A4>-)-.;424'%2A5(;%2 +42,2-*6?.# 1* 52A%?%)%+%.&% ?A7=+ =K 'K>)*%&+ 4 -)E5+42 )43,2AB b 0 0 !"# ∞ ∞  $ %
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 #(!# +" 0    !"#  $ 7       2 7 3  2  f  g 2     a   2  2     52 3   0 3        g (x) 6= 0 Y8#0FZ f 0 (x) x&a g 0 (x) lim : 2  ;   1     5  8   Y8#0SZ Y8#0NZ Y8#0WZ Y8#0IZ 5 > = U 5 95 2 M 2 2 ++ + + % 4 3 * -?AQ ,) % . %* %?(?.< %/A%*P. '%/< -/A 4* / 4*, )-. )-&,? 5 = > 5 2 2 2 2 M R R +4*,2 Q+4,)3C%25+(-?AQ +%2+<+,),U-/AU-* U4)?,2=6.C% + + + 4* 2E-?B 1 ;4E-?4'+ A)%?K34?C-&/ -33- K=4 -33-. &?A%*% (3%?< 4 . .%* ;% > 5 2 2 2 M R + 4 .

&?A%*% (3%? ( ; 2 2 2 R + + + ++ + + + E-? -.&4 4* , 4 ,?-)Q - . *(E(& (. ( % 1 (1) lim f (x) = lim g(x) = 0, x&a  x&a  f 0 (x) f (x) = lim 0 . x&a g(x) x&a g (x) lim 1 (2) lim |g(x)| = +∞, x&a   f (x) f 0 (x) = lim 0 . x&a g (x) x&a g(x) lim   g 0 (x) 6= 0 a a x>y f (x) − f (y) f 0 (ξ) = 0 g(x) − g(y) g (ξ) U?%',?A42%>U-)C-&', = U 5 > >M 5 U K 2# @ 2 @2 2 M + + + ,34?Q & 4 C% ??%.4* ? -. , )4B % /%? 4* 3% , ? < -/A4?%;%* . >5 2 # 1 2 # 1 + + + + %)% 4.-& (3%? %*42.(EE%? 4 C%4 .%;()R*(E(&+(. +(+%2+ 4 E%CP//;(?A&%< / 4*(& ?%'4 2>%%&U%5;+,?62+=%244*4?%.<'%&?6+ U,45??6+%&;42 4 22%6??%< 2 4 ' 4*;4 U 5 > 2 2 2 2 2 2 # + + '34?;4 4 ?6 4 % ??%<4' % % ? '-? 4?4 4 C% %;()?%. Y";(/%)ZB 1;4/A35*-?AQ 5 5 U Y8# 5 2 0 F 2 R R + + + + + + + ,)34? .( %)% % C-/6?. '%/.P ?3 * % ? < 4)*%& ? -/A 4

Z '%)* 2 # + % % ? ;(/ 5 > 5 5 U # 1 2 M2 + + + + B . ? 6 , ) )*%& ? % )* % /%) E-* Q )*,' -* ;4? ; -2A4? E-*5+Q; ),'< U-/A U4 U4 Y8#GTZ 2 # ++ )*,' 4 4 ,2+42,?-)Q5?++%&;4 %/A4*+%+)M2%/%)5?+%&;4&R/Q2 6+%'34?<4.-& 6 '34? 4 4* %/A 5 2 R +;4?%%/A%? +)*-M22A%4?/%)4 ),' ++ + + + + . *(E(& (. ( % )*%& ? )*,'<4'%2A&% () g 0 (ξ) g(x) − g(y)  g g (y, x) 1  δ ! 0 (x) =α ∈R limx&a fg0 (x)  f 0 (ξ) − α < , g 0 (ξ) x ξ ∈ (a, a + δ). (a, a + δ) # y (a, x) ξ ξ ∈ (y, x) ⊂ (a, a + δ) f 0 (ξ) f (x) − f (y) = 0 g(x) − g(y) g (ξ) $ %
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E-?+-)4?B 4 3%35*-?AQ+4?>K Y8#104NZ &%2 42,D5%)%K;4++2%%.;2< 4'5;452%-&?)2#5'%)* 'Q?6) +%2%?< ?%' ;(/+ ;%)*P?.# 1 >K>5 U U M2 M2 2 + + ++ % .%5 2?>P622,R)?)*%&C-/25?42+.+-*?6?.<'%& 4 . %33 4* %)% 3 < 4 4* % % A %+)*M2%/%) E-5+Q; ),'U- ;4? -2A4? E-5+Q; ),'< U-/A U4 Y8#G"Z 5 2 # M2 R + + + + ++ ? %&;4 6 '34? 4(+%4*2 )%&%/A5?+ %/A%? 4* %/A % )* % /%) & /*Q % )*,' 4* 5 2 2 2 R + + + ++ + + +;4?%+)*-M22A%4?/%) ),' ? %&;4 6 '34?< 4.-& 4* , 4 , ?-)Q . *(E(& (. 4* )*,'<4'%2A&% () f (x) − f (y) − α < . g(x) − g(y) y a # f (x) − α 5 , g(x)     x ∈ (a, a + δ), 0 (x) = +∞. limx&a fg0 (x) β δ f 0 (ξ) > β, g 0 (ξ) ! −f  f ξ ∈ (a, a + δ). x # (a, a + δ) y (a, x) ξ ∈ (y, x) ⊂ (a, a + δ) ξ f (x) − f (y) f 0 (ξ) = 0 , g(x) − g(y) g (ξ) 4'53M2 4 8#G" )*%&5?+ 5 U

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

)*%& ?%U%*%33<'%/.P'5?2R+?34*RY8#+%+0?S5Z< 2&%4)*%&,D55?5K?+ +V+RU/A%&-/A+(?2%'3%;( 5 8# 0 F 2 2 # + 4* ' %)* % % ? ;(/%) ;4/A ;(/ 2- A4? E-*B 5+Q; ),'< U-/A # 1 .5 ?>62,) )*%&5?+ +%+)M2%/%) E-5+Q; ),'U- ;4? U4 U4 Y8#G0Z # y f (x) − f (y) > β. g(x) − g(y) a f (x) = β, g(x) x ∈ (a, a + δ),  
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x y (a, a + δ) ! # y x (a, y) ξ ∈ (x, y) ⊂ ξ (a, a + δ) f (y) − f (x) f 0 (ξ) = 0 , g(y) − g(x) g (ξ) 4'%2A3M2 4* Y8#G0Z)%&5?+ U4 Y8#GGZ 5 = > 5 5 2 2 2 2 2 2 2 M 2 R R + + % % ';4 4 ' %? -334 . &?A%*% (3%? 4.P ?3)(/ 4%33%?4*%/A%? % ?)(/3%?4* A U,?A4>-))4 U 5 5 2 2 # @ M2 ++ + % % %)Q %? < 4.-& .()*%? ) % ??(?. 33 4 A DU(,?A4 23K2>.-)5 ?+ >4*62;4;(/%P. 2% 4 2.R+6;%>+?,? M 2 .%* ,+4 4.Q+,)+B f (y) − f (x) − α < , g(y) − g(x) # a < x < y < a + δ. a f (x) g(x) f (y) − f (x) f (x) f (y) − f (x) f (y) − f (x) − f (y) − = + = g(y) − g(x) g(x) g(y) − g(x) g(x) f (y) − f (x) f (y) − f (x) f (y) + − = g(y) − g(x) g(x) g(x)  
1 1 f (y) = f (y) − f (x) + − = g(y) − g(x) g(x) g(x) = f (y) − f (x) g(y) f (y) − . g(y) − g(x) g(x) g(x) JR/*Q+)P. '-)+ 4 3%2,+-++ 4*-?-)),/-+B = Y8#G8Z U >5 U > 9,+-)-)+ UE%4>54*/ 4()Y8#GG4Z

2&%%Q2&,+D54.K3)*%&K2 %52?M+)UR%&2A5)%.%4*+>5-.2;%4))6 2 D5 % < -/A 4 %&%? 4 ,?A4 -) .-& U5 5 2)*,' 2 + < ) *%? ?%' ?4/A-33< ' ? 4* Y8# = > = # L G 8 2 2 # K2 + + + (*P. %*%. 6 ,? 4* Z 4*-?-)),/?4. 4 -33 4 , 4 4* (& (.% -33& Y8# = U 0 W 2 2 M 2 R +>54&+%&%?4* D54UU-<,?A44.-& + + + + + ++ 4* Z 4 4 E ,? 4* % ) 4/ 4& 4 ?6 , -*< '%& & /*Q % < 4 > >5 >5 5 U U = 2 # 1 2 K2 + + + + + +4&+ 4* U-< +%U-),+ E%4 / .-& , -) ',)- 4/)* ? (? 4& 4?6 , -< 44 -33& f (y) − f (x) f (x) f (y) − f (x) g(y) f (y) − = − . g(y) − g(x) g(x) g(y) − g(x) g(x) g(x) x  y   # |α| + ε a x g(y) x a  f (y) − f (x) f (x) − g(y) − g(x) g(x)  = U U U = >5 Y8# 2 K2 # @ 2 G G 2 D5 K + + + + CP//;(?A 4& 4 ?6 , -*< 4 4* -33& 4& 4* -* * 4 4 E ,? E% / <4 Z &% , )*%&5?+ ;4? -2A4? &()*.R&?A%*%+% 4

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β, g(y) − g(x) a < x < y < a + δ. f (x) f (y) − f (x) f (y) − f (x) g(y) f (y) = − + , g(x) g(y) − g(x) g(y) − g(x) g(x) g(x) 4'%2A3M2 Y8# = > U 2 G S 2 M 2 2 2 M D 2 M 2 M R + + + + %+4/%2U4)*?, 4 4 Z%/A%? % ?)(/% < 4 -334 4 . .%* .(EE%? 3% )P % B 4* % ) ; ;% 5 U 5 U 2 M M= M 2 + + + + + + ) (?A%* %?4/A-33<' ? <4',) . (?A%* 4& 4* %*<'%& 4 ?6 , 2+6.<+?%< +4&-*+ +44&?6+<'2 ,U5;-*%2#4 @%. '+4&54+++ 4;4? -2UA%*4? 4 ',)->5. +&()*. 4/<4RU&?A%*% -/A4?+4*% 4%2M33 ',& 5?>.--R&?A%*% 4'%2A3% %)M (&+(.%&% f (x) f (y) − f (x) = g(x) g(y) − g(x)  |g(x)| +∞  g(y) 1− g(x)  + f (y) , g(x) 1 β (a, a + δ1 ) g(y)/g(x) (a, a + δ) x 4'%12 A%22M3%M5)+ 2+,++2+6.= 4 +(2+>%2+25# U + + > 5# 5 5 L 2 2 K K2 + + * % * ( % -334 4 ,&(& (.%&% '-? 6. . ,?;4 4 3 * -?AQ ,)-.3 < A ; U5)-/A# 1='%/C% 5 > 5 U

2 2 M 2 2 2 2 2 + + + + + + + % '%/;, -* 4 ,)-.4 <4*-?-) 43 * -?AQ ,) 434 4 4 ,&(& (.% %)% (3%? > 5 > 5 U 5 > 2 2 2 2 2 2 R + + ++ + + + + -334 () 34 4 4 ,&(& (.%&% . ' -? ( % % . . .%*'(?A%.(? ;% /42'4*U4+=6. 4 U4+,&(&+(%&% ;-?4+-*K +(+%2+<'5?+ %/A)%&: .R;%+%*'(?A+# '%/Cf (x) > β, g(x)    #(! # +" 0     !"#  $ 7     5 2 7 3  2  f  g 2     a   2  2  2 4 3    0  3        g (x) 0 f 0 (x) xa g 0 (x) lim $ %
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0, xa  xa  f (x) f 0 (x) = lim 0 . xa g(x) xa g (x) lim (2) 1  lim |g(x)| = +∞, xa  f 0 (x) f (x) = lim 0 . xa g (x) xa g(x) lim !"#  6  *  1   9   5      x3 − 3x2 + 2 lim . x1 x3 − 4x2 + 3 U U > 5 = 1 2 2 2 # 1 M 2 2 +R + + + 7 + & 4* % %? 4 ,&-*4 4 ?4 4 . ?%;%* %& , 4<4*%/A?( < A ; U = 5 5# 2 2 2 2 2 2 2 R + + + + + + + + %*(& 4 4* M %/A>%&5;.,2&?A%*% % , ;4?% %)P25'?%)*%B %. 4 -)E 4 )*43, A C% ( % % 1 )*,'?%'2,2K?6() ?%;% += 45U,?A4(3%?<>-),?4. x=1 1 0 0 3x2 −8x −5 !"# 3 3x2 − 6x = , 2 x1 3x − 8x 5 %*(&+ 4 '%/U4+,&-4?>K 25'%) 5) # lim  3/5    *  tan x − x =? x0 x − sin x > 5 = 1 +R&++(3%?< 2 2 # 1 M 2 2 + 7 + 4 ?62%*(&4 U+%24A%?)43,U2A4+4,&-4 4 ? 4 4 . ?%;%* %& , 4 ?%' ?6 4 4 %/A ; U = 5 2 2 K# @2 M 2 2 2 K M R.>%&&?A%*% R + . 4 ' 4* 4 )* & )*,'6. . 4

)*,' , () ?%;%* 5;,2+= ,?4. 4 U,?A4>-),+B lim 0 0 0 1 −1 (tan x − x)0 1 − cos2 x 1 + cos x cos2 x . = = = (x − sin x)0 1 − cos x (1 − cos x) cos2 x cos2 x $ %
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  9   5     1 2   U4+,&-*4+24? 424.7 Y'%/,2 4E->,))*%&5?+ CP//;(?A 4 ?6 , 34? U%+M 2+R>4&2+5 .4R24&?A%*% 5 5 2 + + + <'%/U+4%U%+P?.<'%& Z# @*++ =K24 ?6.%*%2 4 =2-33.&4 U+-*(3%?U4+=6.<E-*5U+Q4;#;%))P. 4 /4& '6), < 4' @)*%&5?+ 4 lim (sin x)sin x

x&0 1 (sin x)sin x b 6= 0 00 b0 = ln(sin x)sin x = sin x ln(sin x) = ln(sin x) = > 5 5 U 5 = > 5 2 2 2 2 2 2 # 1 2 2 + + 7 +R + CP//;(?A -334 ' %)*( .% '%/ 4 ,&-*? < 4'% 4 4 . -334 & A > 5 = M= 2 + ?%;%* (?%. 4 %& ;, 4 4 4'%2A ?%' ?62 4 4 %/A =-33-2>425 .R&?A%*%+(3%?# 1 ),'2,2K >%&5;,2+=4E%>5/ 1 sin x ∞/∞ cos x d 1 =− 2 , dx sin x sin x 0 1 cos x d ln(sin x) = sin0 x = . dx sin x sin x 1 )*,'2,2K () ?%;%M >%&5;,2+= ,?4. 4 U,?A4>-)4 cos x . cos x = − sin x, − sin x sin2 x = > 5 5 2 2 2 2 2 + + 4'% ?%.;4? -334 ' %)*%4?6 , 34?<?%;%*% %)%? 4 <%*(& 4* () CP//;(?A%. A C-2A+-?-)),/4 '54++ lim (sin x) x&0 sin x = lim exp x&0 d ln(sin x) dx d 1 dx sin x ! 0 ln = e0 = 1. exp     *  lim (1 + x)ln x = ? x&0 $ %
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 1 CP//;(?A 2-/4&5+'6)4 ln(1 + x) . 1/ ln x 5 YHR 9U4-)+K+),/,%/A',)6 2 2 2 2 # 2 2 2 R + +R + + + + ,? 33)* & 4 . 4 ' 4*6. 4 -)E 4 )*43, *3%? 4 ( % 4 . 4 '4* A + '5?>5/ %2 %?M&5*?5 .%2 Z (ln x) ln(1 + x) = !"#  1/(1 + x) ln(1 + x) x ln2 x = lim = − lim = x&0 1/ ln x x&0 −1/(x ln2 x) x&0 x + 1 lim ln2 x (2 ln x)/x ln x 1/x = − lim = 2 lim = 2 lim = 0. 2 x&0 1 + 1/x x&0 −1/x x&0 1/x x&0 −1/x2 $%U,+ 4*+ .4E+6< U-/A 4 %&%)%++ 25'%)* = − lim  e0 = 1 #          =%*%+?%.)*5?4+5/43%/A4.-&> 1() =4*%2+%?35C%-?A-) D(2= 4< U-/A2=6.4# >@%&*54;,+2,=)%2.4%?2+56< 2U6-/A),?44>-4++'%/C-& + + Q ,), 3%;%*%))P.< + CP//;(?A %)% (3%? '%/ > 5 = # 2 2 2 + + + .%1 2>%&U45;+,,&-*?6?. 4* 4? CP//;(?A CP//;(?A%.% < 4'% %.?% 4 %& , 4 4* A ;4/A A ; 5 5 U > > 5 2 2 2

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f F f  


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x 7 x2  x 7 x2 + 3 x 7 2x x 7 x2 + c x 7 2x c + $ %
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% 6 6.< -/AE-? -)4?4*, 4 ? CP//;(?A?%. 4 %& , 4?6 4 <%*(& ; ; 4 Y8#F0Z %/A%?2M)(/%+ 5/4*-2=6.< U4 '%/'6+4+=6< U-/A  y u(y)   77    2        x  v(x)   d dx Z u(y) dy − ! Z v(x) dx = 0. V U = U 5 5 1K= 4.5()=%24R2+R>*;%%&5;+%,++2,)CP//;(?A 2 # 1 2 2> D + + + + + %2;(/*()(;%>%&25;%,+2,.)5R??A%? 3% , 4 6. 4 ,&-*4 4 ? ? %/&, % ?Q )*43,2A4 424E=,?B y=y(x)  Z d dx d = dy Z u(y) dy y=y(x) − Z ! v(x) dx d d u(y) dy · y(x) − dx y=y(x) dx Z = v(x) dx = 4'514 Y8#2MF"MZ )*%&25?2+ U?62M4# 5 U = 2 M >5 D5 2 2 2 ,?A % %2?3Q+ &)?4.#%&%? , %/A%? %+ +4?6 ',?A-*,),+# 1 R. ;%+*.%*% M C%24+(>+4%+-.% +(%++%2;()*+5?%)+: C-??( +-)),//4 = u(y)|y=y(x) · y 0 (x) − v(x),       *  $ ,-" (# # , "# )/& ,&#(
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 5   Y8#FWZ    = U = U > 9 2 2 M + + + + + %/ %/A%*P.< -/A 4* %/A%? % 3%? )*%&%E )*-&4 4* '6 4 4< -/A 4* %/A% 2 2 R + + + CP//%+2%?P2 U4+K +(?A%*M.?% (E*%%.2= P )*,', () 4* .P ?3)(/% 4 C4'4 &4 A %2# y (A − y) 8#""# ,3&4# ` /R&3% +QE6)4#  $ %
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 &  1 Y8#NWZ# +(+%2 42.42'4*,),U- Q&=6. 4 Y8#FWZ >5 %&%?D5,2%/A%?2%+%+ 4* 0G0  dy 1 =α (A − y)y dt

424.34# 1 +(+%2 )*%&5?+ %33M2 4+ .4E=6< U-/A +%2=%)P2 4 R;%+%*M %/A%?2M)(/ Z dy = (A − y)y y=y(t) Z 1 342-2>425 5?+%/&,2 '%/U4+,&-*,),U- C%2U4)?,2=6. 4 R;%+%*M 4-?-)),/-+  1 1 = (A − y)y A α dt + c. 1 1 + y A−y  Y8#FIZ @??%. 424E= ,? 4 Y8#FIZ %/A%?2M)(/ 4* 42,335 C-&',34Q&U4+KB 1 A Z dy dy + y Z dy dy A−y  . 5 U = U 5 5 5 1.5)*,'Q 2 2 K2 2 2 + + + + 342-2+>,),?, 425 %22)M/-?5?+>%/&, < ' % 4* & C% % % P.< -/A E-* Q < 4 ',) . ? %/&, ; ; 5 U 5 U 2 2 2 # + + + + ? .% 4&&4< -/A 4* E-* Q () ?%/4 Q %/A4&,? % % 4 ; ; <4.-& ln y # A−y A−y >0 U4 E%>5/ y = αt + c. y=y(t) A−y <0 <4.-& Z 1 dy = − log(A − y) + c; A−y Z 1 dy = − A−y y  y A−y Z 1 dy = − ln(y − A) + c. y−A `*,'-2=6?. '-)+ +-;,33 %*%? .(+ %)%+ )*%&5?+B 4 %=%*P. 5 %33M2 4* ;,2+-K+B 4*4 @33M2 %/A)*%&: ),'-2,))42 .4E=6< U-/A 

ln # (A − y) > 0
1 ln y(t) − ln(A − y(t)) = αt + c. A  = αAt + Ac, <4.-& y = eαAt+Ac . A−y AeαAt+Ac A = αAt+Ac −(αAt+Ac) 1+e 1+e 4 <4.-& H5C%=%*;% 4 ;,2+-K+< U4)-?2K4? ),'-2;4<'5?+ 4 %2M33 4+ .4E=6< U-/A y= # (A − y) < 0 y Y8#STZ
1 ln y(t) − ln(y(t) − A) = αt + c. A y(t) = AeαAt+Ac A = . αAt+Ac −(αAt+Ac) e −1 1−e  $ %
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+ + + + + 7 , % /A%? % % .4E 6.<4' 4*% * ( % 3 %? ,&/A4 Q E 6) *%/A)%& 33 % ( )%. + 3% 4 ;((& ;%*%))P. Y8#SFZ () &R;5>Q+()%.%+# @*%.%2 4 =%2R2()%%2 4* Y8#SSZ >5<4%&%?>K>5D5.,<2%U/A%? > 5# Y8# = 5 5 > 5 = 2 2 2 1 S 0 2 2 2 2 + + + + % % .% '%/4? Z C-&'6 4 4 4 E ,?<' % 4 4? %& , 4 4 ; ; -/A Y8#SNZ 1=-33-2>425 U4+,&-*4+24? 5?+%/&,2+ E4&D5,25) 5?+%/&,2,))42 ),'-2U4+=6. 5B y 0(t) +  δ= α−1 γ y(t) = − t β β α−1 β =− γ β y 0 (t) + δy(t) = t  δ δt −δt y(t) = e Z Z eδt e · t dt = · t − |{z} |{z} δ |{z} |{z} 0 δt f g g f  δt e t dt + c . Z eδt teδt eδt − 2. · |{z} 1 dt = δ δ δ |{z} 0 g f $ %
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