Mathematics | Mathematical analysis » Dancs István - Analízis

     

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− i sin(−nα) = rn 2 = r n (cos nα + i cos nα), cos (−nα) + sin2 (−nα) z −n r −n (cos(−nα) 4%/A%?5;%22M4)(/?%/4)*%&+Q5;?+ +%)%%2=%)P+(+2#5) 3%2,++6.# 1 YGZB 1 %/%2M*M Y"Z )%&5?+ 4%2A3M2 4>K>5. 4* ,2 Q+,)# n z0 = 1 %/,2 4E->,)4 z 0 = r 0 (cos 0 + i sin 0) = 1 r1 (cos(α1 − α2 ) + i sin(α1 − α2 )) · r2 (cos α2 + i sin α2 ) = r1 (cos α1 + i sin α1 ), r2   * 
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    ! . K = max{|α|, |β|, M} n0 |an − α| < ε 2K |bn − β| < ε , 2K 1 5 M ε>0 ε/(2K) n0 5 n. |an bn − αβ| = |(an bn − αbn ) + (αbn − αβ)| 5 |an bn − αbn | + |αbn − αβ| = |an − α| · |bn | + |bn − β| · |α|. @33M2 E%>5/<4 %/%2M*M .5%%2+ )-& %/A%?2M+2%?)(/%5+ V/A%2%3% ;(;% 4E=6< U-/A +%U,+ C%??,2 4* ,2 Q+,)# 2 += U 4 3% , 6.< -/A |an bn − αβ| 5 |an − α| · |bn | + |bn − β| · |α| < 1 2    ε ε K+ K = ε, 2K 2K # 1 1 = , n∞ bn limn∞ bn 5 5 2 2 2

2 M M 2 2 2 2 + + + ++ + % 4)*-&4 )-&-*4 %)*(&%3% , % * , Q ,)4 . 4 4*,),;4 ; 54.-& 2 2 # D5 K 2 # 95 2 + + + + + + + + /4*- 6.4*, Q ,) E-*Q,))6.%*(& 4)-&-*4 &%2A4?E&-.,&4;-?4 .-* , Q ,) ;%< U4 <()# U 5 > U + + <%*(& 4 Q )*, -*;4?? %X< -/A ; @33M2 E%>5/D5 5 ++ > ++ 5+54++ U 2 <()5Q/>A U # /A4?D)4. 4 4 24<24;4B- 4* E- Q; ), - =;4?%2M2(-))%A4?2<C%??,?2 <%X<U-/A-/A < U44.-?;%&/%?# ))*%C-/ U4 () @*%.%+ C%2U4)*?,2;4<42.424) ,+424Q+,))42<4E=6 4 R;%+%*M+ lim an bn = an · b1n ! 0 < |β| |β|/2 n0 | |β| − |bn | | 5 |bn − β| |β| − |bn | 5 |β|/2 ε n1  . n1 5 n n2 = max{n0 , n1 } |β|/2 ≤ |bn | U4 |bn − β| ≤ |β|2 ε , 2 bn β |β − bn | < |β|/2 n 5n  0 |bn | = |β|/2 2 |bn − β| < |β|2 ε n2 5 n. 1 β − bn |β − bn | 2 |β|2ε 1 2 − = 5 = ε, = 2 |β − bn | < bn β bn β (β/2)β β |β|2 2 n2 5 n <() %*%2 3%2,++6. 4* ,2 Q+,)+# $

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ank x x an # x x an x x 1/2 an2 1 an1  n1 < n2 an1 , an2 , . ank , n1 < n2 < · · · < nk ank+1 x ank k = 1, 2, . 66 x nk < nk+1 1/k   !"#  $%,. *"#(  )! , -
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?%?%/4 Q 4/ <() ?4/A-33 Q &% 4/-?.(? ?4/A-33< ; ; 5 >M 2 R +E%2U6)*,+ ;(/4 +%2%?U%<5?+U%&;4 + + 7 + 6 -? )* / -& 4? -?-? ? 4* CP//;(?A< () 4& 44 ;%.% V U5 > 5 5 > = 2 2 K + + + + 4& 4 % % ?3%< ) *%? %.-& 4 % ? , )-& ? %? 4/ 4 4* ;(/ Y4 ?62 4>5.4+ 5;(;%Z< +%+U4&,++ 4 ;(/+%2%?3%# 1 433K2 .R;%+%*5.< U-/A P n exp(x) · exp(y) = ∞ n=0 (x + y) /n! = exp(x + y) exp(0) = 1 exp(1) = e exp(n) = en   (n ∈ N) exp(1) = e  n exp(n + 1) = exp(n) · exp(1) = en · e = en+1 n exp(n) · exp(−n) = exp(0) = 1 exp(n) = 1/ exp(−n) = 1/en = e−n exp [−β, β] (0 < β) Pn n x /n! k=0 exp x x [0, +∞) exp x limx−∞ exp(x) = 0 exp(x) exp(−x) = 1 1 L%E-*#5+95Q;;%2 %)%4%+2(3%? 2 M M2 4* %/A%? )(/3 B+ < 5 5 5 U 5 M M R 7 + 33 % .)*%& ? ? . < 4 ? <%*(& )* / -& 4? ;%.)* U U 5 U M M M# 9 + + 7 C-/A< 4 ? < 4*4 C-/A< % , )* / -& 4? ? < 4 4* ? U=R,?+&4433;4?K2< 4*<U-/AU-/A4

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= 1 + x + x2 2! exp(x) = 1 + ))*%C-/242;4 4 %/A%?2M+2%?)(/%.%+B U4 "80 1+ 1!x < xn +··· n! x2 +···+ +··· n!  1 1 +···+ + · · · 5 1 + x + x2 · e. 3! n! +···+ <4.-& 4%2A3M2 @33M2 E%>5/ ,& 4*-??42 .4E=6 4 =-33-2>425 U4+,&(&+(%+B  0<x<1 1 + x < exp(x) < 1 + x + ex2 , %/A%? -)+ 4* ?%/4+Q;<44 ! x 1< exp(x) − 1 < 1 + ex. x # @.-& lim (exp(x) − 1)/x = 1. x0+ x = −|x| exp(−|x|) − 1 1 exp(|x|) − 1 exp(x) − 1 = = , x −|x| exp(|x|) |x| () Q/A  exp(|x|) − 1 1 = lim |x|0+ exp(|x|) |x| 1 exp(|x|) − 1 = lim lim = 1. |x|0+ exp(|x|) |x|0+ |x| exp(x) − 1 = lim x0− x   5 >M 5 1 2 G# 8 T# R 7 + + + * CP//;(?A )* / -& 4? -?-? ? C-?-) CP//;(?A<%*(& 4* ?;%& < ;%.% A 5 2 2 + + + *45KE-) +(5++Q%;2 ),-. )*%&5?+ 2(R+))%))(/% %*5. 4* 5?# ;%&%<4% ?%.4* (& % %*() 4& -,?A44* A @*+ 4 +(?A+ ,2 Q+,)34? 5) &()2%+%P.B  exp exp(R) = R++ *,(!-+#


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YGZB 1* G#8T# 5?;%&5K) +(+%2 .R*;%+2%? .R;%+%*(?A% # Y8ZB 1*< + < 4*-?-)),/() ;%/AP. 4* U%2A%.%?B (5) limx+∞ (log x)/x = 0  limx0+ x log x = 0  log   exp(x + y) = exp(x) · exp(y) (x, y ∈ R) y

= log v (0 < u, v) x = log u 5+6),+ ;(;%U .4E2 =6B # ()YFZB%33M2O%/AP?. <5?>.(4*+ -%2>2)4M2 22-5/4&%)3%? + ++ 4* % % B A Y 5 5 G# F "# F 2 2 D5 K 2 4 4 25%)*# 1 ,)5. 25%))*%2 ++ <() Q / 4* Z %)* &% , 4 ?6 U4)-?2K4? =,&;4%2B A exp(log u + log v) = exp(log u) · exp(log v) = uv, x log u + log v = log uv exp(u) limu∞ log(exp(u))/ exp(u) = limu∞ u/ exp(u) −u = 0, u−∞ exp(−u) lim x log x = lim exp(u) log(exp(u)) = − lim Y U 5 = G# F "# F 2 2 # 1 2 M M M25 R + + + 4()U-2 6/A4?D)4.4* Z, Q ,) 4)*?, 6. *% * 3 ) . ?B # 5 5 U 1 D5 2 M + + + * %XE-?%? , ) CP//;(?A 3 & -.,34? 3%;%*% % 5%/)< 45U)%& -2 4*+P.# +%+)*M2%/%) &R/Q+%++ ;42K) ),# 1 4 () 4< x0+ u−∞ x log x = − log(1/x)/(1/x)  1/x ∞ α   !/")
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x. 7    (8)  cos  sin 2 2π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cos(x + 2π) = cos x   sin(x + 2π) = sin x.  cos x sin x cos2 sin2 cos2  4U-2 ∞ X x2n (−1)n (2n)! n=0 2 cos x = X cn = ! · ∞ 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!)   X (2n)! (−1)n x2n X 2n = .

(2j)!(2k!) (2n)! k=0 2k j+k=n (−1)j j+k=n n 2n (−1) x (2n)! @*+ 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 x = (−1) (−1)k (2j + 1)! (2k + 1)! j+k=n−1 X j (−1)n−1 x2n (2n)! ! = dn = n=0 dn−1 = X = (−1)j+k j+k=n−1 (2n)! (−1)n−1 x2n = (2j + 1)!(2k + 1)! (2n)! j+k=n−1 X ∞ X ∞ X dn−1 . n=1 x2j+2k+2 (2j + 1)!(2k + 1)!  n−1  X 2n . 2k + 1 k=0 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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x2k y 2n−2k+1 x2k y 2n−2k+1 n−k n dn = (−1) · (−1) = (−1) . (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) # sin (x + y) = sin x cos y+  ∞ 2n  1 X 2n k 2n−k (x + y)2n X n (−1) = x y cos (x + y) = (−1) (2n)! (2n)! k n=0 n=0 k=0 ∞ X = ∞ X n (−1)n 2n X 5 L(*P. 2 M # 1 + 43% ) )*6, %??A 5?>%X: +4/-. R))*%/(&%B 3%? n=0 k=0 xk y 2n−k . k! (2n − k)! n > 0, k 2n X 7/A3-?+)6.C%2 4E,&-) 5?>%X: ()E,&4+24? k n n−1 X X xk y 2n−k x2k y 2n−2k x2k+1 y 2n−2k−1 = + . k! (2n − k)! k=0 (2k)! (2n − 2k)! k=0 (2k + 1)! (2n − 2k − 1)! 5 5 U U 1+4/C%?7 )*64 2 +5 C-&62(&4 +(/ + %)% (? ) / 4*< 4 % ( .)*P?.

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, n X 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 −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−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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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 + 4K>-? %/A(&< 4%+%2A&%:%? %/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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x2 < (2n + 1)(2n + 2), n=1 cos x√2 < 6 0<x< 6 ! sin 0<x< x2 x2 x4 − < 1 − cos x < 2! 4! 2! 0 < |x| < √  x− √ 6 x3 < sin x < x. 3! x (−x) 6 1 − x2 < (sin x)/x < 1 cos 0 = 1 cos 2 < 1 − 22 /2 + 24 /4! = 1 − 2 + 2/3 < 0 t ∈ (0, 2) cos t = 0 {0 < t : cos t = 0} 2 · inf{0 < t : cos t = 0} π cos π2 = 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! cos2 π2 + sin2 π2 0<x< =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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  5 > 5 U5 U 5 1)%3%))(//%2C-/4%/A24&()*% 2 2 2 + + ? %??4E ( % P?.?%< ) *%?C44 -)4??(*P.< -/A A A%? 5 > 5 U K # @ 2 2 + + + 4* 46 ?. * 4 ? %??4E C-/4 4 CP//;(?A 4 ,&(& (. C-/4 ,?4. $%/AP.5 4

)%3%))(/( K 4 5>M5E-?5>M++54&/ +4&46+4%)+ M+%)*7+ 4*%/5)#%&%4+%/ (3%? .(&5;+,?U%=+6M %/U4+#,&-*? C%2 6/A4?+5)< U-/A5?4*+ 446+%/A)(/?A UU-))*, 5 > = U M M 2 O R + + + + + ++ 7+ 4 E-? 34?< 4.-& 4 . .%* .(EE%? ,& 4 6?. % B %))*P. 4 %/ % ;% 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(44)*%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 U44++,&(&;%*%++(.%=P+<4 5 5 = 2 2 2 2# 1= 2 2 2 + + + + + + E%&)*% ?% ( %* . C% ( % ?P % % ?C% %*% 3%? 4-) %XE-?, C-/4 3%<() ?(U,?A 42.424*,),+ 5)%&=P. %/# s(t + h) − s(t) h0 h lim    &( ) -(. $( & ,& )-( ") *#0  7  02  f   2     x ∈ R   2          2   ( 5  2 h 7     3   h 6= 0     0  52   7 f 2 x     9  ) f (x + h) − f (x) , h 2 52   3  2  "F"  f     (x + h) "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)  6 &( ) -(. $( & ,& )-" ") *#  & ,(/" ! &  7  02  f 

 2     x        2    h 7 y f (y) − f (x) . y−x x  (x + h) f  )! + )0  2 52   7 f 2 x f (x + h) − f (x) h  h = 0  2  f (x + h) − f (x) h0 h lim    2           2 f 0 (x),     f     x       2   2  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 ++4&?4.+-,?A,?4< &?A%*%2 6+?4. (3%? ;4?3%2)M(&E-? ;42452A%? 5.?R+%&;4 # 4 Y8# = 5 >5 5 5 > 5 > 5 1 0 2 2 2 D 2 R +R + + +: Z % ( )%. &()*3%? & (?% < &4 ?, ) %&% % %.< () ? %/A . ?%. %/;4? U = U 5 = 5 2 M # @ 2 2 2 2 R + + + 44 44/4 % ?A% () , &,?A4 /A % ( )?( 4* 4 C-? -)< -/A ? ( E-? -)4334? C% %*% . +<45L5=%%2%?+-?2%/B+K2 )*,&45. 4* ;4/A <()%??%.C%2%2 %/4* =%2R1 -?>4?5=%52;R422(K)<4 22%(/&(/%33 = 5 > U5 2 K= 2 2 2 R + + 7 + + )?%# @2-?)M E>=5244%/ ?+,)&44*++%<2=U%)-/A5?C-&, ),/?A % ( )?%. , )* . < %;4?%/A ,>5?A-)),/4B Y 5 5 5 2 2 2 D5 + + 4* 4/A %)% % / 4 ,) )*%& ? Z .% 4 %&%? ; ;4 ;%?? U,?A4%/>-)+.%# @2 *-?/-?>>4?-+5=<%U2%-/A?+U%4+ E(2;4/A >,62 4.-&< U > 2@, .-& 2 2 # ++ ++ 4 4CP//;(?A 4* .(E % % 4 5 5 >5 U > 2 D5 2 # + + )*%& ? .% 4 %&%? ,

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)*%&% % ( ) =%2R2() %/A&% ?(E)*%&:33( ;,25.# V V 5 >5 U > > 5 = > 5 1 M 2 2 D5 2 D K 2 D K R + + . .%* , Q ,)%.; 4 % ?)4 %&%? , ,?A4 -) % ?Q ,;4 <()Q / % ?Q ;% ; A4.,& 5>)%2V%?QU%D+5?%K3#K2@2.M5 ?A%B U 5 U 5 2 2 : + + + ++ ?( ,?A3 * -?AQ ,)%/A)*%& 33%??%.4 4)*?, 4 ,;4 < ? 4%/)*-.> U 5 5 >5 U 2 D 2 2 2 2 D5 + + + + + >-) 25%)*(?%.?# 6 ;4 , &,?A4B ? ? )- A4?)*% ( % %) 4& 4 4< ? 4 %&%? 4 ,?A4  2 3 2          9  . ∆x = (x + h) − x = h . ∆f (x) = f (x + ∆x) − f (x) ∆f (x) f (x + ∆x) − f (x) = ∆x ∆x ∆x df  ∆f dx Dx f (x)      !"#  $& ,(/" ! ,, ,!& ,( "-(.  0 7  52  1 02    f 2     x 2 U f        3    7 23 2   7 2  3         2 x φ:U R x     52  3  f (y) = f (x) + φ(y)(y − x), y 4*+ =%2%?+5<

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         9 23    f 7 2          9 3           x 5   5  2     9  1    3    f+0 (x), Dx+ f (x). f−0 (x), Dx− f (x) 5 U > 5 = > 5 U 1U 2 K2 2 2 2 2 2 2 + + + + + + + 4 ,&(& (. C-/4 & 4?6 4.)*%& ? < 4434 4 () -334 4 ,&(& (.%%/ 5 U 5 U U 2 2 M # @ 2 R R + + + + + %/A%*?%.< 4.-& ( %* . 4 . * ) (& (.% %/%/A%* 4 ,&(& (. )*%& ? < 4 %/A % %? 4 A 5 > 5 = 2>425 >2%&,+5U;4,+2+K4.%/%/A%*?%.<4-&2 2 ++ + + 34.R*2R)()(&4+=(.-33-# `*(E%? 4CP//;(?A?%. ( %* . 4 %& , 4<4 ;>,?.-?< M 2 M 2 R + + + %* 4 (?A 4*-? 4 . .%* E-? 34? )*%&%E E( ;% > 5 5 M2 M2 # R R + + + 4U-12 4* CP//;(?A %/A ?%/A% . &3 () %/A * )* ? %) %/A%?%)3 &4. 6. ))*% ; > = 5 = U U = > 5 > 2 2 2 2 2 +(. CP//;(?A + + + 43)*-27+(&+34?< E( , 4 ) 6 4 4< -/A 4 ( %*?%. 4 34 ()

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  ! )##" 0 7 2   2  3  1    7 2    1 >5 %&%?D54U,?A4>-) Y8#"Z 424.=,;42 )*,-2;4 4+ .4E=6< U-/A f    f (y) − f (x) (y − x) . y−x U 5 U 1.%* MURR))%/ =6())-&4 K 2 2 2 2 2 R + + + + + 4 ,&(& (.(&%;-?4 .-* )*,, ) )*43, -.C% 4)*?, , ),;4 4. A ;% # +6?. f (y) − f (x) =  lim f (y) − f (x) = lim (f (y) − f (x)) = yx yx f (y) − f (x) · lim (y − x) = f 0 (x) · 0 = 0 yx y−x # @*E%4 >5U/%4C-2A%?2A# +-?-)),/ >%V?QD5K=4)%&5?+ E-?+-)4?4+ =%2%?+5< +U%-/AU,+ 4* CP//;(?A C-2A+-?-) 2 Q+,))4+ 4U2 -/A4? %2 %?.%*4M5+ &4,?A34?B >4 %&15*%;,22MU4M+K,),/24 ,335 1E(C-2>24A+6-?-)),/?% +4+=4# 5E25.,2=4 Y;-?=44/46+,?Z = lim yx f (x) = limyx f (y) f  x    *     9 

 x 7 |x|       7 2     x = 0  2       4 4 1* 43)-27+(&+(. CP//;(?A >5 %&%?D54U,?A4>-)4    |h| |0 + h| − |0| = , h h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*/ ) 0 # n & , & , /  7   5  2 7 3  2  f 2  x     2 2  :     4 ;   7   3 3 1       0 2     f x         9  2  f 3     3 5      x     f 0                    3 5     1     f 00      f (n−1) (n − 1) 4       x   2 2 4 3   7  3     x           f

2 n 4  2 n 4   3 5 5           x            2     9   n d dn f (n) f (x), f (x), Dxn f (x). f (x), dx dxn   &( ) -(. $ V U >5 > 5 > 5 = = 5 > O 2 D K 2 2 D5 K 2 # + + + +R + + %/AP.()*&%< -/A4* % . %& , % ?Q 4;4.(EE%? % %) ? 6. & (? ; ;4 H55?>62;4 4* %2)M >%&5;,2+3K25)(+%2+ >%&5;,2,))42=6+6?. %2 4* %>5. >%&5;,2+U-*B () V > 5 > 5 > 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 45 C-&,25)4? .P2R?R)%? )*%2(2%+%)B n 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>54/, 4*,2+ %CP//;(?A 4 %)*%& ?( 45?%/+ 4 V > 5 = 5 = 2 M M 2 D K +%/,%)*P?. + . %& * < % %) % ?Q ,34?< ; 2 4E->,))*%&5?+ 4 5) 2%U%+ 4 .5 ?>62K %)%+# f n f (x).  0 n=0 
#     %# 
! %" # #"# >%&5;,2>+ %&.55;)*,5 1CP//;(?A 2 K> 2 2 + + + , ),&4 )*%&%.% C-/6?. 3%;%*% ? < 4% %.?% 4 )%/Q )(/(;% )-. A  = 5 U U 2 DU K# $ 2 2 +  + + , 4 %U4 >4?%&5;.,6)4? %/*;%+24%?P,&-2 4 >4%V?QD54?6K3K2).,/-)4*-?34? ()?(% A.-& 5 U 5 > >= M 2 2 # 2 R + + + + ?% ) % % ,) 4 . ? 6 44 6.%/ %?.-& ; A 4 .P2R?3)(/5 U,?A4>-) %/4>,)4 6+,? U4+,&(&+( %/U4+,&-*,)4 4 C%24>4+6?.#     *  1 B>5 %&%?D54U,?A4>-) ,+424.Q+,)4

4-&< 45-& 4* 4 %/A .R&?A%*%+(3% %)5.< () 1   9    y 6= x 7    x 7 x3 2      x  2 y y 3 − x3 = y 2 + xy + x2 . y−x x $ %
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, -() ./0123456 7889: 5;<=3<>?< 1 @ : : 8 "FN 1 U4+,&(&+(. )*,-2,)4B +%U,+ 45+ &R;5>%?    *  y 3 − x3 = lim (y 2 + xy + x2 ) = x2 + xx + x2 = 3x2 . lim yx y − x yx K>-? 5) Q&U4+6?.# d 3 x = 3x2 , dx (x3 )0 = 3x2 √ 3  1   9  k : x 7 1 − x2 2       2    2   5 :7  5; 3        4 5 > 5 2 2 R + + ,?;42+K<4U.(&-/A>()4 (&CP//;(?A ? %? E-? ?4. 4 %? . &?A%*% (3%? ;4 A >%&%VL?+5A,4*522+;<%(& U > 2 # 1 M 2 2 + + + + + % %) () %)% %.3%? ?% % C% 4 C% 4 4 < ;% = 5 5 => 2 M 2 # H M 2 + + + + 7 + () ?% 3% ) E-? 4 4* (& % %*() 4& -,?A?4. () 33 4 /A , 4 >%&5>;-),2+C-/4 U 5 5 U > 5 5# 2.P,?-)Q2R?3)(/ 2 2

2 2 K2 2 1 +=6.5U4,?A4 + + + 4 < -/A 4* %? %)% %.3%? ) % %))%? %& , & 3%)*( ? A ; CP//;(?A 4 U%2A%? 4* b ∈ (−1, 1) b ∈ (−1, 1) b = −1  √ 1 − y 2 − 1 − b2 , (y 6= b) y−b  p √ 1 − y 2 + 1 − b2  y k b=1 1 −1 y 7− b p 2U4)*?,2;4 4 CP//;(?A# 1 )*,2,2K+ () ?%;%M+ + + 2 52 +%2 )-&-;4< () C% U5 M R + 4*-?-)),/< 4 E-? 4 %? , ?A-) . &?A%*% (3% %) ;4 A %2 %++ U%2A%) 4* 42,335 ,+424.Q+,)# (u + v)(u − v) = u2 − v 2 p b √ 1 − y 2 − 1 − b2 = y−b b2 − y 2 (1 − y 2 ) − (1 − b2 ) p   = = p √ √ (y − b) 1 − y 2 + 1 − b2 (y − b) 1 − y 2 + 1 − b2 /A E%>5/ 4* 4>K>5.< U-/A =
lim yb p (b − y)(b + y) b+y p  = −p . √ √ 1 − y 2 + 1 − b2 (y − b) 1 − y 2 + 1 − b2 ! √ 1 − y 2 − 1 − b2 b+y = lim − p = √ yb y−b 1 − y 2 + 1 − b2 +%U,+ ;(/P2 5) 4*+ .4E+6< U-/A 2b b = − √ = −√ , 2 2 1−b 1 − b2 b d√ 1 − b2 = − √ , db 1 −

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, -() ./0123456 7889: 5;<=3<>?< 1 @ : : "FW U-/A # C%2+(;%< > 5 > U = 5 U 5 5 U U @ M2 M2 2 2 + + + + + 7 ++ 33 4*%&% (?A3 . ? 6 4,&3% , 4 6.4* ) < -/A4*(& ? () E-? -* *; 5 M2 M # )6/,& %& %/%) 4* (& ?+ &% b ∈ (−1, 1)  8#0# ,3&4# 13)*-27+(&+(.<() %/A R))*%+-2>-++ CP//;(?A#  6  *     7 
         3  [−1, 1]   9     g 2  9    √  1 − x2 −1 5 x < 0, g(x) =   0 5 x 5 1. 1 
    7 2  4 g x = 0  2 U > 5 O M2 M2 R R + + + %/AP. ()*&%< -/A 4 / &3( %/A ?%/A% . &3 () %/A * )* ? %) %/A%?%)3 &4. 6. ;Q
V 5 Y8# > U K2 2 2 0 # # 2 R=-33&4 + + + ))*% ,) ,3&4Z -?,))4-)4?2 >%V;%/AP. /A%?(*P. %3%4 4*>5 <%&%?-/AD544UCP//;(?A 44 U%2A34%?&4Y4 ?6% 2;%4 5 > 2 M 2 + + ,) % Q & ? , < () ,?A4 -) %/A U5,?A-)

.R&?A%*%+(3%? (&+%2%;%ZB UU44 >5 U > U >5 @4.-& D5 2 +< U4 >%++ 4.R;%< 4-& + )*%&5?%/%?/% 4 %&%? 4 ,?A4 -) (& (.% ?6 4 < 4 E% +.%*M ,+424.Q+,)%2;(/*()% 4?62 ,?4.%/A U5,?A-)/ R&?A%*%+(3%?#<  g(h) − 1 g(0 + h) − g(0) = = h h 0<h51 √ 0 0  1 − h2 − 1 = h √   √ 1−h2 −1 h 1−1 =0 h −1 < h < 0 . 0<h51 −1 5 h < 0
√
1 − h2 − 1 1 − h2 + 1 √
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, -() ./0123456 7889: 5;<=3<>?< 1 @ : : 7 "FI >(?A%R&?A%*% 5?%. +4(3%?C%2U4)*?, U = U 1CP//;(?A 2 2 M R + + + %>>5/5 4?6 )*,-2 42,%/A)-.U5%&%,?A-) , ),;4 4* Q & 4 6.< -/A 4 . .%* ;% 5 >5 U > D5 %/%/A%* . 4 %&%? 4 ,?A4 -)CP//;(??A%2# UU4 4 4%2A3M2 4* 4>K>5.<;(;% %??% 4 U%2A%++%)Q+()5 (&+(%+< U-/A $%U,+ ;(/P2 5) .5)*,-2+6.<

U-/A # u  u(h) = ( h − √1−h 2 +1 0 −1 < h < 0, 0 < h < 1, h=0 g(0 + h) − g(0) = 0. h0 h 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    *   > U U > 5 U > 1 8# W 2 2 2 2 2 K ++ + * E( ,34? ,& , 6.< -/A 4 ?6 4 % %? ?% %& , 4 4 CP//;(?A< % 4* A ; = = Y8# > U = U U 5 =# 2 0 # 2 2 2 D O 2 K + + ,3&, 4 4%24ME)*,?R& 4 ,3&4Z4 4 % A%? R.Q??A;P :?C%?24)>344+B 4 34?< %)%+%+# /-?@.-&- 44>%&6<5;,2+-/A%/%*%?U4+,&-,)4. ;%/AP. 2%/A%?+Q;# 2%/A%? 4U-2 4*< 4.-&(&+(%+-2-A24?A4?R*%R%2 2;,;,242)4)+;4+-++46. 4*),-U%<+ 4U-/AU%<4U5-/A) E-45+Q5;) ?%/4 4 E%>5/ 4*+ .4E=6< U-/A U = U U 5 @+%*%.% 2 2 2 2 2 R + + + 7 + + ))*%C-/ 4 4%/, 4 EQ 4 6.<

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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  Y45 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* U424&4# x 7 |x|, x ∈ R {0}  s(x) = sgn  1 −1  0<x x<0 R {0} $ %
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# + <() Q / 4 34 4 %& , A ; < +%U,+ 4 =-33-2>425 >%&5;,2+ #  9            4 x = 0  2 1    # −2(0+h)−(−2·0) = −2h h h 0.5(0+h)−05·0 0.5h = = 0.5 h h  %!   
" %!
 = −2 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%%33K>;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))  & &( ) -(. 02  (x + h, f (x + h)) h>0 h<0 7   f 2     x   1      9    23 2 :; 2  (x, f (x))

 7 f 2 (x, f (x))    3 (x, f (x))     :;   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   1  f 2     a    2 2   3    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 )54%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),    &( ) -(. 1  g $ (#,*.  ,(# ,& )* .0            9 2 52 3  g(h) = 0, h0 h lim 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             2  2     + 4 >%V?QD.55K) 54/A4&,*4 /%? )*%2(+2B%+%)44)KU4)?,242%+.(E%*() -+ C-/42+4* %/;42=# %$2R4&2=+P.<4-& 42,?4. ,4+%/(& ( > U)(4*+%@.4E*(/%/A &%? %*;% V = U U > 5 5 U 2 D K 2 + + + 6.< -/A < 4 4 % ?Q

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g # h (  t(h) t(h) g(h)/|h| g(h) = |h|t(h) )h g h  h o(h) h   $ %
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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-/424U4+KB 1U-/A4? %>>5/ 5) +%++P.< 4 5) %/C-/424*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%++-< +44&+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 .4&4 .4&4 %& * , , Z 4 4 4 %& , 4 ),/?4. () 4 %& , ?4. ??%. ; ; 5 5 => 5 D5 K 2 M 2 K 2 2 + + +R + , ?4. ?4/A-? C-? -) % ?A% 4 4 4 33;, -* ) 4?4 Q * ) ?( ;4??4.< 5 U > 5 >= 2 2 2 2 2 2 + + + + + + + + C-/6?. ;5??5# 4E4)* 4 ? <4 - 4 %& ;, C-/4 4 %%? 4 (& % %()%? .%&%)* P 6 6. -;,33        "
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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) = f (x) h x x 0 f (x + h) − f (x) = = lim f (x).  f (x) h0 h f (x) = lim h0 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;%.%  ?(;%? 5) )*-.+, %2%/%+?5# U 5 U 5 U 1 2 2 2 D 2 2 2 2 : + + ++ + 4 ,&(& (.% )*,, ),?, ) , 6.< -/A %/A ( ) *%& .4 . 6 6 ) . 4 4 .Q ,), -* .(+ 2C% 4>4+-+ .%2 %/-2>4?6?B = 5 >= U 5 >5 "# >%&%35;5*,-?AQ 2 2 2 2 + + + 6. 4 C-&, ) )*43, -.4 < 4% %. 4* 4 ,. %/< -/A4? ) % . % . 4 A A ; 5 = 2 2 2 2 2 2 R R++5 : : + + + , ) % % 4CP//;(?A%. . * % %.% CP//;(?A(EQ () % ,&,)-.4 ;% ;% )*%3%?# U,?A 4)E%>%&D55,;2,5)2+CP//;(?A 0# ;(?A%.< H5)*,-4% 2=6.?(2A%?% ,2+=,+2#?@5#*%.3M2 C%2(EQ+U%+M4*-.4CP// =,+ .5 +6>=>6%&5;)*,    $ %
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>5 K 2 2 2 2 + + + + + .-* 6 4 -?),/ 3%;%*% ()(?( 4 % /C-? -)433B . (? ) % . % .4 ;-?4 ; +624=>-?),/ 4C-&,25) )*43,2A-.42 )*%3%?B   6   !"#  $( & ,& )-(" "# 
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1   7 2  f 3 2      f (x)      g ◦ f 7 2     3 ) x    (g ◦ f )0 (x) = d g(f (x)) = g 0 (f (x))f 0 (x) dx   3 7 x    g  4 8   7 5  2   ( ( 2,?D)*43,2A $ %
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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

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) ;4? 25%)*%<45 4 >%V?QD5K )%&5?+ 1 >%&5;,#2+ >%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 =446.<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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     1  28  f g  fg   f (x + h)g(x + h) − f (x)g(x) = h f (x + h)g(x + h) − f (x + h)g(x) + f (x + h)g(x) − f (x)g(x) h f (x + h) − f (x) g(x + h) − g(x) + g(x) . = f (x + h) h h > 5 5 > 5 U 1=54-332 2 2 2 2 2 K ++ + 4 ?4* %)*% % % < % 4* CP//;(?A %& , 4 ),/4 ; ; U Y8# N U 5

=> 5 2 2 # 2 # 12 2 2 2 ++ + + + + + + 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) 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?<() = 5  7 2 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 f f x f −1 Ux x Ux h y ∈ Ux   f  f x y ∈ (x−δ, x+ δ) 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))) f (x) f (x) + h = f (y) Vf (x) (f (x)+ f −1 f (x)



f −1 f (y) − f −1 f (x) 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# 1()*&%< > 5 2 2 ++ + + + + + -? -)*,4. U)*%&-< U?4 44 +4&+C-4A -?-) 4*
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 x 7 1 x 7 ex  66   !"#  $ & -(
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( %   h 6= 0 (x + h)n − xn = h

(x + h) − x (x + h)n−1 + (x + h)n−2 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 = sin h h π 2 < U4 < 4*-?-)),/-+ 1
−x 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 42.424  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 limh0 eh −1 h =1 $ %
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( % log ! x = ey y = ln x d d ln x = ln(ey ) = dx dx 1 = 1 1 = . y e x 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?+ d y e dx   ! 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 cos y = 1 1 − sin2 y |x| < 1 y = arctan x 4*4 y=arcsinx x = tan y p 1 − sin2 y = =p sin . < %*(&+ 4 .5%%2+ )-& 4 R;%+%*M.(EE%? y=arcsin x 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 + x2 @*%2 4 +(+%2 R))%) ,2 Q+,),+ 3%2,++6.# 5 5 > 5 >= 1 2 2 2 M M 2 2 2 + + + + C-&, ) )*43, -. () 4* % * ( % )%/Q )(/(;% ? %?4?CP//;(?A?%. . 6 6. A A 5 > 5 = 5 5 2 2 2 2 2 2 D5 2 M2 + + + )*,? 4 %& , , <4% %. 4C-&, ) )*43, -.

)%/Q )(/(;% 4 )E% , ) CP//;(?A%.3 ; A A U > > 5 K # H5 2 2 M 2 2 2 2 K R R + ++ + + + %/.4E 4 . / -*E( ,.4 4 . .%* )*,, ) )*43, -.4 ))*%C-/ 4 ;% A > # + E-? 34? 4 6?. * 42E-?)%+ U,2(+&42+%2++(;P?.M &()*(3%? 8#8#1,3&,? # %/A C-?+-) ,2 Q+,)+ 2,+6?. 3% 4* %24)+5D5+,)&4< 4%2A%+ 4   6  !"#  3   8 7 5   1 f    2     0 < x   2 2    f   4  3         x  2    x  2       2  02  ln(f (x + h)) − ln(f (x)) h0 ln(x + h) − ln x lim $ %
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( %  3  d ln(f (x)) d ln x   5    8#8# ,3&4# 1* %24)+5D5+,)<5?+ >5 %&%?D54U,?A4>-) 25%)%# = > 5 5 2 2 2 2 2 O R R + + + ++ % ( ) <.( )*%& 4 . 4 4*6. 4 . *;% % CP//;(?A %& , , ) (;% 4* ; 2 + )*43, A, B 453M2 4* =%2R2())%2 4>K>5. 4*

%&%>(?A#    x = eu 1 d ln(f (eu )) 1 = (f (eu ))0 = f 0 (eu )eu u du f (e ) f (eu ) x = ev 

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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 0 (x)g(x) − f (x)g 0 (x) g 2 (x) 1 0 f (x) = (f −1 )0 (f (x)) = (g ◦ f )0 (x) (e ) 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 dx 1 (x ∈ R) 1 + x2 1 √ , x ∈ (−1, 1) 1 − x2 1 √ , x ∈ (−1, 1) 1 − x2 arctan x = d dx arcsin x = d dx arccos x = = 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 )-&

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ln(h(x)) = n(h(x))n−1 h0 (x) = h0 (x)eh(x) = h0 (x) , (0 < h(x)) h(x) "NG 242K ++,342,2,2U*44++K-.#6+,? 5>-2/-*6?.?(U,?A C%24>4+-+< +-;,335 %/-2>-++ 2C% 49>4-)+-.+ 4*4 E(R))%C-/ 2>4+,&34? : : - &-

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( %  6  *  5 > 5 5 5 U > 1 2 2 2 R + + ++ E? --. %& , , )4<4%??A 3 %? 4 E? - 4 ))*%/%.(? 4 < ; ;,?ACP//;(?A%. > 5 5 = > 2 D 2 2 2 R : + &%?%/ .Q4*;P,2+%/A)%& < ()+%+?# .,331* R))%/< )4. 4 C-&, ) % ()()%.U4+%/)*-.,),?4 4%&.%5;,2;((&,)5 )*43, ?(*P.2A4 > 2 4

, ?-) %)% )*,)-&-) ;,?ACP//;(?A 424E= ,?B  9   pn (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 = Pn i=0    ai xi        p0 (x) = nan xn−1 + (n − 1)an−1 xn−2 + · · · + 2a2 x + a1 n X = iai xi−1 .  U4+K.<4()R%*6;%++.%*,?M%C%2;2(/% =2,P.+U144+=>E-6B%&255;?,-- >4 4+U3%K2+ 2,)+< >%5??%>5/U54*/A=%P.<2M33U5-/AE(2%*>,34?5?>52%/Q&+D(42)24%&.&4:< U4-*U-/A4? i=1  6  *   U U 5 2 2 M M 2 2 R + + + + + + + + 44* 4*6 4 . ?(?.< -/A% (/%*;%4 4 4*% * %.3%? ,&/A4 E??-- ;% ; ;,?A-*,) > 5 > 5 = U > 5 2 2 2 7 + 7 42%2??%< 4.-+ ,()2 Q+4??4 4?,?. 4%2M/A<4-& 4* <?%/A% C-. E? - %& , 4%/A 4&4 C-. E; 5 U U 5# 5 > 2 2 R + + + + ++ + .% %)% % / ?% (* %/ 4 ,&-*? 4 ) *-? 4 %/4 4 4 .; > 5 2 2 + + + %/ 4& ;4;(/%*P. % 4 %& ;, ,) <4-& 14*4 # %/A%?2%+3M2 %2)M2(E().(?+ <453M2 < 1   9   #    5  f (x) = (2x − 3)4 − 216x     

   2   f # f 0 (x) = 4(2x − 3)4−1 (2x − 1)0 − (216x)0 = 8(2x − 3)3 − 216. 8(2x−3)3 −216 = 0 x=3 (2x−3)3 = 216/8 = 27  6&  *     99 3  2  2 7 2 x 7 e−x 2 n 4      hn (x) · e−x 7  3               n 4          9 2x−3 = 3   3    hn 1* %2)M >%&5;,2+ 424.=4 ;42K34? 4* ,2 Q+-++ C-&,=7< 5;%2 > 5 = 5 > 5 1 2 2 D5 K 2 2# 1 + + + + + C% 4 4 3 * -?AQ ,), % %) ? 6. % * %)% 3%? />4*62;44 ;4 ;(/%*P. U 5 >= U 2 Q+,)# +4$+%/AP. 2 @ M2 + ,%/6 C% < -/A &% ,& 6 6.< -/A 33 . ? = U 5 5 > 5 >5 > 5 2 2 2 # O 2 7 + + + + 6.< -/A &% ) %? 4 4 . 4 %& , (;% 4* % . %& , 4 4* A ; ; .4E=6< U-/A  0 2 2 e−x = e−x (−x2 )0 = 2 −2xe−x . n (n + 1)  −x2 (n) (e d dx n+1 −x2 e d = dx  d dx ) n n=1 2 = hn (x)e−x . (n + 1) 2 e−x , $ %
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( %   2 0 d 2 2 hn (x)e−x = h0n (x)e−x + hn (x) e−x = dx 2 2 2 = h0n (x)e−x + hn (x)(−2xe−x ) = (h0n (x) − 2xhn (x))e−x . 952%-;2;%4);4 2 E-254*?-+ .4E>%&=6<5;,2U+=-/A4 E-25?-< 3% 5) 2,++6 4* ,2 Q<+,)4U+-<2U54)%? 4E-%225M?-M ),2 + , ) ;(/( M R + + 4 . .%* ;% 5 5 = 2 2 )*,- ,) )43, A )%& ?+ .4E 6B Y8#8Z @Q33+)6.M2 2,+5U4-)+K<+U-/A4* %2U)4M 4U,&-E-2E-5?-25?--%+>#C-.17* ;-%22)+M<+4.-& # ` 4 C-.)*,4 *, U 2 M M + + ,&-* 26.B,+B # 1 .R;%+%*M. 5)*,-2,),?,2 U4)?,2<()=,&6. 44*3%3% 5-?AQ%.3%?+-++ %/Y8#8Z4C-&6 # ))*%C-/242K4? 4+ Q&U4+=6.< U-/A 2 (e−x )(n+1) = hn+1 (x)e−x 2 hn+1 (x) = h0n (x) − 2xhn (x), hn+1 h0 (x) = 1. n hn hn+1  (n + 1) h1 (x) = −2x h2 (x) = h01 (x)−2xh1

(x) = (−2x)0 −2x(−2x) = 4x2 −2 h3 (x) = (4x2 −2)0 −2x(4x2 −2) =  −8x3 + 12x h1 (x) = −2x, h2 (x) = 4x2 − 2, h3 (x) = −8x3 + 12x.   6  *  99 > 5 5 5 > 5 U 5 5 5 @ 2 2 M 2 D 2 # 2 R + ??%. 4 E( ,?4. % ) E 4 ? ,)&4 )% . *% ) ? ) 4 %& , , ) -* % 4 3 ? - , ) ; V U 5 U 5 5 5 5 K 2 2 2 2 2 2 2 M2 2 + + + + + %/AP ( )% 4 , ? C% . % 4 /A% P?.% < -/A 4 3 ? - , ) ( % 3 .% % ?% . 5 = 5?5>?6>2++.(?5#+4O+-24>2K4)*%&%E 5 5 5 5 2 2 M 2 2 + + + %& ;, 6.4*%/A%? )(/ 34?43 2,+B ?- , ) ( % )*%& ? O(;% %*+ 4 E-25?--+ 4 U%2A%?<4+ .4E=6< U-/A 3  2 Pn n k=1 k k
= n2n−1 .  n (1 +x) = n(1 + x) n−1 x=1 n2 45 (EE%? 35*-?AQ+4?>K ;-2+#  6  *  99 1 3  2  n−1 Pn n k=0 k n   X n = kxk−1 . k k=1 
k  x .   n X n = k , k k=1  7     sin(αx)  cos(αx) 2        2  = 4 2K4? 2%U%<+ ()%2=,&?%??%.5 4 4 +CP//;(?A 2,),;4# 2 .4E=6< U-/A

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>5 U 5 @>%33%?4E-? D5 2 2 D 2 2 2 2 + + + + 34?4 %&%? , ) *,Q ,))4 .4E )4 34? ,&- (,;4 C-/6?.C-/ 4 . -*? < 2- A4? &R;5>%?< U-/A %* D)4. 4 E&-32(, C%2;%+()(?% ?%;%*U%+M# 2 K 2 2   O ] + +  ! -* ?4. %/A 4? CP//;(?A( < 4% %/AP. 4*   ;, A A3%?.P2R%/A?C(2%   > 2 2 2 # @ 2 + < E( ,6 B * 4 CP//;(?A 4.(EE%? 4 ;,(&+2(.%+-*5K %5) )%&%E% ;>U 2 2 K # 2 K K ++ + + % ,) ,) CP//;(?A% 4*

-* ?4. 4)-? -? 4 4 ;, 2 K # 2 2 K K2 M 2 + +R + + + CP//;(?A(&% 4 -* ?4. %? 33 -* CP// CP//;(?A%.% 4 4E-)4?# ;, A ;, 5 5 5 2 M D M 2 R : + + + + + C-/6?. * )/, ? 4 .() 33 % .3%?< () -) )4. 4 . .%* %/A)*%& ()*&%;( % %))*P. ; ;% +%++ 2?(&5<()+(. %%2M?A2 %R++) 4>%&5;,=2%=26R2()B%*+ 4 CP//;(?A+< 4.-& 4 3%+: U%2A%++ 4 =%2%+ 5) 4)*-.,)&R/*QU4)?,   log(1 + x) − log 1 log(1 + x) = , x x  log log 1/x x = 1 x = 1  1 %#%" %#   " % !%#  %%% %##"       x x 7 xz + x2 − x2 z x  x2 z z z 7 xz + x2 − #  z # z z d ∂ Dx d ∂ (xz + x2 − x2 z) = (xz + x2 − x2 z) = Dx (xz + x2 − x2 z) = z + 2x − 2xz. dx ∂x U U 5 > 5 5 > 5 2 2 K D 2 7 + + + + + 4?/) -*;4 4* < -/A ?4 .( -* ) % )4. 4* %/A . )*%& ? %& , 6 ?.< 4* A ;, ;4?< ; U 1 2 M MUR 2 K * % * * 4)-? 4? )*-.,) -?>4?5< U-/A B #  z      x               

            9  d ∂ (xz + x2 − x2 z) = (xz + x2 − x2 z) = Dz (xz + x2 − x2 z) = x − x2 . dz ∂z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

@)*??%. 2 D + + Q / )% (& % % ) ? )< % 4 4* -? 6.< -/A 4 A 53K24*6-?%/A%?>%2V%+?Q?%.D5K E%&)*% 5 = U )*%& ?+ 4+ (&+ P.< -/A    dx df   df (x) = f 0 (x), dx dx dx df (x) = f 0 (x) dx    df (x)    f      x  2  $ %
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(& % % )*%& %? ,&- %)% % .P ?3 * +95"%Z+;P?.L5%2?%??% %/B 5 >K = D 2 2 K 2 2 M 2 : + .% 4>M>%&;4/A5;,2+C-/A ?4.K# 4*@.-& ?A ,?;4 4? , 4 ? % % < () %*(& 4 CP//;(?A ; 5)*0Z /@-&/A74?) /A/A?RRR;%.% # %/()* %? .% ;4? 4 >%&5;,2+?4# 1* ?A52;,?;42K4? .(+)*%&%) /AR.<() 4 >%&5;,2+ 7       7 2 h0 (x) = 3ax2 + 2bx + c.  R α α U U U 5 5 > U 5 22 K 2 M2 2 2 + + + + + + 4U244.34Q&<()U4+K?%/4 <4+5Q3;M2< U4*-??4 , 4 < -/A 4* % % %. ? ? %? % %?E-* Q4*;< A ;% A 5 # @ 2 2 M M 7 4 ??%. %/C% % %?4 CP//;(?A )* / -& 4?C-/A ? ;4/A %E-?2+M34?=%2% U)*%&7-5?+++#(&15?+M E,&U%2AU%?64-) ?62 4 4*6/A4?+%?/%4 >2%&A5%;2#,2+< >% 5++ )(2)M(&+(. ?5?D)# 1* 5? %X5K) G42Z4H.34(+ /AQ&UR4%;4? +KB 4 >%&5;,2+?4.# @-& 4 >%&5;,2+=4<5?+ ,)->C-7 E-25?-/AR+(?A%*M) 2+%U%M+.M<%2UM-/A=%2% 4(+ /AR# @33R*M2R++2%%-22;%?.%* M=%2%+# 1 4/AUR-.2

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

5>?%+%&;4 5 5 2 2 2 # + + + + 6 34?< Q / 4* C-?-) ) 4 ? %&;4 6 -? %/A%? 4* %/A % A A 5 U 5 = 2 2 2 # 1 R + ++ + + + + & /*Q % % % % 4* ? %&;4 6 ?4.< 4 4 4/. ) *,Q 6. 2%.(E%*()+ 4 U U >K 5 U 24.34?+K2%&%))P< 2 2 M2 + + + 4)*,-. 4 4* %/A %/ 4 ,&-*4? )*,< 4 C% % % % / CP// 4* () 5 M2# @ + + <() 4* CP//;(?A )%& ? %/A Y8#"GZ = 5# 5 5 = U 2 M 2 2 2 $ 2 M 7 + + + + + + + C-&, % , Q ,) )*%&% ?(?. 4 , ? %&()*% %)%? )% )% 3 * -)Q 4 % &%< -/A 52A%? 544244.7 +4&4 > > U > > 24 ,26?./A4-& 2 2 2 2 # @ +C-/,) + + + ++ + + % 4 44 , 6?.< 4-& %/- - 6 4 C% 4 4 - * 4 %4+5.(,34?4/-# O< %/AP Y8#"8Z 5 5 5 > 5 2 2 2 2 2 2 2 + + + + CP//;(?A . 4 C% %;()%. )*%& ? C-?-) 4* ? %&;4 6 34?<() 3% P %& , A A ; U4+K<%*(&++< 4% 5 5 2 2 2 2 R 4 . *(E(&+(.+(+% )*%& ?+ ;4? - A4? ), 4 ?+%&;4 634?<4% A&% Y8#"FZ f b b f (x) y = f (b) + f 0 (b)(x − b) f f (x) =

f (b) + f 0 (b)(x − b) + o(x − b).  b b 00 f (b) f 0 b c>b f (b, c) ! [b, c] o(b − x) M  o(x−b) f  M · (x − b)2  f0 f0 x [b, c] o(x − b) b x f f (x) = f (b) + f 0 (b)(x − b) + M(x − b)2 . g(y) = f (y) + f 0 (y)(x − y) + M(x − y)2 [b, x] ξ g(x) − g(b) = g 0(ξ). x−b  (b, x) $ %
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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      f 2       [b, c]  9    2  3  2  !"#  $ # 02 (b, c) 1     ξ ∈ (b, x) 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?. -)+ 42424*,).(?+ %/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  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 2  ()      sin00 y = − sin y,     %*(&+ 4 1 00 sin ξ 2 sin ξ(x − 0)2 = x − x, 2 2 $ %
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% 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?%;%*()+#  & &( ) -(. $ &*,& ). % , , 02  n7 2    n4 f &  !# 0    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!      7 5  3 f 2 b    2 2 b     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 +,44&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 2  7     : ;    4  n 4    n b b b # n !     )* . % ,  !# 0  , & & ,&   3   02  n 7 2 f (n + 1) 4      [b, c] (b < c)  9  1      3  x ∈ (b, c)    2 ξ    (b, x)  9  2  !"#  $ * f (x) = f (b) +    4 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 %)%+<45.-& 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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% "IG 4%2A3M2 4>K>5.< U-/A Y8#00Z # 2+ >5 %&%?D5,2,)5 C-&624 5/4*-2,)4 ;4? U,+&4< 45 %/A)%&: 459;%-)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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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,+ %/A52 5-->?,2 5)%33 U53,;42 4>= 4 %/ 4* e R))*%/#   1+ 1 1 1 + +···+ 1! 2! 9! #! "  
  
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++ + ++ + ?%;%*% %)%?4* CP//;(?A () % %?;% (& (.(?%4P ?3)(/% 4 C% % . A ; > U > 5 > 5 5 U > 2 2 2 2 2 K2 R + + + + ++ 44%.(&2A5()< (+-/A()6 6?CP//;(?A % ;4 4%/;, -?2+-*,),?4. 4? 4* -4AU4?,?A4, >4-)4<44 ,?-)Q - .P ?3)(/ ,?A4 -)& <   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# 21634?<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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3 7  1  f g [a, b] R 2      9     2    3     3          2     9    2 a b 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.-&

f (b) − f (a) · g 0(ξ) = g(b) − g(a) · f 0 (ξ).     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< 2U544=>)*%?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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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 52 3   0 3        g (x) 6= 0 f 0 (x) x&a g 0 (x)   2  2     Y8#0FZ 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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4.4E6=+6<K33U5-/A%/A%?2M+2%?)(/3%? 4* (&+(.% +4&+ 4* U- =-33&K2< 4.-& 4 Y8#0SZ 424E= ,? U4 U4 4%2A E-?+-)4?B 4 3%35*-?AQ+4?>K Y8#104NZ &%2 42,D5%)%K;4++2%%.;2< 45;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) 453M2 4 8#G" )*%&5?+ 5 U = Y8# = = K 2 D5 K K2 0 S 2 + + + + + 44*6 33 &% , 34?4* (& (.% 4& 4* -* -33& <4.-&4 Z4 4 E ,?4* .4E 6.< U-/A U4 U4 4%)%+5(3%? E-?+#-)4? 4 3%35*-?AQ+4?>K Y8#0NZ &%2,D5K;42 %.;5;42%?) 4 E26)* ;(/+%2%? U4+,&(&+(. U 5 O 2 2 2 R R + + + + + + + %/AP. ()*&%< -/A 4;(/%) ();(/ % % ?%)% %.?% 4C%? %/.P ?3 * % ()%?%C% ( 5 U > 26/A4? 2 2 K2 # 95 2 M 2 2 : + + + + + + %?P )*P.)(/)*%& <()%/A)*%&&% ) ,&/A4 4 % ??% ? .( %)% 3%?%/A%? % ?)(/% .% 5 5 5 Y 2 2 K 2 M 2 R R + 7 + + + + + ) .%*% ? 43 * -?AQ ,))-&,? 4* 43)*(& (.&%;-?4 .-* %/A%? % ?)(/.( . * ? U = 5 Y >5 2 M 2 K # @ M R + + + + + )(/%) ?)(/&%5)#3-? 4 Z * 4 %/ %/A()P?. ;-?4 -* . 4 ;% %* ,)- .Z 5@3 *-?AQ+%,),&4 2,[Q+,)%/A%? 5 2 2 2 M M R + + + ++ + ??%. 4* , Q ,)?4. 4 3 * -?AQ ,)4 ( ?A%/(3%? 4* % * %.3%? . % %?% ;(;% ;% 5.< D)4+%24*% ;-Y8#2+0# W1Z C%352+(-?AQ+%2 +.,)34? 5U4)*?,5+2+,)45) .(%/A+ %)%5D+)%5++ C-/6?

)*%& ?%U%*%33<%/.P5?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),  
 1  δ 0 (x) =α ∈R limx&a fg0 (x) f 0 (ξ) − α < , g 0 (ξ)  ξ ∈ (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 .R&?A%*%+?%.< U-/A U4 U4 x 7− x (a, a + δ1 ) f (x) − α < 2, g(x) a (a, a + δ) x ∈ (a, a + δ1 ), $ %
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% 45 E-?+-)4?B 4 3%35*-?AQ+4?>K Y8## 0I1Z&%.52?,>D56K2;4,)2 )*%&%.;55;?4+2+%%?)+)*# M2%/%) E-5+Q; ),U- ;4? -2A4? E-*5+Q; ),< U-/A U4 Y8#GFZ ++ )*,4.-&4* 4 ,2+42,?-)Q5?++%&;4-++ 2.6R*(E(& 34?+#(. +4(+%4*2 )%&%/A5?+ %/A 4*+%+)M2%/%)5?+%&;4&R/Q2 6+%34?< +;4?%%/A%? +)*-M22A%4?/%)4 ), )*,<4%2A&% () 4%2A3M2 4 Y8#GFZ)*%&5?+ U4 Y8#GSZ 1 Y8#G8Z 4*-?-)),/ )%&5?+     0 (x) = +∞ limx&a fg0 (x) δ β f 0 (ξ) > β, g 0 (ξ) ! ξ ∈ (a, a + δ). y # (a, a + δ) x (a, y) ξ ξ ∈ (x, y) ⊂ (a, a + δ) f 0 (ξ) f (y) − f (x) = 0 g(y) − g(x) g (ξ) f (y) − f (x) > 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%.(? ;% /424*U4+=6. 4 U4+,&(&+(%&% ;-?4+-*K +(+%2+<5?+ %/A)%&:

.R;%+%*(?A+# %/Cf (x) > β, g(x)        !"#  $ # (!  #  +"  0 7    2 7 3  2  f  g 2     a 3    0  3        g (x)   2  2  52  4 0 f 0 (x) xa g 0 (x) lim $ %
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%       5  1 8   Y8#GNZ Y8#GWZ Y8#GIZ Y8#8TZ ;M &()*(3%?E( R)* &1.%*E-?>=+P.U,%/A+&42(%/A)*%& : E(2>,;42>2#,.4+ D)5?,26?4 -)E5+42 )*43,2A 42.424*,),&4# @2M (1) 1 lim f (x) = lim g(x) = 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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> 1 M 2 2 2 M R + + + + +R + +R + . .%* .3%? .( 4? %)* 4 ,&-*6?. %/<4% . ?% & < % .% & ;% A A ;%* 424.&4 U-*U4+K 4 2-/4&5+6)CP//;(?A )%/Q+)(/(;%2#    * 
  9   5  = x=0    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 dx ln(sin x) d 1 dx sin x ! 0 ln = e0 = 1. exp     *  lim (1 + x)ln x = ? x&0 $ %
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1 −1/2 1/2 p x −x dx dy = = 3 y3 + 1 x = y3 + 1   p 2 1 x3/2 x1/2 2p 3 − +c y + 1 + c. = = (y 3 + 1) y 3 + 1 − 3 3/2 1/2 3 x = y3 + 1 9 Z y5 U = 95%2M++()R))*%C-/242?,?. 4* <%2%(&MM%.%+ 4+8#< %0=#+8)P? K 2 2 2 M2# 95 2 R + + E,& )* %/A % % )(/%) % ( )& % ; 5 U = U # 2 + + ++ + + + E-? 34? , 4. )*%& ? 4* Q & 4 6.< -/A V = 5 5 Y8# U # 9 DU 2 8 G + + + %/

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&  %      S C>0  6  *  2 7 3  2 2   9     2      5    2  t  2     2  2    2   5    4         5      7   7    y(t)   9     2    1 dy = α(A − y)y. dt      2           9 7 2    9   2     9      2
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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 Z 1 342-2>425 5?+%/&,2 %/U4+,&-*,),U- C%2U4)?,2=6. 4 R;%+%*M 4-?-)),/-+ = y=y(t)  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
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