1. Математический анализ (850924), страница 6
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) % & &&, * & && & & 1 * & && & .? , & = f (u()) u { , & && dy = f 0 (u)du du = Tu, & && dy = f 0 (u)du, u { & , du "1 1, * Tu = du + (Tx)Tx xlim!0 (Tx) = 0:$-+ ,&+1. : *: * 0 (c. 9.1).2. 4 # : *: y = f (x) # * x0? (c. 9.3).3. *: y = jxj : *: x 2 (;1 +1)?4. % %, : (a b) *: 2 ? B " %e? (c. 9.2 # 9.4).5. *% # % *: (c. 9.4).3 26. ; #0 06 *: y = 2cos x .7. + *: x2 5Df (x) = ax + b xx >5.1* " *2: a b, " *: f (x) " " : * = 5.958. ; : dy, y = f (sin2 x) + f (cos2 x):9. ) *: y = 5x3 ; 3x2 + 10 : 1) Ff (1)D 2) df (1) , Fx = 0 1.p10.
7e "% 35.3. ;, * ** *: y = jxj # * x = 0 (. 8.1).3 26. y0 = 2cos x ln 2 3 cos2 x2 (; sin x2) 2x = ;6x 2cos3 x2 ln 2 cos2 x2 sin x2:7. ) !, " *: f (x) " : * x0 = 5, %," " 2 * #0. , * x0 = 5 % :lim f (x) = x!limf (x) = f (5)D x!limf 0(x) = x!limf 0(x)x!5;05+05;05+0..
25 = 5a + bD 10 = a:, *: f (x) " : * x0 = 5 a = 10Db = ;25.8. dy = 2 cos x sin x If 0(sin2 x) ; f 0(cos2 x)]:9.10.Ff (1) = 0 925D df (1) = 0 9:ppp1 = 5 11 :35 = 36 ; 1D x0 = 36D Fx = ;1 35 6 ; 121296 10? 7 456. 5@ 7 456 10.1. P f (x) - (a b), " 1 2 *1 (a b),1 < 2, , f (1 ) < f (2 ). P f (x) # (a b), " 1 2 *1 (a b), 1 < 2, f (1 ) > f (2 ). 10.2. P, "1 (a b) &, 1 (c d) # (c d).
10.1. 1 ./ y = f (x) , # (a b). 2 (a b) ) ./ c ./ x = g (y ), - # (a b).! " 10.2.1 ./ y = f (x) # (a b), - 0 . 1 0 f 0 (x0 ). 2 ./ x = g (y ) y0 = f (x0 )1 .f 0 (x0). = 0 1 T. 2 & x = g(y)) " & = f (x) 1 T. <, T 6= 0, & = f (x), T 6= 0. = ,T = 1 :(10:1)T TTxE T . $ & x = g(y) ( . 10.1) Tx ! 0. 6 , - , 97 f 0 (x0) 6= 0.
= , 1 (10.1), " & x = g(y) 0. , g0(y0 ) = f 0 (1x ) :0) " ) , f 0 (x0 ) = tg , { x & & = f (x) 0 = (0 f (x0 )).6 " & x = g(y) * &, % y. + g0(y0) = tg , { * y. 2 ",& % :tg = tg1 = tg ( 1; ) = ctg( 2 ; ):26 " &1) = arcsin xJ ;1 < x < 1J ; 2 < y < 2 :Py = arcsin x " & x = sin y,y 2 (; 2 2 ), + & x = sin y (; 2 2 ) * x0y = cos y , , 10.2.(arcsin x)0 = x10 = cos1 y = q 1 2 = p 1 2 :1;x1 ; sin yy2) y = arccos xJ ;1 < x < 1J 0 < y < :x = cos y ddxy = ; sin y 6= 0:2 ",1 = ;p 11 :p(arccos x)0 = x10 = ; sin=;y1 ; cos2 y1 ; x2y983) P y = arctg xJ x 2 (;1 +1) * " & = tg y.
2 ",(arctg x)0 = (tg1y)0 = 11 = 1 + 1tg2 y = 1 +1 x2 :cos2 y4) P y = arcctg xJ x 2 (;1 +1) * " & = ctg y. 2 ",11 :(arcctg x)0 = (ctg1 y)0 = 11 = ; 1 + ctg=;2y1 + x2; sin2 y) O" & 1 &:1) " ( . . 10.1, a):x ; e;xesh x = 2 J2) " ( . . 10.1, "):x + e;xech x = 2 J3) " ( . . 10.1, ):x ; e;xeth x = ex + e;x J4) " ( .
. 10.1, ):x + e;xecth x = ex ; e;x :P sh xJ th xJ ch x . Pcth x , = 0.O" & , .99O" & " , &, :sh(x + y) = sh x ch y + ch x sh yJch(x + y) = ch x ch y + sh x sh yJch2 x ; sh2 x = 1:?*, &:0 x ;x 12ch2 x = @ e +2 e A = 14 e2x + 12 + 14 e;2xJ0 x ;x 12sh2 x = @ e ;2 e A = 14 e2x ; 12 + 41 e;2xJch2 x ; sh2 x = 14 e2x + 12 + 41 e;2x ; 14 e2x + 21 ; 14 e;2x = 1:;;;;) a))")). 10.1100"0 x ;x 1 0x;x(sh x)0 = @ e ;2 e A = e +2 e = ch xJ0 x ;x 10x ; e;xee+e0A@== sh xJ(ch x) =222 x ; sh2 xch0(th x) = ch x == 12 J2ch xch x ch x !0 sh2 x ; ch2 x(cth x)0 = sh x = sh2 x = ; sh12 x : sh x !0! "R f (x)1 c2x3ex4axf 0 (x) R010f (x)ctg xx;1 11 arcsin xex12 arccos xax ln a 13 arctg x114 arcctg xx6 loga x x ln1 a 15 sh x7 sin x cos x 16 ch x5ln x8 cos x ; sin x 17th x1cos2 x 18cth x9 tg x101f 0 (x); sin12 xp 1 21;x; p1 1; x211 + x2; 1 +1 x2ch xsh x1ch2 x; sh12 x$-+ ,&+1.
)*% ch 2x = ch2 x + sh2 x:2. )*% ch(x + y) = ch x ch y + sh x sh y:1. C* **, 0,ch x = 12 ex + e;x sh x = 12 ex ; e;x 2 2ch2 x + sh2 x = 14 ex + e;x + 14 ex ; e;x =hi hi 2x ;2x= 41 e2x + 2 + e;2x + e2x ; 2 + e;2x = 14 2e2x + 2e;2x = e +2 e = ch 2x:2. ?# *: ch x sh x.102 11B4 446) - &y = u(x)v(x) u(x) > 0 v(x) { & , 1 u0(x)J v0 (x).& y = uvln y = v(x) ln u(x):= ,y = ev(x) ln u(x) :2 ",!!0y = ev ln u v0 ln u + v 1 u0 = uv 1 u0 v + v ln u uuy0 = vuv;1 u0 + uv (ln u)v0 :" & f (x) (a b) && * + . = , * x 2 (a b) 1 f 0 (x), , , & x. 11.1. 8 & f 0(x) && x0 2 (a b), + 0 2 f d & f (x) " f 00 (x0 ) ( dx2 )Jx = x0!2d df d f =2dx x = x0 dx dx x = x0 : 11.2.
n- & f (x) ( n 2) (n ; 1)- & f (x) , 1 f (n) (x) = (f (n;1) (x))0 10301dnf (x) = d @ dn;1f (x) A :dxndx dxn;1 " " , " . "1 & " (cu)(n) = cu(n) J (u v)(n) = u(n) v(n) :/ #(u(x)v(x))(n) = u(n) v + nu(n;1)v0 + n(n2!; 1) u(n;2) v00 + : : : + uv(n)(uv)(n)nX= cknu(n;k) v(k)k=0u(0) = u(x)J v(0) = v(x)J ckn = k!(nn;! k)! : 11.1. ?* &!(sin= sin x + 2 n :(11:1)? .? , (sin x)0 = cos x = sin(x + 2 ), ..
& n = 1. n ; 1, .. *, !(n;1)(sin x)= sin x + 2 (n ; 1). 6 ,!0!d(n;1)(n;1)(sin x) dx (sin x)= (sin x + 2 (n ; 1) = cos x + 2 (n ; 1) =!!= sin x + 2 (n ; 1) + 2 = sin x + 2 n :x)(n)1042 & (11.1) . > &:!(n)(cos x) = cos x + 2 n (ln x)(n) = (;1)n;1 (n x;n 1)! n 1: 11.2. $ e (x3 ex)(100):$ & A", *u(x) = exJ v(x) = x3J (ex)(k) = exJ k = 1 2 : : : nJ(x3 ex)(100) = x3 ex + 100 3 x2ex + 1002! 99 6xex + 100 3!99 98 6ex == ex(x3 + 300x2 + 29700x + 970200):( & f (x), { , *1 (a b). ?&& + & dy = f 0 (x)dx & .
6 * * f 0 (x). $ * * dx 1 + . 11.3. ?&& && & && && + & " d2y (d2y d(dy)):6 * &&:d2y = (f 0 (x)dx)0 dx = f 00 (x)(dx)2 = f 00 (x)dx2 : && ", .. (dx)2 % dx2. 11.4. ?&& n- ( n n 2) && (n ; 1)- &&: d y =d(d(n;1) y):A & n- &&dny = f (n) (x)dxn :105(11:2)( ,n & (11.2) "f (n) (x) = ddxyn :)* " 4 , , , & y = f (x)J x = '(u):$ && dy = f 0 (x) dx:(11:3), & (11.3) u & f (x), && dx:dx = '0(u)du:= ,d2y = d(f 0 (x)dx) = (d f 0 (x))dx + f 0 (x)d(dx) = f 0 (x)dx2 + f 0 (x)d2x:$"1 , d2x 6 0, d2x = '00 (u)du2:= , ../ # .# ./, " , & x = '(t)J y = (t)J t0 t T:*, + & & x = '(t) " & t = Y(x), * .2 y = (Y(x)) * & * .106 11.1.01 1) ./ '(t)J (t) ../# t05 2) ' (t0 ) 6= 05 3) ./ x = '(t) ./ t = Y(x) x0 = '(t0). 2 (x = '(t)Jy = (t) ./ y = f (x), x0 = '(t0 ), ../ x0 , t0 ) :fx0 (x0) = Q('(t0)) 3) 11.1 , & y = f (x) * " * &:y = (t)J t = Y(x):y = f (x) (Y(x)):= , f 0 (x0 ) * &, f 0 (x0) = 0 (t0)Y0 (t0):6 Y0 (x0) " &, &Y0 (x0) = '0(1t ) :02 ",0(t0)0f (x0 ) = '0(t ) ):0 11.1.
11.1 "t 2 ( ), 0fx0 (x) = '0((tt)) " t 2 ( ) x = '(t): 11.3. , &, ?0(t)0P f (x) = '0(t) & f 0 (x) * & c * t:0 (Y(x))0f (x) = '0 (Y(x)) :107.,0101d2f = d (f (x)) = d @ ((x)) A = d @ (t) A d =dx2 dxdx ' ((x))dt ' (t) dx= (t)' ((t') ;(t))2(t)' (t) ' 1(t) = '(';)3 ' :00000000000000000000 1. : 1) y = xx 2) y = eex :2.
!", " y (x), y(x) = x sin x:3. & u = u(x) " x ' " (. d y, y(x) = eu :4. " " d y + y = y(u(x)) y = y(u) " u = u(x):5. & y(x) x(y) "+ " !" ." x (y) y y :6. &"+, y = ch x "(y ; y = 0:7. 0 "+ 1" ,"" "" (c . 11.1).(20)2220000000(ln x + x x1 ) = xx (ln x + 1):2) y = xex = eex x y = eex x (ex ln x + ex x1 ) = exxex (ln x + x1 ):2. !", + u(x) = sin x, v(x) = x . 3'"(x sin x) = (sin x) x + 20(sin x) 2x + 20 2 19 (sin x) 2 == x sin(x + 10) + 40x sin(x + 192 ) + 380 sin(x + 9) == x sin x ; 40x cos x ; 380:1. 1) y = xx = exln x y = ex0lnln xln022(20)(20)2(18)(19)22d y = d(dy) = d(eu du) = eu du du + eu d(du) = eudu + eud u:d y = d(dy) = d(y du) = d(y )du + y d(du) = y du + y d u:!!dd1d15.
x (y) = dy (x (y)) = dy y (x) = dx y (x) ddxy =!d11 = ;y 1 = ;y :=dx y (x) dxy yydy3.4.222000000022000000020108000032 12 ! ". 1) f (x) a b] 2) f (x), , (a b) 3) : f (a) = f (b):"# (a b) (a < c < b), f (c) = 0.%. f (x) a b]. , ! "!#, $% '(# (# $)!, .. *% )% x1 2 a b] x2 2 a b], )f (x1 ) = M = xmaxf (x) f (x2 ) = m = xminf (x)ab]ab]0022 + , x 2 a b] : m f (x) M:1) . m = M , f (x) const = m , (, f (x) 0 (a b). " ) )% c { '+ )% .2) ( m 6= M . $ )% x1 x2 1 (a b) (1 '( ' )% 1 (a b), , %!! , $ , (a b)).(, x1 2 (a b). % %% f (x) f (x1 ) + ,x 2 a b], ' ( ) ) 2x ** 2f = f (x1 + 2x) ; f (x1 ) 0:% '$, ! f (x1) = xlim0 f (x1 + 22xx) ; f (x1 ) == xlim+0 f (x1 + 22xx) ; f (x1 ) 0(2f 0 2x > 0)c ! ,f (x1) = xlim 0 f (x1 + 22xx) ; f (x1 ) 0 (2f 0 2x < 0):10900!!0!;5 ), ) )% x1 1 +(+ $ : f (x1) 0 f (x1) 0.
6 $1 #( ), % f (x1) = 0 %) )% c $( )% x1: %$.7)% 8+ $) *: %!x )% %! , ! +, %!! , )%%!, %! %(+ % ! %! ( 9x(. 12.1).000;. 12.1 12.1. :+ 8+ *. ;,<%=+ y = jxj x 2 ;1 1], + + 1) 3) 8+. : * $ + , $ %) !)% (x = 0). ; %! )% c, %! f (c) = 0, % %%0f (x) = ;1 8x 2 ;1 0) f (x) = 1 8x 2 (0 1]:00 ! !$!%!. 1) f (x) a b]2) f (x) & (a b)."# (a b) , , c, f (b) ; f (a) = f (c):b;a%. (f (a) :Q = f (bb) ;;a? (, f (b) ; f (a) = Q(b ; a):" <%= '(x) = f (x) ; f (a) ; Q(x ; a). 9, ) <%=+ '(x) + + 8+.
@!(,1) <%=+ '(x) $% a b] %% $( , a b] <%=!001102) c* $ + ' (x) %!! (a b)0' (x) = f (x) ; Q003) %=, $% a b] <%=+ '(x) $)+:'(a) = f (a) ; f (a) = 0'(b) = f (b) ; f (a) ; Q(b ; a) = 0:(12:1)" 8+ *, %!! , )% c 2 (a b) %+, )f ( a) :' (c) = 0 ' (c) = f (c) ; Q = 0 f (c) = f (bb) ;;a %$.7)% A 1 $), ) %! ! + %+ )% (c f (c)), %! %(+ ( , (. 12.2).0000;. 12.2 12.2.
@%$+ < f (b) ; f (a) = f (c)(b ; a)(a < c < b) $ < A 1 < %),*!.1 c = a + B(b ; a) 0 < B < 1, )0f (b) ; f (a) = f (a + B(b ; a))(b ; a):0 12.3. C, ) 8+ ( )! )! A 1: % + A 1 '( f (a) = f (b), )ae, ) f (c) = 0. 12.1. @%1, ) 1 + %+ <<=! <%= 1, %!! , %( $ !.1110%.
( $ <<=+ <%=+y = '(x) ( '(x1) = '(x2) = 0, <%=+ '(x) + $% x1 x2] + 8+. @!(, $% x1 x2], % %% '+ <<=+ $% x1 x2] $%. : * $ + (x1 x2), %=, $%a x1 x2] $)+: '(x1) = '(x2 ) = 0: ? (, 8+ *, %!! , )% c 2 (x1 x2) %! ' (c) = 0. 12.2. @%1, ) + ', x1 x20j arctg x1 ; arctg x2j jx1 ; x2j:"$( $( )% x1 x2 ( x1 < x2. <%=+ y = arctg x + $% x1 x2] + A 1: y = arctg x , $) $% x1 x2]. : * $ +y = 1 +1 x2 : A 1, ! + %+ )% c 2 (x1 x2), )arctg x2 ; arctg x1 = 1 :x2 ; x11 + c2? (,j arctg x2 ; arctg x1j = 1 +1 c2 jx2 ; x1j jx2 ; x1j: ! &'.
1) f (x) g(x) a b] 2) c( f (x) g (x) (a b) 3) g (x) 6= 0 x 2 (a b):"# (a b) , , , f (b) ; f (a) = f (c) :(12:2)g(b) ; g(a) g (c)00000(* (12.2) +,.)1120%. : ), ) $( ! ) (12.2) ) +. . ' g(b) = g(a), <%=+ g(x) + ' + 8+ $% a b] 8+ (a b) * ' )% c1, %!g (c1) = 0, ) ) 3) D#.8 ( <%=f (a) (g(x) ; g(a)):F (x) = f (x) ; f (a) ; fg((bb)) ;(12:3); g(a)6 <%=+ + + 8+. " , F (x) $% a b], % %% <%=f (x) g(x) $%. $ + F (x) * (a b), )f (a) g (x)F (x) = f (x) ; fg((bb)) ;; g(a)00000 %=, $% a b] <%=+ F (x) $)+:; f (a) (g(a) ; g(a)) = 0F (a) = f (a) ; f (a) ; fg((bb)) ;g(a)f (a) (g(b) ; g(a)) = 0:F (b) = f (b) ; f (a) ; fg((bb)) ;; g(a)++ 8+, )ae: * %+ )%c 2 (a b), ) F (c) = 0: 9 ; f (a) g (c) = 0:F (c) = f (c) ; fg((bb)) ;g(a)8$ g (c), ) 1 (, %%% g (x) 6= 0 + , x 2 (a b), )f (b) ; f (a) = f (c) :g(b) ; g(a) g (c) %$.
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