!"#$ % &# &%#'()(! $ * +

Transcript

1 ,!"#$ % &# &%#'()(! $ * +

2

3 ,!"#$ % &# &%#'()(! $ * + 6 7

4 57 : - - / :!", # $ % & :'!(), 5 ( -, * + :! ",, # $ %, ) #,

5 '(#,!# $$,',#-, 4 "- /,#-," -$ '# &",,#- "-&)'#45)')6 5! "- /,#-7 ",',8##! -#9,!")) 54! : 54 : *; 7 : *; 96 "- /,#-,'-<<,",#=-)>#45")$#,#- &",,#-<--6,+&-5-!, (,9"-6,+&-5)')6 4!,? *; 5 <,?*; 7 ' 4-9,")9") 5

6 --##!# $ **, % - ; *, ; - ;, ;, ( ;, - - :, - : % : - - -, ;?, : $ - ;, &, -, +, ; :;,, ; - - -, ;? --,, - *; #? - «-», «:», «-» *, - : -, ; ; : - * $ ;,,, * *; ) ; *- )#? 4

7 (,#!#($-(!( #/#!!"$ % f ( x ) f ( x ),? δ - x, x ( x δ ; x δ ) x x < f x f x ( f ( x) > f ( x )) 9 ; -, y = f ( x) x, ;-? 9, f ( x) = f ( x) 5?, ; x y = f ( x) ; - f ( x) ; (;), f ( x) x (), f ( x ) =, f ( x ), f ( x) x,, f ( x ) <, f ( x ) > f x X ;, * - # * 6 f ( x) [ a; b ],, :, -? ( a; b) - ;, f ( a) f ( b ); * *?* : )?;,,?, z = z x, y (,, ); ), - x y ; )?;, z = z x ; ;

8 4) ; ; 5) z = z( x) ; $?, (, ) * x, - (), * (*) ) * * # y = x 6x +, - ; y = x ( ) $ * * (( x, y ), ;? ; - (, ; - ;, y * = ( x * ) 6x * +, ( ( x ) * ; x * 6x * + " d = Ax + By + C A + B 6

9 * y = x?: x + y = ; A =, B =, C = x, y (, d ( ) * * * * * x + x 6x + + x 5x + = = + ( x ) ( x ) * * x 5x * * 5x + =, : D = 5 44 <, * * 5x + + > 9, * x, d ( x ) * * 5x + = * x R 9 d x, ;: * * * d ( x ) * * 9 d ( x ) =, d ( x ) * * x 5 ( x ) d = ;, >, * 5 * x 5 =, * x = 5 * x = d ( x ) -, * 9 * y = = 5 - d = 6 6 M (,4 ) ;,, ;, * 7

10 x a y + =, b a, b, a b, #, S = a + b - (,? ; b: a 4a + = ; = ; = ; b = a b b a b a a b S, ; - : ( a ) 4a S ( a) = a + = a + = a a a a ; a ( ; ) ; S ( a) -- S ' ( a) = 4 ( a ), ' 4 S =, ( a ) a a a =, = 4 = ; = 8

11 a = $ S ( a) = 8 ; S ( ) = 8 = ( a ) ( ) ;, a =, 4 * ; b = = 6 x y + = x + y = 6 6, l,, ;?; &,? *? $;? : R, α ( ) 9? S αr = P = R + L, L, L = Rα, 9 ( α ) P = R + Rα = R +, R l + = l α = R ( α )

12 α?: l R lr S = S ( R) = = R R S ( R ) ; : S ( R ) > l R ; &,, ; # ; S ( R) ( R) S = l 4 l S ( R) = R = <, R =, S ( R) R = l 4 -, * 9 l,? *, 4 4 # *?, x y + = a b $, 4 * ) x *, y * -,, x y x b a b a a * * * * * * + = ; ( y ) = b ; y = a ( x ), ( x * >, y * >, ) ) 9? ABCD * * * * * b * 4b * * S = x y = 4x y = 4x a ( x ) x a ( x ) a = a

13 4 * ; * S x x ( ; a) x * = x * = a ; - * S = S x ; * * 4b * * x 4b * S ( x ) = a ( x ) x a ( x ) a + * = a a ( x ) * * * * ( x ) 4b a ( x ) ( x ) 4b a ( x ) * = = a * a * a ( x ) a ( x ) a ( x ), - * * S ( x ) ;, * a x = a x = ± ; * * * a S ( x )? x = ± a 9 x = - * x - * S x ;, S ( R) - *, S ( x ) ABCD : * # ; a b a AB = = a BC = a b a =

14 5, * - 5 *? R, )$ )$CD a, $+ R ( 5) *,???, S R a π R = +, P = R + a + π R 5 ; 5, R + a + π R = 5, ; a?: a = 75 R π R ; π R π R S = R 75 R + π R = 5R R

15 5 ; R ; + π S R >, &,, π R R R >, 5 5 R ; + 5,, R ; + π # ; S : ( π ) S = 5 4R π R = R 5 ; R = 9 S = ( 4 + π ), 4 + π *, R S * +, - 5 * R = 4 + π 6$ * *?-,,?? +$ = a, )$ = )+ = b ) = R ( 6) 6

16 a 9 HB =,,, b, AK =, AKO? OK b 4 = R ab S = 4R,? S π R = a, b R ' : ABH HB KO AB = AO a R b 4 = b R ; ; b R b 4 b 4R b a = = ; R R S b 4R b = 4R - S, ; b ; ( ; R ) # ( b) ( 4 ) 4 b b R b b S ( b) = 4 b R b + = = 4R 4R b 4R 4R b ( ) 4 R b b b R b = = R 4R b R 4R b, 4

17 , S ( b) = b = R, b = S ( b) b = R # b ( ; R), b ( ; R) 9 b R, S ( b), S ( b)? = - ;, * #- *;? - *? S ( b ) R 4R R R R R = = = ; 4R 4R 4 R S 4 = = S π R 4π 7# x y 8,4 B, ( 7) # ; +;,? )$+ * + = A 7 + x y? - ABC S = AB AC 5

18 # AB AC : AB = {, 4} = {, 4 }; AC = x, y 4 ; { } i j k AB AC = 4 = y 4 4 x k = 4x + y k 9 x y 4 ( ) S = ( 4x + y ) = x + y 6 " : ) x + y 6 >, ; y > 6 x, + - () ( 7); ) x + y 6 < y < 6 x, + (), )$+ *;? + (), * ; S = x + y 6 = 6 x y +, - ; x + y = 8 y = ± 8 x y ; S, ; S x x = 6 ± 8 # * S ( x ) x [ ; ) # -,? : 4x x S = ± = ± 8 x 8 x S ;, : x ± = 8 x x ± = 8 x 6

19 $ : x 8 x = = = = = ± ; x 8 x ; x 8; x 6; x 6 $ ;? y : y y = 8 ± 6 = 6; = 8 ± 6 = 6 9, : M ( 6; 6 ), M ( 6; 6 ), M ( 6; 6 ), M 4 ( 6; 6 ), M - () S? x = ±,, $ S = 6 x y S ( 6; 6 ) = = ; S ( 6; 6 ) = = 6 6 ; S ( 6; 6 ) = = ; S ( ; ) = 6 + = # * S ( 6; 6 ) = 6 + 6, C ( 6; 6 ) * ABC 8 A V &, *?, l - a a, A V = l - l = V, 4 4 4V l = a a S = + al 4 7

20 ; l, a 4V a V S = + a = + a a 9, S - a, * : a > #, : S = a ; 8 V V V a = ; a = ; a = 4 V ; a = 4V a a 4V S = + >, a, * 9, - * - ; a = 4V 9 9 ;, S &, A *? R, H - S S = π RH + π R, ; H = R A π R V = π R H, H, R = S SR V = π R R = π R π R a S ;, R = π S V R R ; π ; V ( R ) >

21 # ; V ( R ): V S = R S ;, R =, 6π S S V = 6 π R, V <, R = V - 6π 6π,, -, * $, ;?; * A: π H S S 6π S S S S S S S = = = = = π S 6π π S 6π π 6π 6π 6π 6π p &-, A, -?,?, *? $ : )+=, )$ = $+ = b ( 8) 8 ; p, a + b = p, - a b = p A V = π R H 9

22 R, )+ a, R = ; H, )$+ 6, BCN : BN = BC CN a a H = p = p pa R H A a π = π = 4 V p pa a p pa ; V a - ( ; p ) 9 * $ ; V ( a) ; π p π a 4 p pa pa V ( a) a p pa a p pa p pa π p a( 4 p 5a) = 4 p pa = + = = ( a) V = a = 4 p 5 a = ; V ( a)? a = ; p $ - 4 p a = V ( a) ;, 5, V ( a) * A *, 4 p a = # ; : 5 a p p b = p = p = α - ( 9) α A *?

23 9 SB = R ;? SB ;,?, SB α = π KB Rα R α = π KB, ; KB = A π V = π KB SK = KB SB KB π,, V π R α R α = R = R α 4π α 4π 4π 4π - ; V ; α α ; π, R α V ( α ) = α 4π α α + 4π = 4π α R α ( 4π α ) α R α ( 8π α ) = = 4π 4 4π α π 4π α

24 ,, ;? - π α = α V α ;, π α =, * V 9, A *, π α = " ) $ ( ) 6 - MN # * s = s + s, a =, b =5, H =, h =5, L = 5 MN O M O x 9 PO = PM + MO = + x, OQ = PQ PO = 5 x = 5 x,, ; s = s + s # s s APO BQO APO : AO = AP + PO s = + + x = x + x + 5 ; BQO : BO = OQ + BQ s = 5 x + 5 = x 5x + 85

25 9 s x x x x = # s x * x 9 O - MN, MN = 5 = 5, *- s( x) x [ ; 5] $ ;: x + x 5 x + x 5 s = + = + x x x x x x x x ; ;, : ( x + ) ( 5 x) x + 5 x = ; = ; x + x + 5 x 5x + 85 x x x x ( x + ) ( x x + ) = ( x) ( x + x + ) ; ( ) x + 5 x + 5 = 5 x x ; ( x ) ( 5 x) 5( x ) 4( 5 x) ( 5 x) ( x) = + + ; ( x + ) = ( x) ( x + x + ) = ( x + x ) ; ; 75x 45x 75 ; x 4x + = + = & ; - ; 5, [ ] s( x) * - s = ; s 5 = = = 5 +, *, - N

26 $ + $ # *, R ( : 4 R 5 R 5 ) 5 5 $ ; l &,? *? l l ( :, ) 4 4 $ -, y = x x = 4, y =, *?; ( : S = 9 ) $ 4 # * A, S S S ( : V = ) 6π $ 5 # * A, R ( ) ( : 5R ) $ 6 $ * * A, * A*? ( : ) $ 7 $ * A - * A,? ( : 9 4 ) $ 8 # * A * * A, * ( : ) $ 9 $ *? *- *?, *? 4 ( : ) + 5 4

27 +* $* s = s t, t + t t v s( t ) s s t = = t t t + ( ) - v t = v t = s t, s = s t, t 6,, : t, - v( t ) = + = s ( t) v t s ( t) =, t * 9 - * s( t ), ; v = v( t) 9 v ( t) a t =, s s( t) a t = s t =, 4! "? α = α t 9 ( t) = ( t) ω α α ( t) a t = 5

28 s( t) = t + 6t + 7t + # *; -, * #: t 9 s t, t + 6t + 7t +, ( t )( t )( t ) + +, : t ( ; ] [ ;] t, t [ ;] #:, t [ ] = = v t t t t t t - ; ; * - ; #: v t = t + t + 7 = 6t +, 6t + =, t = $ [ ;] v = = 49, v = = 7, v = = 4 v = 49 4 s( t) = t + 5t + $;; 5 t = -, ;; t [ ;5] v s ( ) s 5 = = = =

29 ; t = v t = t + 5t + = 6t + 5, v = #,, t s t = te $ v t = te = e te = e t t t t t ( ) #:, ( v( t ) = ) e t ( t) =, t = $, e s = = 7 e 6 6 s( t) =, t $ ( t + ) -, 6 s ( t) = =, ( t ) + ( t + ) 64 a( t) = s ( t) = = t + t + 4 $ a 64 ( ) = = 8 8 ( + ) 4 7

30 7 ; s ( t) = 5t + t + 6 s ( t) = 4t + t ;- ;? $, #: 5t t 6 4t t = + +, t t =, t = t = 4 t = 4 t,, $ -, v t = 5t + t + 6 = t +, v t = 4t + t + 8 = 8t + #: ( 4) 4 4 v ( 4 ) = = 5 v = + =, 8 (,, t α ( t) = t t ()# ;? - t = 7, - 4, : ; t ω t = t t = t $ ; t = 7 ω ( 7) = 7 = 86 ( ) #: ( t),t =, 5 c t = 8 ( ω = )

31 4 * * * ( * * ) ; -, : 9 9 m = 5 h = 45 & k = 5 +, v =, g =,, *;;; mv & E k = ( m = m kt = 5 5t, v = v + gt = t 9 ( t)( t) mv Ek = = = ( 5 t 5 t ) 6 t, : h - gt gt h = vt + =, t, &, Ek h 45 t = = = g 4 Ek = ( 5 t 5 t ), 4 t t 5 5, 9

32 t, t ; [ ] - ; ; Ek Ek ( t) t [ ;] #: ; & 4 E k = ( t 5 t ) t 4 ( t) 5 =, t =, t = = = = * & [ ; ] $ E k E =, k E k = =, E = k Ek, * 7 = E max k = " # h, A - OA = a ( )??: *

33 ?: ( ) sinϕ E = k r #: AB - OAB AB = OA + OB r = a + h #: - OAB - OB h h sin OAB = sinϕ = = 9 AB r a + h E = k h ( a + h ) -, h [ ; ) - ; E = E ( h) * $ ; + + h a h E = k = k == ( a h ) + a + h a + h ( a h ) h ( a h ) h 5 #: a h =, a a h =, h = 9 h = a [ ; ) #: - a E,5 = = >, ( a) ( a) a (,5a) ( a + (,5a) ) a + (,5a) 5 5 a a a E = = < ( a + a ) ( a + a ) 5 5

34 a $ «+»,, - ;, *?: [ ) E ( h) # H, *? L ( ) 9 L = vt, v, t ;? v = gh, g, h v = g H x ', ; s, ( ) gt s =,

35 x = gt, x t = 9 g x L vt g ( H x) ( H x) x Hx x g = = = =, x [ ; H ]- ; L = L( t) * $ ; H x L = Hx x = Hx x # H x =, H x = & [ ; H ]#: - H H H H L = = = >, H H H H H H H H 4 L = = = < 4 H H H H ; «+», - [ ; H ], * H, - n, ;, a -, ( n a ) - 9,,

36 I = naε nr + a r, Ε?, r, R *, - a * - - = * $ ; -, a ( ; n] ; I I ( a) ( ) a nr a r ar a nr a r I n + = Ε = nε = nε + + nr + a r nr a r nr a r #: a nr = nr nr a r = r a r =, ( ;n] #: nr a = r nr nr nr r nr I r = nε = nε = Ε >, r nr nr 9R nr + r r nr nr r nr r nr I = nε = nε = Ε < r nr ( nr) 9R nr + r r 9 ; «+», - ( ;n ], - * nr a = r 4

37 -, - * a b # *, d N = kab, k -, k > -, a + b = d, b = d a 9 N = ka d a = k d a a -, a ( ;d )- ; N = N ( a) * $ ; & ( ) N = k d a a = k d a a d a =, d d = a = - ( ;d ) d #: - 5

38 d d d N = k d = k >, 4 4 d d d N k d = k < d ; N a 9 «+»,, ( ;d ), * d d b = d a = d = a = d $* 4 + # - * a b 9 = a b ; a + b =, b = a - ( ) = a a = a a - = a * - a ( ; ) - ; a ( ;) $ ; #: = a a = a a =, a = 5 & ( ; ) $ - ( a) 6

39 ( 4) = 4 = >, ( 6) = 6 = < ; «+»,, a = 5 9 ;, * - ( a) a = 5 9 b = 5 = : - * #,, * a, a, 54 a a = 54 a + = a a 54 a = 8a 6a #: a ;, a > a > 54 a > a ( ;8) - ; ( a) = * # ; & a a a a = 8 6 = 6 8 ( a) a 6 8 =, a =, a = & a = ( a) = 6 8 = 6 >, = 6 8 = 4 < 7 ;8 $

40 ; «+»,, a = 9 ;8, * a a =, *, 4, , - * # a, b 9 S = a + b - ; b = 5 a S a 5 a a 45 a = + = + #: a ;, 5 a > a > a ( ;5) -; S S ( a) = * - # ; & S a a a = + 45 = a =, a =, a = & a = ( ;5) $ S ( a) S,5 =,5 =, 5 <, S = = 9 > 8

41 ; S ( a) - «+»,, a = 9 a = ( ;5 ), * a = 9 b = 5 = 5 = t t -;, t -; + t + 6 5,? U ( t ), ; t U t t = t t #: t - t U t # t t, t t, t ( t + 6) t, 5 t + 6 t 5 6 ( t ) t t, t [ ;9] ; U ( t) * [ ] ; t 6 U ( t) = t = t ( t 6) - ;9 # -

42 #: 6 =, 5 ( t + 6) 6 = ( t + 6) 5 ( t + 6) = 4, t = 4 t = 6 $ U ( t) [ ;9] U =, 4 U ( 4) = 4 >, U 9 =,,, ; V +,, a,,? -, b *?? S = πr,? S = π rh 9 = π + π a r b rh -, V π = r h ; V h = 9 πr V bv = a πr + b π r = aπ r + πr r 4

43 r ( ; ) - ; = ( r) * $- ; aπ r bv bv π r r = + = a r #: bv aπ r =, r aπ r bv =, a π r = bv 9 4bV = aπ + r aπ >, bv a π r = 9 bv, * a π r = bv 5 V aπ b V h = = π bv a π #*, - * h b V = r a π 4

44 $ + $ 9 m s ( t) = ( t ) ', ;? : ; $ ( - + t x t = # t + t = ( : 65, -5) $ 9,? h ; v, gt h( t) = h + vt, h, t #, *, h =, v = 4 ( g = ( : 45 ) c ) $ 4 # AB, ;? - : A ( p ) B ( q ),,?;, AB = a (?: ) a p ( : A) p + q $ 5 - d & * b h,,, *? ($ ;* - ) ( : b = d, h = d ) $ *,,?, 4

45 F = cx ( a + x ), a, x, c = const x F *? a ( : x = ) $ 7 #, : *; ( : ) $ 8 # -, : ) *; ) * ( : ) ; ) 6 6) $ 9 t t -;, t -; + t +,5 49,? ( : ) +*5 $* * 6 $ ; -, &, *-, - " - 9 & 8? 4%, 5% - ; x ; % -,?? 4

46 $ 8 : () - 8 x 9 8 x + 8 x 4, ; :, ( x) ( x) ( x) ( x) = = 8 4 ' ( 5%): 5 x + x = x + 5x = 5x 9 - x, x 5x 6 5 x 5 x x x p, p % = # : i = p x x p x x x 5x 5x = 5x 5x i = 5x i, 6? ( ) : x S x x x i = 4( 8 ) + 5 ( ) ' * * [;7] ' ;: x x S x = + i = + i 4( ) 5 ( ) 4 5 ( ) 44

47 : : x 5 ( i) = 4 ; x,4 5 = ; i x 4 = 5 ; i 4 4 x = 5 = 5 i i 4, x = 5 9? * i [;7], -, < x < 7 : , 5, i, i 84 i > > i > 5 < 5 =, p i =, p < 84, p < 84, p<84 %,? U ( t ) : U t # $ t t + t : U = U ( t ) U + U 9 t z U = t t z = lim z = lim = U t t t 45 U ( t )

48 z (t) - ; ; ( ln y) = y y ; ; z $, ( ln z) = - z +:, A: u t u t = t + 6t + 6t + 4, t 8, t, #: ), : ; ) $ * " + - ) z( t) ( t) = u ( t) : z z ( t) = t + 6t + 6t + 4 = t + t + 6 # : ( ) = = 4 z + :; z t : ; z t ( t) = ( t + t + 6) =

49 # : ( ) = 4 + = z ' : z z ( t) 4t + = ( t) t + t + 4 : z z ( ) ( ) 4 + = = = )! z( t) = t + t + 6, +, * - * ' : * : b t = = =, z z = a ( ), ( t ) = ( ) = $* M ( ;4) 9-, : ; Ot ' z t ;: t + t + 6 =, t + 6t + 8 =, D = 6 4 ( ) 8 = 68, t = 7 t = ' ; Oz = ( ) = = 6 z t ( t) z : ( 4):, :, 47

50 4 ' x ; y -, x?, y?, y? - x 9 y = y (x) x lim x " ) ;? - :, : ; ; A:- Q P Q ;: Q < ( Q) = 9Q + 4Q + 88Q + 5, P( Q) = 44Q #: ) ;, A:, - ;? ; ), ;? ; ) [; 5]; ) * [; 5] ) ; : : A:? : ( Q) = Q P( Q) C( Q) = ( 44Q + 898) Q ( 9Q + 4Q + 88Q + 5)= = 9Q 576Q + 864Q 5 48

51 - ; *, : : ;: = 9 Q 576 Q = = 576Q 5Q #: ( = ) : 576Q 5Q = : 576, Q + Q 5 =, ± ± 8 Q, = =, Q = ; Q = 5 Q = 5, Q A:,, ; ;: = > ( 4) = <, : #: ;, ( Q) ( ) = = 4 5 #:, : ( ) = = P ) #:, ;? : ( ) = = 4 6 9, ; - : ( ) 46 C = = = 8 49

52 6 ; : $ Q = : ( Q) = 576Q Q ( ) = = 8 64 ) #:,? ) A: ( ), A:- 5 ( ) ) #*: ;: ( Q) = 576Q Q # C ( Q) = : 576Q + 864Q + 88 = : 88, Q + Q + =, ± 9 4 ± Q, = =, 4 4 Q = 5; Q =,?, - - * & Q = 5 Q = [; 5], 5: # C( Q) = = ( 5) = = 77 4,, * [; 5] * 75 *?; - & - -, 5

53 @ ; y = f ( x) * y x : $ % E x ' y x f x x ( y ) = lim : = x y x f x E p ( q ) = dq p dp q ( ) - - ; ; $ dq I EI ( q) = di q (z) & " ; z = f (x, y) (z, ; E x, ) E y $- z x z y Ex( z) =, Ey( z) = x z y z ; z = f ( x, y) x y 6 E x (z) z, x %, y ) - #, P z A E B P B, () ), $ %, ) : q( p) S ( p) : p + 8 q ( p) =, S( p) = p + 5, p +, q S - #: ) ;; 5

54 ) + - ; ) 5% ( ) ", : p p = p + 5; 5 p + 8 = ( p + ) p + = p + p + ; 5 5 p = p + p + 8 = p + p 7 ; p + p 7 = ; p = 7 p = ) E p ( q) p = q p q 5 ( p) p + 8 p + ( p + 8) 6 $ q p = = = p + ( p + ) ( p + ) p 6 p 6 p E ( q) = q = = p q( p) p + p + 8 p + p + 8 p + = p 6 E ( q) = = ( + )( + E p ( S) p = S p S ( p) S p p $ = ( + 5) =

55 E ( S) = 9 E ( q) ( S) p p p = S p = S ( p) p +, 5 E ( S ) = = < E <, : ; % * %, - 8% ) 5% * 5% ( 5 = 5), 5% $ + $ A: u ( t) = t + 4t + t + $ - 4 ( : ; ) 5, - $ A: - Π = x + 4x + # A:-, ( : ) $ ; A: t Q < ( Q) ( Q) = t + 5t + #, A: 5 ( : 7) $ 4 A: - K = x x + # A:, ; 8 - A:, - K 5 = 75 ) ( :, : [5,,, 6; 6, IV, ; 7,, II; 8,, 5, ; 9, 4, 5;,,, 7;,, 6, 68]

56 (,#!8(##9!!()(!&#) + : - - b a ( ξ ) f x dx = Lim f x n, max x i= i n i i $ : - ;? ; ; - ; Q, ;?; : [ ; ] a b, Q, - [ a; b], Q Q, ;? - dq Q, ;? [ x; x + x], dq = q( x) dx ;?- #, ; - ;, ; ; *, dq,, dq Q, ; - dq ; ; [ a; b] b Q = q x dx a ) * 7 :! '? - y = f x,, y = f ( x), x = a x b =, - : b a S = f x f x dx 54

57 $,, x = x( t), y = y( t), t [ t ; t ],, «+» x( t ) x( t ) x( t ) x( t ) t t S = ± y t x t dt, > t > t ; < t > t?,,, ϕ ρ ( ϕ ) ρ ( ϕ ) dϕ, ϕ S = ρ = ρ ( ϕ ), ρ ρ ( ϕ ), ϕ [ ϕ ; ϕ ] = * - $?, y = x x +, y = x! y = x x + * ( ; ),! y = x - ( 5) 5 55

58 #: x x x + =, x 4x + =, x =, x = -,? S = x x x + dx = x + 4x dx = 4 x = + x x = 4#? x = acost, y bsin t =, t [ ;π ]? S,? S 9 S = 4S ( 6) 6 $?, π π π S = 4S = 4 b sin t a sin t dt = 4ab sin tdt = ab cos t dt = π = abt sin t = π ab 56

59 5 $?, ; ( cosϕ) ρ = ρ = cosϕ ρ = ( cosϕ), cosϕ, ; ρ = cosϕ (;) ( 7) 7 -? S S=S, S *? S -? S S, ; : #: ϕ, ( ϕ ) cos = cosϕ, π cosϕ =, cos ϕ =, ϕ =? S (ϕ π ) π π π 4( cos ) ( cos cos ) S = ϕ dϕ == ϕ + ϕ dϕ = + cos ϕ = ( cos ϕ + ) dϕ = = ( ϕ sinϕ + ϕ + sin ϕ) = 4 7 = π 4 π 57

60 ? S (ϕ π π ) π π 4cos ( cos ) S = ϕdϕ == + ϕ dϕ = π π π π = ( ϕ + sin ϕ) = 6 4 π S=(S + S )= 7 π 7π π + = $?, ; x 4x + y =, x 8x + y = x y =, y = x ( 8) ' :?; - x = ρ cosϕ y = ρ sin ϕ x 4x + y = ρ cos ϕ 4ρ cosϕ + ρ sin ϕ = ρ = 4cosϕ, x 8x + y = ρ cos ϕ 8ρ cosϕ + ρ sin ϕ = ρ = 8cosϕ, x ρ cosϕ π y = ρ sinϕ = tgϕ = ϕ =, 6 y = x ρ sin ϕ = ρ cosϕ tgϕ = π ϕ = 58

61 ρ = 4cosϕ ρ = 8cos ϕ, π π ϕ 6 8 6? * - π S = ( 64cos ϕ 6cos ϕ ) dϕ = = ( + cos ϕ) dϕ = π π 6 6 π = ( ϕ + sin ϕ) = π π 6 π, y = y( x), y ( x) [ a; b ], L ;-?? b ( ) L = + y x dx a, L, x = x( t), y = y( t), t [ t; t] x ( t), y ( t),, t L = x + y dt t 59

62 ϕ, L ρ ρ ( ϕ ) [ ϕ ; ϕ ], ρ ( ϕ ), =, ϕ L = ρ + ρ dϕ ϕ 7 $ y = x (;) (4;8) - y = ± x 9 - (;) (4;8) y >, x y = = x y = x = x #: ; 4,, 4 -, x 4 + 9x 4 + 9x L = + x dx = + dx = dx = dx = = + = + = = ( 4 9x) dx ( 4 9x) ( 4 4 ) ( 4 4 8) [ ;] x 8 $ y e + e $ ; x x x x y = e e = e e -, x x x x x x x x =, L = + e e dx = + e e e + e dx = 4 6

63 4 x x + e + e x x x x = + e + e dx = dx = e + + e dx = 4 4 x x x x x x x x x x x x ( e e ) dx ( e e ) ( e e ) = e e e e dx e e dx + + = + = = + = = = e e e e = e e = e e 9 $ = ln cos x + y, x π 6 $ ; y = ( ln cos x + ) = ( sin x) = tgx cos x #:, dx cos x >, x [ ; π 6] tg xdx dx π 6 π 6 π 6 π 6 dx + = = = = = cos x cos x cos x = cos x cos x π 6 x π π π π = ln tg + = ln tg + ln tg + = π π = ln tg ln tg = ln ln = ln 4 4 $ ( ) ( t) x = 9 t sin t y = 9 cos t [ ;π ] $ - xt = t t = t y t = ( 9( cost )) = 9sin t ( 9( sin )) 9( cos ) 6

64 #:, π π L = 8 cost + 8sin tdt = 8 cost + cos t + sin t dt = t cost π π sin =, = 9 costdt = 9 costdt = = t sin = cost t π π t π, π, π t t t = 9 sin dt = 8 sin dt t 8 sin dt = = = sin >, t t sin sin = π t 8 ( ) cos = 6( cosπ cos ) = 6( ) = 7 4$ $ $ x = 5cos t y = 5sin t, t π x t = 5 cost sin t = 5sin t yt = 5 sin t cost = 5sin t π π L = 5sin t + 5sin tdt = 5 sin tdt = π π sin t >, t π, = = 5 sin tdt = 5 cos t = sin t = sin t 5 = ( cosπ cos ) = 5 ( ) = 5 6

65 4$ x = cos t y = sin t 9 ( 9) Ox Oy, :;, 4 t ; π - [ ;π ] π, [ ] 9 $ = = = = xt cos t sin t 6cos t sin t yt sin tcost 6sin tcos t -, : L, L = 4L π π L = 6 cos t sin t + 6sin t cos tdt = 6 cos t sin t cos t + sin t dt = = 6 π cos t sin tdt = 6 cost sin tdt = sin tdt = cos t = π π π = ( cos π cos ) = L = 4L = 4 = 4$ ( ) x = cost cos t y = sin t sin t, t π

66 π $ π ( t) ( ) x t = sin+ sin yt = cost cos t L = 9 sin+ sin t + 9 cost cos t dt = = 9 4 sin t sin t sin t + sin t cos t cost cos t + cos t dt = π 6 sin cos sin cos sin sin cos cos π = t + t + t + t t t + t t dt = = 6 sin t sin t + cost cos t dt = 6 costdt = cos = cos t ( t t) t π π sin >, t π, π t t t = 6 sin dt = 6 sin dt sin dt = = t t = sin = sin t = 4cos π π = 4cos cos = 4 π ( ) = 4 44 $ ρ ( cosϕ ) = 9, L = L, L, ϕ - π 64

67 #: ;, π ρ = cosϕ = sinϕ - L = 4 cosϕ + 4sin ϕdϕ = cosϕ + cos ϕ + sin ϕdϕ = π π π cos cos sin ϕ = ϕdϕ = ϕdϕ = dϕ = π π π ϕ ϕ ϕ π = 4 sin dϕ = 4 sin dϕ 8cos 8 cos cos 8( ) 8 == = = = b π π 45# ρ = a e ϕ, ϕ b b ϕ ab $ ; ρ = ae = e, : π ϕ - ϕ π,5π b b,5π ϕ ab ϕ bϕ a b bϕ L = ae + e dϕ = a e + e dϕ = 4,5π,5π,5π b b + ϕ + ϕ,5π 65,5π 4a a b 4a a b = e dϕ e = = b 4a + a b = e e b bπ bπ 4 4,5π 46 $ ρ 5( sinϕ ) #: ; π = +, ρ = 5 + sinϕ = 5cosϕ $ L = 5 + sinφ + 5cos ϕdϕ = 5 + sinϕ + sin ϕ + cos ϕdϕ = π π = 5 + sinϕdϕ = 5 + sinϕdϕ = sinϕ = cos ϕ = π π

68 π π π ϕ ϕ ϕ π π ϕ 5 + cos d = 5 cos d = 4 π ϕ π ϕ π π π = cos dϕ = sin = sin sin = π π = sin sin = = 4 4 π (;7 '()? S x, ;, Ox x [ a; b] x + x dv = S ( x) dx ( ) ' A: x, A: (- Ox ) V b = S x dx a, ;,, '()!,? Ox, y f x y = f x, x = a x = b, =, = π S x f x f x -, Ox b a V = π f x f x dx 66

69 ,? Oy, Oy b a V = π f y f y dy 47 $ A:, x y z =, z a + b = x y z = a + b -, Oz z =, - xoy,? : : ; z = const, xoy $ x a x za y + b = z, y + zb = 67

70 za zb? (? S z = π za zb = π abz -, A: ) b a z π ab V = S z dz = π abzdz = π ab zdz = π ab = 48$ A:,? Ox y = sin x,, y = sin x, x π ( ) -, π π VOx π 4sin x sin x dx π sin xdx = = = π π ( cos x) dx x sin x = π = π = π 49$ A:,? Oy, y = x x +, x =, y =, ;?, - ( 4)! y = x x +, -, * ( ; )! x =, Oy * y = x x + x = f ( y) - ( x ) y =, x = ± y 68

71 4 9 ;, x = + y ' A: π VOy = π + y dy = y y dy = 4 y 7 = πy y y = π 6 5 $ A:,? - Ox ( sin ) ( ) x = a t t y = a cos t ; t [ ;π ] -, dx = x dt = a( cost) dt, Ox π π ( cos ) ( cos ) π ( cos ) V = π a t a t dt = a t dt = π ( cos cos cos ) = a π t + t t dt = π = a πt sin t + t + sin t sin t sin t = 5a π t 69

72 : :?,? y f x x a; b, Ox =, [ ] b S = π f x + f x dx a, x x( t) =, y = y t,?,? Ox, t S = π y t x t + y t dt, t t t t, ;? 5$?,?- Ox y = x, : x = * y = f ( x) (,? y f x = x = $?, x = x 9 4x + S = π x + dx = π x dx = x 4x π x 4x + = π dx = π 4 + = $? R? Ox y R x,? =, x [ R; R] - 7

73 R x S = π R x + dx = R R x R R R R x R = π R x dx = π Rdx = 4 π R R 5 $?,?- Ox = y = a t x acos t sin ' AB? Ox -? ( 5) #:?,? AB Ox, 5 #: $ A( a;) t =, t = $ B( ; a) x = a cos t = a, cos cost =, π t = $ = = y ( a t = ) = a t t x acos t a cos t sin t sin sin cos x a t = cos =, 7

74 $ π 4 4 sin 9 cos sin 9 sin cos S π a t a t t a t tdt = + = π π 5 4 sin t 6 = 6π a sin t costdt = 6π a = π a 5 5 $? S 5 = π a $ + $ $?, y = x, x =, x = ; Ox ( : S = ) $ $?, ρ = 4( + cosϕ ), ρ = cosϕ ( : S = 8π + 9 ) $ $ A:, x y + =, z = y, z = 4 ( : V = 8) $ 4 $ A:,? Ox, y = x, x + y =, y = ( : VOx = π ) $ 5$ A:,?, y = sin x, - ;? ( : VOy = π π ) 4 π $ 6 $ y = ln cos x +, x 6 ( : L = ln ) 7

75 $ 7 $, x = 5cos t π y = t t 5sin, ( : L = 5 ) $ 8 $?,?- Ox y = x +, ; - x =, x = 7 ( : S = π ) 7 < * * + -? ds dp = γ hds, γ, h - ds 6 ' ") * " 6, - Ox (Ox, *), Oy ; ( 7) 7

76 9 h x x, ( ) 7 ( ) =, ds = y ( x) y ( x) dx, γ x x ( ) P = γ x y x y x dx, dp = x y x y x dx x x, ; y x, ; ;, y x, " 6 - γ, Ox, Oy - ; ( 8) 9 h x =, π ( ) 8 ds = y + y dx k P = πγ xy + ( y ) dx, k, y y( x) =,?- Ox 74

77 54 $, ( 9) 9 $ 6 y ( x) y x, -,, - ; ; 5 : ( y y ) = p( x x ), ( x, y ) * &, * (5, ) - y = p x 5 p 9 (, ), ; ; : ( ) p( 5) 4 = p( 4), =, p =, ( y ) ( x 5) = 75

78 ; ( y ) = 5 x, y = ± 5 x, y = 5 x, y = 5 x + + ' ( γ = ) P = ( ) x x x ( y ( x) y ( x) )dx P = x 5 x + 5 x + dx = x 5 xdx = x 5 xdx = $ 5 x = t,5 x = t, x xdx = = t t tdt = t t dt = x = 5 t, dx = tdt 4 5 ( 5 ) ( ) ( 5 ) ( 5 x) 5 ( 5 x) t t = 4 = x x 5 5 x 5 xdx = 4 = ( 5 5) 5 ( 5 5) ( 5 ) 5 ( 5 ) 46 = 4 = ", :; 76

79 $ + $* (;), - x = p y (-;) : : p = 9 : ( y ) x =, y = x, y = + x ' ( γ = ) P = x x x ( y ( x) y ( x) )dx 4 P = x xdx = 5 56 " γ, :; $ 4 77

80 ",, ;? - P P 4 $ P +,? A ( 5;), B ( ;) ( AB ) B ( ;), C ( ;) ( BC ) $,? : x x y y = x x y y 9 AB : x 5 y = 5 ; x y = 5 BC : x y = ; y = x + x 6 y ( x) y ( x) x = + = x ; x 6 P = γ x xdx = γ = γ

81 $ P +,? C ; ( AC ): A( 5;) 9 x 5 y = 5 ; y = x + 5 x 4 y ( x) y ( x) = x + 5 = x + 4 ; P = γ x x + 4dx = γ x x = γ P = P + P = γ + γ = 4γ $,, - ( 5) 5 $ AD BC 6 79

82 ,? A( ;) D( 5;) (?): : x y = 5 ; y = x 5,? B( ;7) C ( 5;5) x y 7 = ; y = 7 x 5 $ x P = γ x( y ( x) y ( x) ) dx = x7 x x dx = 5 5 x 5 x 4 x 5 = 7 = $ ( γ ),, ;? R "- ; $ 7 7 8

83 ", ;? Ox AB + AB '- ; R x + y = R ; y = ± R x 9 AB ( R ;), AB # y : y = R x y = R x x R R x R P = πγ x R x + dx = πγ x R x dx = R x R x R R xrdx R = πγ = πγ = πγ 59 $ ( γ = ), ;?,?: * 5 $ $ 8 8 8

84 &, ;?- Ox AB +,?- B ; ( ;) A y = ( x ) x, y = 5 9 y = 5 ( x ) ( x ) 9 P = π x + dx = π x dx = = π x( x ) π 5 = 5 + F - s " A A = F s *" 6, - ; γ h : 9 ( Ox, - *) dp dx 9 dp γ dv γ S x dx S x? da = =,,, ;, dp x + h 8

85 9 9 dp x + h, da = dp ( x + h) = γ ( x + h) S ( x) dx x x A = γ x + h S x dx, S ( x )? x, x, x, ;? () 6, *- ; A: m H, Mm F( r) = G, r G, M, r - $: 4 4 8

86 F ( x), x :, a( x ) x F ( x) = ma( x), x = ( ) a( x) = g = 9,8, F = ma = mg +, -, mm mm F ( x) = G, F = G R x R ( + ) mm mm ma( x) = G, mg = G R ( R + x) - M gr = 9 G F m gr mgr ( x ) = G = R x G R x ( + ) ( + ), da,, ; F x x 9 mgr da = F x dx = dx 84 ( R + x) ", * H - :, H mgr A = dx ( R + x) " 6, - ;, s ( - : k ),! F = ks, F, ;? 6 dx da = F x dx 9 ; s s A = F x dx,

87 4 6, - ; R H, PV = PV P A- : V = π R H P( x )? * x ( 4) 4 A: V ( x) π R ( H x), π π PV = P x V x P R H = P x R H x = 9 - P( x) P π R H P( x) = H x? * dx - P π R H da = P( x) dx = dx 9 ;- H x a a P π R H dx π, A = dx = P R H H x H x a,? * 85

88 6 $ ;, γ, R H $: 4 4 S ( x)? x #: r ( x ) " BOC AOD - AD BC OB = H, 9 OA OB = 9 AD r ( x) r x x xr =, r ( x) = R H H x R S ( x) = π r ( x) = π H -, H H x R π R π R H π H H 4 A = x + dx = x dx = =, BC = R, OA = x, 6 /,, ; γ $, ; ' 5, * 4, 4 86

89 $: 4 4? x S ( x ) +, a = 5 #: ; x = 4 y y = 4 x, y = 4 x, b = y y = 4 x 4 x = 4 x $? S ( x) = a b = 5 4 x = 4 x #, A = γx 4 xdx = γ 4 x 4 x = γ 5 6 = +,, *; *, * ( 4, ) 5 $ dx A = π dx = π 56 = x x 5 = π 56 ln x = π 56 ln = π 56 ln 4' 5 87

90 6 $, ;,,, : # #: : -! 5 F = ks, F = s = k = : = -, 5 5 A = xdx = = ' $* " ;, ;? v = v t 9, t t " [ t; t t] : ds = v( t) dt - [ ; ], : + t t, s t = v t dt t + " - [ t ; t ] x = x t y = y( t) 6,,, - : t ( ) ( ) t s = x + y dt 88

91 64 ( - v t = 6t + 4t + c #, ; [ ; ] +, s = 6t + 4t + dt = t + t + t = αt v( t) = te #, β, β, β αt s te dt = - : αt β u = t dv = e dt β αt t αt s = te dt = = e αt du = dt v = e α α β t t β β α αt αt β αβ α α α α α α e dt = e e = + e + 66 $;;, - t ;6, [ ] x = t sin t + t cost y = t cost + t sin t 89

92 6,, - ( ) = + = y = (( t ) t + t t) = t t x t sin t t cost t cost cos sin sin,, t [ ;6] t s = t cos t + t sin tdt = t dt = = 7 $ ;; v s 7 = = = t 6 -* = " ; ; y = f ( x), x [ a; b] ( x) ρ = ρ ( 44), 44 $ dl dm = ρ x dl -, ( ) dl = + y dx, b a m = ρ x + y dx 9

93 , % A? I = mr, m A, r -? + ; - b I = ρ x x + y + y dx, a Ox Oy - x b b ρ, y ρ a a I = x y + y dx I = x x + y dx A M = mr, m A, r - + ; - b M = ρ x x + y + y dx, a Ox Oy x b b ρ ( ), y ρ ( ) a a M = x y + y dx M = x x + y dx 4 - * M x = myc M y = mxc %- C ( x ; y ) C C x C b b ρ x x + y dx ρ x y + y dx a a =, y b C = b ρ x + y dx ρ x + y dx a a 9

94 " ;, ; y f x x a; b ρ = ρ x ( 45) =, [ ], 45 $? ds 5, dm = ρ ( x) ds - - ;; :, b = a m ρ x f x dx 6 + ;, % b a I = ρ x x + f x f x dx, Ox Oy - x b b ρ, y ρ a a I = x f x dx I = x x f x dx 7 + ; ( ), - b a M = ρ x x + f x f x dx, Ox Oy x b b ρ, y ρ a a M = x f x dx M = x xf x dx 9

95 8 & % ( ; ) C x y ; C C x C b b a C b a b a xρ x f x dx ρ x f x dx a =, y = ρ x f x dx ρ x f x dx 67$ - ( ρ = const ) x + y = R 9 y = ± R x, y = R x x, R R x dx = 4 ρ + = 4 R x ρ = R x πρ, I R dx R R R R x dx = 4 ρ + = 4 R x ρ = R x πρ M R dx R R 68 $ - ( ρ = const ), y = 5 x, y = ; ( 46) 46 - Ox, x C = ' yc 8-9

96 y C x 5 dx 5x x 5 5 x 5 x dx 5x 5 ( x ) 5 5 = = = = 5 5 ( ) $ - y = x, x, ρ ( x) = + x ' ' y = x = x, ( ) x m = + x + x dx = + x dx = x + = 6 : ( ) x x 5 + x x + x dx = x + x dx = + = 6, x x + x dx = x + x dx = x + x = $ 4 x C 5 6 = : =, yc = : = $ + $ $ (, 47), :;, γ = ( : 468 ) 94

97 47 $ +,t v = te #, - 4 ( : ) $ $, ;,, ;? ; - ; 5,?:- ;* ( : 8 ') $ 4, *; A: 6 ( 7 (" 6 68, g = ) 9 ( : 89 ') $ 5 $ ( γ = ), ;?, 5,?:* + ; ( : 9π ) $ 6 $ y = ln x, ;: - x =, x = 8, - ( : 9 ) $ 7 $, y x, x, y ρ x = x = = =, ( : 8 5 ) $ 8 #, y = sin x, y =, x π ( : π π ; 8 ) 95

98 7 < 5 - * * 56 " y - y = f x x, ;?,, ; +?; f ( x ) (, 48) 48 f ( x) OA? O fa,? - OAB - : ' *, - ; ' 67, ; ; ' 67,? 7, 5 y =,, $- x ' 96

99 ? fa, :? OAB ( 49) ' S 49 OAf OAB OfAB OAB OfAB OfAB OfAB k = = = = = = S OAB S S S OAB ( S OAB = ab = : : = S S OAB S S OAB S S S OAB S OfAB ) $? OAfB?; - = 5 dx 5 = = = 5 S OfAB dx dx ln x x x x = ( ln ln ) = ln 69 = 4; k = 4 = 94 9 ' *, (;7 + 6 '*6 -, f ( t), t,, f ( t) 9, U T = f t dt ( [ ] ;T 6 * - %,,,,? $ ; () 97

100 9, [ a; b ], c [ a; b], b f c = f t dt = L b a a 6 L f (t) [ a; b ] - 7 $ f ( t) = t + 6t + 7 #: ) A: [;4] ) + [;4] t t, ; - ; ) " ) A: [;9] - U 4 t 6t =( t + 6t + 7) dt = + + 7t 4 = = ) + f ( t) [;9] F ( c) = ( t + 6t + 7) dt = = t -, *: t, 5 4 t + 6t + 7 =, t 8t + =, = 5 t = 45 [ ; 4] f t = t, : 98

101 6 t -, : * f ( t) = t + 6t + 7 ( *, ): b 6 t = = = a ) ' : ; *: ( t ) = f ( ) = ( ) = 6 f, * ( ;6) ; Ot : t + 6t + 7 = ; D = 64 t =, t = 7 #: ( 5)!,? *, - ;? A: 4,?- OABC,, ;

102 7 # K = x + x + 7, - A: A: -, ; " - + K ( x ), A: - m n,, x x l = ( x + x + 7) dx = + + 7x = = 4 6 A:, ;, *: x 4 x + x + 7 =, 6x + 6x + = 4, 6x + 6x =, [ ;] 6 + = x = 48! K = x + x + 7, #: * : b x = = = =, a 4 ( x ) = K( ) = ( ) + ( ) + 7 = K ( 5): K ( ) = =, K( ) = = 6

103 5 #,? ABCD, [ ;],,?* + ;, -, t p ( - ) 9 - ;? P f ( t ), i = 9 K T T it K f t e dt =, N m, f ( t) = N + mt 7 & 4 p=6%, -, lim K T # T

104 $: ; f ( t) = N + mt, N -, m ; -, f ( t) = + t ' 9 K = T f it ( t) e dt, 6 i = p = = 6, u = + t, 6t ( + t) e dt = = t du = dt, dv = e 6t 4 K = 6 4 = 6 6 = v = e 6 6t 6t ( + t) e e dt = ( + t) 6 e 6t e 6t = e + e #: K( T ) lim K T lim : T ( T ) = lim( + t) = lim T T 6 6 e 6t 6T ( + T ) e ( + ) 6 e T e 6T 6 4 e e dt = 6t 6 T =

105 , lim 6 T = 6 e lim = T e T : lim K ( T ) = T = 6 ( 6) = $ + $ &, ;?, : y = x $ ' ( : k=) $ $ f ( t) = t + 6t # A: - [;] ( : 48) $ # K = 6x + 8x, A: 5 ( : ) $ 4 & "=8%, - 8, lim K T # T ( : 555; 575) : [5,, 9,; 7,, III; 9,,, ]

106 (,#!#>>##!=?!" -!#! #/#!>$%#')#(#&%#'$ % ' * - ;? : ), -, ;?; ) ; )? * ; 4) * ; 5) - (, ) - ; 6)? ; 7) $ * --% *! ' *!, f ( x) dx = g ( y) dy & *; ;? - ' --% '-, (, ) y = f x y, f ( x, y), f ( kx, ky) f ( x, y) = y +?; u = ( y = ux, y = u x + u ), u = u( x) x, ; - ;? 4

107 , ;? y + p( x) y = q( x) ' * y = u v ( y = u v + uv ), u = u( x), v v( x) = ) * *, : < $,,? -, ; (? ;? ) /, y = f ( x) - ( 5), y ( x) = tgα, α ; Ox M ( x; y ), Ox #,? M (, ), - k = 5

108 !, y x y( x) x ; M x y x - y = kx y = x - -,? * y = x dx = x + C -, M (, ) y = *, C : = + C C = 8 * : y = x 8, x = y =? 75 #,? M (,4 ), - k = -, y = y dy dx = y ", : y ydy = dx ydy = dx + C = x + C y = x + C ; ; ; 4 -, ;, y = 4 C : 6 = 4 + C, C = * : y = 4x + y = 4( x + ) & - * A(,), ; Ox -, ( 5) 6

109 5 76 # ;,?; M,, ;, ;, * : ( Oy ) 54 M ( x, y)? y y( x) = - Ox α, tgα = y $ MBN MBN = 8 α, MN tg MBN = MN = y, NB, NB MBN AMP, : NB = MB ; NB = NB = PM = x PM AM PM 7

110 MBN, : y y y y tg (8 α) = ; tgα = ; y = ; y = x x x x " - : dy y dy dx dy dx = ; = ; = + ; ln = ln + dx x y x y x C y x C C 9, y =, ;? x y ( ) =, M, : = = ; * : y = x 77 #,? M (, ),, ;, * ; M ( x, y ) $ MON MON α ( 55), ; Ox, tgα = y ' MPN* : MP tg MNP = PN # MNP MON : MNP = 8 9 α = 9 α MP = y ; PN = ON OP = x + x = 8

111 9, y y y tg ( 9 α ) = ; ctgα = ; = ; tgα = ; y = tgα y y 55 ", *?*: 9 y 4x C = + M,, y =, = 4 + C C = 4 56 y 4( x ) * 9 = # - M,, ; Ox, ( 56)

112 - ;?;, ; M MN Oy ; Oy ; 78#;,?; M (, ) 57-9 M x, y : M ( x, y), M ( X, Y ) Y y = y x X x 9 N (, YN ) ( 57); -, N ( ) Y y = y x x Y = y xy +, PN = yn yp = YN y N ; PN =, - Y y = Y = y N Y N, - : N y xy = y y = x

113 -, : dx y = + ln C; y = ln x + lnc x -? * ;,?; M : (, ) = ln+ lnc ln C =, y = ln x + 79 #,? -, ; [, x]?, ;?, 58 + ; x = y( x) y t dt ' x, y = y y, y = y

114 ",? - *: dy = ; ydy = dx ; ydy = dx; y = x + C; y = x + C dx y y =, C =,? : * y = x 8 #;,?; A ( ;), -, ; Oy, - -, ( Oy ) 59 M ( x, y ) ; MN M ; Oy N ( 59) + - ;, ON = OM # OM = x + y, ON Y y = y X x, X =, Y = ON = y xy

115 -, ; y x + y = y xy = ux, : du du dx x + u x = ux x( u x + u) ; x = x + u ; = dx + u x ; ln u + + u = ln C ln x u C x + + u = y u =, x ( ) x = C C y (, ; Oy ) A?*, - = C ( C ) ; C = C = *, C = Oy -, - x 4( y) = 8 #;,, - ;, ; M ( x, y ), M ( X, Y ) - ON,, - M ( 6) ; ON = ± xy, N (, Y N ), : ON = xy YN = ± xy +, * M x, y : Y y = y x X x

116 6 $ N (, YN ) YN y = xy, YN = y xy Y N, - y xy = ± xy y y y = ±, x x y ux,,?*: dx du u x + u = u ± u; u x = ± u; = ± ; ln x = ± u + ln C x u x = Ce ± u y $ ; u =, x, ;? : x Ce ± y x = 8,? P(, ) ;? ;? :?, - - ;, ;, 4

117 6 $ ;? : M ( x, y ) -, M ( X, Y )? OMA ( 6 ) SOMA = OA MB MB = y, OA, Y y = y x X x A: A( X, A ), y y = y ( X A x); X A = x y, OA S OMA : = X ; S =, *;? A OMA S OMA y = y x = y 5

118 : y x y = 4; y = x ; y = y ; dx = xy ; x x = y y y xy 4 dy y y y 4 4 4, x x y x = uv u = u y, v = v y =?; ( ) -: v v = ; uv 4 v 4 y u v + uv = ; u v + u v y y y = y 4 u v = ; y dv v dv dy () = ; = ; ln v = ln y; v = y; dy y v y () u y = ; u = ; u = dy C; u C; y y + = + y y x = uv = y + C Cy = + y y - P (, ), x = C +, C = +, x = xy = y =,, $ + $ #,? M (, ), - k = 5 ( : y = 5ln x + ) $ #,? M (, ), ; k = ( : y = e x ) 6

119 $ #;,, - ; -, ; x = C y + C ) ( : $ 4,, - ; ; y = C x + y ) ( : $ 5 #,? M (, ),?,, -, ( : y x = x ) >* *, : < <:+ + T, T ( t) t, ;? 9 t #; - dq, dt, ;?,, ( ) dq = k T T dt, T > T k ;, dq #, Q = mc( T T ), m ; c ' -,,?, - dq = mcdt +, ( ) mc dt = k T T dt 7

120 , ;; mc, - ;? dt dt k = ( T T ) mc ", dt k = dt T T mc -, **: k t mc T = T + C e, C t = T = T 9,, C = T T - ;? ( ) kt mc T t = T + T T e +, T > T,, ;? ( ) * kt T t = T + T T e, k k = ; mc ), ( ) t ;? ( 6) 8

121 6 8 $ ; T = T = 6 T = 5 # - 5,, T ( t), dt dt * - ;? : dt k ( T T ) dt = dt k ( T ) dt = ", * dt T = kdt, *?*: dt k dt C; ln ( T ) kt C; ln ( T ) kt C T = + = + = ; kt kt T = Ce ; T = Ce 9

122 -, T T =, = 5 -?* : T = Ce = C =, ; C = 8 T = 8e kt ; T e k k k 5 k 5 = 8 = 5, ; 8e = 5, e =, e = 8 8 T ( t) t 5 = 8 8 $ 5 : T 5 5 = , 5 +, - 5 ', ;? dt ( T ) k T dt =, T, T =, : dt k ( T ) dt = ": dt dt kdt; k dt C; ln ( T ) kt C; T = T = + = + T = + Ce kt

123 # C k - T = 6 : T =, T = + Ce = C = 8 T = + 8e kt ; t k k T = + 8e = 6 e = 5, e = 5, T ( t ) = + 8 ( 5) t $, 5 +, * t = 5 t t t 8 ( 5) 5; ( 5 ) ; ( 5) ( 5 ) 4 t = = = ; = 4 t = 8 6 9, m F T ; F # t =, v = v ; t, v, F = f ( v), F g ( v) T = + + #;,,, ; : dv m F dt =, dv, F -, ;? dt $ * F = FT F - * dv m F F T dt =

124 - ; ; ; - S ( t), v( t) dv mv ds = F T F ds dv =, = dv ds = v dv dt dt ds dt ds 85 9, - ; v t # ;, 5 /, 6 / -, *: a = kvt dv dt = kvt ( ;? ) v = 5, v = 6 ( ; ) -, dv dv kt = ktdt; = k tdt + C ; ln v = + C ; v t = C v v kt e?* : kt v = Ce = C = 5 C = 5; v t = 5e ; = = = k k v e e 9 ; 9 v t = 5 t

125 86 ( - ; v = / # ; 4 * v =8/ #;, ; F = kv, ; v = ma = F dv m kv dt = ; v = 8 9 #?* : dv k dv k = dt; = dt + C ; v t = C v m v m k t m e? *, - : v Ce C ; v t e m = = = = ; v = e = e = = 9 k k 9m m ; v t 4 k t 9 $ =, v = ( 4) = 8 (/) t

126 87*; 8 + k = 4 - *; 64 * *;: ma = mg F dv mg 4v = dt m h( t ) *; -, dh dt dv dv dh dv = v( h) = = v, dt dh dt dh *; dv mg 4v v = dh m " : 4

127 vdv = dh vdv ; = dh + C ; mg 4v m mg 4v m 8h m e h 8h ln mg 4 v = + C; ln mg 4v = 8 C; mg 4 v = C ; 8 m m e 8h 8h m m 4v = mg C v h = mg C,, *; v =, C : ;, v = mg C = C = mg v( h) e C,, *; - : = 8 98 = 4 8h h 8 v h e e ' *;, v : v h e e = 4 = 4 4 / 9, *; 4 / 88, P, ( ) + F + W ( W = Pf + kv) #, ; ( f, k - ) 5

128 65 : ma = F W, Pa = F Pf kv dv P F Pf kv dt =, : S, v v( S ) dv dv ds dv = = v, dt ds dt ds = * - dv Pv F Pf kv ds = ", v dv ds F Pf kv F + Pf ds = ; dv = ; F Pf kv P k F Pf kv P F Pf k F Pf k = ds; v + ln( F Pf kv) = S + C F Pf kv P k P - v S =, = ; C : F k Pf ln( F Pf ) = C, 6

129 : F Pf k F Pf v + ln F Pf kv = S + F Pf k P k ln F Pf kv k v + ln = S k F Pf P ; v + ; F = βmv, β, m, v A + ; ' m - F P = mg, g t, h( t ) -, v( t) ; dv dt #;, dv m ( P F ) dt = +, 7

130 , ;, ( 66) - F P, dv m = mg + βmv dt dv dt ( g βv ) = +, dh v( h) dt = dv dv dh dv v dt dh dt dh = =, * dv v = ( g + βv ) 9 t =, v, - ;, * v = v # *, ;? ; v = v " -, : vdv g + βv = dh -: vdv g + βv h = dh + C; ln g + βv = h + C; g + βv = Ce β ; β h Ce g v( h) β = β 8

131 -, C : C g v = = v C = g + βv β * v h h g + v e β g ( β ) = β $ * ( h = H ),, ;, v( H ) =, - *; A: ( β ) e g + v v H = g = ; g + v = g; = g ; β H β H β H ( β ) e e β g + βv g H = ln β g + βv 9 9 m ; -, ( - k )&,, - ; ( - k )#, t = v 67 v( t), dv dt * dv dt = mv + kvt = kt, m kt kvt 9

132 + v = yz y = y t, z = z t, : ( ) z + zt = k k k k m + + = ; + + = ; m m m m k y z yz yzt t y z y z zt t y z = dz k dz k k = zt = tdt z = t z = e dt m z m m k t m k t m () ; ; ln ; ; k k k t k k t k t m m m () y e = t ; y = t e ; y = t e dt = m m m ; ; k m t = p, t = p, m k k m km tdt pdp, tdt dp = = k k p p = = e pdp = p e dp = m m m k k k km km km t k = p e e dp e p C e t C k = + = + k k m p p p m 9, km k km k v( t) = yz = t + C = t + C k m k m k k k t t t m m m e e e - v v = +: km km v = + C = v C = v + k k *, ;? - : km k km km k v( t) = t + v + = t + + v k m k k m k k k t t t m m m e e e

133 9 (* m F ( F = const ), +,,, α ; (α β ; ) # t + β *,, t = v = 68 * v( t ), * dv dt *, dv α m = F v dt t + β αv v + = m t ( + β ) F m v = yz y = y t, z = z t -:, * α z z + = ; α yz F α z F m( t + β ) y z + yz + = ; y z + y z + = ; m( t + β ) m m( t + β ) m F y z = ; m α dz α z dz α α () = ; = dt; ln z = ln ( t + β ) z = ( t + β ) m ; dt m t z m t m ( + β ) ( + β ) α α α F F F () y ( t + β ) m = ; y = ( t + β ) m ; y = ( t + β ) mdt + C m m m

134 ( t + β ) α + m α F F + y = + C = ( t + β ) m + C ; m α + α + m m ( + β ) F F t v( t) = ( t + β ) ( t + β ) + C = + C ( t + β ) α + m α + m α α α + m m m?* v =, C : α + m α Fβ Fβ m v = + Cβ = C = α + m α + m * *: v t α + α α m + F m ( t + β ) Fβ ( t + β ) m F β = = t + β α α + m α + m α + m ( t + β ) m 6 *, 6 6 *, R, I? E + L,? E = Esinω t, E ω 9 t,, t = ;: I = -, I R - IR ' L di L : dt di E = IR + L dt?; E,, I ; - : di R E + I = sinω t ( ) dt L L

135 I * dv R + v =, dt L du E sin t ω = dt Lv t, = uv, u( t) v( t) R t L e ; v = u t :, du dt R E = sinω t e L L t -, R E u = sinω t e dt + C L 9? ( ) : R R t E t L L I = e sinω t e dt + C L # t R Rt L Le ( Rsinω t Lω cosω t) L e sinω t dt =, R L t + ω L E I = ( Rsinω t Lω cosω t) + C R + ω L - ( t = I = ), ; C : e R t L ELω C = R L + ω

136 E I = Rsinω t Lω cosω t Lωe + R + ω L R t L * L ω = tgϕ ( ϕ ;?), : R Rsin t L cos t R L sin t ω ω ω = + ω ω ϕ, I R ELωe t L E sin( ω t ϕ ) = + R + ω l R + ω L +,, : ( sin ω t ϕ ) ( 4 R t L e ); - 9@ ; E= $,? R=5 & L=! # $ - ; t : di IR + L = E dt ;, ;?: I + 5I = I + 5I ; I = u( t ) v( t) ( I = u v + uv ) : u v + uv + 5uv = u v + u v + 5v =

137 , v + 5v =, u v = dv dv v v v dt v t v e dt v 5t + 5 = ; = 5 ; = 5 ; ln = 5 = ; = = = + 5t 5t 5t u e ; u e u e C 9, I t e e C Ce 5t 5t 5t = + = +, t = I =, C : ;, I = + Ce = + C =, ; C =, 5 e I t t = : I 5 e = 9865 ()) <: + ", - ;?,, x( t )?, dx dt hm ( h > ), #;, - d x dx m = k mx hm dt dt d x dx + h + k x =, x( t ) -? dt dt 5

138 @, z + hz + k = z = h + h k z = h h k,, -,, h k <, h k = ω, z = h + ωi, z = h ωi? - *: ( ω ω ) ht x = e C cos t + C sin t, t = dx x = x, = x o dt $ ; h ( ω ω ) x = C cos + C sin e C = x ; dx dt ( cosω sinω ω sinω ω cosω ) = + ht e Ch t Ch t C t C t, ;, : C x + x h ω = 9, ht x + xt x = e x cosωt + sinωt ω *; 6 ht x = ρe sin ωt + ϕ, ωx ρ = x ω + x + x h, ϕ = arctg ω x x h +

139 -, T π π = = ω k h π + T =,, k T > T x x x = 9 = h ht sin ρ = x + e ωt + ϕ ω t t =, t = π π, t, ω = ω,, : ω x = x ; x = x e ; hπ x hπ ω = x e ; hπ ω x4 = x e ; ;, ;, - ;?, - ; ;?; ; ; - hπ ω, ;? - x( t)

140 ", : h k > & z z?- zt zt = + x C e C e ( x = x, dx = x t = ) -, dt C C : C + C = x, zc + zc = x, : C x x z, C x z x = = z z z z * : x x z x z x x t e e z z z z zt = + - *, x : x x z x z x z z x xz zt x t = e + e z t ( z ) z t dx dt ; * dx x x z zt x z x z = z e + dt z z x x z z e ( z ) z t, xz x x x z >, 8

141 x( t ), dt ;?, -, xz x x x z <, dt ", h k = 5- z = z = h - x( t ), dx?* x = C + C t e ht : C = x, C = x + x h $ **: ( ) x = x + x + x h t e ht ),, - t ;? &,, * ;? ', ;? h k > h k =, * - (, - ) 9

142 $ + $ & +?, 5 + +, ln ( : 75 +, t = 5 ) ln 4 ln $ ( ) +? ( : 7 ) $ 9 m ; v =, - # ;? v( t) ; t ( ) ( : v t = mg e t m ) $ 4, # - #, ** 5,, v = /, S = * v =8 / ( : 9 /) $ 5 -, # - 5 /, 4 / #, - * / ( : 5, 5 ) 4

143 $ 6 F m # - v, F = b kv, v ; b, k - # t, - t = F = F = b kv k t m ( : F = Fe ) $ 7 9 m ; -, ( k ) &,, - ( k ) # -, t = ; kt k m m m ( : v = t + k k k e ) $ 8 = R=6 I t, - L=! I I =, % I R t L e ( : I ( t) I, t 7 = = ) $ ;-, ;? E, - R, L ;, ; t = # I R E L ( : I = e t ) R $ 9 96, *, #,, / # $ 5 5t x = 5e 5t + ) ( : 4

144 * 9 $ A?, B ; M M N, - N * #? T x( t) x? t + x t + t, x < t > ( ) A t V ( t ): x V ( t) = B + Mt Nt &? t,, V [ t, t t] +? ; x, x Nx x N t = t V B + M N t? x : t dx dt, Nx dx = dt B + M N t M - ;? > N,?*: dx Ndt dx dt = ; = N + C; x B + M N t x B + M N t 4

145 N ln x = ln B ( M N ) t ln C M N + + x t = C ( ) N B + M N t M N - x = A t =, C - * *: C A = C = A B ( B + ( M N ) ) N M N N M N B x( t ) = A B + ( M N ) t N M N t = T,? T : N M N B x( T ) = A B + ( M N ) T 94 $,? - $ ; 5 /, ; & +? x( t) t $ *?, M = N, * : Nx dx = dt B ;, B =, N = 5, x dx = dt ; 4

146 #?*: dx x dt = = + = t ; ln x t ln C; x Ce ( ) t = x = x =, = Ce C = 9, t = x t =6, e x 6 6 e e 5 = = () 95 = 6 4 -, ;?, -,? '* %-, ;? v -,? h : v = σ gh g = 98 /, σ (- ),? ( σ 6) t *- h dt? dh ( dh < ) A, * - dt 44

147 7 +, A dv A dh, r ( r = ) 9, dv π r dh π r dh = = +, A A,, vdt ( v ), ρ ( ρ = ), = πρ = πρ σ dv vdt gh dt A, - ; σρ r dh = gh dt ", r dh r h dt = t C σρ g h = σρ g t = h =6, h = 6 ; 6r C = σρ g 45

148 9, t h ( 6 h ) r t =, σρ g T, h =: 6r t = σρ g - (r =, T 6 77 ρ =, σ 6, g = 98 / ), 6 (), P t = P 9 P t 9 dp dt, ;*: dp kp dt = ±, k > ; + ; "-, : dp kdt P = ±,,,?* P = Ce ±kt, C 46

149 , t = C = P P = P e ±kt, P kt = P e, P = P e kt ),, - () P # -, - 8, - ; + P( t ) 9 dp dt dp ; = kp (, - dt *) # P = ; 8,, P = 8 9 P ( ) ;? dp kp dt =, P = 8 P =, - dp P dp = kdt; = k dt; ln P = kt + C ; P = Ce P kt 47

150 # C k - : P P = Ce = C, ; C = P = e kt ; k e k k = = 8, ; e 8, e ( 8) = = ; * P( t ) = ( 8) t 9 : 5 P = 8 = , -,, (, ) 59 m( t), dm dt + ;, dm km dt = ' * ;?, kt Ce m t = 48

151 m, m, = m -, 59, m( 59) = m? *, m = Ce = C, C m m t = m e kt ; = 59k m( 59) = me = m, e 59k k 59 = 5, e = 5 59 m m t = 5 t $ m : 59 m = m 5 = 965m 9, m 965m 965%,, %- 965%=85% $ + $ + - * $, 65,, 6 - ( : ) $ $, A,,? $ - ; /, ; - ;,, + *? &? & -? (& - *) ( : 97 ; 68 t = ) 49

152 $ = 6, - - ;?? ( : 84 ) : [,,, 4; ; 4,, IV, ; 6, XIV,, ; 6; 8,, 9, ;,, 5, 5, 5, 55, 56;,,,, ] 5

153 &# & ( $* / ) # & : &--, (, - - / : $ # ),! + % & : '!(), (, - / : ) #, + ) &, % #! * & : '!(), , #$* 6 I, II /, ', )!, 9 7 & ( : $* *, $* / # & ( : %, 47 )!+ /! # % ( : #, /A, B $? : B & CC '- B B B BDE B E " / ( C B, 9 $ &, $ ( & & : B, I, II / ) $,, % ' ( : #, , - -$? : B / $ $, # ' : +, 496 " *; * 6 I, II /, -! ( : $** *, * 6 I, II / ) ( : $** *,

154 !B$ %B &# &%!()(! B$ *A A A+ C B B B B B B (#**) *A: (-#!'? C 5