untitled

f ( ) tan e, > = arcsin a = ae, a = tan e tan lim f ( ) = lim = lim =, arcsin + + + lim f = lim ae = a, y e ( ) = <+ + + S = e d= ( ) de + + + = e + e d = e = yy '' + y ' = y =, y' = = = y = + y = + y' = p dp dp dy dp y'' = = = p d dy d dy yp dp + p =, dp = dy, dy p y ln p = ln y+ C, p y = e = C C

y' =, y =, C = dy =, =, =, =, d py y ydy d y y y = + = + yy = y = yy = C y = C + C '' ' ', y() = C =, y'() = C =, y y = + = + 4 π lim cos cos π nπ + + + + + + cos = n n n n n π π π nπ lim + cos + + cos + + + cos = cosπ d n n n n n + π π = cos d = cos d π = si n π 4 4 λ λ λe A = λ = λ λ λ λ λ λ λ 4 4, = λ = λ λ = π

f u y = f( ) = = y f ' = A B C D5 D y = y', = y' y '( ) =, y = f, y' = f '( ) =, y'( ) = f '() f ' () = = 5 f () () ( A) f t dt ( B) f t dt ( ) + ( ) ( C) t f t f t dt ( D) t f t f t dt F( ) f () t dt f ( ) f ( ) F() f ( ) F () t f () t + f ( t) t f t f t dt () + ( ) () y = y( ) y '' + py + qy = e ln( + ) y( ) = y' ( ) = y A B C D C y'' py qy e + + = y y = ' = y ''() =

ln( + ) lim = lim = lim = lim = = y y y y y ' '' ''( ) 4 y = f ( ) (, + ) lim f = lim f ' = ; + + lim f ' lim f ' = ; + lim f = lim f ' = ; + + + lim f ' lim f ' = B + + sin, f ( ) = f ( + ) lim f = lim f ' = + + cos sin sin f '( ) = = cos, lim f ' lim cos + = + sin A f ( ) = sin, ( + ) f ( ) f ( ) CD B lim =, lim ' =, + + lim f ', lim f ' = l >, + + l f '( ) >, > X +, X +, [ ] ( ξ) f f X + = f ' X, X + < ξ <, l, f ( ) f ( X + ) > ( X + ) f ( ) f f ( ) lim =, lim ' =, + B + α, α, α βα, α, α β 5

α, α, α k α, α, α β β B α, α, α β β C α, α, α β β D α, α, α β β A α, α, α β, k, k β α + k α + kα ( ααα, kβ+ β) ( ααα, kα+ kkα + kkα+ β) ( αα α, β ) r ( αα α kβ β ) ( αα α β ), +, = 4, α α α β β,k + k =,α α α α α α, β BC β α α α β α α α k =, α α α β β D + A π r = cos θ, θ ( θ) = cosθ cosθ = cosθ cos θ y = cos sinθ y = sinθ sinθ cosθ π θ =,, 6 4 4

dy d dy dθ cosθ cos θ + sin θ = = = d sinθ + cosθ sinθ π θ = π θ = 6 6 dθ π θ = 6 5 y + = +, y + = 4 4 4 4 y + =, y 4 + + + = 4 4 4 f ( ) F( ) + < = e,, ( e + ) = f t dt < F( ),, F( ) = t t dt t t + = + = + = () = () + () F f t dt f t dt f t dt t te = t + t + dt + t ( e ) t dt = td t = + e t t + e + e + t de = + = + dt t t t e + e + e + e e + t ( e ) t e e = + ln = + ln + ln t e + e + e + e +, + < = e ln ln, + + e + e +

f ( ) f ( ) f ( ) f ( ), + >, lim = ( + ) f ( ) h f h lim = e, h + f + h y = f ( ) h, ( + ) f h ln y = h f ( + ) ln f + h ln f h f lim ln y = lim = lim = ln f ( ) ', h h h f h h ( + ) f ( ) h f h ln f ( ) lim e ' = h ' ln f = e e, ln f ' =, ln f ' f = Ce lim f =, C =, + f = e dy + y d = y = y( ), y = y( ) =, = dy y = d

d d y = e e + C = + C = + C y = + C =, = 5 7 VC π C d π C C 5 = + = + + 6 5 75 V '( C) = π C+ =, C = 5 4 6 V ''( C) = π >, 5 75 C = 4 75 y = y( ) = y 4 l ABCD 54 h m () h+ P = ρ g h+ y dy y = ρg( h+ ) y = ρgh ρ g h+ P = ρg h+ y ydy = ρg ( h+ ) y y 5 = 4 ρg h+ 5 5

P 5 h 5,, P = 4 = 4 4 h + 5 h=, h= ( ) h = m P = ρgd= ρgh h h+ P = ρ g h+ d h+ t h t t P = g h+ t t dt = g h+ = 4 ρ g h+ 5 4ρ ( ) 4ρ ( ) 5 < <, = n=,,, { } n+ n n n < <, < = + = < k k > < k+ = k ( k) ( k + k) = n >, < n, { n } ( ) ( ) n > = + = n n n n n n n n ( ) n n = + n n

n>, { } n+ n n lim n n lim = a, = ( ) n a = a( a) n n+ n n,, lim n = n a= a= ( ) < a< b, a lnb lna < < a + b b a ab ln b ln a < b a ab a ln ln, a ϕ ( ) = a ( > a> ) ϕ ( ) a a ' =, + = < a a > aϕ( ) ϕ( a) =, ϕ, > a ϕ < a = a ln ln a<, a b> a > b a ln b ln a<, ab ln b ln a < b a ab a lnb ln a < + a b b a

f ( ) = ln ( > a> ), ξ ( ab, ) < a< ξ < b ln b ln a = (ln )' =, = ξ b a ξ a ξ > b > a + b ln b ln a a > b a a + b, f ( ) = ( + a )( a) a a ( > a> ) ln ln, f '( ) = ( ln lna) + ( + a ) a ( a) = ln ln a + > > af ( ) f ( a) =, > a f ( ) > f ( a) =, + a ln ln a a( a) > b> a> a + b ln b ln a a( b a) >, a lnb ln a < a + b b a f () = f f f λ, λ, λ, h ( ) ( ) ( ) λ f h + λ f h + λ f h f h, ', ''

λ, λ, λ, + ( ) + ( ) ( ) λf h λf h λf h f lim = h h + ( ) + ( ) ( ) λf h λf h λf h f = lim h h λf '( h) + λf '( h) + λf '( h) = lim h h λf '' h 4 λf '' h 9 λf '' h = lim h = ( λ+ 4λ + 9 λ) f ''( ) λ, λ, λ, λ+ λ + λ =, λ+ λ + λ =, λ+ 4λ + 9λ = =, 4 9 + + λ, λ, λ, h λ f ( h) + λ f ( h) + λ f ( h) f ( ) h f ( h) = f ( ) + f '( ) h+ f ''( ξ) h ( ξh) h f ( h) = f ( ) + f '( ) h+ f '' ( ) h + '' ''( ) f ξ f h = f ( ) + f '( ) h+ f ''( ) h + o( h )

= + + + f h f f ' h f '' h o h, 9 f ( h) = f ( ) + f '( ) h+ f ''( ) h + o( h ), + ( ) + ( ) ( ) ( λ λ λ ) ( ) ( λ λ λ ) '( ) λ f h λ f h λ f h f = + + f + + + f h+ ( λ+ λ + λ) f h + o( h ) 4 9 '' λ, λ, λ, λ+ λ + λ =, λ+ λ + λ =, λ+ 4λ + 9λ = AB A B= B 4 E, E AE B =, A A B= B 4E AB B 4A =, A E B 4E = 8 E, ( A E) ( B 4 E) = E 8 A E ( A E) = ( B E) 4 8 A= E+ 8 B 4 E, ( B E) 4 4 4 = =, 8 8

A = 4 A = α, α, α, α, α, α, α, α 4 4 4 α, α, α4α α α, βα + α + α + α 4 A = β = = = β, A ( α, α, α, α ) 4 4 4 α + α + α + α = α + α + α + α 4 4 4 α α α ( ) ( ) ( ) + α + + α + α = α, α, α 4 4 4 + =, + =, 4 = = + k, k α, α, α 4 α α α + α 4, A A α α α α + = A 4

= k, k 4 4 A β α + α + α + α ( α, α, α, α ) A = + k, k = β A = β