4 6 4 4 ( n ) f( ) = lim n n +, f ( ) = = f( ) = ( ) ( n ) f( ) = lim = lim n = = n n + n + n f ( ), = =,, lim f ( ) = lim = f() = f ( ) y ( ) = t + t+ y = t t +, y = y( ) dy dy dt t t = = = = d d t + t + t +, dt 4 = = = d dt d d t + ( t + ) ( t + ) d y d dy dt t, d y d < t < = + +, t <, (,) ( t = = (,] t t )
+ d = + d sct tan t sc = t dt = dt= sct tan t d t = = = = t + ( ) dt dt arcsin t t t t z 4 z = z(, y) z = + y, z + z = y z z = + y y z, y z z z = ( ) z z z = ( ) + y y z = + z z z = y + z z z z + + = = z y + z F(, y, z) = + y z = F z = F F = y z z = ( ) F z = = = F (+ ) + z z z z z
F z = y = = F z z y (+ ) + z z z z + = + z z = y + + z dz = (d dz) + dy z z = d+ dy dz z z ( + ) dz = d + dy z dz = d + dy z z + + z = + z z z = z y + z z + = y 6 5 ( y+ ) d dy = y = = 5 y = + 5 dy y = d dy y = d dy = d y ln y = ln + ln c y = c y = c( ) c ( ) + c ( ) ( ) c = c ( ) =
5 c ( ) = d+ C= + C 5 5 y = ( + C) = C + 5 5 6 y = C = 5 = y = + 5 dy y = d d d y = d+ C ln ln = d+ C 5 = d+ C = + C 5 6 y() = C = 5 y = + 5 6 A =, B ABA = BA + E, A A, E, B = 9 ABA = BA + E ABA BA = E, A E B A = E =, ( A E) BA = E,
B = = = = A E A ( ) ( ) 9 A A = A A ABA = BA + E AB A A = B A A + AA AAB= AB+ A A ( A E) B= A A A E B = A B = = A A E 9 8 4, + 7 α = cos tdt, β = tan tdt, γ =,, A α, β, γ B α, γ, β C β, α, γ D β, γ, α sin tdt B γ lim = lim α + + sin cos tdt tdt sin = lim + cos = lim = lim = + + γ = o( α) β lim = lim + γ + tan tdt sin tdt tan = lim = lim = + + sin β = o( γ )
α γ β, B 8 f ( ) = ( ), A = f ( ), (, ) y = f( ) B = f ( ), (, ) y = f( ) C = f ( ), (, ) y = f( ) D = f ( ), (, ) y = f( ) C f ( ) = ( ), < ( ), < < f ( ) = +, < <, < <, < < f ( ) =, < < < <, f ( ), > >, f ( ), (, ) f () =,, f( ) >, =, =, (, ) y = f( ), C n 9 limln n ( + ) ( + ) ( + ) n n n n A ln d B lnd C ln( + ) d D ln ( + ) d B n limln n ( + ) ( + ) ( + ) n n n n n = limln ( )( ) ( ) n n + + + n n n
n = lim ln( ) ln( ) ( ) n n + + + + + + n n n n i = lim ln( + ) n n n i= B = ln( +) d + = t lntdt = lnd f ( ), f () >, δ > A f ( ) (, δ ) B f ( ) ( δ,) C (, δ ) f ( ) > f() D ( δ,) f ( ) > f(), C f( ) f() f () = lim >,, δ >, < δ, f( ) f() > δ > >, f ( ) > f() δ < <, f ( ) < f(), C y + y = + + sin A y = a + b+ c+ ( Asin+ Bcos ) B y = ( a + b + c + Asin + B cos ) C y = a + b+ c+ Asin D y = a + b+ c+ Acos
A y + y = λ + =, λ =± i, + = + = +,, y y ( ) y = a + b+ c i y + y = sin = I ( ), i, m y = ( Asin+ Bcos ) + = + + y y sin ( ) = + + + ( sin + cos ) y a b c A B f u, D {(, y) y y} = +, f ( y) ddy D A d f ( y) dy y y B dy f ( y) d C D sinθ dθ f( r sinθcos θ) dr sinθ D dθ f( r sinθcos θ) rdr D ( y ) f ( y ) ddy = dy f ( y ) d ( y ) + = d f ( y) dy AB = rcosθ y = rsinθ
sin f ( y) ddy θ = dθ f ( r sinθcos θ) rdr, D D y A o, A B, B C, AQ A B = C Q C D D B= A C = B C = A = A = AQ Q = D 4 A, B AB =, A A, B B A, B C A, B D A, B A
A= ( a ij) l m, B= ( b ij) m n A ( A A A ) = m b b b b b b AB = ( A A Am ) b b b m m mn n n ( b A b A b A b A ) = + + + + = m m n mn m B, b i m, j n, b A+ b A + + b A + + b A =, ij j j ij i m m A, A,, Am B B B =, B m AB = a a a B m a a a m B al al a lm B m ab + ab + + a mbm ab + ab + + ambm = = alb + alb + + almb m A a i l, j m ab + ab + ab+ + a B = ij i i ij j im m B, B,, Bm A A mn B ns AB r (A)+r (B) < n AB r (A) >, r (B) >, r (A ) < n, r (B ) < n, A B A 9 94 5
+ cos lim + cos lim = lim + cos ln + cos lim + cos ln = lim ln + cos ln = lim sin lim cos = + sin = lim = + cos 6 = lim + cos ln + cos ln = lim cos ln+ = lim cos = lim = 6 6 f ( ), +, [, ], f( ) = ( 4), f( ) = f( + ), f ( ) [,] ;, f ( ) = < + < f( ) = f( + ) = ( + )[( + ) 4] = ( + )( + 4)
f () = f( ) f() ( 4) f+ () = lim = lim = 4 + + f( ) f() ( + )( + 4) f () = lim = lim = 8 f () = f + (), = =, f ( ) = 7 + f ( ) = sint dt, f ( ) ; f ( ) + + f ( + ) = sintdt, t = u+, + + ( + ) = sin( + ) = sin = ( ), f u du u du f f ( ) sin (, + ), [, ] f ( ) = sin( + ) sin = cos sin, f ( ) =, =, =, 4 4 f( ) = 4 sintdt = 4, 4 5 5 4 4 4 4 f ( ) = sin t dt = sin t dt sin t dt = 4, f () = sin tdt=, f ( ) = ( sin t) dt =,
f ( ),, f ( ) [, ] 8 + y = =, = t( t > ) y =, Vt, () St (), = t Ft () St () Vt () () lim St t + Ft () St () = y + y d t t + + = + 4 d t + = d, t t +, Vt () = yd= d St () = Vt () t t + Ft () = y = = t t + d St () lim = lim t + Ft () t + t t + t t + = lim t + t t t t +
t t + = lim = t + t t 9 4 < a< b<, ln b ln a> ( b a) 4 ϕ ( ) = ln, ln 4 ϕ ( ) = ln ϕ ( ) =, >, ϕ ( ) <, ϕ ( ), < <, 4 4 ϕ ( ) > ϕ ( ) = =, < <, ϕ ( ), < a< b<, ϕ ( b) > ϕ ( a), 4 4 ln b b> ln a a 4 ln b ln a> ( b a) 4 ϕ ( ) = ln ln a ( a), ln 4 ϕ ( ) = ln ϕ ( ) =, >, ϕ ( ) < ϕ ( ), < <, < <, ϕ ( ) 4 4 ϕ ( ) > ϕ ( ) = =, < a < b<, ϕ ( ) > ϕ ( a) = = b ϕ ( b) > 4 > ln b ln a ( b a)
ln [ a, b], ln t lnt ϕ () t =, ϕ () t =, t t lnξ ln b ln a> ( b a), a< ξ < b ξ t >, ϕ ( t) <, ϕ () t, ϕξ ( ) > ϕ( ), lnξ ln ξ > =, ln b ln a> ( b a) 4 9g 7 m / h 6 = 6 g m / h m= 9g v = 7 m/ h t () t vt () dv m v dt = dv dv d dv = = v dt d dt d m d = dv m () t = v+ C m v() = v, () =, C = v, m () t = ( v v()) t
vt () mv 9 7 ( t) = = 5 ( m) 6 6 5 m dv m v dt = dv = dt v m t v C m =, v = v =, C = v, t vt () v m =, t + mv t mv m = vtdt ( ) = = = 5 ( m) + d d m = dt dt d d dt + = m dt r + r = m r =, r = m t C C m = + () =, d C mv = = = C = C = t m v() v dt t= m t= mv t m t () = ( ) t +
mv 9 7 ( t) = = 5 ( m) 6 6 5 m z z z z = f( y, y ) f,, y y z y = f + y f, z y = yf + f y z y = [ f ( y) + f ] + f + y f y y y y y + y [ f ( y) + f ] = 4yf + ( y ) f ++ y f + ( + y) f y y y 9 ( + a ) + + + 4 =, + ( + a) + + 4 =, + + ( + a) + 4 =, 4+ 4 + 4 + (4 + a) 4 =, a,, A, + a + a + a a a = B + a a a 4 4 4 4+ a 4a a a =, ra= ( ) < 4,, + + + 4 = η = (,,,) T, η = (,,,) T, η = (,,,) T,
η η η = + +,,, a, B + a a+ 4 4 a =, r( A ) = < 4,, + =, + =, 4 + 4 =, η = (,,,4) T, = η, + a + a A = = ( a+ ) a + a 4 4 4 4+ a A =, a = a =, a =, A, A = 4 4 4 4 + + + 4 = η = (,,,) T, η = (,,,) T, η = (,,,) T, η η η = + +,,,
a =, A, =, =, 4 = 4, η = (,,,4) T, 9 9 8 A = 7 4 4 4 6 4 9 4 4 = η, 9 4, a, A a 5 A λ λ λ λ 4 = λ 4 a λ 5 a λ 5 = ( λ ) λ 4 = ( λ ) λ a λ 5 a λ 5 λ λ λ = ( )( 8 + 8+ a) λ =, 6+ 8+ a =, a = a =, A,, 6, E A=, λ =, A
λ =, λ 8λ+ 8+ a, 8 + a = 6, a = a =, A, 4, 4, E A=, λ = 4, A