995 + t lim( ) = te dt =. α α = lim[( + ) ] = e, α α α α = t t t t te dt = tde = te α α e dt = αe e, =, e α = αe α e α, α =. y z = yf, f( u) z + yz y =. z y y y y y y z = yf + y f = yf f, y y y y z y = f + y f = f + yf, y y y y z + yz y = yf y f + yf + y f 3 ( ) y = yf = z f ln = +, f ( ) =. + e + C t ln = t, = e, = + f ( ) = + e, t f () t e,
( ) f ( ) = + e d= + e + C. 4 A =, A A ( A ) =. 3 4 5. 5 5 3 5 AA = A E, A = A = A= A=. A A 5 5 3 5 ( ) ( ) 5 X DX ( ) =. + f( ) < 6 + E ( X) = f( ) d = f( ) d + ( ) d =, + D( X ) E ( X ) f( ) d = = = ( + ) d + ( ) d. = 6 f() f( ) f ( ) lim =, y = f( ) (, f () )
.... ( A) ( B) ( C) ( D) ( D ). f (), f () f( ) f( ) f() lim = lim = (), f = f () =. A d. B d. sin ( ) ( ) + C e d. D d. ln ( ) ( ) ( A ). = sin lim sin =, sin d. (3) n α =,,,, E n AB ( ) T A. B E C E (D) E + α α (C) AB E E E T T T T T α α α α α α α α α α 4 T T T Eα α α αα α T T = E + αα i αα = E. A R( A) = m< n, m m n E m
( A) A m. ( B) A m. ( C ) A ( E ) (D) A ( D ). m, O. = b. A, r( A) = m, ( Ab), m n m, r( A) = r( Ab ),, A =b, m< n,. ( A) ( B) ( ) C ( E O). ( D). 5 X N ( µ, σ ) σ P{ X µ < σ} ( A). ( B). ( C) ( D) ( C ). ~ ( µ, σ ) X N µ Y = X ~ N (, ), σ { µ < σ} = { < } P X P Y σ µ.. m, ( 6 ) ( cos ), <, f( ) =, =, ( ) cos tdt, >. f =.
sin lim ( cos ) lim, = = cos lim cos tdt lim, = = + + lim f( ) = = f(). f ( ) =, f ( ) =. ( ) cos f () = lim f ( ) cos sin = lim = lim 3 3 cos sin = lim = lim =, 6 3 + () = lim cost dt + costdt lim lim + + = = sin = lim = + f ( ) cos = ( ) f =. ( 6 ) : (rcsin ) d. rcsin d = d (rcsin ) (rcsin ). rcsin = + d (rcsin ) ( )
= + (rcsin ) rcsin d = + + C (rcsin ) rcsin. : u = rcsin, = sin u, d = cos udu. d = u udu = u d u (rcsin ) cos sin = u sin u + u cosu cosudu = u sin u+ ucosu sin u+ C = + + C (rcsin ) rcsin. ( 7 ) f ( ) g( ) [, ]( > ) g( ) f ( ) f ( ) + f ( ) = A A. ( ) ( ) = ( ) f g d A g d; π sin rctn ed. ( ) ( ) = ( ) ( ) + ( ) ( ) ( ) ( ) π f g d f g d f g d, f g d t = ( ) ( ) = ( ) ( ) f t g t dt f g d. ( ) ( ) = ( ) ( ) + ( ) ( ) f g d f g d f g d ( ) ( ) ( ) ( ) = f + f g d= A g d. f ( ) = rctn e, g( ) = sin, =. f ( ) g( ) g( ) π π π,
( e e ) rctn + rctn =, rctn e + rctn e = A, = rctn e = A, π A. f f π = ( ) + ( ) =. π π π π sin rctn e d= sin d= ( cos ) =. π π π ( 6 ) Q = Q( P), R = PQ, P Q Q( P). P Q dr dq Q= Q dr = >, dp P = P = c <, EP = b >. P Q. R= PQ, Q dp dr dp = P+ Q = P+ P ( P) = P, dq dq dq Ep Q dr dq Q Q = P =, = b b P =. b R= PQ, P dq dr d ( ) ( p ), d = + Q d = Q Q P Q d Q = Q E P P P P
dr dp P= P c Q =. b ( b) = Q = c, ( 5 ) f ( ) [ b, ] ( b, ) ( b, ) ξ bf ( b) f ( ) ' = f( ξ ) + ξ f ( ξ). b : F( ) = f( ), f ( ) [ b,, ] ( b, ) ξ, Fb ( ) F ( ) ' = F ( ξ ), b ' F'( ) = f( ) + f ( ), bf ( b) f ( ) ' = f( ξ ) + ξ f ( ξ). b : ξ, bf ( b) f ( ) ' = f ( ) + f ( ), b bf ( b) f ( ) ' [ f ( )]' =. b bf ( b) f ( ) F( ) = f( ). b F( ) [ b,, ] ( b, ), F ( ) = Fb ( ) =,, ξ ( b, ), F ' ( ξ ) =, bf ( b) f ( ) ' = f( ξ ) + ξ f ( ξ). b ( 9 )
z = f(, y) = y(4 y) + y = 6 y D. ' f (, y) = y(4 y) y= ' fy (, y) = (4 y) y = =,( y 6) (4,),(,). (4,) = ( y 6) D. (,) f y y y " = 8 6, f y " y = 8 3 4, f " yy =, (,) A= f = y y y = = < = " 8 6 6, y B= f = y == = " y 8 3 4 4, y C = f = = = = B " yy 8, y AC = 6 48 = 3 <, (,), f (,) = 4, = ( y 6) y = ( y 6) f(, y ) =, + y = 6, y = 6, f ( y, ) z = 3,( 6), z' = 6 4= =, = 4. z" = 4 = 4>, = 4 = 4 (4,), f (4,) = 64. f (,) = 4, f (4,) = 64.
( 8 ) λ+ + 3 = λ3 + λ + 3 = + + 3 = λ. : λ λ3 λ A = λ λ λ λ λ λ 3( λ) λ λ λ. ( λ+ )( λ) 3( λ) i. λ λ, ra ( ) = ra ( ) = 3,. ii. λ =, ra ( ) = 3, ra ( ) =, ra ( ) ra ( ),. iii. λ =, A., ra ( ) ra ( ) = < 3,. =, 3 = 3 =, u = (,,) T =, 3 T v = (,,), v = (,,). T,
= u + cv + cv = + c + c, c, c. ( 8 ) T A Aα i = iα i( i=,,3) α = (,,), α = (,,) α3 = (,,) T A Aα = iα ( i=,,3) i i T A( α, α, α ) = ( α, α,3 α ), 3 3 AP = B, P = α, α, α3 = =7, P, A = BP, P =, 9 7 4 6 3 3 5 A = 4 3. 9 = 3 3 6 3 3 ( 8 ).7.3.8.. n( n ). α ;
β ; 3 θ. { } B { } A= = { } AB { }. A= = B = A+ AB, ( ) P( B A) P A =.3, =.8, ( ) P( A) P( B A) P AB = =.3.8 =.4. P( B) = P( A) + P( AB) =.7 +.4 =.94. X n X n ( n,.94) n { n}.94, α = P X = = n { } β = P X = n = C n.94.6, { } { } { } θ = P X n = P X = n P X = n = n n n.94.6.94. ( 7 ) X.Y X = e (,). e, >, X F( X) =,, y ln( y) =, =, Gy ( ) Y,
Gy PY y P e y ( ) = { } = { },, y, = PX { ln( y)},< y<,, y,, y, = y,< y<,, y.,y (,). 996 y = y y dy =. d ( + ln) ln = yln y, = y ln y+ y y, y y = ( + ln), dy = y d = d ln ( + ). f ( ) d= rcsin + C, f ( ) + C 3 ( ) 3 d =.
f ( ) d= rcsin + C, ( ) f =, ( ) ( ) ( ) f ( ) d = d = d = + C. 3 ''' 3 y = ln( + + ), y =. = 5 3 y ' = i ( + ) =, + + + + 3 y" = ( + ) i =, 3 ( + ) 3 3 3 5 5 3 y"' = ( + ) + ( + ) i = ( + ) ( ), ''' y = = 3 5. 3 4 D = =. + + 3 4 5.., D = ( ) D + D = ( )[( ) D + D ] + D 5 4 3 3 3 = + + [( ) ] D3 ( ) D = + + + ( )[( ) D ( )] ( ) D
= + + + + + ( )[( )( ) ( )] ( )( ) = + + 3 4 5. 5 3 i pi = ( i =,,3) i + 4 X 3 PX= { } =. A i i, PX { = } = PAAA ( ) + PAAA ( ) + PAAA ( ) 3 3 3 3 3 = i i + i i + i i =. 34 34 34 4 ' '' '' f ( ) = f ( ) =, f ( ) > A f ' ( ) f ' ( ) B f ( ) f ( ) C f ( ) f ( ) D (, f( )) y = f( ) (D) AB(C f = =,(D) 3 ( ),., f f "( ) f "( ) f "( ) "( ) = lim = lim >,, f "( ) >. f "( ) <,; f "( ) >,, (, f( )) y = f( )
() f ( ) ( A) ( ) ' lim f ( ) + =+. lim f( ) = + + ' B lim f( ) =+ lim f ( ) = + ( C) ( ) + ' lim f ( ) + = lim f( ) =. ' D lim f( ) = lim f ( ) =. (A) f ( ) f ( ) = e, =, lim f( ) ± lim ' f ( ) lim e, + (C),(A). = ±, f '( ) =,(B),(D). = = lim f( ) = lim e =+, (3) n A ( n, ) A A, n ( A)( ) ( B)( ) n+ A =A A. A =A A. n ( C)( ) ( D)( ) n+ A =A A. A =A A. ( C ) A, AA = A A= A E. A = AA, ( A ) = ( A A ) = A A ( A A ) ( ) n A A A A A. A n = = (4) n α,, αm β,, β m, λ,, λm k,, k m, ( λ k ) β, + = m m m ( ) ( k λ ) ( k λ ) ( λ k ) + α + + + α + β + m m m A α,, αm β,, β m.
( ) B α,, αm β,, β m. ( ) α + β,, α + β, α β,, α β. C m m m m ( ) α + β,, α + β, α β,, α β. D m m m m ( D ), λ λ,, λm k,, k m,, ( α β ) λ ( α β ) k ( α β ) k ( α β ) + + + + + + + = m m m m m m α + β,, α + β, α β,, α β. m m m m (5) A, B A BPB, ( ) > A PA ( ) < PAB ( ) B PA ( ) PAB ( ) C PA ( ) > PAB ( ) C PA ( ) PAB ( ) (B) A BPB, ( ) > PAB ( ) PA ( ) PAB ( ) = PA ( ), PB ( ) = PB ( ) (B). ( 6 ) ( ) ( ) g e, f =,, = () f ( ); () f ( ) (, ) +. (), g( ), g( ) g ( ) = =.
( ) + ( ) + ( ) ( ) + ( + ) g e g e g g e f ( ) = = =,, ( ) ( ) ( ) ( ) g e g + e g e g f ( ) = lim = lim = lim =. ( ) ( ) ( ) g g + + e,, f ( ) = g ( ), =. () =, ( ) ( ) ( ) g g + + e lim f ( ) = lim g ( ) + g ( ) g ( ) + e ( + ) e = lim f ( ) ( ) g e = lim = f ( )., f ( ) (, ) +. ; ( 7 ) f y t f f ( y, ) = e dt, + y y y y f f f = ye = e y y y f f 3 y 3 y = y e = ye f y ( ) = y e y f f f + =e y y y y y.
( 6 ) + e ( ) + e e d. e d = d = d + e + + + ( + e ) ( + e ) + + + = + d = + e + e + e d. e = t, = ln td, = dt. t e + + d = ( + e ) t( + t) dt t = dt ln ln = = t t+ t+ + + e d d d + e + e + e = = ( + e ) e e = d = ln ( + e ) + C. + e + e + e + e ( + e ) e d = lim ln ( + e ) + ln. + + e e e lim ln ( + e ) = lim + ln ( + e ). + + e + + e e = lim + ln = + = + + e + e + e ( + e ) d = + ln= ln.
( 7 ) p Q Q = c, p+ b b c > bc. p. p R R = PQ = p c, p+ b ( + ) ( p+ b) b c p b R =. b b R = p b ( bc) =. c = c > b p ( bc) < < R >, c p. b p ( bc) > R <, c p. b p ( bc) = R c b R b c bc c b c ( ) m = =. ( 9 ) A(,), B (3,)
. y S A B O S A, B y = ( )( 3) 4 S = ( )( 3) d= ( 4 3) d + = 3 3 4 S = ( )( 3) d= (4 3) d = 3 S = S V = π [( )( 3)] d 4 3 = π [( ) 4( ) + 4( ) ] d ( ) 4 38 = π + = 5 3 5 5 4 3 [ ( ) ( ) ] π 3 V = π [( )( 3)] d 3 4 3 = π [( ) 4( ) + 4( ) ] d
( ) 4 6 = π + = 5 3 5 5 4 3 3 [ ( ) ( ) ] π V V = 9 8 ( 5 ) f ( ) [ b, ] ( b, ) b f ( d ) = f ( b ) b ( b, ) ξ f ' ( ξ ) =. f ( ) [ b, ] ( b, ) b c f ( d ) = f ( )( b ) b f () c = f() d= f() b b f ( ) [ cb, ] (, cb) (, cb) ξ f ' ( ξ ) = ξ (, cb) (, b). ( 9 ) + 3+ 34 = + 63+ 44 = 3+ + p3+ 74 = 63 4 = t p, t A
3 4 6 4 A = 3 p 7 p+ 8 6 t t+ t ( A) ( A) t = ( A) = ( A) p =8 4 = + c + c ( c, c ) b p 8 = + c ( c ) ( 7 ) T 4 A E+ A =, AA = E, A < E 4 A A *. * * AA = A A= A E, A * = A A -. λ A, A <, λ, A = λ,. * A A = A A = Ai =, A * A λ, λ λ, A A,. E+ A = A- E =, A λ =
T 4 AA = E = E = 6 T A A = A =6 A = 4 A <, A A λ = α Aα = α A Aα = ( ) A α A α = α A A α α α α * = AA = ( 4)( ) = A * ( 7 ). 5 5. X 5 X ( 5,.) k k 5k { = } = ( = ) P X k C..8 k,,,3, 4,5, 5 P{ X = } =.8 =.38, 4 { } C 5 5 P X = =..8 =.4, 3 { } C P X = =..8 =.5, 5 { } { } { } { } P X 3 = P X = P X = P X = =.57. Y
, X =, 5, X =, Y = f ( ) =, X =,, X 3. E( Y ) =.38 + 5.4 +.5.57 = 5.6 ( ). ( 7 ) λ > T. Xi ( i =,,3) i X, X, X 3 λ e, > F( ) =, Gt () T t Gt () = t > G() t = P{ T t} = P{ T > t} = P{ X > t, X > t, X > t} 3 = PX { > t} ipx { > t} i PX { > t} 3 = [ Ft ( )] 3 = 3 e λt. 3λt e, t > Gt () =, t T 3λ.