Высшая математика. Том 2

! C$ @ ! A 9 # <

' $+ =: A(±a,0,0).

# ! # + &C '

2

y

2

2

z

2

y

2

z

2

x

−

= 1,

x

−

= 1,

−

−

= 1,

b2

c2

b2

c2

a2

a2

z = 0.

y = 0.

x = 0.

+ " > – ! C " # C

3 («&"$ !+»).

b = c C&<# ! C$ @ ! A 9 A "

x2

−

y

2 + z2

= 1.

a2

b2

1.35.2. ) # /

+ &< ! A 9: ) ! 3$ ! A 9; A) @ ! A 3$ ! A 9. A 9 # D ' E + # 9.

+ # / – E ! > ", " < '+" " "#

2z = x2 + y2 , p > 0 , q > 0 .

pq

?" " > ', % " 3 E <9 ! > z ≥ 0 . E <9 !-

> ! % # z = h , h ≥ 0 – !+, ! 3 #& ! h = 0 !+ A < '+" & 3 & (+. 2.36).

! 3$ ! A 9 (+. 1.36) # < @ ! % (! % + # 9) x = 0 y = 0 . ! > E # ! % # – ! A

x2	= 2 pz,		y2 = 2qz,
	y = 0			x = 0.

+. 1.36. ! 3$ ! A 9

+. 1.37. ! A 3$ ! A 9

51

! % #, ! ' # @ # – ! " ! A. -

x2

h2

z =

+

2q .

!

2 p

y = h

p = q C&<# ! A 9 A ".

# + ! A 9 E ! > ", " 3 < '+" " "#

x2

y2

2z =

−

, p > 0 , q > 0 .

p

q

! % # z = h ,

h ≠ 0 – @ ! A

x2

y2

−

= 2h , z = h .

p

q

h > 0

@ ! A = # A(± 2 ph, 0, h);

h < 0 – '" @ ! A;

h = 0 & ! = ! ! "# >, % ! D '+"

	x	−	y	= 0 , z = 0			x	+		y	= 0 , z = 0 .
	p
			q				p			q
! A 3$ ! A 9 (+. 1.37) # < @ ! %: x = 0
y = 0 . @ # ! % # – ! A
			x2		= 2 pz,		y2 = 2qz,
					y = 0				= 0.
							x

J> @ ' D = D + &C ' ! 3 . > " @ &<

+.

! % #, ! ' # @ # – ! A. !,

				y	2		h	2
	! ! % # x = h – ! " ! A	z = −				+
				2q			2q .
			x = h

1.35.3. 4 + $+ 0 %. . %

B 3 D ! > D (+. 1.38) < '+" ! > ", & &> # ! "# 9 ( 9), % ! > ' 3 & 3 & (= & &+) ! < & &.

B 3 " " + / 0 &@ @ ! " & # < @ "

		x2	+	y2	−		z2	= 0 .
		a2		b2			c2
= D &+	+ &C '			! 3 .					$+, " % 3
M (x, y, z) C ' E$	! >,					9$ C '			3 M ′(λ x,λ y,λ z),

−∞ < λ < +∞ .

52

3 M ′ C ' ! "#$ % , C, !- > " + < '+" ! "# >, % ! > " ' 3 ! 3-. E' #& ! % z = c ! < !-

	2		2
> D ! !+&		x	+	y	= 1,
				b2
	a2
			z = c.

+. 1.38. B 3 ! > "

C, ! > " # C A& & &> # ! "- # 9, % ! > ' 3 ! 3 ! < E$ !+. % a = b , ! > " < ! > D A-

	"	x2 + y2	−	z2	= 0 A z = ±	c	x2 + y2 .
		a2		c2		a

* + 0 – E ! > &@ @ ! " &, & &> # ! "# 9 ( 9), % A @ < ! + $ $ ! "# ! < & & (+! "#- &D3& E 3 9 ! >).

+ &< E 3 ! > &@ @ ! " & (+. 1.39. , A, ):

) ! 3$ E

x2

+

y2

= 1; " % a = b , E < '+"

a2

b2

&@ #.

x2

y2

A) ! A 3$ E

−

=

1.

a2

b2

) A 3$ E,

y2 = 2 px .

A

+. 1.39. / 3 ! >:

− ! 3$ E; A − @ ! A 3$ E;

− ! A 3$ E.

53

1.36. ) 0 & %. . %

"#$D D ! > < '+" ! "#, % E # - = ! >. >, " # C& ' A& & &> # ! "#$9 9, D '+" $3 #. B # 3 > E 3 > ! > ' &- @ @ ! " &, $3 # ! > "# &@ @ ! " & < ! C$ @- ! A 9 @ ! A 3$ ! A 9.

) 5 . . # /

" " ! C @ @ ! A 9

x2

+

y2

−

z2

=

1 # C ! +

a2

b2

c2

x

y x

y

z

z

& @ "

−

+

= 1 −

1 +

.

a b a b

c

c

? + <, % C ! "# >

x

−

y

= λ

1 −

z

a

b

1

c

,

(1.1)

x

y

z

+

=

1 +

a

b

λ

c

λ –

'$

! # ,

C '

@ ! A 9 &.

$+, " % 3

(x0, y0 , z0 )

C ' ! "#$ (2.1), x0 , y0, z0

' "D ' A #

" "# (1.1). ? > "3 9> A& , C&<# + ', +! ! D3& - & ' # # x, y, z & " @ ! A 9 x0 , y0, z0 . C,

A& '-" 3 (x0, y0, z0 ), % C ' ! "#$ (1.1), C ' @ ! A 9 &,

A ! "# E # C ' $#&.

# #& λ " " (1.1) 3 D ' + #$+ ! "# >, % -

C ' @ ! A 9 & (+ #$+ ! "#$>

>).

! C$ @ ! A 9 # < + #$+ ! "#$> >.

&@ + #$+ # C 3 + + # D " '

x

−

y

= µ 1 +

z

b

a

1

c

(1.2)

x

+

y

=

−

z ,

1

a

b

µ

c

# #, % " "

> ! ' = ! +& & @ "

α

x

−

y

= β 1 −

z

,

α

x

−

y

= β 1 +

z

,

a b

c

a b

c

x

y

z

x

y

z

β

+

= α 1 +

β

+

= α 1 −

.

a b

c

a b

c

/ " " 3 D ' % ! $

$ C @ + #$+

(! α = 1, β = 0 ). C @ + #$+ A C +. 1.40.

54

+. 1.40. #$+ ! "#$> > ! C @ @ ! A 9

										)
			?$ ! "#$ ! >							> x2 − y2 + z2 = 1, " ! > " ' 3-
		3 & M1 (1,1,0).									@ " x2 − y2 = 1 − z2 A
(			N " "				! > # C			! + &
	x − y		)(	x + y	)	(	)(	)	. " " > + #$+		> # < @ ":
						= 1 − z	1 + z

α1 (x − y) = β1 (1 − z),			α2 (x − y) = β2 (1 + z),
	β1 (x + y) = α1 (1 + z)			β2	(x + y) = α2	(1 − z).

+ "D3	3 M1, C # β1 = α1 ,
" >, " ! > " ' 3 M1	, # D ' @ " x − y = 1 − z,
	x + y = 1 + z

β2 = 0 . "-

x − y = 0,

1 − z = 0.

! ': x = 1,	x = y, O
y = z	z = 1.

C + , % 3 C & 3 & ! C @ @ ! A 9 M1 (x1, y1, z1 ) ! > ' ! $ $ C @ + #$+ . $+, + '

3 M1 C ' @ ! A 9 &, +! + '

x

y

x

y

z

z

1

−

1

1

+

1

= 1

−

1

1

+

1

(1.3)

a b a b

c

c

? 3 " ! # λ ! A $ + + # " ' # -

# #

x

y

z

1

−

1

= λ

1

−

1

b

a

c

(1.4)

x

y

1

z

1

+

1

=

1

+

1

λ

a

b

c

?$# λ ! = @ " " (1.4). / 3 " λ ' " #

55

&@ #& " D (1.4), + ' ! = A 3 + " " (1.3) ! 3 + ! = @ " " (1.4), C # &@ " " (1.4).$#& 3 ! # λ + + # (1.3) 3 ' =& & &. # C 3 # # C $ $ & + #$+ (1.4), " ! > ' 3 3 & M1.

) . # + . # /

" " @ ! A 3 @ ! A 9 # < @ ":

2z = x2 − y2 , p > 0 , q > 0 . p q

! ! ' #& , # C $ + #$+ ! "#$>

>

x

y

= 2β z

x

y

= 2β z

α

−

α

+

p

q

p

q

A

x

y

x

y

β

+

= α

β

−

= α

p

p

q

q

+ #$+ ! "#$> > A C +. 1.41.

+. 1.41. #$+ ! "#$>> @ ! A 3 @ ! A 9

1.37. 3 % 5 & + / . / $

&3 + # = + &< 3 & @ # D " - 3 " + > E$ & + C& #& ! E +, ! A 3 " ! # 3 > !, + 6 E 9 A’< ! @ &! #, %. # - ' !.

' 0 [@., net present value, NPV] "< C$A '= & @ ' & > + & & ' & + & ", A $@ E-$ 6 A+ D $ +. 3 + # ! # > # < '+" - E" # C ! # $+ 9 + + D 3 + @ @ = @ ! & ! +! & E 9 + E$@ ! & $ + D + E$> $ - @ E D. >& E' @ ! ! #& $+ +- E$> $+ D< '+" 6 #& D:

56

T	In − Ot
NPV =	t	t	,
	(1	+ r )t
t=0		t

T – @ '$ ! + E$@ ! &, t – ! + E$@ ! &, Int, Ott –– ! > t-#& ! + E$@ ! &, rt – # + &.

E <9 6& E 9 ! &<, % < $D $ C ' & ! =& 3 @& ). # A '=, # &C3 +! < NPV. C C + ' !- # & rt. /" -! ! E$ C + ' ! +. 1.42

, / / % 3 < '+" " + ' 9 > + + E$@ ! 6 D E > ! !:

E(R	p	) = R	+	[E(RM ) − RF ]		SD(R ) ,
		F		SD(RM )		F
RM , RF – ! > + ' >					A > E > ! !,
E(RM ) – + " > + ' > E > ! !, SD(RM ) , SD(RF ) – !-
E 9 + @ > " >					A > E > ! !.
E' @ " " ! &< $& C + ' + ' 9 > +
! # RM , RF , SD(RF )		! A 3 & – ! # & SD(RM ) .

@ 6 &, #& +. 1.43, # C A 3 , " > + ' # D- < '+" $@ (> 9) $@ (C " ") 6 # .

+. 1.42. #! +! & NPV C +3 +& 6E < + &"

3
2,8
2,6
2,4
2,2
2
1,8
1,6
1,4
1,2
1
1	2	3
+. 1.43. ?# 3 ' + ' 9
> + ! 6 D

4 ' – E +& &! + ' # , % "D ' + 6 &-A @ # +! + C ", C " > ! +&< '+" A # ! 3 > # > *1,*2, ..., *m. D + @ & < & " @ &! +> C > # C + A D A'< , " !$" + #.

57

B + $ ! ' 3 + @ A " @ &! & >&- "# &+ > &@ &!& ". E' #&, " !, 3 # C - C 9 @ &!, C # !, + ' C @ &! E ' +- C&$ +& &! + .

+ @ & "D ' =& + &! 3:

• ! " + 6 E 9 A'< & >& "# , % A C D ' +& ', ! & A'< . = " 9 3, " !, ! ' ! @ A- " ' ! +& &! + ' A'< , % + 6 &D '+";

• ! ! !&% ', % +& D '+", ! " + ' " 9 + & &+& &! + A'< , % 3 < '+", A ! =& + &D3 9 + & &;

• ! A& > + 6 E$ " + A 3 > " %, A> + " + ' '" &+ +& &! + $ +! A& ! + 9

+& & &.

" @ &! &< '+" = "> # 3 " + @ A’< E @ &! # # 3 9 @ # 9.

> !, # E = + &D ' 6&- E 9 A @ ' > # >, @ 6 " > # C ! A& & $, C, ' - + +& " ! C ' 3 9 @ # 9 "< ! + 9> .

1.38. 2 '" #% . , ,%$

6 Maxima A& &D '+" ! # @ D ! plot2d. @ '- #& @ " E$ ! # < > C3 6 # :

plot2d (expr, x_range, ..., options, ...); plot2d ([expr_1, ..., expr_n], ..., options, ...);

plot2d ([expr_1, ..., expr_n], x_range,..., options, ...),

expr, expr_1, ..., expr_n # C& ' A& A # # 3 # #, A - #, A +! + # 3 ' 6& E 9.

# A C " ( % $, # +$, ! + @ 6 > .!.) # C # ! # @ D + &! > !E$:

•[y, min, max] + &< '+" " 3 " ! & ' 9 +; "- % + &< '+" E$ , @ 6 ! C !$ !, ' "- % 6& E " +"@ < +' @ E' @ ! &. , " % '$ - ! 3$ & set_plot_option, @ E ' 9 + A& & ' &+ # 3.

•xlabel + &< '+" " @ ' 9 +; " % E$ -

+ &< '+", #'"# @ ' 9 + A& #, 3 x_range.

! C #& ! & #'" # 9 A& = ! C #.

•ylabel + &< '+" " ' 9 +; " % A& &D ' @ 6 '-

@ & $ !E " ylabel + &< '+", #'"# ' 9 + A& #, 3 y_range.

•[logx] [logy] ! " ' @ ' $ ' + @ 6# 3$ #- += A.

•legend + &< '+" " @ 6; " % A& &D ' @ 6 " '-

58

> , @ 6 A& A C " C @ . % E" !E " + &< '+", Maxima + ' .

• style + &< '+" " + D 9: & # > !, A " ! $. !, [style, [points,5,2,6], [lines,1,1]]. ? 3 " ' > 3 + ! + " «points» !E 9 <: 5 – &+ !, 2 – , % A& +- & +" (3$), 6 – A'< , % A& & ' + & +" (6- A ). 3 + ! + " !E 9 «lines» D ' % & 9 ( 3) (1 ! < + ' #&).

+ &< ' @ A ' > !E$ & +! + & plot_options, % # C& ' A& #-6& E <D set_plot_option; C E > @ A ' > !E$ # C A& - + 3 # !E "# # plot2d.

" @ % A 6& E D <9 # 9 ! A ! + " & ! #- & plot2d + & #'" 6& E 9. & > &C > #'" #- 9, 99 # # ' $ # + # ' 3 ", # # #. !-" ' 9 + E A’" !E " + &< '+" & !- >, # D '+" 6& E$. B # E' @, + & !E D nticks & 3 D 200. / 3 <, % " ! A& @ 6 6& E 9 A& >-200 3.

	)
A& & @ 6 6& E 9 y =	1	.
	sin x + cos x

(%i1) plot2d(1/(sin(x)+cos(x)),[x,-5,5],[y,-20,20],[nticks,200], [style,[lines,4,0]])$

+. 1.44. A& @ 6 6& E 9 y =	1	.
	sin x + cos x

59

% ! A ! A& & 3 + @ 6 ' > 6& E$, ! - A + 9> # & > &C >, "D3 # #. ? # + ' & " A3 + " 6& E 9 # C + 99 ! 3 ", " % A& ! ! ' .

)

A& & @ 6 6& E$ y = 2sin x y = cos x . (%i2) u: 2*sin(x)$

(%i3) v: cos(x)$

(%i4) plot2d([u,v],[x,-2*%pi,2*%pi],[style,[lines,4,0]])$

+. 1.45. A& @ 6 6& E$ y = 2sin x y = cos x .

, " " @ A& &< '+" @ 6, # C A& $ & + $ A ! # 3$ 6 #, + #, " +! +, % ! 3 < '+" + «discrete» A «parametric». % 6& E " + , + # discrete ! A&+! + 3 ', 9 C, % " "D ' + A D @ ' $ - ' 3. ' , C 9 3 # C& ' A& = +! + # 3 "# A + 3 ! A& &+ = @ +! + &: [discrete, [x1, ..., xn], [y1, ..., yn]] A [discrete, [[x1, y1],

..., [xn, ..., yn]] .

)

A& & @ 6 6& E 9, % $ A E D 3 ':

x	10	20	30	40	50
y	0,6	0,9	1,1	1,3	1,4

(%i5) xy:[[10,.6], [20,.9], [30,1.1], [40,1.3], [50,1.4]]$

(%i6) plot2d([discrete, xy], [style, [points,5,0,1]])$

60