D(x, y) - Oxy. (x, y) D z Î Z Ì R, , z x y. x y , , D - , , , Z - . z x y z = f(x, y), z = z(x, y), z = F(x, y) .. , V = pR2 R , .. V = f(R, ), , R , V.
- , z - , x - , y - ( z y , x - -); A, a, b - . - : z = f(x, y). z = f(x, y) x = xo, y=yo zo= f(xo, yo). D , , . D D, - . , y = f(x) , z = f(x, y). R 3 Oxy D. M(x, y)ÎD Oxy z = f(x, y). . , , , z = f(x, y) . x y Oxy M(x, y), z = f(x, y) - P(x, y, z) . , z M(x, y) z = f(M).
f(M) A, , f(M) - A , r = MoM 0 M Mo (, ).
f(x, y) Mo, .
, . , R , (a) (b) , R = /(a+b), .. R R = f(, a, b). R 3, , ( , , ..), f(x,y).
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n
z = f(x1, x2,..., xn).
D Ì R n. . ( - , - ) . z = f(x1, x2,..., xn), ( ) z, xi .
, , . z = f(x, y) x x y. x : .
z = f(x, y) y y x. :
.
z = f(x, y) . , , x, .
z = f(x, y) D Mo(xo, yo) . , f(x, y) Mo(xo, yo) (),
(xo - d, xo + d; yo - e, yo+ e),
f(x,y) £ f(xo,yo) (f(x,y) ³ f(xo,yo)).
() , , . . , ( , ). , . c . Mo(xo, yo) - z = f(x, y), .. ,
z = f(x, y) Mo(xo, yo). . :
1) D > 0, z Mo: A < 0, A > 0;
2) D < 0, Mo ;
3) D = 0, .
28. z = y4 - 2xy2 + x2 + 2y + y2 .
. : = - 2y2 + 2x, = 4y3 - 4xy +2 +2y. : .
, Mo(1,-1) - , . : , , A=2, B=4, =10, D = 4, .. D > 0, Mo - (A>0). z min = (-1)4 - 2×1×(-1)2 +1 - 2 +1 = -1.
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, , .. , . . , . x y, , . n :
x | x1 | x2 | ... | xn |
y | y1 | y2 | ... | yn |
x y. . , , ( ) . , x y `y= ax+b, a b - , ,` y - . , b a=tg a, . a, b .
ax + b -`y=0. , , , . x ` y xi yi, ,
- ei=`yi -yi; :
ax1 + b - y1 = e1, - ax2 + b - y2 = e2, -
axn + b - yn = en. e1, e2,..., en, , . . , .. a b. a b , . , a b , u = . , . u ei .
u = (ax1 + b - y1) 2 + (ax2 + b - y2) 2 +... + (axn + b - yn)2, u = u(a,b),
xi, yi , a b - ,
. a b , u(a,b)
. , . :
= 2(ax1 + b - y1)x1 +... +2 (ax1 + b - y1)xn, = 2(ax1 + b - y1) +... +
+ 2 (ax1 + b - y1). :
.
. a b `y = ax + b. , x y :`y=ax2 + bx + c, . u = = . u = u(a, b, c) - a, b, c. , , :
.
`y = ax2 + bx + c, , .
, - . . , , . ..
, .
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29. y .
x | ||||||||
y |
, y x : y = ax + b, a b.
. : .
, , , : a 15,93; b 110,57. , :
y = 15,93x + 110,57.
3.1.
F(x), X, f(x), f(x), x Î X :
F¢ (x) = f(x). (3.1)
. f(x) f(x); -
ò f(x) dx.
F(x) - - f(x),
ò f(x)dx = F(x) + C, (3.2)
- .
:
1) d ò f(x)=f(x)dx,
2) ò df(x)=f(x)+C,
3) ò af(x)dx=aò f(x)dx (a=const),
4) ò(f(x)+g(x))dx= ò f(x)dx+ ò g(x)dx.
1. ò xm dx = xm+1/(m + 1) +C (m ¹ -1).
2. = ln êx ê +C.
3. ò ax dx = ax/ln a + C (a>0, a¹1).
4. ò ex dx = ex + C.
5. ò sin x dx = cos x + C.
6. ò cos x dx = - sin x + C.
7. = arctg x + C.
8. = arcsin x + C.
9. = tg x + C.
10. = - ctg x + C.
, , .
f(z) [a, b], z=g(x) [a,b] a £ g(x) £b,
ò f(g(x)) g¢ (x) dx = ò f(z) dz, (3.3)
z=g(x).
:
ò f(g(x)) g¢ (x) dx = ò f(g(x)) dg(x).
:
1) ;
2) .
u = f(x) v = g(x) - , . , ,
d(uv)= udv + vdu udv = d(uv) -vdu.
d(uv) , , uv, :
ò udv = uv - ò vdu. (3.4)
. udv=uv'dx vdu=vu'dx.
, , ò x cosx dx. u = x, dv = cos x dx, du=dx, v=sinx.
ò x cos x dx = ò x d(sin x) = x sin x - ò sin x dx = x sin x + cos x + C.
, . , ,
ò xk lnmx dx, ò xk sin bx dx, ò xk cos bx dx, ò xk e ax dx
, .
. [a, b] f(x). [a, b] n a = x0 < x1 <...<xn = b. (xi-1, xi) xi f(xi)D xi,
D xi = xi - xi-1. f(xi)D xi , l = max D xi 0, , f(x) a b :
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f(xi)D xi. (3.5)
f(x)
[a, b], a b .
:
1) ;
2) ;
3) - ;
4) , (k = const, kÎ R);
5) ;
6) ;
7) f(x)(b-a) (xÎ[a,b]).
.
f(x) [a, b].
ò f(x) dx = F(x) + C
-, c :
F(b) - F(a). (3.6)
: , y= f(x), x = a x = b Ox.
() . I - , :
. (3.7)
, f(x) [,+¥), f(x) [,+¥). , , .
(-¥, b] (-¥, +¥):
.
. f(x) x [a,b], , f(x) , II f(x) a b :
,
. :
= . (3.8)
30. ò dx/(x+2).
. t=x+2, dx=dt, ò dx/(x+2) = ò dt/t = lnïtï+C =
= lnïx+2ï+C.
31. ò tg x dx.
. ò tg x dx = ò sin x/cos x dx = - ò d(cos x)/ cos x. t=cos x, ò tg x dx = - ò dt/t = - lnïtï+C = - lnïcos xï+C.
32. ò dx/sin x.
.
33. .
. =
34. ò arctg x dx.
. u=arctg x, dv=dx. du = dx/(x2+1), v=x, ò arctg x dx = x arctg x - ò x dx/(x2+1) = x arctg x + 1/2 ln(x2+1) +C;
ò x dx/(x2+1) = 1/2 ò d(x2+1)/(x2+1) = 1/2 ln(x2+1) +C.
35. ò ln x dx.
. , :
u=ln x, dv=dx, du= 1/x dx, v=x. ò ln x dx = x lnx - ò x 1/x dx =
= x lnx - ò dx = x lnx - x + C.
36. ò ex sin x dx.
. u = ex, dv = sin x dx, du = ex dx, v=ò sin x dx= - cos x Þ ò ex sin x dx = - ex cos x + ò ex cos x dx. ò ex cos x dx : u = ex, dv = cos x dx Þ du=exdx, v=sin x. :
ò ex cos x dx = ex sin x - ò ex sin x dx. ò ex sin x dx = - ex cos x + ex sin x - ò ex sin x dx, 2 ò ex sin x dx = - ex cos x + ex sin x + .
37. J = ò cos(ln x)dx/x.
. dx/x = d(ln x), J= ò cos(ln x)d(ln x). ln x t, J = ò cos t dt = sin t + C = sin(ln x) + C.
38. J = .
. , = d(ln x), ln x = t. J = .
39. J = .
. : . =
=
= .
40. - ?
. , . -, . , = .
f(x) = > 0 , , . , f(x) = x = 4, . , - .
41. .
. , , F(x)= .
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: = .
-,
= F(b) - F(0) = + = ;
= = .
3.2.
42. , ,
f(t) = 3/(3t +1) + 4.
. f(t) t, , t1 t2
V = .
V = = ln 10 + 12 - ln 7 - 8 = ln 10/7 + 4.
43. , , f(t) = 2t + 5.
. :
V = .
44. (.6.1) d t = f(t),
S = P ex d t dt,
P = S ex(- d t dt).
, c, d t :
d t = d o + at, d o - t = 0, a - ,
d t dt = (d o + at)dt = d o n + an2/2;
ex(d o n + an2/2). d t = d o at, d o - , a - ,
d t dt = d o at dt = d o at /lna = d o(an -1)/lna;
ex(d o(an -1) / lna).
, 8%, 20% (a=1,2), 5 . ex (0,08 (1,25-1) / ln1,2)
ex 0,653953 1,921397.
45. , R . , , , - .
R t = f (t), n .
0 n :
S = .
A = .
: Rt = Ro + at,
Ro - , , . A, :
A = = + .
A1 = , A2 = .
: A1 = = - Ro/d ê = - Ro/d( -eo) = - Ro/d( -1) =
= Ro( -1)/d. A2 = .
: u = t, dv = dt Þ du = dt, v = = - /d, = - t /d + 1/d = - t /d (t+1/d) +C. ,
A2 = -a t /d (t+1/d)ê = ((1- )/d - n )a/d.
,
A = A1 + A2 = Ro( -1)/d + ((1- )/d - n )a/d.