3 35. f (x), x dy y, lim dx x (fluxion).,, dy dx (differential quotient), (differential coefficient)., dérivée. y = f(x), y/ x (x, y) (x + x, y + y),

Transcription

1 2 3 x y = f(x). x, x y, y, x x = x, y x = y y x x y x x. y y = y. x, x, y lim x 0 x, y = f(x) x, dy dx., x, y. h x, dy dx = lim h 0 f(x + h) f(x). h, y = f(x) x. f(x) x, f(x)., dy dx x, f(x), f (x). dy dx = f (x). dy dx (Leibniz). f (x) (Lagrange). y f(x), y ẏ f (x). ẏ (Newton). D x y D x f(x).,, D. dy dx = f (x) = y = ẏ = D x y = Df(x)..,,. (derived function, derivative) f(x)

2 3 35. f (x), x dy y, lim dx x (fluxion).,, dy dx (differential quotient), (differential coefficient)., dérivée. y = f(x), y/ x (x, y) (x + x, y + y), dy (x, y) dx ( 2 ). x, y., dx, dy (X, Y ), dx = X x ( x ), dy = Y y ( y ), (x, y). 2 dy = f (x)dx () dx, dy, ()., (x, y) (), dx x (differential), dy y., dy, dx. dy dy, (x, y) y = f(x). dx dx, y x. lim y x = f (x), y x = f (x) + ε, x 0, y = f (x) x + ε x. (2) ε x x, x, x 0, ε 0., x 0, y = A x + ε x. (3), A x x, ε x x. x 0 y y, ε 0. (3), x 0, = A + ε A, A = lim x x., f (x) x, A = f (x). (3) f (x)., (3) (2)., f (x) (2). (2) f (x) x x ( x, x)., x 0, x ε, ε x ε x x., x 0, (2) f (x) x y

3 36 2., y f (x) x y = f(x) x, dy.,, x x, x =, dy = f (x) x. (4) dx = x., x x ( x ). (4), dy dx. dy = f (x)dx. (5) dy dx = f (x), (6) dy dx dy,., dx,., (5), f (x) dy dx,. x, dx = x,, dx x. [] (2), ε x x ε = ε(x, x), x 0. x 0 ε 0, ε x = 0, x = 0, ε = ε(x, 0) = 0., x, ε x x = 0. (3), x = 0 ε = 0, ε x = 0. (2) (3), x = 0., x = 0, ε x u, v, ( ) (u ± v) = u ± v. (2 ) (uv) = u v + uv. ( ) u (3 ) = u v uv v v 2 (v 0). [] ( ). (2 ) (uv) = (u + u)(v + v) uv = u v + u v + u v,

4 , x 0, (uv) x = u x v + u v v + u x x. u x du dx, v x dv, u 0. dx d(uv) dx = du dx v + u dv dx. (uv) = u v + uv. ( ) u (3 ) = u + u v v + v u v u u v = v v(v + v),, x 0, (u/v) x u x du dx, d dx = u x v u v x v(v + v). v x dv, v 0. dx ( ) du u = dx v u dv dx v v 2. ( ) u = u v uv v 4 37 v 2. u, v,, w, ( ), (2 ). (2 ) (uvw) = u (vw) + u(vw) = u vw + uv w + uvw, (2 (uvw) ) uvw = u u + v v + w w. uvw 0. c x, c = 0., (2 ), (cu) = cu., x x, x =., (2 ) n x, d(x n ) dx = nxn., ( ), (2 ), (3 ), x.. x = 2h, sin(x + x) sin x x sin h x 0 h 0., h ( 9), = sin h h cos(x + h).,, cos(x + h) cos x. 5 D sin x = cos x.

5 38 2, D cos x = sin x.,, D tan x = ( cos 2 x. x nπ + π ) 2, n = 0, ±, ±2,. y = f(x) x, x 0 y 0, f(x) [] f(x) = x sin, f(0) = 0, f(x) 0, x = 0 x. h 0,. f(h) f(0) h = sin h, x = 0. (872),., x, y lim x +0 x, lim x 0 y x.,. D + y, D y., y. [] y = x, x = 0, D + y =, D y =. [] f(x) [a, b], f(x) x = a, x = b., f(x) x = a, x = b. ±, f(a + h) f(a) lim = ± h 0 h, f (a) = ±,. f (a) = ± lim h 0 f (a + h) = ±,. [] y = signx. ( 8 [ 5]), x = 0, Dy = +. 5 y = f(x) x 0 x x x, ϕ(t) t 0 t t t. ϕ(t) x 0 x, y = f(x) ϕ(t) x, y

6 5 39 [t 0, t ] t., y = f(ϕ(t)) = F (t). f(x) ϕ(t), t t 0, ϕ(t) ϕ(t 0 )., f(ϕ(t)) f(ϕ(t 0 )), F (t) F (t 0 ). y t. 7 f(x) ϕ(t), F (t) = f(ϕ(t)), F (t) = f (x) ϕ (t), dy dt = dy dx dx dt. (). [] y, t t,, x x, x, y y t = y x x t. () t 0, x t dx dt., x 0,, y x dy dx. y t dy dx dx dt. F (t) = f (x)ϕ (t).,. x, x x 0,, x t, t, x = 0., ().,.., t t, x y x, y, y = f (x) x + ε x, x = ϕ (t) t + ε t. t 0, ε 0, x 0., t 0, x = 0. 3 [], x = 0, ε = 0, t 0, ε 0., y = (f (x) + ε)(ϕ (t) + ε ) t = f (x)ϕ (t) t + [εϕ (t) + ε f (x) + εε ] t, [ ] ε, y = f (x)ϕ (t) t + ε t, ε = εϕ (t) + ε f (x) + εε,

7 40 2 t 0, ε 0, dy = f (x)ϕ (t)dt., dx = ϕ (t)dt dy = f (x)dx. ( 3).,, y x, x t, t u. dy du = dy dx dt dx dt du. [] (infinitesimal) 0. :. x 0 sinx, x 0 x 2, x + e x, x a, y b (x a) 2 + (y b) 2, α β, β 0, β α. α β = εα, ε 0. α, α oα., εα, ε, ε 0.,. [ ] β = oα, γ = oα, β + γ = oα oα + oα = oα., oα,, oα α., β α 0, γ β + γ 0, 0. α α [ 2] u, uoα = oα, o(uα + oα) = oα. : uεα, ε 0, uε 0;, ε(uα + ε α) = (εu + εε )α, ε 0, ε 0, εu + εε 0. [ 3] 7, y = f (x) x + o( x), x = ϕ (t) t + o( t), o( x) = o( t). [ 2]

8 6 4 y = f (x)ϕ (t) t + f (x)o( t) + o( t) = f (x)ϕ (t) t + o( t). [ 2, ] α, β, ω = β/α, α, β = Oα. ω 0 β = oα, oα Oα,., ω a, a 0, β = Oα, α = Oβ., α, β. β α n, β α n. [] oα, Oα, α. x, ln x = o( n x), ln n x 0., ε, ε = o(). x, oα ε, ε o, εα, ε α, ε 0, ε 0., Oα ω O, ω.,. o, O (order). o, O. 6 y = f(x) a x b. y p, q ( 3), y p y q ( 2)., y = f(x) (), y, x. f(x), x < x 2 < x 3, y < y 2 < y 3 y > y 2 > y 3., y < y 2, y 2 > y 3, η y 2 > η > Max(y, y 3 ), (x, x 2 ) (x 2, x 3 ), x, η = f(x) ( 2 2)., y p y q,, x y = f(x)., x y. x = ϕ(y), ϕ f., f ϕ, f ϕ x y, y, x y.. y x, x y, [] dy dx dx dy =.., y = f(x), x = ϕ(y), y = η x = ξ. ϕ(y). {y n } η,

9 42 2 {x n }, {x n } λ. f(x), f(λ) = η. λ = ϕ(η) = ξ. y n η, x n ξ, ϕ(y n ) ϕ(η). ϕ(y). / x y y = x. y 0, x 0, lim x / y = lim y / dx dy, x dy = dx ( 2 3),, dy x dx = 0., ±. dx y dy = ±, ( 2 4). 2 3 dy dx = tan θ, = tan ϕ 2 4 dx dy. ( ) arcsin x. y = sin x π 2 x π 2 (2n ) π 2 x (2n + )π 2 (n = 0, ±, ±2, ),, y ( 2 5). y y = sin x, x = arcsin y, x. 2 5 y = sin x

10 6 43 x, arcsin [., arcsin, π 2, π ] 2,, Arcsin ( 2 6). x, y, x, y, y = Arcsin x ( x ). x = sin y ( π 2 y π ) y = Arcsin x y = sin x, d sin x = cos x, dx d arcsin y dy = cos x = ± y 2., π 2 x π, cos x 0. ± +. x, y 2, D Arcsin x = x 2. (2 ) arctan x. y = tan x π 2 < x < π + ( 2 7). 2 tan arctan (). x, < x < +, ( 2 8) y = Arctan x, π 2 < y < π y = tan x 2 8 y = Arctan x Arctan 0 = 0, Arctan (±) = ± π 4,

11 44 2 Arctan (± ) = lim Arctan x = ±π x ± 2., y = tan x, dy dx = cos 2 x = + y2,, [] D Arctan x = + x 2. arccos, arccot,, Arcsin Arctan,. [ y = arcsin x y = arctan x, y π ] 2, +π, 2,.,. [ ] y = arcsin x 2 x 2 = sin y. x 2 = cos 2 y, x = ± cos y. x y 2 9. arcsin, ( ABC, B 0, π ). y, A BC ABC 2 (arccos x arccos( x) ). y ( 7 ), dy dx = x ( x2 ) = x x 2 x = x 2 x 2 (x 0), x = 0 D + y =, D y = +., y = arcsin 2x x 2. [ 2] y = arctan x., x = 0. ( 2 0). arccot x

12 a >, y = a x < x < +, 0 < y <. log a x 0 < x <, + ( 2 ). a x,,. a x, x = 0 Da x. d(a x ) dx = lim a x+h a x h 0 h = a x a h lim, h 0 h lim h 0 a h, h 2 h > 0. a h >. a h = +, t > 0. t ( 0), h 0, a h, t. ( h = log a + ), t a h h h 0 t, =, log a ( + t ) t log a e. ( t log a + ) = t ( log a + ) t. t ( + t ) t e ( 9). log a, h 0 a h lim = h 0 h log a e = log e a. h > 0. h < 0, h h, a h h h 0, a h, = ah h a h h a h (h > 0). log e a.

13 46 2 d(a x ) dx = ax log e a. () 0 < a <, a x, ()., a = e, log e e =, ( 8), d(e x ) dx = ex. (2) d log a x = dx x log e a (a > 0, x > 0), (3) d log e x = dx x (x > 0). (4) () (2), (3) (4), e. e (natural logarithm). log nat ln., ln x = log e x = log nat x. De x = e x. Da x = a x ln a (a > 0). D ln x = x (x > 0). D log a x = (a > 0, x > 0). x log a [] ln x x > 0,, x > 0, D ln x = x. x < 0, D ln( x) = x =, x, D ln x = x (x 0). x u, v, w x., u, v, w 0 x, ln uvw ( []), D ln uvw = D(ln u + ln v + ln w ) = u u + v v + w w. D ln uvw = (uvw) uvw. (uvw) uvw = u u + v v + w w (u 0, v 0, w 0). ( u ) / u v v = u u v v. 5. ln a x = x ln a (a > 0),

14 8 47 Da x = ln a. ax Da x = a x ln a., (). x a (x > 0). a, ln x a = a ln x, Dx a x a = a x, Dx a = ax a.. 9 () 8 f(x) [a, b], (a, b). f(a) = f(b), (a, b) f (x) 0. ξ, a < ξ < b, f (ξ) = 0. [] f(a) = f(b) = 0., f(x) 0,. f(x) [a, b], [a, b] f(x). f(ξ) ( 3)., f(ξ) > 0, a < ξ < b., x = ξ, f 0. f (ξ) = 0. x > 0, f x 0, f (ξ) 0, x < 0, f x 0, f (ξ) 0. f(x),. f(a) = f(b) = k 0, f(x) k. [], f(x), a, b,.,, [a, b], (a, b),.,,, f(x) (a, b), x = a ( x = b ), f(a) = lim f(x) = x a+0 lim f(x)., x b 0 lim f(x) f(x) x a+0 x = a, x = a, f(x) [a, b]. f(x), 9.,.

15 () ξ, [] f(b) f(a) b a f(x) [a, b], (a, b). = f (ξ), a < ξ < b. F (x) = f(x) Ax, A, F (a) = F (b). f(a) Aa = f(b) Ab, A = f(b) f(a). b a, F (ξ) = 0, a < ξ < b. F (x) = f (x) A. f (ξ) = f(b) f(a). b a f(a) [a, b], f (ξ).. y x, x () f(x), g(x) [a, b], (a, b). (a, b) ξ, f(a) f(b) g(a) g(b) = f (ξ) g (ξ), a < ξ < b., ( ) g(a) g(b); (2 ) f (x), g (x) 0. [], g(a) g(b). [], (a, b) g (x) 0., F (x) = µf(x) λg(x), λ : µ, F (a) = F (b). µf(a) λg(a) = µf(b) λg(b), µ{f(b) f(a)} = λ{g(b) g(a)}. λ = f(b) f(a), µ = g(b) g(a), F (x) = {g(b) g(a)}f(x) {f(b) f(a)}g(x). F (ξ) = 0 ( 9), {g(b) g(a)}f (ξ) = {f(b) f(a)}g (ξ).

16 8 49, g (ξ) 0. : g (ξ) = 0, ( ), g(b) g(a) 0, f (ξ) = 0, (2 ). {g(b) g(a)}g (ξ), f(b) f(a) g(b) g(a) = f (ξ) g (ξ).., t, x = g(t), y = f(t), a t b. t = a, t = b, t = ξ A, B, P, AB, P. A, B P AB ( 2 2). f (x) g (x) 0, 2 2. g(b) g(a) 0, g(b) g(a)., f(b) f(a) 0 g(b) g(a) 0(A B). 22, f (x) > 0, f(x), f (x) < 0, f (x) = 0, f(x), f(x). []. a < b, f(a) < f(b) f(a) > f(b), f(a) f(b) f(a) f(b). [] ξ,., a, b a < b, f(a) < f(b).. [] f(b) f(a) b a = f (ξ) > 0. x = a, f (a) > 0 ( f (a) < 0), f(x) ()., x a, x a, f(x) f(a) ( f(x) f(a)). f(x) f(a), (). 22, f (x). f (x) x = a, f (a) > 0, a f (x) > 0., 22, f(x). f (a) < 0, f(x). f (a) = 0, f (x), f(x).

17 ,. f(x), f (x) = 0 ( 2 4)., : f (x) 0, f(x) (),. f (x) 0. [] f (x).. [] f(x) = x 2 sin, f(0) = 0. x x 0,, f(x) [a, b], f(x), f (x) = 2x sin x cos x., x 0, x = 0. x = 0, x f (0) = lim h 0 h2 sin h 0 h = lim h 0 h sin h = 0. lim x 0 f (x) = f (0), f (x) x = 0., x a f (x) f (a). lim x a f (x).. 23 f(x), a, f(x) a, lim f (x) = l, f (a) = l. f(x) a, f (x) a. x a [], f(x) f(a) = f (ξ), a ξ x. x a x a, ξ a., f (ξ) l, [] f(x) f(a) lim = l, f (a) = l. x a x a a (), f (a) ()., (). 24 f(x) [a, b], µ f (a) f (b), ξ, a < ξ < b f (ξ) = µ. [] F (x) = f(x) µx, F (a) = f (a) µ < 0, F (b) = f (b) µ > 0. ξ, a < ξ < b F (ξ) = 0., F (x) [a, b], x = a x = b ( []). ξ, ξ, F (ξ), a < ξ < b. F (ξ) = [] 23 24,.

18 9 5 9 y = f(x) f (x), f (x) f(x), f (x). n f (n). f (n) y (n), y (n) x D (n) x y. f (x) x, d ( dy ) dx dx d 2 y dx 2. d n y dx n = f (n) (x)., dx 2 (dx) 2, d 2 y d(dy), y. 3, dy = y xdx,, d(dy), d(dx) d 2 y, d 2 x, d 2 y = y x(dx) 2 + y xd 2 x. (). x, dx = x x,, d 2 x = d( x) = 0, d 2 y = y xdx 2. d2 y dx 2 = f (x). x = ϕ(t) t, y = f(x) t. d 2 x = x t dt 2, () d 2 y = y xx 2 t dt 2 + y xx t dt 2. d 2 dt 2 f(ϕ(t)) = f (ϕ(t))ϕ (t) 2 + f (ϕ(t))ϕ (t), () t, x y.. u, v x, d n dx n (u ± v) = dn u dx n ± dn v dx n. uv,. d n ( ) ( ) (uv) n n dx n = u (n) v + u (n ) v + + u (n k) v (k) + + uv (n), k ( ) n.. (u + v) n k

19 52 2. u/v. 20, f(x). f (x) f(x). f(x), y = f(x ), y 2 = f(x 2 ). y = f(x), A = (x, y ), B = (x 2, y 2 ) AB, f(x) (). y...,, (), : x < x < x 2, x x x 2 0. () y y y 2, P = (x, y),, AP B, AP B ( 2 3), () () ( 0)., (). (), x < x < x 2, y y y 2 y x x x 2 x. ( ) 2 3 x x > 0, x 2 x > 0,, y y y 2 y y 2 y x x x 2 x x 2 x. ( ), AP () P B, AB. [x, x], AP.,,. 25 f (x), ( ) f (x) 0, f(x). (2 ) f(x), f (x) 0..,. () >,,.

20 2 53 [] ( ) ( ) f (ξ ), x < ξ < x. f (ξ 2 ), x < ξ 2 < x 2. ξ < ξ 2. f (x) 0, f (x) (), f (ξ ) f (ξ 2 ). ( ). f(x). (2 ) f(x), ( ). x x, ( ) f (x ). f (x ) y 2 y x 2 x. x x 2, ( ) f (x 2 ). y 2 y x 2 x f (x 2 ). f (x ) f (x 2 ), f (x) (). f (x) 0.,., ()., ( ), x, x, x 2 x,, x < x < x 2, y y y 2 y x x x 2 x. y 2 y x 2 x y 2 y lim x 2 x x 2 x = D+ y y y x x D + y., D y y y = lim,, x x x x D y D + y.,. D + y, D y, y. D + y, D y. 2,,. z = f(x, y), z x = lim x 0 z x, z y f(x + x, y) f(x, y), x z y = lim y 0, x, y. f(x, y + y) f(x, y). y

21 54 2, z x = f x(x, y) = D x f(x, y), ( z ) = 2 z y x x y = f xy(x, y),. [ ] x 2 + y 2 = r, ( z ) = 2 z x x x 2 = f xx(x, y), z y = f y(x, y) = D y f(x, y). ( z ) = 2 z x y y x = f yx(x, y), ( z ) = 2 z y y y 2 = f yy(x, y),. f(x, y) = ln r = 2 ln(x2 + y 2 ). x f x = x 2 + y 2 = x r 2, f y = y r 2, f xx = x 2 + y 2 2x 2 (x 2 + y 2 ) 2 = r 2 2x2 r 4, f xy = 2xy (x 2 + y 2 ) 2 = 2xy r 4 = f yx, f yy = r 2 2y2 r 4. f 2 f x f y 2, f = 2 f x f y 2 = 2 r 2 2(x2 + y 2 ) r 4 = 0. [ 2] x 2 + y 2 + z 2 = r, f(x, y, z) = r = (x2 + y 2 + z 2 ) 2. x f x = = x (x 2 + y 2 + z 2 ) 3 2 r 3, f y = y r 3, f z = z r 3, { } f xx = r 3 x 3 r 4 r = x r 3 + 3x r 4 x r = r 3 + 3x2 r 5, f yy = r 3 + 3y2 r 5, f zz = r 3 + 3z2 r 5,

22 22 55 f = 2 f x f y f z 2 = 3 r 3 + 3(x2 + y 2 + z 2 ) r 5 = 3 r 3 + 3r2 r 5 = 0, ( f xy = x 3 r 4 r ) = 3 xy y r 5 = f yx,.,,. 22 P = (x, y) z = f(x, y). z = f(x + x, y + y) f(x, y), z = A x + B y + ερ, (), A, B x, y, (x, y) ; ρ (x, y) (x + x, y + y) (ρ = ( x) 2 + ( y) 2 ); ε x, y, ρ 0, ε 0. 5, ερ = oρ. z (x, y). (), y = 0 ρ = x, z x = A ± ε. x 0 ε 0, z z (x, y), A., x y, B. (x + x, y + y) (x, y), α, x = ρ cos α, y = ρ sin α, z = A cos α + B sin α + ε, ρ z z z lim = A cos α + B sin α = cos α + sin α (2) ρ 0 ρ x y z lim ρ 0 ρ x = ρ cos α, y = ρ sin α. (), (2). z, z, x, y z z x + x y y z, dz. z = x z = y, dx = x, dy = y ( 3). dz = z z dx + dy. (3) x y z, dz = z z x + y (x, y) z = f(x, y). x y

23 56 2 X, Y, Z, dx, dy, dz X x, Y y, Z z, Z z = z z (X x) + (Y y). x y., (x, y) (3). z = f(x, y),., f(x, y). 26 z x, z, z. y [] h, k x, y, z = f(x + h, y + k) f(x, y) = {f(x + h, y + k) f(x, y + k)} + {f(x, y + k) f(x, y)}. x, f(x + h, y + k) f(x, y + k) = hf x (x + θh, y + k), 0 < θ <., f x, h 0, k 0, ε 0. y, k 0, ε 0. f x (x + θh, y + k) = f x (x, y) + ε, f(x, y + k) f(x, y) = kf y (x, y) + ε k, z = hf x (x, y) + kf y (x, y) + hε + kε. h ρ, k ρ (ρ = h 2 + k 2 ), hε + kε ( ε + ε )ρ, z. [] z = hf x (x, y) + kf y (x, y) + oρ. 26., f x. z x, z y, (x, y), z. 23 f(x, y) x, f x (x, y). f x y, f xy. f xy f yx.,, f xy f yx.,. z x, z y z x, z y.

24 f xy, f yx, f xy = f yx. [] (a, b)., = f(a + h, b + k) f(a + h, b) f(a, b + k) + f(a, b). () 2 4, = f(p 3 ) f(p ) f(p 2 ) + f(p )., ϕ(x) = f(x, b + k) f(x, b). (2), ϕ(a) = f(p 2 ) f(p ), ϕ(a + h) = f(p 3 ) f(p ), 2 4, f x (a, b), (2) = ϕ(a + h) ϕ(a). (3) ϕ (x) = f x (x, b + k) f x (x, b). (4) x = a x = a + h, ϕ(x), (3) (4), ϕ(a + h) ϕ(a) = hϕ (a + θh) (0 < θ < ). = h{f x (a + θh, b + k) f x (a + θh, b)}. (5), f xy (a, b), y = b y = b + k, (5),, f xy (a, b), x, h y, k, = hkf xy (a + θh, b + θ k) (0 < θ < ). lim (h,k) (0,0) hk = f xy(a, b). (6) lim (h,k) (0,0) (6) (7), (a, b),, hk = f yx(a, b). (7) f xy = f yx. []. (6),, f x, f xy, f xy (a, b)., f y, () hk = { f(a + h, b + k) f(a + h, b) h k f(a, b + k) f(a, b) }, k, P P P 3P 2. h, k,.

25 58 2, k 0, (6), k 0, h 0, hk h {f y(a + h, b) f y (a, b)}. hk f xy(a, b). h 0, lim h {f y(a + h, b) f y (a, b)} = f xy (a, b)., f yx (a, b), f yx (a, b) = f xy (a, b)., 27. f x, f y, f xy, f xy, f yx, f xy = f yx ( (Schwarz) ). x y. f yx., f x, f y, f xy, f xy, 27, f xy = f yx. f xy = f yx. f x, f y (, ), f xy = f yx ( (Young) ). (5). (5) h = k, f x ( 22), f x (a + θh, b + h) = f x (a, b) + θhf xx (a, b) + hf xy (a, b) + oh, f x (a + θh, b) = f x (a, b) + θhf xx (a, b) + oh. (5), = h 2 f xy (a, b) + oh 2. lim h 0 h 2 = f xy(a, b). x, y, f xy (a, b) = f yx (a, b). f x, f y, f xx, f xy, f yx, f yy.., f x, f y, f xy f xy, f yx ( f xx, f yy ).., 27. f xy = f yx.,. f(x, y) = xy x2 y 2 x 2, (x, y) (0, 0), + y2 f(0, 0) = 0.

26 24 59 (x, y) (0, 0), x, y f y. f x (x, y) = 3x2 y y 3 x 2 + y 2 2x2 y(x 2 y 2 ) (x 2 + y 2 ) 2. f xy (x, y) = x2 y 2 x 2 + y 2 + 8x2 y 2 (x 2 y 2 ) (x 2 + y 2 ) 3. (x, y) (0, 0) f xy, f yx., f xy (0, y) =, y 0, lim y 0 f xy (0, y) =. f x (0, y) = y, y 0, f x (0, 0) = 0, f x (0, y) y = 0. f xy (0, 0) = ( 23)., f yx (x, 0) =, f yx (0, 0) =.,,., f xxy = (f x ) xy = (f x ) yx = f xyx,, f xyz = f xzy = f zxy = f zyx = f yzx = f yxz, f xxyy = f xyxy = f xyyx = f yxxy = f yxyx = f yyxx,.,, f x 2 n n + = 2 f x 2,. f xy = 2 f x y, f y 2 = 2 f y 2, f x r y s = n f x r. (r + s = n, s = 0,, 2,, n) ys 24 u = f(x, y) du = u u dx + x y dy, u x, u y, du x, y d2 u. x, y (h = dx, k = dy ), d 2 u = d(du) = ( u x x h + u )h y k + ( u y x h + u ) y k k = 2 u x 2 h u x y hk + 2 u y 2 k2 = 2 u x 2 dx u x y dxdy + 2 u y 2 dy2.

27 60 2,,. d 3 u = 3 u x 3 dx u x 2 y dx2 dy u x y 2 dxdy2 + 3 u y 3 dy3. d n u = n u x n dxn + + d n u =,, ( ) n n u k x k y n k dxk dy n k + + n u y n dyn. ( x dx + y dy ) nu d 2 u = 2 u x 2 dx2 + 2 u y 2 dy2 + 2 u z 2 dz u x y dxdy u x z dxdz u y z dydz +, d n u = ( x dx + y dy + z dz + ) nu. u x, y, x, y t, u t. u x, y [], x, y t [], u t [] ( 5). du dt. t t,, o 5. (2) (), u lim t 0 t u = u x x + u y y + o( x 2 + y 2 ). () x = x t + o( t), y = y t + o( t). (2), u = (u x x + u y y ) t + o( t). du ( dt = u xx + u y y x = dx dt, y = dy dt, d 2 u dt 2 = du x dt x + u x x + du y dt y + u y y = (u xx x + u xy y )x + (u xy x + u yy y )y + u x x + u y y = u xx x 2 + 2u xy x y + u yy y 2 + u x x + u y y. ).

28 25 6 d 2 u, x, y. x, y ξ, η, u ξ, η., u ξ, u η, x ξ, y ξ x η, y η x, y. u ξ = u x x ξ + u y y ξ, u η = u x x η + u y y η. u ξξ = u xx x 2 ξ + 2u xy x ξ y ξ + u yy yξ 2 + u x x ξξ + u y y ξξ, u ξη = u xx x ξ x η + u xy (x ξ y η + x η y ξ ) + u yy y ξ y η + u x x ξη + u y y ξη, u ηη = u xx x 2 η + 2u xy x η y η + u yy yη 2 + u x x ηη + u y y ηη. [] u = u(x, y), x, y. x = r cos θ, y = r sin θ, r = x 2 + y 2, θ = arctan y x. u r, θ, r, θ x, y. u xx = u rr cos 2 θ + u θθ r 2 u yy = u rr sin 2 θ + u θθ r 2 28, f(x) = f(a)+(x a) f (a)! r x = x r = cos θ, r y = y = sin θ. r θ x = y r 2 = sin θ r, θ y = x r 2 = cos θ. r u x = u r r x + u θ θ x = u r cos θ u θ r u y = u r r y + u θ θ y = u r sin θ + u θ r sin2 θ 2 u rθ r cos2 θ + 2 u rθ r sin θ, cos θ, cos θ sin θ + u r r sin2 θ + 2 u θ cos θ sin θ, r2 cos θ sin θ + u r r cos2 θ 2 u θ cos θ sin θ. r2 2 u x u y 2 = 2 u r u r 2 θ 2 + u r r. 25 f(x) n.,, a, x +(x a) 2 f (a) + +(x a) n f (n ) (a) 2! ξ = a + θ(x a), 0 < θ <, f (n) (ξ) (n )! +(x a)n. n! ()

29 62 2 ξ a x. (Taylor).., f (n) a, a x ξ., R n,. [] F (x) = f(x) { R n = (x a) n f (n) (ξ). (2) n! f(a) + (x a) f (a)! F (x) R n., F (x) n, F (a) = F (a) = = F (n ) (a) = 0, } + + (x a) n f (n ) (a) (n )!, (3) F (n) (x) = f (n) (x). (4) 2 F (x) G(x) = (x a) n. F (a) = 0, G(a) = 0, F (x) F (a) G(x) G(a) = F (x) (x a) n = F (x ) n(x a) n, x a x., F (a) = 0, G (a) = 0, F (x ) n(x a) n = F (x 2 ) n(n )(x 2 a) n 2, x 2 a x, a x. F (n), (4), ξ a x, F (x) (x a) n = F (n) (ξ) n! = f (n) (ξ), n! F (x) = (x a) n f (n) (ξ). n! (5) (3) F (x) (). (), n =, f(x) = f(a) + (x a)f (ξ).., 28., f (n) (x) (5), f (n) (x) a., n f (n) (a) x = a,, a n,,.

30 f (n) (a), f(x) = f(a) + (x a) f (a)! [] f(x) x = a n, x = a + (x a) 2 f (a) 2! + + (x a) n f (n) (a) n! + o(x a) n. (6) (3) F (x),, F (x) (x a) n = F (n ) (ξ) n!(ξ a), a ξ x. F (n) (a) = f (n) (a), F (n ) (a) = 0, x a ξ a,. F (n ) (ξ) F (n ) (ξ) F (n ) (a) lim = lim = F (n) (a) = f (n) (a). x a ξ a ξ a ξ a lim x a F (x) (x a) n = f (n) (a), n! F (x) = (x a) n f (n) (a) n! + o(x a) n, 28, f (n) (x), f (n) (x) x = a, R n (2), 29., f (n) (a), n =, f(x) = f(a) + (x a)f (a) + o(x a). f (a), (f (a)!). 29.,, f (a) f (i) (a) (x a) 2 (x a) n f(x) = f(a) + A (x a) + A A n + o(x a) n 2 n! A i... a x < b 28 29, f (i) (a). b < x a,.. [] y = f(x) x x ( x 0), y y = f(x) = f(x + x) f(x) y (difference). y x, x,

31 64 2 y, 2 y., n 2 y = f(x + x) f(x) = {f(x + 2 x) f(x + x)} {f(x + x) f(x)} = f(x + 2 x) 2f(x + x) + f(x). n y = n f(x + x) n f(x) ( ) n = f(x + n x) f(x + (n ) x) ( ) n + f(x + (n 2) x) + + ( ) n f(x). (7) 2, g(x) = ax n + n, x = h, g(x) = nahx n +, 2 g(x) = n(n )ah 2 x n 2 +, n g(x) = n!ah n, n+ g(x) = 0. (8) 29, x, n f(x + k x) = (k x) v f (v) (x) + o( x) n. v! (7), n y = k=0 = v=0 n ( ) n ( ) n k k v x v f (v) (x) + o( x) n k v! ( n x v f (v) (x) n ( ) n ( ) )k n k v + o( x) n. (9) v! k k,v=0 v=0 k=0 n ( ) { n 0 (v = 0,,, n ), ( ) k k v = k ( ) n n! (v = n). y = x n (9)., (8), n y = n! x n, n y = ( x) n f (n) (x) + o( x) n, n y lim x 0 x n = f (n) (x). y x f (x), 29 f (n) (x), lim n y x n, f (n) (x)., f (n) (x).

32 f(x, y) n, A = (x, y), h, k, B = (x + h, y + k) AB, F (t) = f(x + ht, y + kt) 0 t ( AB ) t, 28. ( ) F (t) = h x + k f(x + ht, y + kt),, y ( ) n F (n) (t) = h x + k f(x + ht, y + kt). y F (t) = F (0) + tf (0) + + t =, tn (n )! F (n ) (0) + tn n! F (n) (θt), 0 < θ <. f(x + h, y + k) = f(x, y) + df(x, y) + 2 d2 f(x, y) + d v f 24, + (n )! dn f(x, y) + n! dn f(x + θh, y + θk). df(x, y) = hf x (x, y) + kf y (x, y), d 2 f(x, y) = h 2 f xx (x, y) + 2hkf xy (x, y) + k 2 f yy (x, y),., () x + θh, y + θk x, y., n =, f(x + h, y + k) f(x, y) = hf x (x + θh, y + θk) + kf y (x + θh, y + θk). (x + θh, y + θk) AB.. 29, A = (x, y) n, f(x + h, y + k) = f(x, y) + df(x, y) + d2 f(x, y) 2! ρ = h 2 + k dn f(x, y) n! + oρ n,, F (t) = f (x + hρ t, y + kρ t ), F (t) = F (0) + tf (0) + + tn F (n) (0) n! + ot n

33 66 2 t = ρ., ot n /t n AB () , f(x), x, f(x) = lim lim R n = 0, n n v=0,, n (x a) v f (v) (a) v! f(x) = f(a) + (x a) f (a) + (x a) 2 f (a) + + (x a) n f (n) (a) +. (0)! 2! n! f(x)., a = 0, (Maclaurin).., 28. ( 5 ),. [ ] f(x) n, f (n+) (x) = 0, (0). f(x) (x a). [ 2] f(x) = e x. n f (n) (x) = e x. a = 0, f (n) (0) =, x, lim x n n! R n = xn n! eθx, 0 < θ <. R n < x n n! e x. = 0 ( 4 [ 3]). < x <, e x = + x! + x2 2! + + xn n! +., x =, e = +! + 2! + + n! + R n+, () e e θ R n+ = (n + )! < 3 (n + )!. ( e lim + n) n,,., () 7. n = 0, n! e, n 3 0 7,

34 26 67 R < 3! = 3 0! < 0 7,. e = ! + 2! = 2.5 = ! = ! = ! = ! = ! = ! = ! = ! e e., (), e, e = m,, m, n. n!e n. n!r n+ = eθ > 0 (0 < θ < ) n + eθ n + < 3 n +, n + < 3, n < 2, n =. e= m, e. 2 < e < 3,. [] sin x = x x3 3! + x5 5! +, cos x = x2 2! + x4 4! +. x. R n f(x) x = x 0 f(x 0 ), x 0, f(x 0 ) x 0 x f(x) [], f(x 0 ) [], x 0 f(x)

35 68 2 []., x 0. x 0 f(x), δ, 0 < x x 0 < δ f(x) f(x 0 ) > 0. >, f(x 0 ).., f(x) f(x), ( 2 5)., (im Kleinen, local). 30 x 0 f(x). ( ) f(x) x 0, f(x) x = x 0, f (x 0 ) = 0. f (x 0 ),. (2 ) f(x) x 0, x 0 x 0. f (x) x = x 0, 2 5 f(x 0 )., x x 0, f (x) +, f(x 0 ), +, f(x 0 ). (3 ) f(x) x 0 f (x 0 ), f (x 0 ) = 0 f (x 0 ) > 0, f(x 0 ) ; f (x 0 ) = 0 f (x 0 ) < 0, f(x 0 ). [] ( ) 9, f(x) x 0, f (x 0 ), f (x 0 ) = 0. (2 ) x < x 0 f (x) > 0, f(x). x > x 0 f (x) < 0, f(x) ( 22)., f(x) x 0, f(x 0 )., f(x 0 ). (3 ) f (x 0 ) 0, f (x) x 0 ( 8 [])., f (x 0 ) = 0, f (x) x = x 0.. [] f (x 0 ) = 0,., f (x 0 ) 0, f(x) f(x 0 ) = 6 (x x 0) 3 f (x 0 ) + o(x x 0 ) 3. ( 29) x x 0,, f(x) f(x 0 ) x = x 0. f(x 0 )., f(x 0 ), x 0. (stationary), f(x) f(x 0 ) x x 0 (), f(x) x 0., f (3) (x 0 ) = 0, f (4) (x 0 ) 0, f (4) (x 0 ) 0, f(x 0 ).

36 26 69, f (x 0 ) = 0, f (x 0 ) = 0,, f (k ) (x 0 ) = 0, f (k) (x 0 ) 0, k, x 0, k, x 0. [] A, B. P c, c 2, P A B. [], P, A, B.. ( 2 6)., x A = (0, h ), B = (a, h 2 ), a > 0. P = (x, 0) x, AP + BP P. c c 2, O M., 0 x a, h 2 f(x) = + x 2 h (a x) 2 c., f(x)., f (x) = x c h 2 + x a x 2 c 2 h (a x). 2, f (x) = 0 (). f (x) [0, a] x., x. c 2 c c x h 2 + x = sin α. () 2 c a x h (a x) = sin β (2) 2 c 2 () (2), f (x) [0, a]., x = 0 f (x) < 0, x = a f (x) > 0. f (x) (0, a) 0, f(x). x = x 0 f (x) 0, (0, x 0 ) f (x) < 0, f(x), (x 0, a) f (x) > 0, f(x). f(x 0 ). x 0 sin α sin β = c c 2. [].. P 0 = (x 0, y 0 ),

37 70 2 P 0 P = (x, y) f(p ) < f(p 0 ) [ f(p ) > f(p 0 )], f(p 0 ) []., f(x 0, y 0 ), x y, f(x, y 0 ) f(x 0, y) x = x 0 y = y 0, f x (x 0, y 0 ) = 0, f y (x 0, y 0 ) = 0.., f(x, y) f(x 0, y 0 ) = 2 {a(x x 0) 2 + 2b(x x 0 )(y y 0 ) + c(y y 0 ) 2 } + oρ 2. (3), a = f xx (x 0, y 0 ), b = f xy (x 0, y 0 ), c = f yy (x 0, y 0 ), ρ = (x x 0 ) 2 + (y y 0 ) 2. ρ, (3) ax 2 + 2bXY + cy 2 (X = x x 0, Y = y y 0 ).. ( ) ac b 2 > 0.. a (c ),. f(p 0 ), f(p 0 ). (2 ) ac b 2 < 0.., P 0, f(p ) > f(p 0 ) f(p ) < f(p 0 ), f(p 0 ). (3 ) ac b 2 = 0..,,.,, (x 0, y 0 ) = (0, 0), y 2 ( a = 0, b = 0, c = ),. [ ] z = y 2. (0, 0) z 0. (x, 0) z = 0,. [ 2] z = y 2 + x 4. (0, 0). [ 3] z = y 2 x 3. (0, 0). 2 7 (x, y) z < 0, z > 0. [ 4] z = (y x 2 )(y 2x 2 )., 2 8., (x, y) (0, 0) z., (0, 0).. f(x, x 2,, x n ), A = (a, a 2,, a n ), f x = 0, f x2 = 0,, f xn = 0., A, f xix j (A) = a ij, f(a + ξ,, a n + ξ n ) f(a,, a n ) = 2 Q(ξ, ξ 2,, ξ n ) + oρ 2,

38 z = y 2 x z = (y x 2 )(y 2x 2 ) n Q(ξ, ξ 2,, ξ n ) = a ij ξ i ξ j, i,j= ρ = n ξi 2. i= a ij a a 2 a n D = a 2 a 22 a 2n (a ij = a ji ) a n a n2 a nn A. Q,,. D a a 2 a k D k = a 2 a 22 a 2k (k =, 2,, n) a k a k2 a kk, D k, D k ( ) k. Q, f(a). D 0 D k,. D = 0,. : 304.

39 72 2,.,.,., ( 3),. (Steiner).,.,,.,.,. [ ] []. 2p, x, y, z, S, S f(x, y), f(x, y) = p(p x)(p y)(p z), z = 2p x y (4) x, y, (K) 0 < x < p, 0 < y < p, p < x + y. (5) y,, x = z. f x = 0., f y = 0 y = z, x = y = z = 2 p. (6) 3 ().,., f(x, y), (5),..,,. K, ( 2 9) [K] 0 x p, 0 y p, p x + y., 0., [K] f(x, y)., [K] f = 0, [K] 2 9 f > 0. [K]. [K], (K) (6), (6). (6) f(x, y),, a = c = 2 3 p2, b = 3 p2, ac b 2 > 0. (6).,,.

40 26 73 [ 2] [ (Hadamard) ] a b l D = a 2 b 2 l a n b n l n, n a 2 i + b 2 i + + l 2 i = s 2 i (i =, 2,, n) (7) (, s i ). D s s 2 s n. D n 2 a,, l n. (7), n(n ). P i n (7), P i, P = (P, P 2,, P n ). D P, D = D(P ) P, P,., D, D. D = a i A i + b i B i + + l i L i,, A i, B i,, L i a i, b i,, l i, D i. (7), D b i, c i,, i k, D a i = A i + L i l i a i = A i L i a i l i = 0. A i a i = B i b i = = L i l i. a k A i + b k B i + + l k L i = 0, a i a k + b i b k + + l i l k = 0, (8) (7) (8) a,, l n, D. (7) P, P (, ).

41 74 2 (8), s D 2 = 0 s = (s s 2 s n ) 2, 0 0 s 2 n D = ±s s 2 s n. D s s 2 s n [ s s 2 s n ], []. 27,..,.. O P = (x, y, z) OP v = (x, y, z), x, y, z v (). v v, v = x 2 + y 2 + z 2. u = (x, y, z ), v = (x 2, y 2, z 2 ),. ( ) uv = x x 2 + y y 2 + z z 2,. u, v, x u, y u, z x 2 u v, y 2 v, z 2 v., u, v θ, cos θ = x x 2 + y y 2 + z z 2. u v uv = u v cos θ.. u, v, uv = 0. uu = x 2 + y 2 + z 2 = u 2.. uv = vu, (u + u 2 )v = u v + u 2 v., () (s i),. (8) s i, s k.

42 27 75 x, y, z t, OP = u = (x, y, z) t., t + t OP = u + u = (x + x, y + y, z + z), u, P P ( 2 20) u = ( x, y, z). x, y, z, t 0, ( u x t = t, y t, z ) t. u = du dt, u = (ẋ, ẏ, ż). 2 20,, d (uv) = uv + u v. dt, u ( u = ),, uu =, u u = 0., u 0, u u. (2 ) u v u = (x, y, z ), v = (x 2, y 2, z 2 ), u, v ( ) x y z, x 2 y 2 z 2 w, w = (y z 2 y 2 z, z x 2 z 2 x, x y 2 x 2 y ), w u, v, u v.., w = (x, y, z), xx + yy + zz = 0, xx 2 + yy 2 + zz 2 = 0, wu = 0, wv = 0,,.

43 76 2 w u v ( 2 2). x y z x y z = x 2 + y 2 + z 2 = w 2 x 2 y 2 z 2 u, v, w., u, v, w (). w 2, w u, v., u, v, u v = 0.,, 2 2 u v = v u., (u + u 2 ) v = u v + u 2 v, v (u + u 2 ) = v u + v u 2., u, v t, d (u v) = u v + u v. dt (3 ), (u, v, w) u = (x, y, z ), v = (x 2, y 2, z 2 ), w = (x 3, y 3, z 3 ),, x y z (u, v, w) = x 2 y 2 z 2. x 3 y 3 z 3 i, j, k,., i 2 = j 2 = k 2 =, i i = j j = k k = 0, ij = ji = 0, i j = k = j i, jk = kj = 0, j k = i = k j, ki = ik = 0, k i = j = i k, (i, j, k) =. C t., C P = (x, y, z) x, y, z t, P OP v, v = (x, y, z) t. C t + δt P = (x + δx, y + δy, z + δz), OP

44 27 77 v + δv., δv P P, δv = (δx, δy, δz). x, y, z, ( 29), δx = ẋδt + ẍ δt x δt3 6 + oδt3, δy, δz, δv = vδt + v δt v δt3 6 + oδt3. () v = (ẋ, ẏ, ż), v,... v, o δt 0 o 0. () v, v,... v C P. v = (ẋ, ẏ, ż) v = ẋ 2 + ẏ 2 + ż 2, C P cos α = cos β = cos γ ẋ ẏ ż = v., v = 0 ẋ, ẏ, ż 0 ().., t C s, s, δv = v δs2 δs3 δs + v + v oδs3. (2), P P P P P P 0 dx dy = cos α, ds δx 2 + δy 2 + δz 2 δs 2. dz = cos β, ds = cos γ. ds x 2 + y 2 + z 2 =, v =. s, v s. v =, v 0, v v (). P v C (principal normal), v, v (osculating plane)., P l(x x) + m(y y) + n(z z) = 0 (l 2 + m 2 + n 2 = ),,, C P.

45 78 2 p = (l, m, n). P = (x+δx, y +δy, z + δz) lδx + mδy + nδz, p δv,, δv = (δx, δy, δz). (2), pv δs2 δs + pv 2 + oδs2, p v, v (pv = 0, pv = 0). v, v, δs 2.. P P δα, δα v v + δv ( 2 22). v, δs 0, δv δs v, δα δv. δv δs v. δα δs v, dα ds = v dα, C, P, ds ρ, (d ρ = dα 2 ) 2 ( ds = x d 2 ) 2 ( y d 2 ) 2 z v = ds 2 + ds 2 + ds 2. (3) P (binormal). i, j, k,, i = v j = ρv k = i j k = i j + i j. i = v j, i j = 0, k = i j. k i. k =, k k. k j. ( k 0.) v, k s, (4)

46 s,. k j, j =, k = ± k j, k = τ j, C ().. τ P C, (i, j, k) τ,. ai + bj + ck i, j, k, j, j = k i, j = k i + k i. i = v = ρ j, k = j, j i = k, k j = i, τ j = ρ i + τ k. i, j, k, i = ρ j, j = ρ i + τ k, (5) k = τ j. (Frenet). i, j, k, s. (4), v = j. (5), ρ v = ρ ρ 2 j + ρ j = ρ 2 i ρ ρ 2 j + ρτ k. (), (v, v, v ρ ρ ρ ) = (i, j, i 2 ρ 2 j + ( = i, ) ρ j, ρτ k = ρ 2 τ s. ) ρτ k ( i, j, k ) = ρ 2 τ. x y z x y z τ = x y z ρ2 (v, v, v ) = x 2 + y 2 + z 2 (6)

47 80 2 t ρ τ., t, v = v ds dt, ( ) 2 ds v = v + v d2 s dt dt 2, (7) ( ) 3... ds v = v ds d 2 s + 3v dt dt dt 2 + v d3 s dt 3. v v,, v ( 2 23). v d2 s dt, C 2 ( ) 2 ds, dt, d2 s dt 2 ( v = ), v C, ρ ( ds dt ) 2 ( v = ρ t, v,. v 2 = ẍ 2 + ÿ 2 + z 2 = ( ) 4 ( ds d 2 ) 2 s ρ 2 + dt dt 2. ( ) 2 ds t = ẋ 2 + ẏ 2 + ż 2, dt ρ 2 (7), ( ds dt (6) (8), ) 6 ( ) 2 ds = v 2 dt ds d 2 s = ẋẍ + ẏÿ + ż z. dt dt2 ( ds dt ) 2 ( d 2 ) 2 s dt 2 ). = (ẋ 2 + ẏ 2 + ż 2 )(ẍ 2 + ÿ 2 + z 2 ) (ẋẍ + ẏÿ + ż z) ẏ ż ż ẋ ẋ ẏ = + +, ÿ z z ẍ ẍ ÿ ( v, v,... v ) = 2 23 v v = ρ v 3. (8) τ ( ) 6 ds (v, v, v ). dt = ( v, v,... v ) v v 2. (9) t, (8) (9).

48 27 8 [] x = a cos t. y = a sin t. z = ht (a > 0). ẋ = a sin t. ẏ = a cos t. ż = h. ẍ = a cos t. ÿ = a sin t. z = x = a sin t. y... = a cos t. z = 0. v = ẋ 2 + ẏ 2 + ż 2 = a 2 + h 2 v v = a 2 h 2 sin 2 t + a 2 h 2 cos 2 t + a 4 = a a 2 + h 2. ( v, v,... v ) = a 2 h. ρ = a a 2 + h 2, τ = h a 2 + h 2.. τ h, ( 2 24) z = 0,, δv = v δs2 δs + v 2 + oδs2. v =, θ x ( ), v = dθ ds s,.,, ρ = dθ ds, ρ. s, ( v = (x, y ) = v = (x, y ) = (cos θ, sin θ), sin θ dθ ) dθ, cos θ = ( ) sin θ, cos θ = ds ds ρ ρ ( y, x ). (0)

49 82 2,, (7), ẋ ẏ ẍ ÿ = x y ( ) 3 ds x y = x y dt y x ρ ρ (3 ), ρ = x y = y x. () ( ds dt ) 3 = ρ ( ) 3 ds. dt ẋÿ ẍẏ = ρ ṡ 3. (2) ( v, v) = ρ v 3. ( ṡ > 0, t.),,,,, x, ρ = dxd2 y d 2 xdy (dx 2 + dy 2 ) 3 2 ρ = y = f(x) ( + d 2 y dx ( 2 dy dx (3) ) 2 ) 3. (4) 2 P C, C, ρ, (ξ, η). ξ = x ρ sin θ, η = y + ρ cos θ. (5) dθ (0), ds ρ,., (4), ρ d2 y dx 2, ,. C (ξ, η) E, (5) C E., C s, E ξ = x ρy, η = y + ρx. s, (), ξ = x ρ y ρy = ρ y, η = y + ρ x + ρx = ρ x,

50 ξ x + η y = 0. C E. C E. E C ( 88). E σ, ( ) 2 ( dσ dξ = ds ds ) 2 + ( ) 2 dη = ρ 2 (x 2 + y 2 ) = ρ 2, ds σ = ±ρ. E, ρ 0 σ = ρ. ρ 0 σ 0, σ σ 0 = ρ ρ 0., E C. E, P P C ( 2 27). C E (involute), E C (evolute). C,, E C. [ ] 2 27, (cycloid).., () (). a, t, x. t = 0 P, t. x = a(t sin t), y = a( cos t). dx = a( cos t)dt, dy = a sin tdt, ds = dx 2 + dy 2 = 2a 2 ( cos t)dt = 2a sin t 2 dt, d 2 x = a sin tdt 2, d 2 y = a cos tdt 2,

51 84 2 dxd 2 y dyd 2 x = a 2 cos t sin t sin t cos t dt3 = a 2 (cos t )dt 3 = 2a 2 sin 2 t 2 dt3, ds 3 ρ = dxd 2 y dyd 2 x = 4a sin t 2, ξ = x ρ dy = a(t + sin t), ds η = y + ρ dx = a( + cos t). ds., AB, B C BC, AB ( 2 28). t = 0, t = π ρ = 0, ρ = 4a, AB a. ABC 8a [ 2],. x = a cos t, y = b sin t ρ = (a2 sin 2 t + b 2 cos 2 t) 3 2, ab ξ = a2 b 2 cos 3 t, η = a2 b 2 a b t (aξ) (bη) 2 3 = (a 2 b 2 ) 2 3. sin 3 t. 2 29, (asteroid)., (cusp) ( 86 [ 2]). E,. [ 3]., ξ = cos t, η = sin t 2 29

52 85, ( ) x = cos t + t sin t, y = sin t t cos t () f(x) [a, ) lim x f(x) = f(a), ξ, ξ > a, f (ξ) = 0. (.) d n (2) a > 0, dx n ( + x 2 ) = P n(x), Pn(x) n, n, a ( + x 2 ) a+n P n (x). [] (). (3) [ ] dn dx n e x 2 = ( ) n H n(x)e x2, H n(x) n [ (Hermite) ]. H n(x). [2 ] e x d n dx n xn e x = L n(x) [ (Laguerre) ]. (4) f(x) (a, b) n, x a+0 f(x) l, f (x) l,, f n (x) l n. f(a) = l,, f (a) = l,, f n (a) = l n. [] 23. (5) f x, f y, (a, b) f xy ( (a, b) ). f xy(a, b) = lim y b f xy(a, y) = lim y b f yx(a, y), f yx(a, b) = lim f x a xy(x, b) = lim f x a yx(x, b)., lim, lim. [] ( 23 []).

53 86 2 (6) f(x, y) = x 2 Arctan y x y2 Arctan x x = y = 0. ( x = 0 y = 0 y f(x, y) lim x 0 lim y 0.) []. f xy(0, 0) =, f yx(0, 0) =. (7). F (u) u = ϕ(x), d n n! dx F (u) = n n X k= X «i!i 2! i F (k) ϕ i «ϕ i2 «ϕ (n) in (u). n!! 2! n! i, i : i 0, i 2 0,, i n 0, i + i i n = k, i + 2i ni n = n., 0! =. [] F (u), u u 0 = ϕ(x) ϕ(x 0) ϕ (x x 0) n. (8) [a, b] f (x) > 0, f(a) > 0, f(b) < 0, a = a f(a) f (a), f(a) a2 = a f (a, ), a < a 2 < < a n < f(x) = 0 [a, b] (). f (x) < 0 f(x) f(x). f(a) < 0, f(b) > 0, b = b f(b) f (b), [] f (x),. ξ. f(x), a i ξ. lim a n = λ, f (a ) < f (a 2) < < f (λ) < 0, λ = ξ. a n+ = a n f(an) f(λ), λ = λ, f(λ) = 0. f (a n) f (λ) [], 0 = f(ξ) = f(a n) + f (a n)(ξ a n) + f (µ) (ξ a n) 2, f (a n) 2, ξ a n+ = f (µ) 2 f (a (ξ n) an)2 < f (µ) 2 f (a (b n) an)2., a n+ ξ, (a n < µ < ξ). [], cos x = x. f(x) = x cos x, f (x) = sin x +, f (x) = cos x., [0, π/2]. f(0) < 0, f(π/2) > 0, b, b,. cos , a = , b = cos a = , cos b = f(a) = , f(b) =

54 87 a b ξ, (sin b = ) b = b ξ. b ξ < 2 (b a)2. ( []) (9) f(x) = P (x)/q(x), a lim x a f(x) =. (0) f(x, y) = x 4 + y 4 + 6x 2 y 2 2y 2 [ ]. [2 ] f(x, y) < 0. () ABC,.. [] () 20, ,. [],.