IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (

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1 IA 2013 : :10722 : 2 : :2 :761 : ) : : / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! = 1

2 e.g. 0 = , π = , 1 7 = n {}}{ n {}}{ = 1 10 n {}}{ < 1 10 n n n = 0 1 = real number) )real line) 2.2 notation, symbols) R: Q: rational number) m n ratio ) Z: integer) 0, 1, -1, 2, -2...) 2.3 A A e.g. 2 = { 2 } = {x R x > 2} e.g. e.g. { 12 } { x R 12 } e.g. 0.1 Z Q R 2

3 ˆ ) ˆ 2 2 x < y x + z < y + z x < y, z > 0 zx < zy x, y x > y z < 0, x < y zx > zy * e.g. x > 0 x < 0 * x * 0 = x + x > x + 0 = x Q: Z: C: complex number) e.g. 2 n / Q % e.g. 2 2 = = N 3

4 2.5 - fx) ) a n a f a) = lim n 0 f a n ) f a) f a n ) = 1 n e.g = ) ε > 0 ε ) N > 0 Z n < N f a) f a n ) < ε N f a ε ) 1. a n a ) ε > 0 ε ) N n > N a a n < ε e.g. a a n a n Q) a a n < 1 10 n ε > 0 ε > 1 10 N > 1 N a a n < ε 2. a R f x) a x a f x) f a) ) ε > 0 δ > 0f a ε ) x a < δ e.g. f x) = x 2, a = 2 = f x) f a) < ε f x) f a) = x 2 a 2 = x + a) x a) 1.4 < a < 1.5, 1.4 < x < 1.5 a x < ) ε > 0 a x < ε f x) f a) < ε 3 δ = ε a ) 3 ε δ ε δ e.g. e x = 1 + x + x xn n! +... e = e e = n! +... e ! e.g. sin1 = 1 1 3! + 1 5! 1 7! !

5 2.6 ) continuity of the real numbers) cut of the real numbers) A, B < R 1. a A, b B a < b A B = ϕ 2. A B = R A<B A<B 1. A A x x aa B ) 2. B B x b xb A ) 1 A = { x Q x < 2 }, B = { x Q x > 2 } A B = Q, A B = ϕ, A < B A B 2 Z A, B, A = {x Z x < 0}, B = {x Z x 0} A =-1, B =0 2.7 A R A =A M =a A a M) A / least upper bound supremum) sup A supremum of A) A A / greatest lower bound inmum) inf A sequence) a 1, a 2, a 3 N = {0, 1, 2, 3 } = { } N a n) a n) = a n 5

6 2.8 a n), {a n } monotone increasing) m n a m a n ) a n = n 2) a n = a n) = a n a n a ε > 0 N n N a n a < ε {a n } ) a N a N = a N+1 = a N+2 = ) 1 ) a m a m < a n 2 ) a = a N ) a a > a n n A a n sup A = a a ε > 0 a ε < a A a N a ε < a N a N+1 a n a n N a n a a N a < a N a ε) = ε a n 6

7 2.9 subsequence) ν : N N n < m ν m) < ν n) b n = a νn) ) ν n) = 2 n a n = a 1 n + 1 b n = a 2 n = n + 1 2) ν n) = n a n 2.10 Bolzano-Weierstrass {a n } {a n } { b n = a νn) } b ν 0) < ν 1) < ν 2) < ) 2.11 accummulation point) {a n } c R {a n } ε > 0 a n c < ε n ) {a n } c c {a n } 2) a n = 1) n 1 1 ) 1, 1 a n n + 1 3) a n = sinn [ 1, 1] 4) a n = 1) n 1, B = { x R a n x n } A = { x R a n x n } 7

8 R = A B x y y A y x y B B c c B, c / B ) c c ε, c + ε) a n c < c + ε B c + ε a n n ε n = 1 2 n 0 c ε n, c + ε n ) a n a n c ε n, c + ε n ) n c a n < 1 2 n a n c limsup {a n } superior limit) liminf {a n }inferior limit) a n = n ) 2.12 series) {a n } S n = n k=0 a k S n k=0 a k a n ) n=0 x n n! S n = n k=0 x k k! 8

9 S n e x ) 2) 1) n x 2n n=0 2n)! k=0 1) k x 2k 2k)! S n cosx) {S n } a 0 = S 0 a 1 = S 1 S 0 a 2 = S 2 S 1. S n = n k=0 a k S n dierence sequence) n=0 a n 0 n a n 0 S n = n k=0 a k {S n } n S n+1 S n a n+1 = S n+1 S n series with nonnegative terms) a n 0 n = 0, 1, 2 ) n=0 a n S n = n k=0 a k {a n } 9

10 2.14 absolutely conuergent) n=0 a n n=0 a n n=0 a n n=0 a n n=0 a n n=0 a n ) n=0 a n 2.15 Cauchy {a n } ε > 0 N n, m N a m a n < ε c = lima n n, m N) a m c < ε 2, a n c < ε 2 a n a m a n c + a m c < ε a N a N a M < ε n N ) a n a N ε, a N + ε) {a n } {b n } c c a n c b n + b n a n b n = a νn ) ν n ) n n, n N b n a n < ε 2, c b n < ε 2 c a n < ε a n c 10

11 {a n } a n a n a ε > 0 N n N a n a < ε ) {a n } Cauchy ε > 0 N m, n N a n a m < ε 2.16 {S n = n k=0 a k} n=0 b n n=0 c n 2.17 n=0 a n {a n }, {a n} 2 n a n a n 1. n=0 a n < + n=0 a n ) n=0 a n n=0 a n 2. n=0 a n a n + ) n=0 a n n=0 a n a n < cr n, 0 r < 1, c n=0 a n 2. l 2, f n) 0 n 0) f x) = x l + a k 1 x l a 0 1 l 1 n=0 f n) 1 n=0 x l l 2 11

12 n k=0 crk = c 1 r ) n+1 1 r r < 1 r n+1 0 n=0 crn = c 1 r 1 n + 1) n + l 1) 0 2. S n = 1 1 n + 2) n + l 1) n ) = n + l a n = S n+1 S n = n k=0 a k = S n+1 S 0 k=0 a k = S 0 1 n=0 n + 1) n + l) a n = a n+n 1 n + N + 1) n + N + l) fx) l x l = 1 ) N f n) n + N + 1) n + N + l) l 1 n + 1) n + l) 3 A, B f : A B mapping) A )a fa) B ) A = R 3 = { x, y, z) x, y, z }) B = R f x x, y, z) = x 2. A R 3 A = { at + a, bt + b, ct + c ) R 3 t R } B = R 3 A at 0 + a, bt 0 + b, ct 0 + c ) B f:a B 3. A: id: A A ida)=a identity mapping) 3.2 f: A B B = R, C f function) real-valued) complex-valued)) 12

13 3.2.1 A,B: f: A B 1. f injection)/1:1 one-to-one mapping) a a A f a) f a ) 2. f surjection)/ onto mapping) fa) B b B a A f a) = b 3. f f R 3 R, x, y, z) x : ) 2. A= B=R 3 x, y, z) x, y, z) A B 3. id: A A A,B,C 3 f: A B g: B C g f: A C a A) g f x)) C) f g compsite mapping) f, g g f 2. f, g g f 13

14 a a A f a) f a ) B f ) g f a)) g f a )) C g ) g f) a) g f) a ) g f 2. c C b g b) = c g ) a f a) = b f ) g f a)) = g b) = c g f f: A B g: B A 1 g f = id : A A g g f inverse mapping) g = f g b B g b) f f a) = b a f a) = b a g b) = a g f) a) = a g f = id f g) b) = b f g = id : B B a = g b) f a) = b g 1. a A, a = g f) a), b = f a) a = g b) g 2. b b g b) g b ) g b) = g b ) = a b = f a) = b 14

15 3.4.3 A = a, a ) R : a, a ) = {x a < x < a } B = b, b ) R f: A B x < y f x) < f y) strictly monotone increasing) f g = f 1 : B A a > 1 f x) = a x f : R 0, + ) = {x R x > 0} g : 0, + ) R g = log a a x a n = n {}}{ a a a a n m = m a n a x = lim x x a x x Q, x R) { x ax x a x a n + n ) a = 1 + h h > 0 a + h) n b b = ε ε > 0) b ε < 1 + h) n n b < b ε < 1 + h)n 1 + h 15

16 b < 1 + h) n f a A a ) continuous) x a fx) fa) >0 >0 x a < δ f x) f a) < ε fx) x=a >0 x a < δ f x) f a) ε x x x 0 1. f x) = 1 x = 0 x=0 0 x Q 2. f x) = 1 x Q fx) a A 2. a A a i A f a i ) f a) {a i } a i a i a i a < δ f a f a i ) f a) < ε >0 x a < 1 2 i f x) f a) ε x x a i 16

17 a i a < 1 2 i 0 {a i } a f a i ) f a) ε f a i ) f a) : a i a f a i ) f a) a i x / Q f a) = 1 x Q a i = 1 i f a i ) = 1 f 0) = 1 2 a i = i + 1 a i 0 f a i ) = 0 f 0) = A = R, x n, x, sin x f: A R) R A ) continuous function on A)) a A A R f, g A a ) 1. f ± g) x) = f x) ± g x) a ) 17

18 2. f x) g x) a ) 3. f x) > 0 1 a ) f x) f, g a 1. x a < δ f x) f a) < ε 2, g x) g a) < ε 2 f x) + g x) f a) g a) f x) f a) + g x) g a) < ε 2. f a) α > 0, g a) β > 0 f x) g x) f a) g a) = f x) g x) f x) g a) + f x) g a) f a) g a) f x) g x) g a) + g a) f x) f a) ε, ε δ ) x a < δ f x) f a) < ε, g x) g a) < ε f x) α f x) g x) f a) g a) < α ε + βε ε > 0 ε = ε 2α, ε = ε 2β 3. α = f a) > 0 x a f x) > α 2 1 f x) 1 f x) f a) f a) = f x) f a) f x) f a) 2 < α 2 α = f x) f a) α2 x a < δ f x) f a) < α2 2 ε 1 f x) 1 f a) < ε R ) 2. A R f x), g x) f x) > 0 x A g x) f x) A f x) = c f x) = x x 18

19 2. 1 f x) g x) f x) = g x) 1 f x) A [a, b] R fx) [a, b] fx) maximum) minimum) α, β [a, b] x [a, b] f α) f x) f β) B ) [a,b] fx) [a,b] f a) < f b) y R f a) < y < f b) f x) = y x [a, b] A B A R f:a f B) = {f x) x B} f [a, b]) R f[a,b]) c n [a, b] f c n ) [a, b] {c n } c R 19

20 c n c a c n b a c b c [a, b] f c n ) + c n c f c n ) f c) f [a, b]) f [a, b]) R 1. f x) = y f [a, b]) f β 2. f [a, b]) d n = f c n ) d n β {c n } [a, b]) c [a, b] 1. a,b) fx)=x 2. a,+ ) fx)=x B A [a, b] A = {x [a, b] f x) y} A a, A b [a, b] A = B b A [a, b] A c [a, b] f c) = y {c n } A, c n c 20

21 1. c n A f c n ) y f c) y 2. c<b c = b c n b f b) > y f c) = f b) y 3. B d n c d > c d B) f d n ) y f d) = lim f d n ) y A+ B f: [a, b] f [a, b]) = [α, β] f image) ) ) f α [a, b] g f x) [c, d] g f x)) = g f) x) 21

22 <0 g fx) ' y f x) < δ g y) g f x)) < ε f '>0 x α < δ f x) f α) < δ g f x)) g f x)) < ε [a,b] c, d [a, b] f c) f x) f d) x [a, b] f: [a, b] f f x<y fx)<fy) x y fx) fy) f f c, d, e [a, b] i) f c) < f d) > f e) ii) f c) > f d) < f e) i) y f c), f e) < f d) f [c,d] f x 1 ) = y c < x 1 < d f [d,e] f x 2 ) = y d < x 2 < e 22

23 f x 1 ) = f x 2 ) a < x 1 < d < x 2 < b) : i) cos x R cos x + δ) cos x) = cos x cos δ sin x sin δ = cos δ 1) cos x sin δ sin x ii) t>1 m t m m>0 t m m t m t f x) = t m 1, + ) t t m s s 1 m [1, + ] 1, + ) t 1 m : [a, b] [c, d] c a, b) f: a,b) f c dierenciable at c) f x) f c) x c α x c c x c α ε < f x) f c) α x c) f x) f)c x c < α + ε 23

24 ε > 0 min {α + ε) x c), α ε) x c)} f x) f c) max {α + ε) x c), α ε) x c)} x c x=c f x) x = c α x c) + f c) α = f c) = df c) dirivative) dx f,g f ± g, fg d f x) g x)) c) = f c) g c) + g c) f c) dx ) f x) d g x) = g x) f x) f x) g x) dx g x) f: a,b) f: a,b) c,d) g: c,d) R h x) = g f) x) = g f x)) f e a, b) g f e) c, d) h x=e h e) = f e) g f e)) >0 α ε) x e) f x) f e) α ± ε) x e) x-e 0 β = g e) β e) f x) f e)) g f x)) g f e)) β ± e) f x) f e)) α ± ε) β ± ε) x e) g f x)) g f e)) α ± ε) β ± ε) x e) h x) h e) = g f x)) g f x)) αβ x e) 24

25 h = e sin x ) = cos xe sin x e y ) = e y y = sin x sin x) = cos x f ) f 1 = g ) g f x)) = x 1 = dx dx = g f x)) f x) f: f: a,b) c,d) ) g = f 1 : c,d) a,b) e a, b) f e) 0 g fe) g f e)) = 1 f e) α ± ε) x e) f x) f e) α ± ε) x e) f x) f e) α ± ε f x) = y f e) = ê x = g y) e = g ê) x e y ê y ê g y) g ê) α ± ε α ± ε f x) f e) α ± ε g ê 1 α 25

26 f x) = e x = y e x ) = e x g y) = log y g y) = 1 e x = 1 y log x) = 1 x f = sin : g = arcsin g y) = arcsin x) = π 2, π ) 1, 1) 2 1 sin x) = 1 cos x = 1 1 = 1 sin x 1 y x f = tan : π 2, π ), + ) = R 2 g = f 1 = arctan y = tan x = f x) g y) = 1 f x) tan x) = g y) = y 2 arctan x) = x 2 ) sin x cos x cos x sin x sin x) = cos x cos 2 = 1 + tan x) 2 = 1 + y 2 x Rolle f: [a, b] a, b) f a) = f b) c a, b) f'c)=0 26

27 f x) = f a) = f b) f a) = 0 2. f x) f a) = f b) f x) f a) = f b) max f x) > f a) = f b) f c) f x) f c) 0 x<c x>c f x) f c) x c f x) f c) x c 0 0 x c f'c) f c) 0, f c) 0 f c) = Cauchy f x), g x) : [a,b] a,b) g b) g a),x a, b) g x) g a) 0 g x) 0 f b) f a) g b) g a) = f c) g c) c a, b) F x) = g b) g a)) f x) f a)) f b) f a)) g x) g a)) F a) = 0, F b) = 0 Rolle F c) = 0 g b) g a)) f c) f b) f a)) g c) = L'h pital f x), g x) x=c f x), g x) g c) 0 f x) f c) lim x c g x) g c) = f c) g c) 27

28 b>c f b) f c) g b) g c) = f e) g e) f c) g c) b c g f f e) f c), g e) g c) a<e<c f a) f c) g a) g c) f c) f e) = g c) g e) = f e) g e) f c) g c) fx) a,b) f'x) f x) = x 2 sin 1 x 0) x2 x 0 x=0 x 2 f x) x 2 fx) x=0 f'0)=0 f x) = x 2 sin 1x 2 ) = 2x sin 1x 2 + x2 sin 1 x 2 ) = 2x sin 1 x 2 2 x cos 1 x 2 sin 1 ) x 2 = cos 1x ) 2 2x ) fx) x=c f'c)>0 >0 c δ < x c f x) < f c) c < x < c + δ f x) > f c) 28

29 3.8.2 f x) f c) = f c) x c)x c ) c fx) fx) a,b) f'x) f'c)>0 x=c fx) f'x) fc)>0 x c δ, c + δ) f x) > 0 α, β c δ, c + δ) α < β α < γ < β f β) > f α) f β) f α) β α = f γ) > 0 f x) = x + x 2 sin 2 1 x 2 f 0) = 0 f'x) f 0) = 1 > 0 f x) = 1 + 2x sin 1 x 2 2 x cos 1 x 2 x=0 f'x)>0 x x=0 29

30 3.8.5 Taylor f: n+1 x a ) f x) = f a) + f a) x a) + + f n) a) x a) n + f n+1) y) x a) n+1 y a x f C n+1 f n+1) y) = f n+1) a) n Taylor f x) < f a) + ε) x a n+1 n + 1)! f x) n n f k) a) k=0 x a)k k! Taylor f : a, b) c, d) g : c, d) R h x) = g f x)) f,g: n + 1 x = α, y = β = f α) ) h + n + 1 f x) n Taylor f α) + f α) x α) + + f n) α) x α)n n! g x) n Taylor h = g f x)) Taylor g β) + g β) y β) + + g n) β) y β)n n! y β x α) f α) + f α) x α 2 ) n) x α)n f n! x α) n+1 30

31 3.8.7 f x) f x) 0 f x) > x: f x): f x): velocity) ) speed) f x): acceleration) f x) 0 f x) g = f x) complex) f x) > 0 f x) g = f x) y = f x) a < b < c f b) f a) b a f c) f b) c b a, f a)) c, f c)) b, f b)) f x) 0 f x) a < b < c 31

32 f b) f a) = f α) b a a < α < b f c) f b) c b = f γ) b < γ < c α < γ f α) f γ) y = f x) concave f x) C 2 f a) = 0 f a) > 0 f a) f x) x a x a f x) > f a) f x) f x) f a) x a = f y) y x a a < x a < y < x 0 = f a) < f y) f x) f a) > 0 x < a 32

33 N p3ewton f a) = 0 a f x) = x 3 10 f a) = 0 a f x) = x 5 + ax 4 + bx 3 + cx 2 + dx + e ) f C 2 f x) a: fa)=0) f x) 0 f x) 0 f ±f x) x y = ±x f x) > 0 f x) ±f) x) = ±f x) ±f) x) = ±f x) y = x f x) = f y) = g y) g y) = f x) 33

34 g y) = f x) a a 1 >a) a 1, f a 1 )) y = f x) x a 2 a 2 a < a 2 < a 1 ) f a 1 ) a 1 a = f a 1 ) a 1 a 2 = f a 1) f a 1 ) f a 1 ) f a) a 1 a = f b 1 ) a < b 1 < a 1 a 1 a = f a 1) f a) f b 1 ) a 1 a 2 = f a 1) f a) f a 1 ) 1 0 ) a 2 a = f a 1 ) f a)) 1 = f b 2 ) a 1 a) f f b 1 ) 1 ) f a 1 ) ) b 3 ) b 1 a 1 ) = f b 2 ) f b 3 ) f b 3 ) a 1 a) a 1 b 1 ) a < b 2, b 3 < a 1 f, f f b 2 ), f b 3 ), f b 3 ) f b 2 ) f b 3 ) f b 3 ) 2 M f a) f a) M 0 a 2 a M a 1 a) 2 34

35 0 M a 2 a) M a 1 a)) 2 a 2, a 1 a 1 a 2 a 3 a 1, a 2, a 3... M a n a)) M a n 1 a)) 2 M a n 2 a)) 4 M a 1 a)) 2n 1 M a 1 a) 10 1 M a n 1) 10 2n n = = 1024 M a n a) n = n = R n = {x 1,..., x n ) x k R} n Eucliden space of dimension n) P = a 1, a 2,..., a n ) R n P = a 1,..., a n ) Q = b 1,..., b n ) P Q distance) d P, Q) = a 1 b 1 ) a n b n ) 2 P = a 1,..., a n ) P r openball) {Q = x 1,..., x n ) R n d P, Q) < r} = {x 1,..., x n ) R n x 1 a 1 ) x n a n ) 2 < r 2} 35

36 n = 2 r opendisc) ) ε > 0 p - -neiborhood) p D R n p R n p D inside D) ε > 0 p ε D p D interior point) p D o D) ε > 0 p ε D p D exterior point) p D on the boundary of D) ε > 0 p ε D R n \ D p D boundary point) p R n D R n = D R n \ D) Q R n Q D Q R n \ D Bε = p ε p {p} D {B} R n \ D) B ε D = ϕ B ε R n \ D) = ϕ p Dp D ϕ) B ε D ϕ ε > 0 B ε D ε < 0 B ε R n \ D) = ϕ p R n \ D) = ϕ R n : n P = a 1,, a n ) 36

37 Q = b 1,, b n ) d P, Q) = a 1 b 1 ) a n b n ) 2 P, Q P R n B r P ) = {Q R n d P, Q) < r} P r : P r A R n : A O = A = { Q R n ε B Q) A } A A C = R n \ A A C ) O A C A compliment of A) A = { Q R n ε > 0 B ε Q) A ϕ, B ε Q) A C ϕ } A A O ) O = A O p A O B ε P ) A, B ε/x P ) A B ε/e Q) B ε P ) A Q A O B) B n = 1 Q R Q = R Q) = ϕ 37

38 R n = A O A C) O A A O A A C ) O A C A O A A O A A = A O A: A C ) A = A O A A: A C) O = A C, A C ) 3.9 {P k } k = 0, 1, 2, ) sequence) P k R n {P k } P R n d P k, P ) 0 P k = a k1, a kn ) R n P = a 1,, a n ) a kj a j 0 j = 1,, n ) {P k } A P k A 38

39 3.9.1 {P k } P R n P A O A ) P A C) O ε > 0 B ε P ) A C P k A d P k, P ) > ε P k P P A O A P A {P k } ) ε > 0 B ε P ) A ϕ ε 1 > ε 2 > > 0 B εk P ) A P k A d P k, P ) < ε k 0 P k P A A {P k } P P A R n d P k, 0) M 39

40 P k a jk a kj M a 2 kj nm a 2 kj M a kj M Bolzano-Weierstrass) R n ) n n = 1 n = 1 P k = P k, a k ) a k a R a k a P k : P k P P k P, a k a P k P = P, a) A A {P k } A A sequentially compact) A {P k } A A R n 40

41 A A : : ) 1. P k A P k + P k : 2. A A k A A C P k B εk b) A ϕ ε k 0 {P k } A P k b / A A R n A A P f P ) R f P A ε > 0 δ > 0 d Q, P ) < d f Q) f P ) < ε A R n ) f A f P 1, P 2 A Q A f P 1 ) f Q) f P 2 ) 41

42 f f Q) f Q 1, Q 2 Q 3, A f Q k ) + Q k Q A f Q k ) f Q) = sup f Q) = α Q A inf f Q) = β Q A sup f Q 1 ) α Q q A Q 1 P 2 A f Q 1 ) f P 2 ) = α α A R n f: A P A f P ) dierentiable) f P P = a 1,, a n ), Q = x 1,, x n ) Q α 1,, α n R 42

43 f x 1,, x n ) f a 1,, a n ) + α 1 x 1 a 1 ) + + α n x n a n ) α k = f x k a 1,, a n ) ) n f x 1,, x n ) = f a 1., a n ) + α k x k a k ) + g x 1,, x n ) k=1 g Q) d P, Q) 0 d P, Q) 0 ) ) d P, Q) max x 1 a 1 d P, Q) n d P, Q) 0 f x 1,, x n ) f a 1,, a n ) = df x 1,, x n ) x k a k = dx k df x 1,, x n ) = f x 1 a 1,, a n ) dx f x n a 1,, a n ) n = 2, e xy P = 1, 1) y = 1 y e x e + e x 1) x = 1 y e + e y + 1) 43

44 n = 2 f x, y) P = a, b) x partially dierentialable in x) f a, b) x x = a y f a, b) y = b f a, b) A x f x a 1,, a n ) P x x 1,, x n ) f A C f A n = 2 f x, y) f a, b) = f x, y) f x, b) + f x, b) f a, b) f x, y) f x, b) y f x, y) f a, b) = f x, c) y b) y c y b x x a f x, b) f a, b) = f x x, b) x a) f x, y) f a, b) = f f f a, b) x a) + a, b) y b) + x y x x, b) f ) ) f f a, b) x a) + x, c) a, b) y b) x y y 44

45 x a < d P, Q) y b < d P, Q) f x x, b) f a, b) x 0 f y x, c) f y a, b) 0 dp,q) 0 P f x, y) f a, b) α x a) + β y b) C 3.10 Taylor df = f f x dx + y dy 2 ) df = n f i=1 x i dx i n ) x 1,, x n ) = a 1,, a n ) f x 1,, x n ) f a 1,, a n ) = f x i a 1,, a n ) x 1 a 1 ) + + f x n a 1,, a n ) x n a n ) + d P, P 0 ) 0 f x i f C 1 ) f n = 2 ) A R n : x 1,, x n ) B R m : y 1,, y m ) 45

46 R l : z 1,, z l ) F : A B F = f 1 x 1,, x n ),, f m x 1,, x n )) F f i ) f i C 1 f i C 1 G: B R l G = g 1 y 1,, y m ),, g l y 1,, y m )) H x 1,, x n ) = G F x 1,, x n )) = G F x 1,, x n )) H x 1,, x n ) = h 1 x 1,, x n ),, h l x 1,, x n )) h i x 1,, x n ) = g i f 1 x 1,, x n ),, f m x 1,, x n )) ) F P = a 1,, a n ) G F P ) = b 1,, b m ) H = G F a 1,, a n ) h i x j a 1,, a n ) = m k=1 f k x j a 1,, a n ) g i y k b 1,, b m ) 2) F, G C 1 H = G F C n = 1 f: A R) B R) 46

47 g: B R h = g f x)) dh dx = dg df dy dx f x) f a) = df dx a) x a) + g y) g b) = dg dy b) y b) + h x) h a) = g f x)) g f a)) = dg dy = dg dy b) b) f x) f a)) + ) df a) + + dx x a 0 h x) h a) = dg df b) a) x a) + dy dx ) 2) C 1 k i x j x) = f k x j x) g i y k F x)) k i x j x) C 1 h i x 1,, x n ) h i a 1,, a n ) = g i f 1 x 1,, x n ),, f m x 1,, x n )) g i f 1 a 1,, a n ),, f m a 1,, a n )) = m g i b 1,, b m ) y k b k ) y k k=1 f k x 1,, x m ) f k a 1,, a n ) m j=1 f k x j a 1,, a n ) x j a j ) h i x) h i a) = m n k=1 j=1 n m j=1 k=1 g i g k b) f k x j a) x j a j ) f k x j a) g i y k b) ) x j a j ) 47

48 dh i = n m ) f k a) g i b) dx j x j y k j=1 k=1 = h i x j a) x = x 1,, x n ) f f df x) = dx dx n x 1 x n = ) dx 1 dx n f x 1.. f x n ) df 1 df m = ) dx 1 dx n f 1 x 1. f 1 x n f m x 1. f m x n ) dg 1 dg l = ) dy 1 dy m g 1 y 1. g 1 y m g l y 1. g l y m y i = f i dh 1 dh l ) = = = dg 1 dg l ) ) df 1 df m ) dx 1 dx n g 1 y 1. g 1 y m f 1 x 1. f 1 x n g l y 1. g l y m f m x 1. f m x n g 1 y 1. g 1 y m g l y 1. g l y m dh h x) h a) = g f x)) g f a)) = g y) g b) = dg x = r cos θ, y = r sin θ k r, θ) = g r cos θ, r sin θ) 48

49 h r = h r = x h r x + y h r y = cos θ) h x + sin θ) h y h θ = h θ = x θ h x + y θ h y = r sin θ) h x + r cos θ) h y cos θ) h r sin θ) h θ = h x r sin θ) h r cos θ) h θ = h y r x = cos θ r sin θ r θ y = sin θ r cos θ r θ r = cos θ x + sin θ y = x x2 + y 2 x + y x2 + y 2 y = r sin θ θ x + r cos θ y = x x + y y 3.11 f x 1,, x n ) C 1 f x 1,, f x n f x i C 1 C 2 n C n f C 2 x j ) f x i = ) f x i x j C n n 49

50 n = 2 y ) f x = x ) f y f x, y) f a, b) = f x, y) f x, b) + f x, b) f a, b) ) ) f f = y x, y ) y b) + x x, b) x a) y y b x x a f x, y) f a, b) = f y x, y ) f ) y a, y ) y b) + f f y a, y ) y b) + x x, b) f ) a, b) x a) + f a, b) x a) x x = ) f a, b) x a) y b) + y x = ) f a, b) x a) y b) + x y y ) f = x x ) f y ) C 3 x y )) f = x y x )) f x 3 f x 2 y C m m f x m 1 1 x m n n m = m m n ) 50

51 x 2 2 x y 2 ) f = x + 2 y 2 ) Laplacian) ) f + x y ) f y 2 x x 2 = n x = cos θ r sin θ r θ y = sin θ r cos θ r θ 2 x 2 f = x = cos θ r ) f θ cos θ f x sin θ r cos θ f r sin θ r = cos 2 θ 2 f sin θ cos θ r2 r ) f sin θ θ r cos θ f θ r sin θ r 2 f cos θ sin θ f sin θ cos θ + r θ r 2 θ r ) f θ 2 f r θ + sin2 θ f r θ + sin2 θ r 2 2 f sin θ cos θ f + θ2 r 2 θ = 2 x y r θ ) ) f x 1,, x n ): C 2 ) f x j x i = ) f = x i x j 2 f x i x j i = j) 2 f i = j) x 2 i f x 1,, x n ): C r r r f x r 1 1 xr 2 2 r xr n r n = r) n 51

52 2) Taylor f x 1,, x n ): C r+1 f x 1,, x n ) = f a 1,, a n ) + r 1 + +r n =r r f x r 1 1 xr n n x 1 a 1 ) r1 x n a n ) rn r 1! r n! + 1) r = 22 ) y f x, y) x ) a, b) = x ) f a, b) y a, b) α, β 0 φ x) = f x, b + β) f x, b) ξ y) = f a + α, y) f a, y) dφ dx = f x dξ dy = f y f x, b + β) x x, b) f a + α, y) y a, y) = f a + α, b + β) f a, b + β) f a + α, b) + f a, b) φ a + α) φ a) = dφ dx a + θα) α = ξ b + β) ξ b) 0 < θ < 1 α dφ [ f a + θα) = α dx x = αβ y f a + θα, b + β) + x ) f a + θα, b + θ β) x ] a + θα, b) 0 < θ < 1 = αβ y αβ = y f x a + θα, b + θ β) ) f a + θα, b + θ β) x 52

53 ξ b + β) ξ b) αβ = x ) f a + η α, b + ηβ) y 0 < η, η < 1 α, β 0 C 2 αβ y ) f a, b) = f a, b) x x y f a) = 0 f a) f a) 0 f a) = 0 f a) f a) 0 f x) x = a f x) f a) + f a) x a) + f a) x a)2 2 + C 3 f a 1,, a n ) f x i a 1,, a n ) = 0 2 Taylor f x 1,, x n ) f a 1,, a n ) + n i=1 2 f x i a 1 ) 2 x 2 + i 2 i<j 2 f x i x j x i a i ) x j a j ) i,j n a ij x i x j 53

54 a ij = a ji ) n = 2 a 11 x 2 + 2a 12 xy + a 22 y 2 ) a 11 a 1n.. a n1 a nn U t UHU = λ λ n n s i = u ij y j j=1 ) ) aij uik y k uil y l = n = 2 n λ i yi 2 i=1 H = αx 2 + 2βxy + γy 2 a 0 α x β ) 2 α y + ) γ β2 y 2 = α x β ) 2 α α y + 1 αγ β 2 ) y 2 α αγ β 2 = det α β β γ ) γ 0 γ y β ) 2 γ x + 1 αγ β 2 ) x 2 γ α = γ = 0 β 0 54

55 H = 2βxy = β {x + y) 2 x y) 2} f x 1,, x n ) C 3 f x i a 1,, a n ) = 0 p = a 1,, a n t U 2 f x 2 1. p) 2 f x n x 1 p) 2 f x 1 x n p). 2 f x p) 2 n = λ λ n λ 1,, λ n > 0 f a 1,, a n ) ) ) f x 1,, x n ) f a 1,, a n ) i,j 2 2 f x i y j p) x i a i ) y j a j ) x i a i = u ij y j f x 1 y),, x n y)) f a 1,, a n ) λi y 2 i λ i > 0 y i = 0 λ j > 0 y i = 0 λ i ) n = 2 Q x, y) = 1 2 α x a)2 + β x a) y b) + 1 γ y b)2 2 55

56 x = 0, y = b ) α, γ αγ β 2 > 0 Q x, y) x = a, y = b 2) α, γ αγ β 2 > 0 Q x, y) x = a, y = b 3) αγ β 2 < 0 Q x, y) x = a, y = b ) 1) α > 0, αγ β 2 > 0 x = x a y = y b αx 2 +2βx y +γy 2 = α x β ) 2 α y + 1 αγ β 2 ) y 2 > 0 α 2) 3) α 0 α 1 α αγ β 2 ) b 1,, b n ) 0 g t) = f b 1 t,, b n t) d 2 g t) dt 2 0) = = a i,j n 2 f x i x j O) b i b j ) b 1 b n H b 1.. b n 56

57 ) H = 2 f x i x j O) x 1,, x n ) x i = u ij y j ) H = λ i > 0 λ λ n d 2 g t) dt 2 0) = λ 1 b λ n b 2 n dg dt 0) = 0 0 < t 1 d 2 g dt 2 0) > 0 g t) > g 0) f b 1,, b n ) > f 0,, 0) = 57

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