I : ( 2 ; 10:40-12:10) 1,, 2, 3,, 4, 1 (Limits and Continuity) (Real numbers) (Sequences)

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1 I : ( 2 ; 10:40-12:10) 1,, 2, 3,, 4, 1 (Limits d Cotiuity) (Rel umbers) (Sequeces) (Limits of Fuctios) (Cotiuous Fuctios) (Derivtives of Fuctios) (Differetil Methods) (Tylor s Theorem) (Applictios of Differetil Clculus) (Itegrls of Fuctios) (Itegrl Clculus) (Properties of Itegrls) (Applictios of Itegrl Clculus) (Ifiite Series d Differetil-Itegrl) (Ifiite Series) (Sequece of Fuctio d Series of Fuctio) (Power Series)

2 1 (Limits d Cotiuity) 1.1 (Rel umbers) [ ] = All, Ay, Arbitrry =, = Exist =, ; = s.t. = such tht = [ (Sets)]. X ( ), P (x) ( ), A = {x X : x P (x) } A. x A: x A (elemet), x / A. : (empty set);. A B: A B (subset) [x A x B] x A, x B. ; A:, A. A = B A B, A B [x A x B]. A B: A B (uio); x A B x A x B. A B: A B (itersectio); x A B x A x B. A \ B: (differece); x A \ B x A, x / B. A : ; x A ; x A. A : ; x A, x A. A c := X \ A: A (complemet); x A c x / A. [, b]; x b, (, b); < x < b, [, b); x < b, (, b]; < x b.,, 1.1,,,,. (Nturl umbers, Itegers, Rtiol umbers, Irrtiol umbers, Rel umbers) [ N] ; 1, 2, 3,...; 1 N, N + 1 N. [ Z] ; 0, ±1, ±2, ±3,... Gze Zhle: [ Q] ; m/ 0, m, ;,. [ ] ; π, 2,... ;. [ R]. Q c = R \ Q ,, (1) (2) (3).,, b R + b, b, b, /b(b 0) R, c R, (1) + b = b +, b = b (2) ( + b) + c = + (b + c), (b)c = (bc) (3) (b + c) = b + c. 2. 2,, (1), (2) (3). (1) < b, b < c < c (2) < b, c R + c < b + c (3) < b, c > 0 c < bc. 3. (Weierstrss) S sup S, i.e., c R; x S, x c = sup S <. 1

3 , 3., 3.,,,,,, : S R. S c; x c ( x S). c S ( ). S S. S : = mx S S, x ( x S). S : = sup S S : = mi{c; c S }. S ( S ).,,,.. (,. ) 1.3 = sup S (1) x S, x, (2) ε > 0, x = x(ε); ε < x ( ) (Archimedes (, )) [ ]. c ( N) c. i.e., c 1, N N; c < N. [ ] = sup N <. N N; 1 < N, i.e, < N + 1. N , b > 0, N N; b < N [ ], b > 0, b > 0 b/ ( ) 2 :, b; < b, r Q; < r < b. [ ] 1 b (> 0), N N; 1 < N(b ), i.e., N < N + 1 < Nb. M Z; N < M N + 1 ( N < M M ). r = M/N < r < b. 1.10, b. (1) ε > 0, < ε 0. ε > 0, < b + ε b. (2) c <, b c b ( ) 2

4 1.2 (Sequeces) { } N α { } α (coverge), α (limit). lim = α α ( ), ( ): α ( ) α 0 ( ) ε > 0, N = N(ε) N; N, α < ε., { } α. [ ] α ( ) ϵ 0 > 0; k N, k k; α ϵ ( α, β α = β.) / 0. ( ) { } (diverge). (±1.) { } lim = or ( ) M > 0, N = N(M) N; N, > M. { }, { },. lim = or ( ) { } (mootoe icresig): ( ) 1 2. { } (mootoe decresig): ( ) 1 2. ( ) (mootoe sequeces).,,. { } M R; N, M ( M). { } M > 0; N, M { } ; ( ) M; M ( ),. [ ] 1.18 lim = α, lim b = β c, : (1) lim( ± b ) = α ± β ( ), (2) lim b = αβ, lim c = cα, (3) lim /b = α/β,, b 0, β 0. 3

5 [ ] (3) 1/b 1/β. N 1 ; N 1, b β < β /2 b > β /2 ε > 0, N 2 ; N 1, b β < β 2 ε/ α R ( )/ α. α = ±. [ ] ε > 0, N 1 ; α < ε/2 ( N 1 ). α,..., N1 α } N 1 ( 1 k α 1 N1 ) k α + k α k=n 1 +1 L = mx{ 1 N 1L + N 1 ε 2 N 1L + ε 2. ( ) 0 ( ), ε N 2 ; N 1 L/ < ε/2 ( N 2 ). N := mx{n 1, N 2 }, ( ) ε/2 + ε/2 = ε. α = ± ,. ( ), M; M α; α ( ). [ ] α = sup lim = α.. 1, > 0 lim = lim ( = > 1, k 0 lim / k =. 4. > 0 lim /! = lim 1 + ) 1 =: e. 1. > 1, 1 lim =: α 1. α > 1 h := 1 α > 0 > α = 1+h, i.e., > α = (1 + h) > 1 + h. = 1. 0 < < := 1 0,, = (1 + ) ( 1) 2 /2 > ( 1) 2 /2, i.e., ( 1) 2 = 2 < 2/( 1),, 1 < 2/( 1) lim +1/ =: r. 0 r < 1 0, r > 1. lim =: r. 0 < r < 1 r < r 1 < 1, ε = r 1 r > 0, r > 1 r > r 2 > 1, ε = r r 2 > 0,. 3, 4. +1/ = (/( + 1)) k, +1/ = (/( + 1)) k. 5. O.K. 2 < e < 3. (1 + 1/) ( 1 1 ) + 1 ( 1 1 ) ( 1 2 ) ( 1 1 ) ( 1 2 ) ( 3!! = ! /2! + 1/3! + + 1/! /2! + 1/ /2 1 < 1 + 1/(1 1/2) = ) ( ) ( )! 1 = k k!( k)!, k = 1 k k!,. O- 1 (1) lim(1 1/) =? (2) lim(1 1/) 2 =? ( 1 1 ) ( 1 2 ) ( 1 1 ) (k = 0, 1,..., ) 4

6 O- 2 1 := N limn 0, 0 < r < 1 =1 1, r > 1. (. r = 1.) 1.22 ( ) [, b ] [, b ] [ +1, b +1 ] ( 1), b 0 ( ) c; c b, c, b c ( ). (or 1 c [, b ] = {c}.) [ ] (Bolzo-Weierstrss ). { } 1 ; L ( 1) α, { k } { }; k α (k ). [ ] I 0 = [ L, L] 2, { } I 1. (.), {I k }. k I k { } (k 1) k, [ ]. { }; L ( 1) α ( ) [ { k } { }; k β], β = α. [ ] ( ). ( ). α α,., ϵ 0 > 0; k 1, k k; k α ϵ 0, k. { k } { kj }; kj β, β α, { } (Cuchy sequece) ε > 0, N;, m N, m < ε ( ) =. m 0 (m, ), i.e., α ( ) m 0 (m, ). [ ] ( ). ( ), [ ] ( ) m α + α m 0 (m, ). ( ) N 1 ; m, N 1, m < 1, 1, mx{ N + 1, 1,..., N 1 } < { }. Bolzo-Weierstrss { k } { }; lim k (=: α). ϵ > 0, N;, k N, k < ϵ. k α ϵ (upper limit) lim sup := lim sup N N (lower limit) lim if := lim N if N = sup = if sup. N 1 N if. N lim if lim if. lim if = lim if lim = lim if = lim if ( N 1, N 2 1, N := N 1 N 2 if N1 N sup N2 ) N 1 5

7 1.3 (Limits of Fuctios) D x 1 y (2 ) (fuctios), f : D R; x y, y = f(x). (D, f(d) = {y R; y = f(x), x D} ) y = x 2. y = x. x R, y x 2 + y 2 = 1 y = ± 1 x 2. 2,, 1 1,, R, δ > 0 U δ () = ( δ, + δ) (δ-) (eighborhood).,,. f(x) x =. f(x) x = (cotiuous) lim x f(x) = f() ε > 0, δ = δ(, ε) > 0; x < δ, f(x) f() < ε. x. f(x) x = (x = ). x f(x) α x f(x) ε > 0, δ = δ(ε) > 0; 0 < x < δ, f(x) α < ε. L > 0, δ = δ(l) > 0; 0 < x < δ, f(x) L. ( x =, α = f() x =.) f(x) (, ). x f(x) α x f(x) ε > 0, M = M(ε) > ; x M, f(x) α < ε. L > 0, M = M(ε) > ; x M, f(x) L x, x > x + or x + 0, x, x < x or x 0. [ ] f(+) f( + 0) := lim x + f(x) ε > 0, δ = δ(ε) > 0; 0 < x < δ, f(x) f(+) < ε. f( ) f( 0) := lim x f(x). ( ) f x 1 < x 2 f(x 1 ) < f(x 2 ). ( ) f x 1 < x 2 f(x 1 ) f(x 2 ) (strictly icresig), (o-decresig). f, f. 6

8 (bouded fuctio). f(x) M ( x D) (± ) (.) (1) 0 < x < π/2 si x < x < t x. (2) x 0 si x < x. [ ] (1). (2) x π/2 si x 1 < π/2 x (1) x > 0 O.K. x < 0 si x = si( x) < x = x si x (1) lim = 1, (2) lim x 0 x x 0 x = 1 ( > 0), ( (3) lim 1 ± 1 x = e x x) ±1 ( ). [ ] (1) 0 < x < π/2 cos x < si x/x < 1, π/2 < x < 0 si x/x = si( x)/( x) O.K. (2) 0 < x < 1. > 1 = [1/x] [1/x] < + 1 1/(+1) < x 1/ O.K. = 1. 0 < < 1 1/ O.K. 1 < x < 0 x [x] x < [x] + 1, O.K.: = (1/) x O.K. (3) x [x]. ( ) [x] ( < x ( < 1 + [x] + 1 x) 1 ) [x]+1. [x] ([x] Guss : x ; [x] x < [x] + 1.) log(1 + h) e h (3) x. lim = 1, lim = 1. h 0 h h 0 h 1.36 lim x f(x) > 0 f(x) > 0 ( ) 1.37 (1) [ ] lim x f(x) {x }; x (x ), lim f(x ) (2) [Cuchy ] lim x f(x) ε > 0, δ > 0; 0 < x 1 < δ, 0 < x 2 < δ, f(x 1 ) f(x 2 ) < ε. [ ] (1) ( ). ( ) lim f(x ) {x }., x, x (x, x ), {x } x α := lim f(x ) {f(x )}, {f(x )} {f(x )} α. lim x f(x) α ε 0 > 0; δ > 0, x = x(δ); 0 < x < δ, f(x) α ε 0. N, δ = 1/ x := x(δ) 0 < x < 1/, f(x ) α ε 0.. lim x f(x) = α. (2) ( ). ( ) {x }; x, x {f(x )},, (1) (2). (1) , x, f(x) α, g(x) β, f(x) ± g(x) α + β, cf(x) cα (c: ), f(x)g(x) αβ, f(x)/g(x) α/β (g(x) 0, β 0). 7

9 1.4 (Cotiuous Fuctios) f(x) x =. f(x) x = (cotiuous) lim x f(x) = f() ε > 0, δ = δ(, ε) > 0; x < δ, f(x) f() < ε. f(x) x = f(x) x = (right cotiuous) ε 0 > 0; δ > 0, x δ ; x δ < δ; f(x δ ) f() ε 0. ε > 0, δ = δ(, ε) > 0; 0 < x < δ, f(x) f() < ε... f(x) I,. lim x +0 f(x) = f() f(x) I., I I I, ε > 0, δ = δ(, ε) > 0; x I; x < δ, f(x) f() < ε. (discotiuous) x + 2 x x, si x, cos x, x ( > 0) (, ). (.,..) 1.40 f(x) f() > 0 f(x) > 0 ( ) ( ),., f, g x =, f + g, αf (α R), f 2, f,. fg = {(f + g) 2 + (f g) 2 }/4. f 0, 1/f 1/f(x) 1/f(), f(x) f() /2, f(x) f() < f() 2 ϵ/ ( ). f(x) [, b], f() f(b) (, b) f() f(b). f(x) coti. o [, b], f() < f(b) y (f(), f(b)) c (, b); f(c) = y. c = sup{x [, b]; f(x) y} f(c) = y.,,, {x }; x c, f(x ) y. f, f(c) y., c < b, x := c + (b c)/, x (c, b],, f(x ) > y., x c f, f(c) y ( ). f(x) coti. o [, b] c, d [, b]; f(c) f(x) f(d) (x [, b]). [ ] M = sup{f(x); x [, b]} {x } [, b]; f(x ) M. Bolzo-Weierstrss d [, b], {x k } {x }; x k d. M = f(d). f(x). 8

10 f(x) I (uiformly cotiuous) ε > 0, δ = δ(ε) > 0; x, y I; x y < δ, f(x) f(y) < ε. f(x) I ε 0 > 0; δ > 0, x δ, y δ ; x δ y δ < δ; f(x δ ) f(y δ ) ε [ ] I, ε 0 > 0, 1, x, y I; x y < 1/, f(x ) f(y ) ε 0. Bolzo-Weierstrss c I, {x k } {x }; x k c. y k c. f(c) = lim f(x k ) = lim f(y k ). 0 = lim f(x k ) f(y k ) ε 0 > ,. (si 1/x, 1/x = 2π, 1/y = (2 + 1/2)π.) f(x) D f(d) 1 1 (i.e., f(x) = f(y) x = y) y = f 1 (x) (x f(d)), f (Iverse fuctio). 1.47,. f; coti, f (or ) o [, b] f 1 ; coti, f ( ) o [f(), f(b)] (or [f(b), f()]). 1.48, I = [, b] f(x) f() f(b), I (f. < x 0 b, f(x 0 ) = f(x 0 )., f(x 0 ) < f(x 0 ). x 0 < b f(x 0 +) = f(x 0 ).) [ ] ( ) si 1 x = rcsi x, cos 1 x = rccos x, t 1 x = rct x si x o [ π/2, π/2], cos x o [0, π], t x o ( π/2, π/2) ( ),,. S, T R, f, g f : S R, g : T R f(s) T. g f(x) := g(f(x)) g f : S R f g. 1.50, i.e., f, g; coti, g f well-ied g f; coti, lim x g f(x) = g( lim x f(x)) si x, cos x h 0, si( + h) si = 2 cos 2+h si h 2 si(h/2) 2 2 h 0. cos( + h) cos = 2 si 2+h si h,. 2 2 x ( > 0): x r = m/ Q = Z/N, y = r y = m. r. r s = r+s, ( r ) s = rs, (b) r = r b r (, b > 0, r, s Q) > 1, r > 0 r > 1. x, > 1 x = sup{ r ; r < x, r Q}. = 1 x = 1, 0 < < 1 x = 1/(1/) x.. > 1 1/ 1 ( ) h < 1/ 1/ < h < 1/ h 1 (h 0), x c, x = c x c c 1 = c.. 9

11 2 1 (Derivtives of Fuctios) 2.1 (Differetil Methods) f(x) x =. f(x) x = ( ; differetible) f(x) f() f( + h) f() lim = lim (=: f ().) x x h 0 h f () f(x) x = ( ). x ± 0 or h 0 ± 0 ( ). f(x) I I. (, I.) f (x) f(x). y = f(x). f (1) (x), df(x), df (x), d f(x), dy (x), y. f (x), C 1, C 1 ( ). ( 1, 1.) f (x) f(x) 2, (f ) (x) f (x) = f (2) (x), 2. f(x), f () (x)., f (1) (x) = f (x), f (2) (x) = f (x),.... f (0) (x) := f(x). C, C ( ). ( ( ).) N, C C ( ). ( ( ).) f(x) C f () (x). 2.1, i.e., f (x) f coti. t x. 2.2 f, g: (1) (f + bg) () = f () + bg () (, b R). ( (2) (fg) () = k k=0 ) f ( k) g (k) ( )., 2.3 ( ) ( ) = k (1) u = g(y), y = f(x), u = g(f(x)) (g(f(x)) = g (f(x))f (x), i.e., du = du dy dy.! ( k)!k!. (2) I y = f(x) f(i) x = f 1 (y) y = f(x) (x; f (x) 0) df 1 (y) dy = 1 f (x) (y = f(x)) i.e., dy = 1 dy/. 10

12 , (1) (e x ) = e x (2) ( x ) = x log ( > 0) (3) (log x ) = 1/x (x 0) (4) (log x ) = 1/(x log ) ( > 0, 1, x 0) (5) f (log f(x) ) = f (x)/f(x) (x; f(x) 0) (1) (si 1 x) = 1/ 1 x 2 ( 1 < x < 1) (2) (cos 1 x) = 1/ 1 x 2 ( 1 < x < 1) (3) (t 1 x) = 1/(1 + x 2 ) ( < x < ) (1) (x ) () = ( 1) ( + 1)x,, (1/x) () = ( 1)!x 1 (2) ( x ) () = x (log ) ( > 0), (e x ) () = e x (3) (log x ) () = ( 1) 1 ( 1)!x (x 0) (4) (si x) () = si(x + π/2) (5) (cos x) () = cos(x + π/2) = 0, 1, 2,... H (x) = ( 1) e x2 d e x ( ) x = f(t), y = g(t) x = f(t) f (t) 0 y x dy = dy / dt dt = g (t) f (t) (Tylor s Theorem) x = f(x) x = ( ) δ > 0; x ( δ, + δ), f() > f(x) (f() < f(x). f() ( ),. ( ), ( ), ( ),. 2.6 f(x) x =, f () = 0. (,, f () = 0 x =.), ( = ) 0, ( (Rolle) ) f(x) [, b], (, b). f() = f(b) c (, b); f (c) = 0. [ ] f. f(x) f() = f(b)., x (, b); f(x) > f()., c (, b); f(c) = mx [,b] f > f(). f(c) f (c) = 0. f(x) f() = f(b) f(x). 11

13 2.8 ( ) f(x) [, b], (, b). c (, b); f(b) f() b = f (c). [ ] φ(x), φ() = φ(b) = 0, : ( ) f(b) f() φ(x) := f(x) (x ) + f(). b. 2.9 f = 0 o (, b) f = cost. o (, b). [ ] [, b ] (, b), f( ) = f(b ) 2.10 (Cuchy ) f(x), g(x) [, b], (, b), g (x) 0. c (, b); f(b) f() g(b) g() = f (c) g (c). g g(b) g() 0. [ ] φ(x) : φ(x) = [g(b) g()][f(x) f()] [f(b) f()][g(x) g()]. 0 = φ (c) = [g(b) g()]f (c) [f(b) f()]g (c) O.K., 0, 0/0, ± /, 1, ( ) lim f(x)/g(x) = lim f (x)/g (x). x x : f, g U g 0, [, f() = g() = 0] [ lim g(x) = ], lim f (x)/g (x) [, ]. ± 0, x x ±. = [ f, g x g 0, f( ) = g( ) = 0, i.e., lim x f(x) = lim x g(x) = 0]. [ ] (1) [, f() = g() = 0]. x, c = c(x); x < c < or < c < x, x. f(x) f() g(x) g() = f (c) g (c). (2) [ lim g(x) = ]. lim f (x)/g (x) = γ (, ) x + 0 x x. ε > 0, x 0 > ; < x < x 0, f (x) g (x) γ < ε. x, c = c(x); ( <) x < c < x 0, f(x) f(x 0 ) g(x) g(x 0 ) = f (c) g (c). 12

14 g(x) 0, f(x) g(x) = f (c) g (c) ( 1 g(x ) 0) + f(x 0) g(x) g(x). x + 0 g(x) f(x) g(x) f (c) g (c) = f ( (c) g g(x ) 0) + f(x 0) 0. (c) g(x) g(x) δ > 0; < x < + δ ( c), f(x) g(x) γ f(x) g(x) f (c) g (c) + f (c) g (c) γ < 2ε. x 0. γ = ±. x (or ) x = 1/y y +0 (or 0) (1) lim x ex /x ( : ) x si x (3) lim x 0 x 3 (2) lim x x / log x ( > 0) x b x (4) lim (, b > 0) x 0 x [,, 1/6, log(/b)] 2.13 (Tylor ) f(x) I., b I; < b (or > b), c (, b) (or (b, )); [ ] f(b) = 1 k=0 f (k) () k! (b ) k + f () (c) (b )! = f() + f ()(b ) + f (2) () (b ) 2 2! + + f ( 1) () ( 1)! (b ) 1 + f () (c) (b ).! 1 F (x) = f(b) k=0 f (k) (x) (b x) k, G(x) = (b x) k!,. F (b) = G(b) (= 0). [ ] R = f(b) φ(x) = 1 k=0 ( 1 k=0 φ() = φ(b) = f(b) ) f (k) () (b ) k, k! f (k) (x) (b x) k (b x) + R k! (b ). c (, b) or (b, ); R = f () (c) (b )! 13

15 (0, 1); 2.14 f(x) x = U x U, θ = θ(, x, ) f(x) = 1 k=0 f(x) C f (k) () k! R (x) 0 (x ) k + f () ( + θ(x )) (x ).! R (x) := f () ( + θ(x )) (x )! f(x) = =0 f () () (x )!, = 0 ( ) (e x ) () = e x e x = =0 (si x) () = si(x + π/2) si x = =0 (cos x) () = cos(x + π/2) cos x = x! = 1 + x + x2 2 + x3 eθx + ( 3!! x ). ( 1) (2 + 1)! x2+1 = x x3 3! + x5 5! =0 ( 1) (2)! x2 = 1 x2 2! + x4 4! x > 1 (log(1 + x)) () = ( 1) 1 ( 1)!(1 + x) =1 si(θx + π/2) ( x ).! cos(θx + π/2) ( x ).! ( 1) 1 log(1 + x) = x = x x2 2 + x3 3 + ( ( 1) 1 (1 + θx) x ). R ((1+x) ) () = ( 1) ( +1)(1+x) (1 + x) = =0 ( ) x = 1 + x + ( 1) x 2 + ( 2! ( ) ( 1) ( + 1) =! ( ) (1 + θx) x ). [ ], : si x si x x lim = 1, lim = 1 x 0 x x 0 x 3 6, lim cos x x = 1 x 0 x 2 2, lim log(1 + x) x = 1 x 0 x 2 2. e. ( θ (0, 1); 0 < e ! ) e θ =! ( + 1)! < 3 0 ( ). ( + 1)! e e = m/ ( m, Z).! e θ /( + 1)! = e θ /( + 1). 1 e θ /( + 1) < 3/( + 1) 3 > + 1 = 1 e Z. 2 < e < 3. 14

16 2.3 (Applictios of Differetil Clculus) 2.15 f(x) [, b], (, b). f 0 ( 0) o (, b) f ( ) o [, b]. ( ). [ ] (, ). (1) x > 0 x x3 3! (3) x > 0 x x2 2 < si x < x (2) x 0 1 x2 2! < log(1 + x) < x x2 < cos x < 1 2! + x4 4! 2.17 f (x), C 1, 1 f () > 0,. f(x) = x + 3x 2 si(1/x) (x 0), = 0 (x = 0) f (0) = 1 0., f(x) C 1. f () > 0 (< 0) x = f(x) ( ). f (x) x = x = f > f(x) C 2. f () = 0, f () > 0 (< 0) x = f(x) ( ). f(x) f() = f ( + θ(x ))x 2 /2 ( θ (0, 1)) C 2 f f > 0, f..,., I f(x) ( ) x, y I, p, q 0; p + q = 1, f(px + qy) pf(x) + qf(y) x = y ( ).,, I f(x) 2, x 1,..., x I, p 1,..., p 0; p p = 1, f(p 1 x p x ) p 1 f(x 1 ) + + p f(x ).. o I I f(x) C 2. f > 0 o I f 2.22 ( ) log x o (0, ). x i > 0, log x x 1 (log x log x ) x x x 1 x > 1 x k 0 (k = 1,..., ) : (x x ) 1 (x x ). (,.) 15

17 3 1 (Itegrls of Fuctios) 3.1 (Itegrl Clculus),,,,. ( ) [ ] [, b] F (x), F (x) = f(x) ( ) f(x) := F (x) + C (. C ).. (, ). log x = x log x x, t(x + b) = 1 log cos(x + b) ( 0), x2 + = log x + x 2 + ( 0), x 2 2 = 1 2 log x x + ( 0), 2 + x 2 = 1 x t 1 ( > 0), 2 x = x 2 si 1 ( > 0) : 3.2. (1), b, [f(x) + bg(x)] = f(x) + b g(x). (2) ϕ (t), x = g(t) f(x) = f(ϕ(t))ϕ (t)dt. b f(x) = F (b) F (), ( ),., ( ) x,. f(x) I = [, b], i.e., m f(x) M. I : = x 0 < x 1 < < x = b, = mx{x k x k 1 ; k = 1,..., }. k = 1,..., ξ k [x k 1, x k ], R( ; {ξ k }) :=. f(ξ k )(x k x k 1 ) (Riem ) 16

18 ,, R( ; {ξ k }) {ξ k }, f(x) I = [, b] (Riem) (or ), f(x) I = [, b], b f(x). α R; = {[x k 1, x k ]} ; I, ξ k [x k 1, x k ] (k = 1,..., ), ( ) α = lim R( ; {ξ k}) 0 =: b f(x) 3.3 I = [0, 1] f(x), 0, 1 Riem., [x k 1, x k ] ξ k Riem 1, 0., M k = sup [xk 1,x k ] f, m k = if [xk 1,x k ] f ( ), S( ) = M k (x k x k 1 ), s( ) =,, m k (x k x k 1 ). S = if S( ), s = sup s( ) m(b ) s( ) s S S( ) M(b )., 2 1, 2, = 1 2, s( 1 ) s( ) S( ) S( 2 ). 1, 2, s( 1 ) S( 2 ). if 1, sup 2. (,, S = s Riem.,.) 3.4 ( ) f(x) I, 0, S( ) S, s( ) s. S( ) (s( ) ). M = sup I f, ε > 0. S, ε ; S( ε ) < S + ε. ε K, δ. ; < δ ε 2KM, S( ) S < 2ε ( 0 S( ) S ). = { x k = [x k 1, x k ]}, ε = { y j = [y j 1, y j ]} S( ) S( ε ) = k sup x k f x k j sup f y j y j 17

19 . < δ, [x k 1, x k ] (x k 1, x k ) ε 1. ε 1 [x k 1, x k ] = [y j 1, y j+1 ] x k = y j + y j+1 sup yj f, sup yj+1 sup xk f M, sup f x k (sup f y j + sup f y j+1 ) 2M ε 2M. x k y j y j+1 [x k 1, x k ] = [y j 1, y j ], S( ) S( ε ) 2KM < ε. S( ε ) S S( ε ) S < ε,. 3.5 s( ) s. 3.6 I f(x) Riem S = s 0 S( ) s( ) 0 ( 0). s = S s( ) R( ; {ξ k }) S( ) R( ; {ξ k }) S = s ( 0) Riem. ε > 0. = {[x k 1, x k ]}, k = 1,...,, ξ k, ξ k [x k 1, x k ]; f(ξ k ) < m k + ε b, M k ε b < f(ξ k). R( ; {ξ k }) < s( ) + ε, S( ) ε < R( ; {ξ k}). Riem 0, b f(x) s + ε, S ε b f(x). ε > 0, ε 0,, s = S = b b f(x) s S b f(x). f(x) [, b] f(x) Riem.. S( ) s( ) (f(b) f()) 0 ( 0). 3.8 [, b] f(x) Riem. 18

20 f(x) I = [, b], ε > 0, δ > 0; x, y I; x y < δ, f(x) f(y) < ε. < δ, M k m k ε. S s S( ) s( ) ε 0 S = s. (M k m k )(x k x k 1 ) ε (x k x k 1 ) = ε(b ). 3.9 f(x) [, b], b lim f ( + k ) (b ) = b f(x). [, b] = [0, 1], 1 lim f ( ) k = 1 0 f(x) F (x) = f(x); [, b] ).., F (x) = ). x b f(x) = F (b) F () (= [F (x)] b f(u)du F (x) = f(x) ( [ ] : = x 0 < x 1 < < x = b,, x k 1 < ξ k < x k ; F (x k ) F (x k 1 ) = f(ξ k )(x k x k 1 ). F (b) F () = f(ξ k )(x k x k 1 ) = R( ; {ξ k }) b f(x) ( 0) lim [ ] 3.12 lim 2 + k 2 = k (k/) x 2 = [t 1 x] 1 0 = π k 2. [t 1 2] 19

21 3.2 (Properties of Itegrls) (,,, ) f(x) [, b] [c, d] [, b]. [ ] [c, d], [, b] \ [c, d], [, b]. 0 S( ) s( ) S( ) s( ). 0, 0 0, 0, S( ) s( ) 0 ( 0), [, b] f(x), g(x). (1) b (αf(x) + βg(x)) = α (2) c [, b], (3) f(x) 0 b b (4) f(x), [ ] (1) b b f(x) + β f(x) = c b c f(x) + g(x) (α, β ). b f(x) 0. f(x) g(x) f(x) = b f(x) lim 0 f(x). b f(x) b b f(x). [x k 1, x k ] (k = 1,..., )) ( ). g(x). f(ξ k )(x k x k 1 ) ( = {[x k 1, x k ]}, ξ k (2) < c < b,. (3), f g 0 (1). (4) f f f (3) f(x) [, b], f(x) 0. ( x [, b]). b f(x) = 0 f(x) = 0 [ ]. x 0 [, b]; f(x 0 ) > 0, δ > 0; f(x) f(x 0 )/2(=: m.) for x [x 0 δ, x 0 + δ] [, b]. b f(x) m [x 0 δ, x 0 + δ] [, b] = m {(b (x 0 + δ)) ( (x 0 δ))} > ( ) f(x) [, b]. c (, b); b f(x) = f(c)(b ). [ ] f., f, M, m m < b f(x)/(b ) < M., c (, b); f(c) = b f(x)/(b ). 20

22 3.17 ( ) f(x) [, b]. [ ] F (x) = x d x f(t)dt = f(x) f(t)dt 2, F (x + h) F (x) = x+h c x x + h. d x x ( x (, b)). f(t)dt = f(c)h f(t)dt = F F (x + h) F (x) (x) = lim = f(x). h 0 h 3.10 F (x) = f(x); [, b] ( ). b f(x) = F (b) F (), 3.18 ( ) f(x) [, b], φ(t) [α, β] or [β, α] C 1 ; φ(t) b, φ(α) =, φ(β) = b b f(x) = β α f(φ(t))φ (t)dt. [ ] F (x), (F (φ(t)) = f(φ(t))φ (t) β α dt, ( ) = F (b) F () = F (φ(β)) F (φ(α)) = ( ) ( ) f(x), g(x) [, b] C 1, b f(x)g (x) = [f(x)g(x)] b b f (x)g(x). [ ] (f(x)g(x)) = f (x)g(x) + f(x)g (x) π/2 cos x si 2. t = si x dt = cos x π/4 x 3.21 f(x) [, ]. f(x) ; f( x) = f(x) f(x) ; f( x) = f(x) f(x) = 2 0 f(x) = 0. f(x) , π/2 π/2 si 2 x = cos (2 1) π x = π/2 π/2 si 2+1 x = cos x = (2 + 1) 21

23 3.3 [ ] ( ) cos x = 1 cos 1 x si x + 1 cos 2 x si x = 1 si 1 x cos x + 1 si 2 x t 1 x = 1 t 1 x t 2 x ( 2) (log x) = x(log x) (log x) 1 ( 1) 3.23,. [ ] f(x) = P (x)/q(x)., P (x), Q(x), (P ) < (Q ), f(x) = Q(x) = α k m i i=1 s=1 k (x i ) m i i=1 l (x 2 + b j x + c j ) j (b j 4c j < 0). j=1 A i,s l (x i ) s + j j=1 t=1 B j,t x + C j,t (x 2 + b j x + c j ) t ( ) (.): I = (x 2 + ) I = 2( 1)(x ) 1 2( 1) I 1 ( 2, 0). ( (x + 1) 2 (x 2. log x ) x ) 2 log(x2 + 1) 3.25 x ( 1 3 log x log(x2 x + 1) + 1 ) 1 2x 1 t 3 3 [ ]. R(t), R(x, y). R(t) R(e x ) = dt (e x = t = 1t ) t dt. R(si x, cos x) = ( 2t R 1 + t 2, 1 ) t2 2 ( 1 + t t 2 dt t x ) 2 = t. si 2 x, cos 2 x, t x R ( si 2 x, cos 2 x, t x ) = R ( t t 2, t 2, t ) dt 1 + t 2 (t x = t) (t 1 + cos x. x ) 2 ( ) e x. 2 (x log(ex + 2)) 22

24 [ ] ( ) x + b R x, = cx + d ( dt ) b (d bc)t 1 R ct, t ( ct ) 2 dt ( ) x + b cx + d = t., d bc 0, 2. R(x, x 2 + bx + c) > 0 x 2 + bx + c = t x x = t2 c 2 2( t 2 + bt + c), = t + b (2 dt. t + b) 2 < 0 α < β (, x 2 + bx + c 0 ) x α β x = t (β α)t x2 + bx + c = 1 + t 2, x = α + βt2 2(β α)t, = 1 + t2 (1 + t 2 ) 2 dt. ( ) 2 x 2 x = si t ( t π/2) 2 x 2 = cos t, = cos tdt. x2 + 2 x = t t ( t < π/2) x = sec t, = sec 2 tdt. x2 2 x = sec t x 2 2 = t t, = t t sec tdt x 4 1 x. ( 445 (5x + 4)(1 x)5/4 ) 3.29 x + 2 x. x + 1 ( 1 x (2x2 + x 1) x log( x ) x + 1) [ ] ( ). (1) x (x + 1) 2 (2) 1 x (3) si x (4) t x (5) x (6) x 5x 6 x 2 [ ] ( (1) log x x + 1 (2) log x 2 + ) 2x + 1 x 2 2x [t 1 ( 2x + 1) + t 1 ( 2x 1)] 2 2 log 2 si x + 3 cos x + 3x (3) = t x sec x 1 (4) 1 + t(x/2) 13 (5) 2 x 2 log(1 + x 2 x) (6) 5 t 1 3 x ( ) x 2 5x 6 x 2 t =. 3 x 3.4 (Applictios of Itegrl Clculus) < b <, (.b] f(x), ( ) b lim f(x) f (, b], ( =: b 23 b f(x). ) f(x).

25 ,,.) < < b, [, b),, (, b), < c < b, b 3.30 c f(x), b f(x). 1 0 c f(x), (, b) x = [ x ] 1 0 = 1, 1, (0, 1],. 1 0 x = lim 0+ 1,. 0 x = [log x]1 0 = [ ] 1 = lim x = lim (1 ) = 1 x { x α = 1/(1 α) (α < 1) (α 1), x(log x) α log x. 1 { x α = 1/(α 1) (α > 1) (α 1). [ (log 2) α 1 α 1 ] (α > 1), (α 1),,. < b,, b b f(x) =, b f(x) <, b [ 1] f(x) f(x), < b (, b) or (, b] or [, b) f(x), φ(x); f(x) φ(x), b [ ]. b f(x) φ(x) <, f(x). b f(x) b φ(x) < (1) si x/x (0, 1]. si x (2),. 1 x si x (3). x 0 24

26 3.36 Γ(t) := 3.37 B(p, q) :=. 0 e x x t 1 (t > 0) well-ied. [1. lim x e x x t+1 = 0 x 1 e x x t 1 Kx 2 ] 1 0 x p 1 (1 x) q 1 (p, q > 0) well-ied [ 0 < p < 1 x 1/2 x p 1 (1 x) q 1 (1 (1/2) q 1 )x p 1, 0 < q < 1 x 1/2 x p 1 (1 x) q 1 (1 (1/2) p 1 )(1 x) q 1 ] [ ] C : (x, y) = (g(t), h(t)) ( t b), l, l := sup L( ),, L( ) [, b] : = t 0 < t 1 < < t = b ; : L( ) := (g(tk ) g(t k 1 )) 2 + (h(t k ) h(t k 1 )) C 1 C : (x, y) = (g(t), h(t)) ( t b), l l = b g (t) 2 + h (t) 2 dt. [ ] : = t 0 < t 1 < < t = b,, k = 1, 2,...,, ξ k, η k (t k 1, t k ); (g(tk ) g(t k 1 )) 2 + (h(t k ) h(t k 1 )) 2 = g (ξ k ) 2 + h (η k ) 2 (t k t k 1 ). {τ k }; t k 1 < τ k < t k, Riem R( ; {τ k }) =, : lim L( ) = lim R( ) = 0 0 b, 2 + b 2 + b, g (τ k ) 2 + h (τ k ) 2 (t k t k 1 ) g (t) 2 + h (t) 2 dt. (1) g (ξ k ) 2 + h (η k ) 2 g (τ k ) 2 + h (τ k ) 2 g (ξ k ) g (τ k ) + h (η k ) h (τ k ), L( ) R( ) [ g (ξ k ) g (τ k ) + h (η k ) h (τ k ) ] S(, g ) s(, g ) + S(, h ) s(, h ) 0 ( 0). (1). L( ) L( ), L( ) L( ) b g (t) 2 + h (t) 2 dt ( 0). 25

27 ,, L( ) : b l = sup L( ) l L( ) b g (t) 2 + h (t) 2 dt. b g (t) 2 + h (t) 2 dt. g (t) 2 + h (t) 2 dt ( 0) C 1 C y = f(x) ( x b), l l = b 1 + f (x) C x = r cos θ, y = r si θ, C 1 r(θ), r = r(θ) (α θ β), l l = β α r(θ)2 + r (θ) 2 dθ. [ ] g(θ) = r(θ) cos θ, h(θ) = r(θ) si θ, g = r cos θ r si θ, h = r si θ + r cos θ, g 2 + h 2 = r 2 + r 2. 4 (Ifiite Series d Differetil- Itegrl) 4.1 (Ifiite Series), ( ) : { }, := lim =1 N N =1 (if exists). 0,. (,,,,.),, <,, =,,. : 4.1 lim 0. ( lim = 0.) ( S =, S S R = S S 1 S S = 0.) = r 1, r < 1 = /(1 r), r ρ := lim +1 / or lim (if exists). 0 ρ < 1 ( ), ρ > 1 =, i.e.,. 26

28 , [ ε > 0, N; N, +1 / ρ < ε, i.e., ρ ε < +1 / < ρ + ε. (1) ρ < ρ 0 < 1 ρ + ε = ρ 0, i.e., ε = ρ 0 ρ +1 < ρ 0 < ρ N 0 N. (2) 1 < ρ 1 < ρ ρ ε = ρ 1, i.e., ε = ρ ρ 1 +1 > ρ 1 > ρ N 1 N.] ρ = (1) 1 [ 1 x p, 2 1 p 1 p > 1, p 1. (log ) p (2) 2 x(log x) p (f(x) = 1 (p > 0) x > 1 ).] x(log x) p > 0, ( 1) , ( 1) 1. [ ] S = ( 1) 1, S 2 1 S 2. 0 S 2 S 2, S 2 1, S 1 (= 1 ). (, ( 1) 1 = ( 1 2 ) + ( 3 4 ) + = 1 ( 2 3 ) ( 4 5 ).) 0 S 2 1 S 2 = 2 0,. 4.5 ( 1). 4.6 ( ),,. [ ],. S, S S, S ( ). 1, S { k} m m S S m S. S S. S S S = S. ( S <, S S. S S. S =, S <, S S <. S = S =.) 4.7 ( ), b, c = j+k= jb k, c ( ), c = b. [ ], b, c A, B, C. (1), b. C A B C 2,. (2). =, b = b, c = j+k= jb k, A, B, C,, c = b, c c, c <. C A B A B C 0 c = b. 27

29 4.2 (Sequece of Fuctio d Series of Fuctio) I f (x), f f o I x I, f (x) f(x) x I, ε > 0, N = N(x, ε) 1; N, f (x) f(x) < ε. f f o I sup x I f (x) f(x) 0 ε > 0, N = N(ε) 1; N, x I, f (x) f(x) < ε. 4.8 I = [0, 1] 1, y = f (x) 4 (0, 0), (1/(2), 2), (1/, 0), (1, 0), f 0 f f f I. = 1, 2,..., f (x) = 1 x o ( 1, 1), lim 1 x f (x) = 1 ( ), 1 x 0 < < 1, [, ], ( 1, 1). ( f ( 1, 1).) ; [I f : coti., f f f: coti.] (,,.,. 4.8.) [ ] x 0 I, ε > 0. f f o I, 0 ; f 0 (x) f(x) < ε/3 ( x I). f 0, δ > 0; x I; x x 0 < δ, f 0 (x) f 0 (x 0 ) < ε/3. f(x) f(x 0 ) f(x) f 0 (x) + f 0 (x) f 0 (x 0 ) + f 0 (x 0 ) f(x 0 ) < ε ( ),. I = [, b] f : coti., f f lim [ ] b b f b b f = b f. f f f (b ) sup f f 0. I ( ) I f C 1, f f {f } f C 1, f f. I f C 1, x 0 I; f (x 0 ), {f } f C 1, f f, f f. ( x ) x [ ] f(x) = lim f (x) = lim f (x 0 ) + f (t)dt = f(x 0 ) + lim f (t)dt. f x 0 x 0 C 1, f (x) = lim f (x), i.e., f f. f (x 0 ) α, f (x) = f (x 0 ) + f (t)dt α + lim f (t)dt, x 0 x 0 f(x) f(x 0 ) = α, f (x) = lim f (x) OK. f (x) = f (x) ( F (x) = f k (x).) =1 I f (x), x x 28

30 f (x) o I f (x) o I = 0, 1, 2,..., f (x) = x F (x). F. o (, ), 0 < < 1, [, ] i.e., ( 1, 1) (Weierstrss ) =0 x = 1 1 x o ( 1, 1), I {f }, {M }; f (x) M, M < f. [ ] f (x) f (x) M <, f (x) (=: F (x) ). F (x) = f k (x), F (x) F (x) f k (x) M, i.e., sup F F M 0. I k +1 k +1 k ( ) I = [, b] {f }, f b b f (x) = f (x). [ ] F, ( ) I C 1 {f }, f ( x 0 I; f (x 0 ) ), f f C 1, ( ) = f (x) f (x). [ ] F,. 4.3 (Power Series) (x x 0 ) : x 0 ( ) x 0 = 0 x x (1) x = α 0, x; x < α, x ( ). (2) x = β, x; x > β, x. [ ] (1) x = α (x/α) r = x/α < 1, x α r,, α, α 0 { α }, i.e., M; α M. x M r <. 0 < ρ 0 < α, r 0 = ρ 0 / α < 1, x ρ 0, x Mr 0,. 29

31 (2) x 0 ; x 0 > β x 0,, β. x r [ x < r { ( ). x } > r ]., r := sup. x ; x ( ) sup S, S,. if x r = 1/ρ ;, ρ = lim +1/ or = lim if exists, (0 ρ, 1/0 =, 1/ = 0.) [ ] ρ = lim +1/. +1 x +1 / x ρ x,, () 0 < ρ <. x < r = 1/ρ, ρ x < 1,, x > r = 1/ρ, ρ x > 1,. (b) ρ = 0 x, +1 x +1 / x 0 < 1,. (c) ρ = x, +1 x +1 / x > 1,. ρ = lim ( x ρ x ).,, ( x < r), (,, 0 < r 0 < r, x r 0 ),, ( ) f(x) = x r > 0. (1) 1 x 1 r, x < r, f (x) = 1 x 1. (2) f(x) ( r, r), k 1, f (k) (x) = k ( 1) ( k + 1) x k. = f () (0)/!. [ ] (1). 1 x 1 r. x x r r., x < r, 1 x 1. x < r 0 < r r 0, M; r 0 M, x = r 0 x/r 0 M x/r 0. 1 x M 1 x/r 0 <., 1 x 1 = x 1 1 x ( ) f(x) = x r > 0, x 0 f(t)dt = x+1 ( x < r). [ ] r, x < r. ( ),,,. R (x) := f () ( + θ(x )) (x )! 30

32 R (x) 0 f(x) = =0 f () () (x )!, = 0,.,. ( ) < x <, si x = =0 cos x = 1 < x 1, e x = =0 x! = 1 + x + x2 2 + x3 eθx + ( 3!! x ). ( 1) (2 + 1)! x2+1 = x x3 3! + x5 5! =0 =1 ( 1) (2)! x2 = 1 x2 2! + x4 4! si(θx + π/2) ( x ).! cos(θx + π/2) ( x ).! ( 1) 1 log(1 + x) = x = x x2 2 + x3 3 + ( ( 1) 1 (1 + θx) x ). R, x < 1, [ ] e x (1 + x) = =0 ( ) x = 1 + x + ( ) ( 1) ( + 1) =! ( 1) x 2 + ( 2! R eθ x x e x!! x 0 ( ) ( ) (1 + θx) x ). (x) = e x x /(!) +1 (x)/ (x) = x /( + 1) 0.. si x, cos x. log(1 + x) x < 1 1/(1 + x) = 0 ( 1) x, 0 x, x < 1. x = x = 1 x + x2 1 x + ( 1) 1 + x, 0 < log 2 = ( 1) x 1 + x < x 1 + x x = 1 0 ( 0) + 1 log 2 =

33 , x = 1. ( ) (1 + x) = +1 = ( ) 1. f(x) = ( ) x f (x) = ( ) x (1 + x)f (x) = + { ( ) ( )} ( + 1) + x = f(x) x ( x < 1), f (x) f(x) = 1 + x f(0) = 1,, f(x) = (1 + x). log f(x) = log(1 + x), i.e., f(x) = ±(1 + x) 32

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

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