3 6 I f x si f x = x cos x + x x = x = /π =,,... x f x = f f x = f..4. [a, b] f a, b fb fa b a c.4 = f c, a < c < b.5. f a a + h θ fa + h = fa + f a +

Transcription

1 I 6 I. I. f a I 3. fx = fa I a. fx fx 45 + I f I I I I f I a 6 fx fa. x a f a f a I I 7.. f a f a F, G F x = α, Gx = β F x ± Gx = α ± β, F xgx = αβ * 5 /5 5 5 a iterval; a ope a closed iterval. a fuctio. 3 cotiuous; a cotiuous fuctio. 4 it; right-had it; left-had it. 5 IV 6 differetiable; the differetial coefficiet; the derivative. 7 a theorem; a corollary; a propositio; a lemma; a proof. 8 fx fa fx fa = x a fx fa = x a fx = fx fa + fa x a x a = f a =. = fx fa + fa = + fa = fa.... x x fx = x = x x <, gx = 3 x f g I f I x x f f x f : I x f x R f 9.3. I f x si fx = x + x x x =. 8 IV 9 derivative.

2 3 6 I f x si f x = x cos x + x x = x = /π =,,... x f x = f f x = f..4. [a, b] f a, b fb fa b a c.4 = f c, a < c < b.5. f a a + h θ fa + h = fa + f a + θhh, < θ <. h = θ h > f [a, a + h]..4 b = a + h fa + h = fa + f ch a < c < a + h c θ = c a/h a < c < a + h < θ < the mea value theorem I 6 4 h < [a + h, a].4 fa fa + h a a + h = fa + h fa h = f c a + h < c < a c h < c = a + θh < θ <.6. fx = x a = 9, b = = c c c > 9 = 3 +, 9 < c < c < = < 3.7. c < > 3 + > = > = < < 3.7 3, I I f x = f I I a a x I fx = fa x > a [a, x].4 fx fa x a = f c, a < c < x the square root. a approximatio. 3 a decimal fractio; the first decimal place.

3 5 6 I c a x I c I f c = fx = fa x < a [x, a].8. I F, G f 4 Gx = F x + C C F, G f F x = G x = fx Hx = Gx F x I H x =.7 I.9. a, b f a, b f a, b 5 a, b x, x x < x [x, x ].4 fx fx x x = f c a <x < c < x < b c f c > f c < x x > fx fx > fx fx < x < x fx < fx fx > fx f.. f f 6 f c 7 f c > c I f I f f c > c I f x > I IV 4 a primitive; a costat. 5 mootoe icreasig decreasig; positive; egative. 6 C - 7 c f I f c f c > c f.3 f f = / > I f.4 I-4.. [a, b] f c b a fx dx = b afc, a < c < b..4 IV.3. [a, b] f [a, b] I f c I 8 x I fx fc fx fc f I c I I c I 9 c I I = [a, b] c a, b I a, b I.4. I f I c f c f c = c I δ c δ, c + δ I f c 8 the maximum; the miimum. 9 a iterior poit f c = h fc + h fc h

4 7 6 I f c h < δ h fc + h fc fc + h fc h { < h < δ δ < h < h f c.5. [a, b] F a, b F a = F b F c =, c a < c < b F [a, b].3 c, c [a, b] F c c c, c a, b F c = F c F a, b F = c, c a, b c.4 F c =.4. F x = fx fa fb fa x a b a.5.6. [a, b] f, g a, b ga gb a, b g x c fb fa gb ga = f c g c, a < c < b. Michel Rolle 65-79; Fr; Rolle s theorem. Augusti Louis Cauchy , Fr;.4 ; Joseph-Louis Lagrage , It. I 6 8 F x = fx fa.5 I. fb fa gx ga gb ga I R f f f f f f 3 k k k I f k k f k k k f k f k x, d k dx k fx, d k y dx k. y = fx y.7. fx = x f k x =... k + x k k > f k x = fx = e x k f k x = e x. 3 fx = cos x k f k x = k cos x, f k+ x = k+ si x m f m x = cosx + mπ.8. I f I f C - 3 the secod derivative; k the k-th derivative.

5 9 6 I I f f C - I k f k f k C k - k C k - C f a I + a + h I h. fa + h = fa + f ah + f ah + +! f ah + R + h = j= j! f j ah j + R + h, R + h = θ 5 [, ] f k a + th F t := t k h k k! h+ +! f + a + θh, < θ < + t + fa + h f k a h k k! F = F = fa+h.5 I-9.. fx = x a = 9, h =, =.9 = , < θ < 9 + θ 4 Sir Brook Taylor , E 5. j = h h = I 6 θ θ, < = = = > = = = < < 3.64 = , 4,... I-.9.. f a, b I +.3 fb = fa + f ab a + f ab a ! f ab a + R +, a b c R + = b a+ f + c +!

6 6 I I I- 5. si., ta.. radia? I- IC 346.8km IC km I-3. I-4. I-5.3,, [, ] I I-7 6 fx, gx [a, a + h a, a + h g x a < x < a + h fa = ga = f x + g x I-8 si x x x ta x x, x + 5 x 3 x x fx + gx 5 x 3 x,. x + x I 6 I- α k α αα... α k + =, k k! α = α 7 k + x = + x + x + + x = k fx f k = k =,,..., fx I-. fx = x, a =, =. fx = e x, a =, = ;. fx = e x, a,. fx = cos x, a =, = ; = k k fx = si x, a =, = 3; = k k fx = ta x, a =, = 3. fx = ta x, a =, = 4 fx = log + x, a =, = 3; fx = + x α, a =, = 3; α I-. 3 I-3. fx = x a =, h =., =.9 R 3 h R 3 h. = 3 x k I Guillaume Fracois Atoie, Marquis de l Hôpital, 66 74, Fr; l Hospital 7 biomial coefficiets

7 II 6 4 II. II..9 R + h. I-, I-3.. fx a C + -. fa + h = fa + f ah + +! f ah + R + h R + h h h =....9 h. h h. f I := a δ, a + δ δ > C + - h < δ h a + h I f I C + - f + I.8.3 f + I I := [a δ, a + δ ] m m M := max{ m, m }.9 h I R + h := fa + h k! f k ah k = h+ +! f + a + θ h h θ h < θ h < a + θ h h I R + h h+ M, R + h +! h M h +! M h +! R +h h M h +!. h * 5 8 / remaider the pricipal terms max{a, b} a b.3. e x a bx x x a, b. fx = e x, a =, h = x, = e x = + x + R 3 x x + R 3 x, x x = e x a bx x = a x + b x + + R 3x x x X := a x + b = a + x b x x x a X x a = X = b/x b = a = b = e x x x x = x + R 3x x =..4. f, g. fx gx =.3 fx = o gx x a o o 34 gx x a. fx gx.3 3 Edmud Gerorg Herma Ladau; , De. 4 Ladau s symbol; o O

8 5 6 II x a fx gx 5 fx gx = o hx x a.4 fx = gx + o hx x a..5 II-4. fx = o x a fx = m, x m = ox x m > cos x = + ox x.6. fx = o gx x a. o gx x = ox, x 3 = ox x x x 3 =..7. fx a C fa + h = k! f k ah k + oh h f a I + a + h I h.9 R + h.6 R + h = h+! 5 order u f + a + uh du. II 6 6 x = a + h fx fa = x a f t dt = x a t x f t dt = [ t xf t ] x t=x t xf t dt t=a a x = f ax a t x f t dt a [ t x = f ] t=x x ax a f t x t + f t dt t=a a = f x x a t x ax a + f 3 a + f t dt 6 =... = k! f k ax a k +! k= t = ua + ux II. R + h : =! = x a x a+! t x f + t dt a x u f + ua + ux du a t x f + t dt.. h h.9. fx = e x a =, h = x,.9.7 e x = + x +! x + +! x + R + x, R + x = +! eθ x x + < θ <

9 7 6 II θ f II-5 < θ < e x x e θx x < x < < e x = e x x R + x e x x + =,,,.... +!. x R +x =.7 x.8 e x = + x +! x + 3! x3 + =. II-6. x.9. cos x =! x + 4! x4 6! x = si x = x 3! x3 + 5! x5 7! x = k! xk k k! xk, k k +! xk+... fx = log + x < x.9 a =, h = x k f k x = k+ k! +x k.9 log + x = x x + 3 x3 + + x + R +, x + R + = + + θx + < θ < θ x. R + x II 6 8 < x <.8 h := x < h < R + x + u + ux u = h + uh + du = u h+ uh du uh = h+ + du s hs ds. s = u/ uh s hs h < h <. R + h + s h ds h + = + h + h...3 log + x = x x + x3 3 = k= k+ x k < x k.3 x > x x > x f a C -.8. R h I h R h = h I.4 fa + h = fa + f ah +! f ah + = k! f k ah k f a 6.4 a = the Taylor expasio. 7 the Maclauri expasio; Coli Maclauri , Scotlad. 8.9 fa + h h fa + h

10 9 6 II y x y = e /x x > ; x y = e /x x < ; x..3..8,.9,.,.3 e x, cos x, si x, log + x.. a C - f a I.4 f a 9 f f C ω - C -.3. f e /x x > fx = x. fh f h + h fh f h.3 h e /h = h + h = h h = f fh f = = h h = u + ue u =, y x II 6 f x = x e /x x > x.. f f C - k P k e /x x >.5 f k x = x x P k t t P t =, P k+ t = t P k t P kt k =,,,... II-9 f C - x fx = k! f k x k = k! xk =, x > x fx > C - ±. e x x < gx = x.4. α k α αα... α k + = k >, k k! α = 9 real aalytic; a aalytic fuctio. C ω - of class C-omega. the biomial coefficiet

11 6 II.5 II-8. =, =, =, =,..., = k. k = 6,.... =, = 8, 3.6. k k k > k =.7. α k α + α α = +. k k k α α + = k k = = αα... α k + k! + αα... α k + k! αα... α k + k + α k + k! α + αα... α + k + k! = α + k.8. α α α α + x α = + x + + x + ox x o.4 fx = + x α f k x = αα... α k + + x α k.7 C k k α α C k. II 6.8 x I- α I α + x α = + αx +.. II.3 + x = x k = k α x + = α x k < x <. k k x k < x <... x x /! = x N < x N N > N x! = xn N! = xn N! N + N x N N N... N + xn N! N + N N N N = C N + N + C := xn N! N N + < N/N + <.. P x P x x + e x =. N

12 3 6 II II 6 4 P x N.9 fx = e x, a =, h = x >, = N + e x = + x +! x + + θ < θ < N +! xn+ + eθx N +! xn+ N +! xn+. P x = p N x N + p N x N + + p x + p p N x > P x N +! P x e x = x N+ N +! x p N + pn + + p x + x x N.3. a I a I \{a} = {x I x a} f + fx = A II- II- fx = A, II fx = A fx x fx = fa + f ax a + +! f ax a = k! f k ax a k. fx = x 5 3x 3 + x x + 4 f + f. a =, a = e x x. x x x cos x + x x 4. x si x x x 3. x 3 ta x 3x x 3 x 5. II-3 II-4 II-5 II-6 II-7 II-8 II-9 log + x x + x. x x 3 x si x ta x x 3. si x x x ta 3 x. a, b.5 ta x a si x + bx. x x 5 e fx = e x.9 fx = e x, a =, h =, = e θ < e < θ <.6 < e < 3 3 e e = m/ m, 4.9 fx = e x,a =, h = 5! 6.9,. cos X, si X cosh x, sih x x =.5.3.5

13 III 6 6 III. III. I f a x fx fa fx fa 3.. fx = x 4 x = R C - k f k = e / x x fx = x = II.3 x = 3.3. f a ε f < x a < ε x fx < fa fx > fa a x fx < fa fx > fa 3.4. fx = x x = f x = x fx = x =. fx = x 3 3x x = x = * 6 5 /8 6 8 the maximum the miimum. a maximal; a local maxima; a miimal; a local miima: a extremal f x = a C - 3 A. fx x = a f a = B. A f a fx x = a C. f a =, f a > f a < fx x = a 3.6. fx = x 3 3x f x = 3x x + f x = x = x = 3.5 B, f f x = 6x f >, f < 3.5 C fx x =, x = A fx = x C B m = f a m >. m = f a R h fa + h = fa + mh + R h = h h R h h mh h fa + h fa mh h 4 m > h > h < h fa + h > fa h > ; fa + h < fa h < 3 A, B f a C f 4

14 7 6 III ε < h < ε fa+h > fa, < h < ε fa + h < fa f x = a 3.5 B m > R h/mh h δ h < δ m > R h mh < III 6 8 R 3 h h mh h fa + h fa mh h m > h h fa + h > fa fx x = a m < h < δ m h < R h < m h III. h < δ mh m h < fa + h fa < mh + m h < h < δ h = h fa + h fa > mh mh = mh >, > h > δ h = h fa + h fa < mh + m h = mh < ε h < ε fa + h fa 3.5 C m = f a m > f a =, f a = m fa + h = fa + mh + R 3 h h R 3 h h = R R := {x, y x, y } = {x, y x, y R} =. a, b R ε U ε a, b = {x, y R x a + y b < ε } a, b ε- 5 R U a, b A ε U ε a, b A R U 6 P, Q A U R ε- a ε-eighborhood a ope set. 6 coected; pathwise coectedess R 7 a domai.

15 9 6 III D R f a, b D ε x, y U ε a, b x, y a, b fx, y < fa, b fx, y > fa, b , f x, y = a, b C - 3. fa + h, b + k = fa, b + f f a, bh + a, bk x y + f x a, bh + f x y a, bhk + f a, bk + R y 3 h, k R 3 h, k h,k, h + k = a, b h, k F t = fa + th, b + tk F [, ] C - F.9 F = F + F + F + 3! F θ < θ < θ 8 F = fa + h, b + k = fa, b, F = f f a, bh + a, bk x y F = f x a, bh + f x y a, bhk + f a, bk y F θ = 3 f x 3 h f x y h k f x y hk + 3 f y 3 k3 8 the chai rule, 3., 4 III 6 3 a + θh, b + θk f C - f 3 f h,k, x a + θh, b + θk = 3 f a, b 3 x3 h, k = r cos t, r si t r > h, k, r 3 f x 3 a + θh, b + θk h 3 h + k F θ = 3 f x a + θh, b + θkr 3 cos3 t h,k, F θ/h + k = f 3. h, k 3. fa + h, b + k = fa, b + f f a, bh + x y a, bk + R h, k, h,k, R h, k h + k = h, k f f h h a, b, a, b = dfa, bh h = x y k k dfa, b = f x a, b, f y a, b a, b f 9 h, k fxx a, b f xy a, b h 3.3 h, k = t h Hess fa, bh, f yx a, b f yy a, b k fxx a, b f xy a, b Hess fa, b := f yx a, b f yy a, b 9 the total differetial 3

16 3 6 III t h h Hess fa, b f a, b 3.. R D C - f a, b D f x a, b = f a, b = y f a, b ε h +k < ε fa + h, b + k > fa, b h < ε fa + h, b > fa, b F h := fa + h, b h = 3.5 F = f x a, b Gk = fa, b + k f y a, b = 3.. R D C - f a, b D := f x a, f b y A := f a, b x f x a, b = f a, b = y f a, b a, b x y = det Hess fa, b, > A > fx, y x, y = a, b > A < fx, y x, y = a, b < fx, y x, y = a, b 3 a symmetric matrix. the Hessia matrix; Hesse, Ludwig Otto, 8 874, de. III h k φh, k := Ah + Bhk + Ck A, B, C h, k, φh, k > A > AC B > h, k, φh, k < A < AC B > φ AC B < AC B = φ φ = h, k, A h + B A k + AC B k A A φh, k = C k + B C h + AC B h C C Bhk A = C = fa + h, b + k fa, b = φh, k + R3h, k, R 3h, k h,k, h + k = A := f xxa, b, B := f xya, b, C := f yya, b φh, k := Ah + Bhk + Ck h + k R 3 h, k φh, k fa + h, b + k fh, k φh, k 3.3

17 33 6 III III.3 R D C - f x = t x,..., x fx fa + h = fa + dfah + t h Hess fah + R 3 h, f dfa = a,..., f a, x x f x a... f a x x Hess fa = f f a... x x x a R 3h h h = 3.4. f a dfa = dfa = Hess fa f a dfa = Hess fa f a x,..., x φx,..., x = a ij x i x j i,j= III 6 34 x i x j = x j x i a ij a ji φx,..., x = a ij x i x j, a ij = a ji. i,j= x = t x,..., x A = a ij 3.5 φx = t xax A A φ 3.5. A A P µ... t µ... P AP =.. t P P = E = µ µ,..., µ A 3.5 X = t X,..., X := t P x φ = µ X + + µ X N µ,..., µ x φx > 3.5

18 35 6 III µ,..., µ x φx < 3.5 µ,..., µ φx III III 6 36 x, y 3. fx, y x, y = /3, /3 /7 III- fx, y = ax + by e x y a, b 74 III- III-3 fx, y = x 4 + x y + y 4 x 3 + y 3 R D 3 f f xx D f D III- fx = x 4 x = 3. III- C - f x = a... f x = a 3. 3 fx = x x = 3.4 fx = x 4 x f III A B 3.7 III-4 III-5 III-6 III-7 III C 3.7 fx = x 4 + px 3 + qx p, q 3 f x = B m < 3.5 C 3.5 f a =, f a = R R, {x, y R y > }, {x, y R y }, {x, y R x + y }, {x, y R x + y < } III-9 III- 3.3 fx, y = x 3 xy + y 3 f xx, y =, f yx, y = x, y 3

19 IV 6 38 IV. IV. x x x = x, x < x = x x x x, y 4. x, x x, x = x, x = x, xy = x y a δ 4. x a < δ a δ < x < a + δ. 4.. x, y a x + y x + y, b x y x y. x, y =, x, y, 4. x + y x + y x + y + x + y = x + y x + y x + y + x + y > a a = x + x y + y x + y = xy xy x = y + x y y + x y, y = x + y x x + y x = x + x y b {a, a, a,... } {a } {a } 4.. {a } α ε N 3 N a α < ε * 6 4 /9 the triagle iequality. AB + BC AB + BC {a } α A sequece {a } coverges to α. diverge 3 a = α a α α {a } {a } 4.3. {a } 4 M N N a > M a < M a = {a } M a M 5 {a } α N N a α a 3 {a } {/a } : {a } α 4. ε N a α < N N M := max{ a, a,..., a N, α, α + } 6 M : 4. ε α/ > N a α < α/ N N 3: ε 4.3 M /ε N a > /ε N N /a < ε 4.5. c a = c {a } c {a } α {b } β a a a + b α + β, b a b αβ, c α b β β : ε N = N a c = c c = < ε a N, N N a α < ε N b β < ε N = max{n, N } N 4. 4 to diverge to the positive egative ifiity. 5 {a } 6 max{... } {... }

20 39 6 IV IV 6 4 a + b α + β = a α + b β IV. a α + b β < ε + ε = ε. b 4.4 a M M ε N a α < ε ε, b β < N β M β = a b αβ = a b a β + a β αβ = a b β + βa α c b /b /β 4.4 N N b β/ b β N b β < β ε/ N N = max{n, N } 4.6. {a }, {b } α, β a b α β {a }, {b }, {c } a c b {a }, {b } α c = α 3 {a } { a } {a } 4 {a }, {b } a b {a } {b } : β < α ε := α β/3 N a α < ε N b β < ε N, N N = max{n, N } ε α ε < a N b N < β + ε 3 α + β 3 3 β + α α β. 3 : a α c α b α c α max{ a α, b α } =,,,... {a }, {b } α ε N N a α < ε, b α < ε N N c α < ε 3: a a a 4: M N a > M N N N b a > M 7 R A R x A x M x M M 8 M A 4.7. A A 9 sup A if A 4.8. α A x A x α a < α a < x α x A α A α a A 4.9. A M A M A Q 4. A := {x Q x < } = ,.4,.45,.443,.44,... A Q 7 the set of real umbers. 8 bouded; bouded from above; bouded from below. 9 the supreimum, the ifimum. cotiuity of real umbers. a axiom

21 4 6 IV {a } {a } a j a j+ a j a j+ j =,,, {a } 4. {a } α a α 4.8 ε α ε < a N N 4.8 > N α ε < a N a α a α < ε {a } α 4.3. {} {} 4. α 4. ε N α < N α < < α + N M M < 4.5. = + = M 4.4, N > M N N M < N < {} I = [α, β] {p } < <... {p j } j= I Archimedes, B.C. 87 B.C.; Gr. 3 IV 6 4 k Q j := {p j, p j+,... } 4. q j := sup Q j Q j+ Q j q j+ q j Q j α q j α {q j} 4. γ {p } γ = j j Q j + q j + p m > q j + /j m m j + m j+ q j+ /j < p j γ {q j +} {q } γ j {p j } γ IV I a I f x a α ε δ < x a < δ x I fx α < ε fx = α fx α x a ε δ 5 < x a < δ x I fx α < ε x a f α fx = α + II a I a I \ {a} = {x I x a} f fx = α, + fx = α fx = α ε δ, δ < x a < δ fx α < ε δ < x a < fx α < ε δ = mi{δ, δ } < x a < δ fx α < ε 4 ε-δ Augusti Louis Cauchy, Fr

22 43 6 IV 4.9. I a f x a α δ < x a < δ x I fx > 4.7 ε = α/ < x a < δ x I fx α < α δ < x a < δ fx α > α, fx > α >. 4.. I a I f x a M δ < x a < δ x I fx > M fx = + fx + x a b, + f x + α fx = α x + ε m > b x > m x I fx α < ε 3 b, + f x + fx = + x + M m > b x > m x I fx > M x 4.. I a I I \ {a} f fx = α a = a, a I \ {a} =,,,... {a } fa = α fx α x a {a } fa α ε fx α < x a < δ fx α < ε δ a a δ N a a < δ IV 6 44 N a a N N < a a < δ fa α < ε ε fa α fx = α {a } fa α ε δ < x a < δ fx α ε x {a } fa α ε δ = / < a a < fa α ε a {a } fa α ε {fa } α f x a α {a } 6 IV.4 a = a, {fa } α I f I a fx = fa I I f a C -. f a > f I a I C - f x f x = f a > 4.9 < x a < δ f x > δ.9 f a δ, a + δ a {a } {fa } α {fa } α {a }

23 45 6 IV I = [α, β] f I f Y := {fx α x β} A y y Y y Y y = fx x [α, β] 4.6 {x j } γ [α, β] j y j = fx j j + j + 4. f fx j fγ f Y η := sup Y 4.8 x [α, β] fx η, η / < y = fx x [α, β] {x } γ [α, β] {x j } 4.6 η j η = j j fx j j η = η. fγ = j fx j = η f γ IV [α, β] f fα <, fβ > fγ =, α < γ < β γ Y := {x [α, β] fx } γ = sup Y γ x Y γ / < x γ {x } γ 4. fx fγ x Y fx < 4.6 fγ γ < β fγ < 4.9 f γ δ < x < γ + δ fx < δ γ x x Y γ fγ = 7 the itermediate value theorem. IV P, Q, R 9 P Q P Q P Q P Q P P P Q P Q 4.3 P Q P Q.. P Q P Q P Q P Q P Q P Q. x P x, Qx x P x x P x P x x x P x ad x P x or ad, or. x P x x P x x P x x P x. P = {a } α {a } α 4. P ε [ N { N a α < ε }] 8 de Morga s laws; Augustus de Morga, 86 87, 9 true; false P Q P ad Q; P Q P or Q; P ot P ; P Q P implies Q.

24 47 6 IV IV 6 48 IV- IV- IV-3 P ε [ N { N a α ε }] {a } α ε N N a α ε fx = α ε δ < x a < δ fx α ε x IV r {r } r > r = 3 < r < 4 r I- + = = 9 {p } a := p + p + p + + p = p k k =,,,... {a } p.p p p 3... IV-4 IV-5 IV-6 IV-7 IV-8 IV-9 IV- s { s } s > s = 3 s < {a } 4. a = + =,, 3,.... e I a I \ {a} f, g x a α, β fx fx + gx α + β, fxgx αβ, gx α β β α c f x α si x fx = x c x = f f 3 f α c = α c 4.6 c α f [a, b] Y := [fa, fb] y Y fx = y x [a, b] y fi R Y y fx = y x R f f f f fi 4.5 fγ = η = sup Y f γ

25 V 6 5 V. V. {a } 5. a + a + a + = s = a k = a + a + + a =,,,... {s } {s } {s } c 5. c = a + a + = {a } {p } q = p + =,,,... {q } V- 5. {s } {t = s +} c = c c = t s = t s = a + = a {a } 4 5. * 6 9 / a series; a ifiite series; 5. a j + 5. s a IV a a = s := = k= k m m m m s s m = k = l k k= l= k= l m l m l = l= k= l l= l l = m. m {s } r r r < r = r < r IV- r {r } 5.3 r < IV- r k = r+ r r. {a } a the harmoic progressio; a geometric progressio ; a arithmetic progressio. 7 a oegative-term series;

26 5 6 V 5.6. {a }, {b } a b b a a s := b a k, t := s t =,,,... : b β t β s t β {s } s : a {s } 4.6 {t } N a b 5.8. p 5.3 p = + p + 3 p = p 5.3 p < fx = x x : p p = / b k V 6 5 p : p k= p = =. kk = = < p < : p = q q, p+ p+ = q q = q + q.9 θ, + q = + q qq + + θ q + + q q q q = + q + < q <, p q p+ p+ p < k p+ k p+ = p+ q q q k= q q + q 8 9 p = = π Leohard Euler, , Sz. / 5.7 a alteratig series.

27 53 6 V 5.9. {q } a = q =,,,... a = q = q q + q q q = q > s := a j a j := s j, b j := s j b j a j = s j s j = j q j = q j > a j+ a j = s j+ s j = j+ q j+ + j q j = q j q j+ b j+ b j = s j+ s j = q j+ q j+, a < b b... b {a } 4. α {b } β β α = b a = b a = q = α = β ε N j N a j α = s j α < ε, b j α = s j α < ε N = N N s α < ε {s } α = +. =..3 x = log V-4 V = + = π. 4 {[a, b ]} V = 3 + VI V = 9 3π + 3 log. {a } A := {a, a +,... } = {a k k } =,,,..., a + := sup A, a := if A =,,, {a } a + = + =,,,... {a } {a + } 3 {a } {a } {a + } : {a,..., a } A A a + = + : {a } A a + A A + a + A + a + + a+ 3: {a } α α a a + α =,,,... {a + } {a } 5.. {a } 5.5 {a + } {a } + 4. sup a := a+, if a := a {a } 3 3 the it superior; the it iferior sup if

28 55 6 V 5.3. { + } { } 5.4. α {a } ε N N a < α + ε ε N m N α ε < a m m 5.5 {a + } α ε > N a + α < ε N a + N < α + ε N a sup A N = a + N ε N a + N = sup AN a+ N ε x x A N 4.8 A N 5.4 x = a m m N m α, ε > m N a + m α < ε/ N N N a + = sup A 4.8 a + ε/ < a m m m a + ε < a m α + ε, a+ α < ε α ε < a m m m a m A a m sup A = a + α ε < a m a + α a + < ε a + α < ε ε N a + α 5.5. {a } {a } α β β = α ε N N a α + ε a β < ε ε < a β α β + ε ε < α β. ε, N α ε < am m N α β ε < am β < ε α β < ε. α β < ε ε ε = /m m m α = β β α 5.4 ε > N N a α + ε, α ε a N N a α ε < ε {a } α V {a }, {b } a = α >, sup b = β supa b = αβ. {a b } αβ 5.4 {,. : ε ε ε = mi, } ε α+ β a α N a α < ε N β = sup b 5.4 N b < β + ε N N = max{n, N } N a b < α + ε β + ε αβ + α + β ε + ε αβ + ε. { } : ε ε ε = mi, ε, α N 4α β 3 a α < ε N 3 N 5.4 m max{n, N 3 } m β ε < b m αβ ε < a m b m α > a m b m αβ + ε a m β ε αβ + ε a m αβ a m ε + ε a m α β α + ε ε + ε ε β αε + ε > {p } 4 ε N m, N m, p m p < ε 5.8. {p } p ε N p p < ε N m, N p m p = p m p p p p m p + p p < ε + ε = ε. 4 a Cauchy sequece.

29 57 6 V 5.9. {p } 5.7 ε = m, N p m p < N m = N N p p N < k p k M M = max{ p, p,..., p N, p N + } V-7 5. {p } 5.9 α := if p, α + := sup p k m, N p m p < /3k N α N p < α + + /3k N m N α + /3k < p m m α N 3 p > α /3k N 3 m N 3 α + /3k > p m m N = max{n, N, N 3 } m, m N m, m α + < pm < α+ + 3k 3k, α 3k < p m < α + 3k p m p m < /3k α + α < k k α + = α 5.5 {p } 5.. a ε N N m +m a k < ε. k= 5. V a a a 5. N N m N N +m a k k= +m k= 5. a a k < ε. +m k= a k < ε a a a 5.6. {a }, {b } N a b N b a a {a } N a cr c r < r < a {a } N a c p c > p < a absolute covergece; to coverge absolutely. 5.4

30 59 6 V VI 5.9. {a } α := sup a α < a α > a α < ε := α/, r := +α < 5.4 N a α + ε = r N a r 5.7 a α > ε = α / >, r = + α/ > N N a > α ε = r a > {a } 5. a α = 5.8 α = V coditioal covergece; to coverge coditioally. V 6 6 V V- a a = a = a V- {p } c q = p + =,,,... {q } c 4. V V = = ta x V-5 r a = r =,,,... r < {a } r = /+h h > +h +h+ h. r {a } 3 a V-6 = + +, = log. V-7 V-8 {a } α = a +/a α < a α > a α = V-9 r < r > p r. p = α r. α

31 VI 6 6 VI. {a } x 6. a x = a + a x + a x +... x 6. I x I 6. fx = a x, x I = {x R 6. }. II x 6. f.4 fa + h h f a x = a+h.4 fx = a x a a 6. VI x = r x < r x 6. x = r x > r x 6. * 6 6 / 9 a power series, 5. a r N N a r < N a x = a r x r ρ ρ := x < r 5.7 a x x > r x x = r r := sup C, C := { x 6. } r r = + r r i ii x < r x > r 6. r = + x 6. r = x C, r i x < r r x = ε > 4.8 r ε < s s C s 6. x = r ε < s ii x > r x x + r/ > r r r i, ii ii r C i r C r = sup C the radius of covergece.

32 63 6 VI r sup a = r a x = x a sup a x = x sup a = x r. 5.9 α = x /r a = r a + r V a = x + x x 3 + = x. x x < x > a = sup a = sup =. 3 Cauchy, Augusti Louis, ; Hadamard, Jacques Salomo, d Alembert, Jea Le Rod; VI a = + = =. a + x + x x 3 + = x = + x x + x! + x3 3! + = x! x < 6.4 e x x +!x + 3!x 3 + =!x pt px 3 pt, qt p q x q x x3 3 + x5 5 x7 7 + = a = m= m x m+ m + = a x = m = m + ; m

33 65 6 VI a b + := sup{ k { } a k k } = sup k k, k k IV- sup a = b+ = = 6.4 s 3 s + 5 s + = m= m s m m s < s > s x x 6.5 x < x > ! x+,! x + si x, cos x. r 6. x = ±r 6. VI-3. x + x x 3 + = x x = ± x x + x3 3 x4 4 + = x x < x > x = log. x = 5.8 VI x = ± π 4 4 +x+ x + x3 3 + = x x = ± 5.8 VI I 6. I f f f 6.6 fx = f x x I; f x = a k x k r r, r J sup f f =. J J := [a, b] r, r δ := mi{r b, a r} > J [ r + δ, r δ] J J J = [ r + δ, r δ] x = r δ a r δ N a r δ N N x J f x fx = k=+ k=+ a k x k x r δ k=+ k ρ+ sup J f f a k x k = a k r δ k x k r δ k=+ ρ := x ρ r δ < 6.3. r 6. f r, r

34 67 6 VI α r, r fx = fα x α d := mi{r α, α + r} > α r + d, r d J := [ r + d, r d] ε 6. N N f x fx sup f f < ε J 3 N f N x α < δ f N x f N α < ε 3. δ δ x α < δ fx fα = fx f N x + f N x f N α + f N α fα f α x J. fx f N x + f N x f N α + f N α fα < ε 6., 3, 4 r r, r 6.3 α = ±r r x = r x = r 6. fx = fr fx = f r, fx := x r x r+ a x f r, r r > 6. f x r < x < r x ft dt = a x + a x + a a 3 x3 + = x. = 5 Abel, Niels Herik; VI f 6. x ft dt x x sup [ x,x] x f t dt = ft f t x dt ft f t dt ft f t dt sup ft f t x [ x,x] f x x x f t dt = k= a k k xk = = a x r > 6.6 f r, r 6.7 f x = a + a x + = 5.6 IV- sup + a + x r < x < r. a = sup a r g 6.5 x r, r x gt dt = a x = fx f f x = gx r < x < r = itegratio differetiatio by term ad term.

35 69 6 VI fx = x x3 3 + x5 5 x7 7 + =, x + + < x < f x = x + x 4 x 6 + = + x < x <. fx = x f t dt = x dt + t = ta x < x < x = = x + = + x ta x = π 4 x= 6.4 x r > x = rt r x = = r x = u u = 6.8. a x x = fx = X x {a } σ = fx := a x, X := a. a k = a + a + + a VI 6 7 {σ } 4.4 {σ X} 6.9 σ X A =,,,... A ε {σ } X M 6. M σ X < ε A 6. M 6. δ = ε 4M + A 6. M N > M + N a = σ σ < x < x N N N N a x = σ + σ σ x = σ + σ x σ x + = = N N = = x N σ x σ x + = N σ x x + + σ N x N = x N σ Xx + M = x σ Xx + N N =M+ Xx N + xx x + σ N x N M = x σ Xx + N =M+ σ x + σ N x N + σ N x N σ Xx σ Xx + X x xn x + σ N x N M N = x σ Xx + σ Xx =M+

36 7 6 VI + X + σ N Xx N. VI 6 7 VI < x < 6.9, 6., 6. N a x X M x σ X x + < xa M x + x N =M+ N =M+ σ X x ε 4 x + ε 4 xn xam + + x ε x N M 4 xm+ + ε x 4 x ε 4δ + ε 4 + ε 4. N fx X fx X ε 4 + x δ < x < < x < δ x + σ N X x N VI- VI- VI-3 VI-4 VI = = 5. 3 = x 3 + = dx + x = 3 9 3π + 3 log fx X 3 4 ε < ε ε > fx = X x

37 I I- fx = x, a = 4, b = = c, 4 < c < 5 c 5 = + c < + 4 = + 4 =.5, 5 > + 5 > +.5 = > + 5 =.. < 5 <.5 fx = si x, a =, b =..4 si. si. = cos c si. =. cos c < c <. c [, π] cos x si. =. cos c <. cos =., si. >. cos. =.. si. >.. =..99 >.99. < y < y > y.99 < si. <. si..9 fx = ta x, a =, b =..4 ta. si. = + ta c ta. =. + ta c < c <. c [, π] ta x ta. =. + ta c >. + ta =., si. <. + ta. <. +. =.. =... < ta. <. ta.. I- + x IC fxkm f I-3 x a/x a = / / /fa fa = / / / + X = X I-4 f [a, b] a x b x [a, x] x F x := ft dt a x b a F [a, b] b F x = fx, F a =, F b = fx dx a F.4 F b F a b a = F c a < c < b c c b fx dx = b afc a I-5 fx = x ; fx = x; fx = a < c < b. { x < x < x =, I-6.4 F x := fx fa fb fa x a f [a, b] b a a, b F F a = F b =.5 F c = a < c < b c c = F c = f fb fa c b a c.6 gb ga F x := fx fa fb fa gb ga gx ga f, g [a, b] a, b F F a = F b =.5 = F c = f c fb fa gb ga g c, a < c < b c g c c I-7 < u < h u f, g [a, a + u].6 f c u fa + u fa fa + u g = = c u ga + u ga ga + u, a < c u < a + u c u u + c u a + f x/g x + I-8, log 5 3, +.

38 I-9 t F F t := f k a + th t k h k k! + t + fa + h f k F F t : =. = = hf k+ a + th t k h k + k! f k a + th k t k h k k! + t fa + h f k+ a + th t k h k+ k! k= f k a + th t k h k k! + t fa + h f k+ a + th t k h k+ k! f k+ a + th t k h k+ k! + t fa + h = f + a + th t h +! + t fa + h f k a F = h k + fa + h k! F = fa + h f k a h k k! f k a h k k! f k a h k k! f k a h k k! f k a h k. k! f k a h k = fa + h k! F F θ =, < θ < θ θ F θ = + θ [ f + a + θh h + +! θ θ I- fx fx := + x k x k x = f = =. x = f x = + x f = m =,..., f m x =... m++x m k= ] f k a h k fa + h. k! k x k k = =. k=m f m =... m + mm... m kk... k m+ x k k m... m + =... m + m! =. m! f = f = = f = fx I- a + h = a + a h 8 a 3 h a+θh 5 h 3 < θ <. e h = + h + h + 6 eθh h 3 < θ <. e a+h = e a hk k! + ea+θh h + +! < θ <. cos h = k m m= m! hm + k h k cosθh < θ <. k! si h = k m m= m+! hm + k h k+ siθh < θ <. k+! ta h = h + h ϕθh < θ <. ϕt = t + t + 3t.

39 ta h = h h3 3 + θh +5θh 4 5+θh 5 < θ < log + h = h h + h3 3 h 4 4+θh 4 < θ < log + h = k+ h k k= + h + k ++θh + < θ < +h α = +αh+ αα h + αα α h αα α α 3h4 + 4 θh α 4 < θ < + h α = α k h k + α h + + θh + α < θ <. I- 3.6 I-3. = θ 8 6 5/ < θ < < R 3. < <. < II 3 II- k > f k x.9 fx = 8 + 5x + 64x + 37x 3 + x 4 + x 5 f + = , II- /;/; /6; /5; /3; /; /6 II-3 a =, b =, /6 II-7 cosh x = xk k!, sih x = xk+ k+!. III 35 III- x fx = x 4 = f f 3. e / x x > e / x x > fx := x =, gx := x =, e / x x < e / x x < f g C - x = f f g 3 x fx = x > = f III- x = ± x = III-3 A f a = f a fx = x 3 f = f C f a f a =, f a > fx = x 4 f f = f = III-4 9p 3q < III-5 p, q = x = 3p/4 9p 3q =, p x = 9p 3q >, p >, q > x = x = 8 3p 9p 3q x = 8 3p + 9p 3q 9p 3q >, p <, q > x = 8 3p 9p 3q x = 8 3p + 9p 3q x = 9p 3q >, q < x = 8 3p 9p 3q x = x = 8 3p + 9p 3q p = q = x = m = f a < m >. R h fa + h = fa + mh + R h = h h R h h mh h fa + h fa mh h m < h > h < h fa + h < fa h > ; fa + h > fa h < ε < h < ε fa + h > fa, < h < ε fa + h < fa f x = a III-6 f a = m > f a =. fa + h fa = m R 3 h h + R 3 h h h = h R 3 h/h δ < h < δ R 3 h h < m 4 δ δ, δ h fa + h fa = m h + R 4 h m h R 4 h > m h m 4 h = m 4 h4 >.

40 f f a = m <. fa + h fa = m h + R 3 h R 3 h h h = h R 3 h/h δ < h < δ R 3 h h < m 4 δ δ, δ h m < fa + h fa = m h + R 4 h m h + R 4 h < m h m 4 h = m 4 h4 <. f III-7 III-8 III-9 f k a = k =,,...,, f a. Yes/Yes/No/No/Yes. A h + B A k + AC B k A A φh, k = C k + B C h + AC B h C C Bhk A = C = A >, AC B > AC B /A > φh, k = A h + BA k + AC B k. A h + B/Ak = k = h, k =, h, k, φh, k > h, k, φh, k > A = h φh, = A h φh, = Ah > A > k φ B/Ak, k = AC B k /A > A > AC B > A <, AC B > AC B /A < φh, k = A h + BA k + AC B k A h, k =, h, k, φh, k < h, k, φh, k < A h φh, = Ah < A < k φ B/Ak, k = AC B k /A < A < AC B > AC B = A = C = B = A h + B A k A φh, k = C k + B C h C A = C = φh, k A φ B/Ak, k =, C φh, B/Ch =, A = C = h, k φh, k = AC B < A AC B /A A φh, k = A h + BA k AC B + k A φh, A φ B/Ak, k A C A = C = φh, k = Bhk AC B = B < B φt, t = Bt φt, t = Bt AC B φh, k 3 III- f xx, y = 3x y, f yx, y = x + 3y f x = f y = x, y =,, 3, 3 f x = f y =,, 3, f 3. 3 f xx = 6x, f xy =, f yy = 6y 3. x, y =, f xx f yy f xy = = < f x, y = 3, 3 fxxfyy fxy = = 3 > xx = > f f 3, 3 = 7 III- f x x, y = f y x, y = x, y =,, ±,,, ± a, b >, < a < b ±, a/e

41 < b < a, ± b/e III- f xx, y = f yx, y =, x, y =, 3 4, 3 4,,,,,, 3 4, 3, 7/56 4, 3., 3. fx, x = x 3 x 4 x =, x, y fx, y > f, fx, y < f, f, III-3 fx, y f xx + f yy = f yy = f xx f xx f xxf yy f xy = f xx f xy < 3. f xx IV 47 IV- h := r I- IV- r = + h = h k k + h + h + + h h r + r = 4.5 r 3 r = r = 4.5 r r < r < / r > IV- / r r r 4 fx = x {r } α {fr } fα = α fr = r r > IV- {fr } {r } {r } 4.3 M = N r > N r N > r < r N+ = r r N < {r } I- IV-3 + = = p + + a + = a + p + + a. {a } p 9 a 9 k = 9 / + / 9 = {a } 4. {a } IV M N a M N = N = s > M b M > N = M /s + N s N s = M /s + s > M /s s = M 4.3 s = s = 3 {/ s } IV-5 {a } a =, a = 9/4 > = a 3 α := + +, β := + α < β m m α m+ β m = α m + β m = α m β m αm + + = α βα m + α m β + + β m + αm + = + αm + α m β + + β m + αm + + mβm + αm + + βm + αm + = + αm β m.

42 a + a a + a > {a } {a } I- a = + = k k k... k + = k k! k = k! k. a + k= k = + {a } 3 4. {a }... k k! + = 3. IV-6 a I \ {a} {x } x a fx α gx β 4. fx = α, gx = β 4.5 fx + gx = α + β {x } 4. fx + gx = α + β IV-8 fx = x α x β β = max{α, } α = x = x β f = α <, fβ = { α α = αα > α > α > < α < fc = c < c < β IV-9 c = α, c = α c, c > c = c = α α = c c = c c c + c c + + c k, l c k cl > 6 9 y Y fa y fb y y = fa y = fb x fa < y < fb F x := fx y a x b x x fx = fx = y x < x fx < fx x x x > x fx > fx x x x = x y [fa, fb] f y x = f y ε > x := x ε, x + = x + ε y := fx, y + := fx + f y < y = fx < y + δ = mi{y + y, y y } y y δ, y + δ f y x ε, x + ε f IV- Y = {fx α x β} fγ = η η Y η = sup Y Y y y η Y y y = fx α x β x fx η = fγ f γ V 6 V- 5. a = / =,, V- ε {p } c N N p c < ε N = N N + N = N q c = p + c < ε V-3 p < p + < p x p x N N N N p = + p = + p dx = = = N N + x p dx = + x p dx = + = p + N p+ = + p + N p+ p + + p +.

43 V-4 N s N = N = p {s N } p {s N } 4. {s N } < p < p + > p x p x + N N N + p = + p dx = = N + N+ x p dx = x p dx = N + p+. = p + N p = x + x N = N+ dx = logn + x N. p p ++ = + N N = + ++ /4. N 3 N x x + x 4 + N x N = x N+ + x = + x x N+ + x. + x = x + x 4 + N x N + x N+ + x. [, X]. N+N+ ta X = X 3 X3 + + N X x N+ N + XN+ +R N X, R N X = + x dx. X x N+ R N X + x dx X X N ta X = x N+ dx = XN+ N +. + X+ X X X = π 4 = = +. V-5 r = / + h h > I- +h = +h+ h + + h +h+ h h. a = r = + h h = h. r a = r + {a } 3 r s := a k = kr k rs = kr k kr k+ + = kr k k r k k= = + kr k k r k r + k= k= = r + + k k r k = r + + r k k= = r + + r r r. r < r s = r r r + r r r. a = s = r r. k= s = r r + r+ r r + r r r r r r = + r + + r r r.

44 + r + + a r = s = a k = + + V-6 =,, 3,... = / log log 9 = log 3 log e =. log V-7 V-4 V-43 s π/4 5.8 {s } V-8 α < ε := α/ 4. N N + a α < ε N a + a < α + ε = + α r r < a + r a r a... r + N a N, A a Ar N. r < 5.7 α > ε = α / N N a + a α < ε N a + a > α ε = + α. r r > N a Ar N A a a 3 α = a = p α = 5.8 a V-9 a = p r a + a = + p r + p r = + p r r V-8 a = p r a + α a = + r + α r = αα... α +! α = r r + V-8 VI 7 VI- VI- a x! αα... α + r a /a + =!/ +! = / m pt = p m t m +p m t m + +p t+p p m p m N p N a a = p + p + = p m m + p m m + + p + p p m + m + p m + m + + p + + p s s 3! + s 5! = +! s s s = x x x 3! + x4 5! = x x x3 3! + x5 5! = +! x +! x+

45 x + VI x = ± 5. VI x =. log x = 5.8 p = x = x = 3 s / x = ± x = ± V-4 V x = ± 5.8 p = + x = x x + 3 x3... = 6.4 fx < x < 6.5 f x = x + x = x = + x f = < x < 6.4 x a = x 3+ /3 + a + a = x 3 x 3 V-8 x < x > x = 5.9 gx < x < 6.5 g x = x 3 + x 6 = g = x dt gx = + t 3 = x 3 = x dt 3 + t + x 6 x 3 = + x 3 < x < dt x + t + tdt t + t + 3 tdt t + t x dt t 3 + = 3 log + x + 6 log x + x + [ ta 3 t ] x 3 = 3 log + x + 6 log x + x + 3 ta x + π. 3 6 < x < x = = gx = x 3 log + π 3 3. x fx = x f t dt = dt = log + x < x < + t 5.9 x = = = fx = log + x = log. x x VI x3+ = x x4 4 + x7 7...