Calculus i

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  1. 1. - . . . . 1 158 url: http://www.aegean.gr 832 00 Copyright , All rights reserved
  2. 2. , , . . . R - . . . . 2 158 1 1.1. , , N : N = {1, 2, 3, . . .}. . 0 , Z : Z = {0, 1, 2, 3, . . .}. , - . Q , m n , m Z n Z{0}. : Q = { m n | m Z, n Z{0}}.
  3. 3. , , . . . R - . . . . 3 158 , , - . 1.2. 1.2.1 k (+), () : 1. x, y k, x + y k. 1(i) x, y, z k,(x + y) + z = x + (y + z). 1(ii) k, 0, : x k, 0 + x = x + 0 = 0. 1(iii) x k, k, x, : x + (x) = (x) + x = 0. 1(iv) x, y k, x + y = y + x. 2. x, y k, x y k. 2(i) x, y, z k,(x y) z = x (y z). 2(ii) k, 1, 1 x = x 1 = x. 2(iii) x k x 0 k, x1 , : x x1 = x1 x = 1. 2(iv) x, y k, x y = y x. 2(v) x, y, z k, x (y + z) = x y + x z. 1.2.2 k - : 3. P k, k, : 3(i) x k : x P, x P, x = 0.
  4. 4. , , . . . R - . . . . 4 158 3(ii) x, y P, x + y P x y P. 1.2.3 3(i), 3(ii) k : x < y y x P. 1.2.4 Q . 1, 1(i), 1(ii), 1(iii), 1(iv), 2(i), 2(ii), 2(iii), 2(iv), 2(v), 3(i), 3(ii). 1.2.5 A k a k x A , x a. a A. a A , a A ( a = max A). 1.2.6 A k a k x A , a x. a A. A , a A ( a = min A). 1.2.7 supremum A a A : a a a A ( a = sup A). 1.2.8 inmum A a A : a a a A ( a = inf A). supremum, , . , a, a suprema A a a a a, a = a . inmum.
  5. 5. , , . . . R - . . . . 5 158 1.2.9 A , sup A = +. 1.2.10 , R : (a) R . (b) R supremum ( - ). 1.2.11 : A = {0, 1, 3}, B = {x R | x < 0}. A, B . 3 . max A = 3. sup B = 0. x B, x 0 0 B. a B, a < 0 ( a 2 < 0, a 2 B, a 2 < a, ( a < 0) 1 2 > 1, ). 0 a, . 1.2.12 A R. sup A A, a < sup A x A a < x < sup A, supremum A maximum, A .
  6. 6. , , . . . R - . . . . 6 158 1.3. - . 1 : 1.3.1 2 . : N Z Q R. 1.3.1. 1.3.2 . 1.4. - : : a, b R n N, : (a + b)n = n 0 an + n 1 an1 b + n 2 an2 b2 + + n n bn = n i=0 n! i!(n i)! ani bi ,
  7. 7. , , . . . R - . . . . 7 158 n k = n! k!(nk)! , k! = 1 2 3 . . . k 0! = 1. Bernoulli: () n 2 x > 1 x 0 : (1 + x)n > 1 + nx. () n x > 1 : (1 + x)n 1 + nx. 1.4.1 x R n N x 0, n + x > 0 : 1 + x n n < 1 + x n + 1 n+1 . 1.4.2 ( ) > 0 R, n0 , n N n n0 < n. , R n < n. 1.4.3 ( ) > 0, n0 N n > n0, 1 n < . 1.4.4 , , 1 n 0. 1.4.5 , .
  8. 8. , , . . . R - . . . . 8 158 1.5. R , R, , R : I1 = (, ) := {x R | < x < } . I2 = [, ] := {x R | x } . I3 = [, ) := {x R | x < } . I4 = (, ] := {x R | < x } . : sup I1 = , sup I2 = max I2 = , sup I3 = sup I4 = max I4 = . 1.5.1 - . = (, ) = ( ) [, ] = {} = { } (). 1.5.2 R R : J1 = (, ) := {x R | x < } . J2 = (, ] := {x R | x } . J3 = (, +) := {x R | < +} . J4 = [, +) := {x R | +} . sup J1 = , sup J2 = max J2 = sup J3 = sup J4 = +.
  9. 9. . . . . . . . . . . . . - . . . . 9 158 2 2.1. 2.1.1 - N. : N R. n N n ( (n)). - (n)nN (n), 1, 2, 3, . . . , n, . . . - n
  10. 10. . . . . . . . . . . . . - . . . . 10 158 n N. 2.1.2 (n)nN n = 2(n + 1) 3 - : 1, 3, 5, . . . , 2(n + 1) 3, . . . 2.1.3 (n)nN , n N n < n+1. (n)nN , n N n n+1. (n)nN , n N n+1 < n. (n)nN , n N n+1 n. . . 2.1.4 , , . .. n = (1)n = 1, 1, 1, . . . , (1)n , . . . . 2.2. . - - . . :
  11. 11. . . . . . . . . . . . . - . . . . 11 158 2.2.1 (n) R R, > 0 n0 : n N, n n0 |an | < . limn+ an = an . . 0 . (n) . 2.2.2 (n) + ( ), M R n0 N n N n n0 an > M (. an < M). limn+ an = + (. ) an + (. an . : + , . 2.2.3 an = 1 n 0.
  12. 12. . . . . . . . . . . . . - . . . . 12 158 y x - 6 . . . . . . n0 = 1 + 1 0 1 2.2.4 (an), an = k . limn+ an = k. 2.2.5 an = n R* . (an) : lim n+ an = 0 , = 0 + , > 0 , < 0 2.2.6 . 2.2.7 (an) (. ) M R an M n N. (an) m R m an n N. (an) .
  13. 13. . . . . . . . . . . . . - . . . . 13 158 2.2.8 1) (an) (. ) {an | n N} (.) . 2) (an) M > 0 |an| M n N. 2.2.9 (an) . 2.2.10 (an) limn+ an = , limn+ |an| = | |. 2.2.11 1) . , n = (1)n . limn+ |n| = limn+ |1| = 1, (n) . 2) = 0. lim n+ an = 0 lim n+ |an| = 0. 2.3. . - : 2.3.1 (an) (bn) limn+ an = 1 limn+ bn = 2. lim n+ (an + bn) = lim n+ an + lim n+ bn = 1 + 2.
  14. 14. . . . . . . . . . . . . - . . . . 14 158 2.3.2 . an = (1)n bn = (1)n+1 an + bn = 0, limn+(an + bn) = 0, (an) (bn) . : 2.3.3 (an) (bn) limn+ an = 1 limn+ bn = 2. lim n+ (anbn) = ( lim n+ an)( lim n+ bn) = 1 2. 2.3.4 (an) limn+ an = 0. k 0 n0 N |an| k n n0. 2.3.5 (bn) limn+ bn = 0. lim n+ 1 bn = 1 limn+ bn = 1 . - . 2.3.6 (an) (bn) limn+ an = 1 limn+ bn = 2 0. lim n+ an bn = limn+ an limn+ bn = 1 2 .
  15. 15. . . . . . . . . . . . . - . . . . 15 158 2.3.7 - . 2.3.8 1) . 2) . 2.4. . 2.4.1 (an) limn+ an = 0. , n0 N n n0 an . : 2.4.2 (an) (bn) k N an bn n k. lim n+ an lim n+ bn. 2.4.3 () (an) (n) limn+ an = limn+ n = . k N n k an bn n. lim n+ bn = .
  16. 16. . . . . . . . . . . . . - . . . . 16 158 - . 2.4.4 a > 0 . lim n+ n a = 1. 2.4.5 (an) > 0 . limn+ an = a > 0 lim n+ n an = 1. 2.4.6 : lim n+ n n = 1. 2.5. (n) (kn) , kn N N kn+1 > kn n N. akn = ( k)(n) k1 , k2 , . . . , kn , . . . (n) (kn )nN. (kn )nN (n).
  17. 17. . . . . . . . . . . . . - . . . . 17 158 2.5.1 n = 1 n n N, 1, 1 2 , 1 3 , 1 4 , . . . , 1 n , . . . . kn = 2n. kn : 1 2 , 1 4 , 1 6 , . . . , 1 2n , . . . 2.5.2 (n)nN , (kn )nN . 2.5.3 an = an , n N, a R. lim n+ an = 0 , |a| < 1 1 , a = 1 + , a > 1 a 1 (an). 2.5.4 (n)nN supremum inmum . 2.5.5 (Bolzano-Weirstrass) . 2.5.6 (an) . > 0 n0 N n, m n0 |an am | < , .
  18. 18. . . . . . . . . . . . . - . . . . 18 158 - Cauchy. Cauchy . , Cauchy, - |an am | |an a|+|am a|. Cauchy. 2.5.7 ( e) e, Euler, : n = 1 + 1 1! + 1 2! + + 1 n! , n = 1 + 1 n n , n N, . 2.5.8 Euler A. de Moivre x R eix = cos x + i sin x. x = e : ei + 1 = 0. 2.6. R {} . .
  19. 19. . . . . . . . . . . . . - . . . . 19 158 an bn = (an + bn) an a R bn = (an + bn) an bn = (anbn) an bn + = (anbn) an a R bn = (anbn) , a > 0 , a < 0 an a R bn = an bn 0 an a R{0} bn 0 (bn > 0) = an bn + , a > 0 , a < 0 an a R{0} bn 0 (bn < 0) = an bn , a > 0 + , a < 0 an + xn + = a xn n + an + xn = a xn n 0 an a xn + = a xn n + , a > 1 0 , 0 < a < 1 an 0 xn + = a xn n 0 an + xn x = a xn n + , x > 0 0 , x < 0 : (an + bn), limn+ an = + limn+ bn = . (an + bn) (+) + (). (anbn) limn+ an = 0 limn+ bn = . 0 . an bn limn+ an = limn+ bn = ( ) limn+ an = 0 limn+ bn = 0 ( 0 0 ). a xn n limn+ an = 1 limn+ xn = + ( 1+ )
  20. 20. . . . . . . . . . . . . - . . . . 20 158 , , , , . 2.6.1 1) an = 5n bn = n. limn+ an = +, limn+ bn = limn+(an + bn) = limn+ 4n = +. an = n bn = (1)n n limn+ an = +, limn+ bn = , (an + bn) = (1)n . 2) an = 1 n bn = 3n. limn+ 1 n = 0, limn+ bn = + limn+(anbn) = 3. an = 2 n3 bn = 3n, limn+ an = 0, limn+ bn = + limn+(anbn) = limn+ 6 n2 = 0.
  21. 21. - . . . . . . . 21 158 3 - 3.1. - , f : A R, A R. A f . A f (A) A. f (A) := {y R | x A f (x) = y}. f : A R , f (A) = B. f : A R 1-1 , : x1, x2 A, x1 x2 = f (x1) f (x2)
  22. 22. - . . . . . . . 22 158 , : x1, x2 A, f (x1) = f (x2) = x1 = x2. f1, f2 A B . f1(A) B f2 f1 : f2 f1 : A R (f2 f1)(x) = f2(f1(x)). 3.1.1 f1 f2 f1(x) = 1 x2 f2(x) = 3x + 2 A = [1, 1] B = R . f2 f1 f1 f2. : f1(A) R f2 f1 : (f2 f1)(x) = f2(f1(x)) = 3 1 x2 + 2 x A. f2 B = [1, 1 3 ] f2(B )