x2 + 2x 15 x 2 + 4x ) f(x) = x 2 + 2x 15 x2 + x 12 3) f(x) = x 3 + 3x 2 10x. x 3 + 3x 2 10x x 2 + x 12 10) f(x) = log 2.

Transkrypt

1 Příklady k 1 zápočtové písemce Definiční obor funkce Určete definiční obor funkce: x + x 15 1 f(x x + x 1 ( x + x 1 f(x log x + x 15 x + x 1 3 f(x x 3 + 3x 10x ( x 3 + 3x 10x f(x log x + x 1 x3 + 5x 5 f(x 8x + x + x 1 ( x + x 1 6 f(x log x 3 + 5x 8x + 7 f(x x 3x + 1 3x + 7x 6 ( 3x + 7x 6 8 f(x log x 3x + 1 x 3x 10 9 f(x x + 5x + 6 ( x + 5x f(x log x 3x 10 x3 x 11 f(x 3x + 18 x 5x + 6 ( x 5x f(x log x 3 x 3x + 18 Funkce inverzní K dané funkci f(x určete inverzní funkci, D(f, D(f 1, H(f a H(f 1 : Rozklad, kořeny a znaménko polynomu Rozložte v R na součin, najděte kořeny a určete znaménko polynomu: 1 6x 5 31x + 3x 3 + 6x 8x 8 3x 5 5x 11x 3 + 9x + 16x + 3 x x + 36x 3 + 6x 6x 9

2 x 5 x 0x x + 18x 9 5 x 5 x 9x x + 8x 1 6 3x 5 11x + 5x x 8x 7 3x 6 8x 5 + x 3 + x + 1x + 8 x 6 x 5 8x + 11x 3 + x + 8x 1 9 x 6 + 3x 5 x 13x 3 3x + x x 5 9x 0x x + x x 5 9x 3x 3 + 6x + 8x 1 x 6 x 5 11x + 3x x 8x x 5 x 13x 3 x + x 8 1 x 5 8x 5x 3 + 0x + x x 5 13x 0x x + 6x 1 16 x 6 7x x 81x x 17 x 5 6x + x x 3x x 5 x 10x 3 + 7x + 19 x 5 x + 3x 3 3x + x 0 x 5 + 9x 13x + 10x 8 1 x 6 x 5 + x 3x 3 3x x 5 + x 3 + 3x 3 x 6 + x 5 x + x 3 3x + 3x x 5 + 8x 3 6x 7x x 5 7x x + 8x x 5 8x + x 3 + 3x 15x 9 7 x 5 + 9x 6x 3 9x x 5 + 1x + 8x 3 13x x 5 + x 3x 3 1x + 36x x 6 + 5x + 7x + 3

3 31 9x 5 x 5x x 8x x 6 x x + Znaménko racionálně lomené funkce Určete znaménko a D(f funkce: 1 f(x x 3x + 1 x + 3x f(x x + 3x 18 x + 3x 10 3 f(x x3 + x 5x x + x 6 f(x x3 + 3x 9x x x 5 f(x x + x 15 x 5x f(x x3 3x + x 3x + 7 f(x x3 x + x x 5x 1 8 f(x x3 x 6 3x 5x 9 f(x 6x x 1 x 3 + x + x f(x x x 8 x x f(x x 8x + 16 x + x f(x x + 3x 1 x x + 13 f(x x x x 3 x 1 f(x 1 x x x 5 x 6x 3 15 f(x x3 + x x + 3 x 3 + x + 8x f(x x3 + x + 5x 6 x x 17 f(x x + x 8 x 3 + 3x 6x 8

4 18 f(x x3 + x x x 3 + x x 19 f(x x3 + x + 17x + 15 x x 5 0 f(x x + 3x + x f(x x 3x + x x f(x x x x + 3 f(x x5 + 1 x 1 f(x 1 x3 x f(x x + x 3 + 6x 15x + 9 x 3 x 3x f(x x3 3 x 9x + 3 x + 5x 6 7 f(x 8 f(x x 3 8 x x 1 5 3x x x 3 3x 15x f(x x3 5x + 96 x 3 3x 3x 30 f(x x3 + x 6x + 3 3x f(x (x + x 1(x + 1 x + x + 3 f(x (x 1(x + 1 x + x 6 33 f(x (x 3x 3 (x + x + 8 x f(x 3x3 5x + x x + x 5 35 f(x (x 3x + (x + 3 x + x f(x (x (x + 1 x + 5x + 6 Rozklad na parciální zlomky

5 Rozložte na součet polynomu a parciálních zlomků a určete definiční obor funkce: 1 f(x x + x 3 16x + 16x x + x 15 f(x x5 x 9x 3 + 1x + x + 11 x 3 x 8x f(x x5 3x 16x 3 13x x + 6 x 3 x 7x f(x x3 x x + 1 x 5 x + x 3 5 f(x 6 f(x 7 f(x 5x + 13x 3x 3 + 6x 3x 6 3 x x 3 x + 1 x + x 1 8 f(x x + x 3 x 9 f(x x + 16 x f(x 11 f(x x + x x + 1 x x x + 1 f(x x + 5x x 3 + x Limita funkce Určete limitu funkce: 1 lim x 1 x 1 x 3 x lim x 3 x 3 x x 3 3 lim x x + 5x + x x + 3 lim x 0 5 lim x sin x sin(x x + 5x + x + 6 lim x x + 5x 1 x 3 x lim x 3 x + 3x 9 x + 3 Derivace funkce

6 Zderivujte funkci a určete její definiční obor: 1 f(x (x 1 e x f(x cos x 3(1 + sin x 3 f(x e x cos(3x f(x cos x sin x 5 f(x log 1 x + 1 x 6 f(x ln x + 1 x 1 7 f(x x3 sin x cos x 1 Tečna ke grafu funkce Napište rovnici tečny a normály ke grafu funkce f(x v bodě T : 1 f(x x + x + 8, T [0,?] f(x arctg x, T [?, π ] 3 f(x sin x, T [ π 6,?] f(x e (x 1, T [?, 1] 5 f(x x, T [3,?] 6 f(x x + x, T [,?] 7 f(x x, T [1,?] (x f(x e x, T [0,?] 9 f(x ln x, T [e,?] NUMERICKÁ MATEMATIKA Metoda bisekce (půlení intervalů S chybou menší než ε najděte přibližné řešení rovnice: 1 0 Metoda regula-falsi Interpolační polynom Nalezněte interpolační polynom procházející body

7 1 [-, 3], [-1, -], [1, ], [, -] [-, 3], [-1, ], [1, ], [, -3] 3 [-, 0], [-1, -5], [1, ], [, -] [0, 0], [-1, -], [1, ], [, -] 5 [1, -], [-, -1], [, 1], [-, ] 6 [-5, -], [-, 0], [, 10], [5, ] 7 [, -10], [-, 0] [-, -], [0, ], [, 0] 8 [-3, ], [1, ], [-1, ], [, 3] 9 [3, -], [1, ], [-1, ], [-3, ] 10 [-1, ], [-5, ], [0, ], [1, -] 11 [1, -], [0, ], [-5, ], [-1, 8] 1 [-1, ], [-5, ], [-5, ], [0, ] 13 [-1, ], [ 1, 1 ], [0, ], [-, -]

8 Řešení Definiční obor funkce 1 f(x (x 3(x + 5, D(f (x (x + 6 ( (x (x + 6 f(x log (x 3(x f(x (x 3(x +, D(f x(x (x + 5 ( x(x (x + 5 f(x log (x 3(x + 5 f(x (x (x 1, D(f (3 x(x + ( (3 x(x + 6 f(x log (x (x 1 (x 3+ 5(x f(x, D(f (3x (x + 3 ( (3x (x f(x log (x 3+ 5(x 3 5 x 5 9 f(x (x + (x 5 (x + (x + 3, D(f x + 3 ( ( (x + (x + 3 x f(x log log (x + (x 5 x 5 (x 3 11 f(x (x + (x 3(x + (x 3(x, D(f x ( ( (x 3(x x 1 f(x log log (x 3 (x + (x 3(x + Rozklad polynomu 1 (x 3 (x + 1(3x + 1 (x (x + 1 (3x (x + 3 (x + 1 (x 1 (x + 3(x 3(x + 1(x 1(x 1 5 (x + 3(x (x + 1(x 1 6 (3x + 1(x (x + 1(x 1

9 7 (3 x + 1(x (x + 1(x (x 1(x (x + 3(x + x (x 1(x (x + 3(x + (x + x (x 6(x 1(x + ( x + x (x 6(x 1(x + (x (x (x (x 1(x + (x (x (x (x 1(x + (x (x (x 8(x + 1(x (x 1(x + 15 (6x 1(x + 1(x (x 3(x + 16 (x 1 3 (x ( x (x 1(x (x + 1(x 5(x (x (x + 1 (x 1(3x (x 1(x + (x (x + (x 1(x + (x x x (x + 3(x 1+ 5 (x 1 5 x(x + 3(x x(x + 3(x + 1(1 x (x + 3(x + 1(x 1 ( x 5 (x 3 (x + (x 1 6 (x 3 (x 1(x (x + 3(x + 1(x (x (x (3x 1 (x + x + 3(x (x 3 (x + 1 (x (x + 3(x (3x 1 (x + 1(x 3+ 5 (x (x (x + (x (x + (x + Znaménko racionálně lomené funkce

10 1 f(x f(x 3 f(x 3+ (x 5 (x 3 5 (x 1(x + (x (x + 5 (x 3(x + 6 x(x 1(x + 5 (x (x + 3 f(x (x 1 (x + 5 (3 x(x + 5 f(x (x 3(x + 5 (x 3(x x + 5 x 6 f(x (x 1 (x + (x 1(x 7 f(x (x 1(x + (x (x f(x (x (x + x + 3 (x (3x f(x 10 f(x (x 1(3x + 1 (x + 1(x + x + 3 (x (x + (x + 1 (x 1 11 f(x (x + (x (5 x(x + 1 (x 1(x + x x + x + 3 3x f(x 3+ (x + 17 (x 3 17 x x + 13 f(x (x 1 3(x x 3 x 1 f(x (x (x + 1 x 3 (x 3(x + 15 f(x (x x + 3(1 x (x + (x + 16 f(x 17 f(x 18 f(x 19 f(x (x + (1 x(x 3 (x + 1(x (x (x + (x + 1(x (x + 1 x + 1 x(x 1(x + (x + 1(x 1(x + (x + 3(x + 1(5 x (x + 1(x 5 0 f(x (x + (x + 1 (x + (x x + x 3

11 1 f(x (x (x 1 (x + (x (x + 1 f(x (x (x + (x + (x (x + 3 f(x (x + 1(x x 3 + x x + 1 x x 3 + x x + 1 (x 1(x + 1(x + 1 (x 1(x + 1 [čitatel lze ještě rozložit, ale zbývající kořeny jsou všechny komplexní; viz graf funkce x 5 + 1] f(x (1 x3 (1 + x + x x + 1 [jmenovatel lze ještě rozložit, ale zbývající kořeny jsou všechny komplexní; viz graf x + 1]; ( porovnání koeficientů: x + 1 (x + ax + b(x + cx + d (x + x + 1(x f(x (x 3x + 3(1 x(x + 3 (x + (x 3x f(x (3x 1(x 3(x + 3 3(x 1(x f(x 6(x (x + x + (3x (x f(x 3+ (x + 9 (x (x 1(x + 3(x 1 (1 x(x + 3 x + 9 f(x 30 f(x (x 6(x (x + 8 x(x 3 1(x 3+ 1 (x 3x + 3(1 x ( 3 3x + 3 ( 3 9x 3 6x f(x (x + x 1(x + 1 x + x + (x 1 (x + 1 (x + 1 x + x + (x 1(x + 1 x + x + (x + x 1 : x 1 1 ± ± (x + x + nelze rozložit v R, protože D 1 16 < 0 NB: 1; 1 D(f R f(x (x 1(x + 1 x + x 6 NB: 3; 1; 1; D(f R { 3; } (x 1(x + 1(x + 1 (x (x (x 1(x + 1 (x (x + 3

12 33 f(x (x 3x 3 (x + x + 8 x + 3 x3 (x 3 (x + x + 8 x + 3 (x + x + 8 nelze rozložit v R, protože D 3 < 0 NB: 3; 0; 3 D(f R { 3} f(x 3x3 5x + x x + x 5 (3x 5x + x x + x 5 3 (x (x 1x (x 1(x + 5 (3x (x 1x (x 1(x + 5 (3x x x + 5 (3x 5x + : x 1 5 ± ± NB: 5; 0; 3 ; D(f R { 5; 1} f(x (x 3x + (x + 3 x + x + 1 (x (x 1(x + 3 x + x + 1 (x + x + 1 nelze rozložit v R, protože D 1 < 0 NB: 3; 1; D(f R f(x (x (x + 1 x + 5x + 6 NB: 3; ; D(f R { 3; } (x (x + 1 (x + (x Rozklad na parciální zlomky 1 f(x x x x 3, x 5, x 3 x + 5 f(x x 1 1 x + 3 (x +, x 3, x x f(x x + x 1 (x + 1 +, x 1, x x f(x 1 x + x + 3 x 1 3 x, x 0, x (x

13 5 f(x 1 x 1 3(x + +, x, x 1, x 1 3(x f(x 3 x x 3 3 (x 3(x + 1, D(f R { 1; 3} x 1 : 0 A + B A B x 0 : 3 A 3B 7 f(x 3 (x 3 3 (x (x 3(x + 1 A x 3 + B x Ax + A + Bx 3B } 3 B 3B 3 B B 3 A 3 / (x 3(x + 1 x + 1 x + x 1 x + 1 (x + 6(x, D(f R { 6; } x 1 : 1 A+B A 1 B x 0 : 1 A + 6B 5 8(x (x x + 1 (x + 6(x A x B x x + 1 Ax A + Bx + 6B } 8 f(x x + x 3 x x + x (x, D(f R {0; } x : 0 B + C x 1 : 1 A B x 0 : A A 1 1 x 1 x 1 x / (x + 6(x 1 (1 B + 6B 1 + B + 6B 3 8B B 3 8 A 5 8 x + x (x A x + B x + C x / x (x x + Ax A + Bx B + Cx } B B 1 9 f(x x + 16 x 16 x + 16 (x (x +, D(f R {±} } x + 16 (x (x + A x + B x + C 1 / (x 16 x + 16 Ax + A + Bx B x 1 : 1 A + B } x 0 : 16 A B/ : 1 + B A B A + B 3 B B 3 A (x 3 (x +

14 10 f(x x + x x + 1 x + (x 1, D(f R {1} x + (x 1 A (x 1 + B x 1 / (x 1 x 1 : 1 B x 0 : A B } A 3 x + A + Bx B 3 (x x 1 11 f(x x x x + x (x, D(f R {} x (x A (x + B x / (x x 1 : 1 B x 0 : 0 A B } A x A + Bx B (x + 1 x 1 f(x x + 5x x + 5x x 3 + x x (x + x : B + C x 1 : 5 A + B x 0 : 1A A 1 1 x + 3 x 1 x + x + 5x x (x +, D(f R { ; 0} A x + B x + C x + / x (x + x + 5x Ax + A + Bx + Bx + Cx } B B 6 B 3 C f(x x 8 x x 6 x 8 (x + (x 3, D(f R { ; 3} x 8 (x + (x 3 A x + + B x 3 / (x + (x 3 x 1 : 1 A + B / 3 x 0 : 8 3A + B 1 x + 3 x 1 x + } 5 5B x 8 Ax 3A + Bx + B B 1 A 1 B Limita funkce x 1 1 lim x 1 x 3 x 0 0 lim x 1 x 3 lim x 3 x x lim x 3 (x 1(x + 1 x (x 1 x 3 (x 3(x + 1 lim x 3 lim x 1 x + 1 x 1 1 x + 1 1

15 3 lim x x + 5x + x x + 3 sin x lim x 0 sin(x x + 5x + 5 lim x x + lim x 0 0 lim sin x x lim x sin x cos x lim x 0 x (1 + 5 x + x x (1 x + 3 x (x + 1(x + x + (x + 5x + : x 1 5 ± lim x x + 5x 1 x 3 x + 1 x + 3x 9 7 lim x 3 x + 3 lim 0 0 lim x 3 x (x + 3x 9 : x 1 3 ± lim x 1 x x 1 9 lim x 1 1 x x 1 [mm] Derivace 1 1 lim 0 0 lim x 1 Zderivujte a určete D(f a D(f : 1 f(x (x 1 e x f (x 1 1 e x + (x 1 x ex 1 e x 1 cos x 1 5 ± x + 1 lim x 1 x 3 ( x + 5 x 1 x 3 x 3 (1 1 x + 1 x 3 (x 3 (x + 3 x + 3 x 1 x (x 1(x + 1 lim x 1 3 ± x 3 lim x x (x 1(x + 1 lim x 1 x x e x (1 ex x e x + e x 1 e x ex + e x x e x 1 e x 1 e x D(f: 1 e x 0 1 e x 0 x D(f (, 0 1 D(f : 1 e x > 0 1 > e x 0 > x D(f (, 0 f(x cos x 3(1 + sin x ( sin x(1 + sin x cos x(cos x f (x (1 + sin x (1 + sin x 1 3 sin x sin x cos x 1 (sin x + 1 (1 + sin x 3 (1 + sin x

16 D(f: 1 + sin x 0 sin x 1 x 3π + kπ -1 D(f D(f R { 3π + kπ} 3 f(x e x cos(3x f (x e x ( x cos(3x + e x ( sin(3x 3 e x ( cos(3x 3 sin(3x D(f : e x 1 e x e x 0 (platí vždy D(f D(f R f(x cos x sin x f (x sin x sin x + cos x sin x cos x sin x sin x + cos x + cos x sin 3 x 1 + cos x sin 3 x sin x ( sin x + cos x sin x π π D(f D(f R {kπ} 5 f(x log 1 x + 1 x f (x 1 1 x ln 1 x x x x (1 x ln 1 x D(f: 1 x > 0 1 > x x ( 1; 1 x 0 1 x > 0 (platí vždy D(f : x ±1 x 0 D(f R {0; ±1} D(f ( 1; 1 {0} D(f 6 f(x ln x + 1 x 1 f (x x 1 (x 1 (x + 1 x 1 x 1 x + 1 (x 1 (x + 1(x 1 x 1 D(f : x+1 x 1 > 0 x 1 0 x 1 7 f(x x3 sin x cos x D(f : x ±1 f (x (3x sin x + x 3 cos x (cos x 1 + x 3 sin x (cos x 1 D(f ( ; 1 (1; D(f

17 3x sin x cos x 3x sin x + x 3 cos x x 3 cos x + x 3 sin x (cos x 1 3x sin x(cos x 1 + x 3 (cos x + sin x x 3 cos x (cos x 1 3x sin x(cos x 1 + x 3 (1 cos x (cos x 1(3x sin x x 3 x (3 sin x x (cos x 1 (cos x 1 cos x 1 D(f: cos x 1 x kπ 1 D(f D(f R {kπ} 8 f(x log 1 x log 10 1 x f (x 1 1 x ln 10 1 x 1 x x (1 x ln 10 D(f: 1 x > 0 1 > x x ( 1; 1 1 x > 0 (platí vždy D(f : 1 x D(f ( 1; 1 D(f x ±1 D(f R {±1} Tečna ke grafu funkce 1 T [0, 8], t : y x + 8, n : x + y 0 T [1, π ], t : y 1 x + π 1, n : y x + π + 3 T [ π 6, ], t : y x + π, n : y x + + π T [1, 1], t : y x 1, n : y 1 x T [3, 9], t :, n : 6 T [, ], t : y 10x 16, n : y 1 10 x T [1, 1 ], t : x + y 0, n : x y f(x e x, f(0 e 0 1 T [0, 1] f (x e x k f (0 e 0 t : y x + q T t : q q 1 t : y x f(x ln x, f(e ln e 1 T [e, 1] f (x 1 x k f (e 1 e n : y 1 x + q T n : q q 1 n : y 1 x + 1 y x + x + y 0

18 t : y 1 e x + q T t : 1 1 e e + q q q 0 n : y ex + q T n : 1 e e + q q 1 + e n : y ex e t : y 1 e x ey x x ey 0 Interpolační polynom 1 P (x 5 x3 1 6 x + 13 x P (x 1 x3 3 x + 1 x P (x 3 x3 1 6 x + 5x 3 P (x x 3 + 3x 5 P (x 1 10 x x x 15 6 P (x 1 10 x3 5 1 x x P (x 1 1 x x 1 x + 8 P (x 1 15 x x 1 15 x P (x 1 8 x3 3 8 x x P (x 1 x3 3x 5 x + 11 P (x 1 x3 5 x + 1 P (x x 3 6x 5x + 13 P (x 6x 3 15x + 9x +