Maths g lykeiou_raptis

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  1. 1. . 1.600
  2. 2. . 2
  3. 3. . 3 1. . 4 2. .. . 11 3. . 14 4. . 18 5. . . 26 6. .. . 33 7. ...35 8. De L Hospital .. 39 9. . 40 10. . 43 11. BOLZANO . 47 12. . . 55 13. .. . . 59 14. ... 63 15. Rolle .. 66 16. . .. 71 17. ... .. . 76 18. ....81 19. . 85 20. . 92 21. . 96 22. . .97 23. . . . 107
  4. 4. . 4 Domain of definition domain 1.1 . f(x) = 2x 1 x2 + 2x 3 . f(x) = x + 4 |2x 3| 5 . f(x) = x2 3x 4 . f(x) = ln(x2 + 3x + 10) . f(x) = 5x 4 3 |x + 1| . f(x) = ex ln(x 2) A. 1.2 . f(x) = 5x x 3 x 2 2x . f(x) = x + 1 + 3 x . f(x) = x 5 x + 2 . f(x) = 5 3 |x 2| 1.3 . f(x) = 5 |x + 1| . f(x) = |2x + 1| 7 . f(x) = ln x + 2 x 4 . f(x) = x + 1 x3 3x2 + x + 2 1.4 . f(x) = 1 x2 x 2 + 5 x 3 . f(x) = x2 4 . f(x) = 12 x x2 . f(x) = ln(1 x2) 1.5 . f(x) = x2 e2x ex 2 . f(x) = 4 ln(x 1) 1 . f(x) = ln(ex 1) . f(x) = ex 1 ex 2 1.6 . f(x) = 4 |3 x| lnx . f(x) = log(|x| 3) . f(x) = ex e x e x . f(x) = x 1 x 2 1.7 ) f(x) = x 7 x2 9x + 8 ) f(x) = x3 8 2x 16 ) f(x) = x 2 x2 9x + 8 +ln(81 x2) 1.8 ) f(x) = ln x 1 x + 2 ) f(x) = x2+ 1 lnx 1 ) f(x) = ln(x 3) Leibniz 1694. Euler , 1748, (function) f(x). LEONARD EULER (17071783) 18 1.
  5. 5. . 5 1.9 f(x) = x2 1 . ) f f(3) f(f(2)). ) f(x) = 8 . ) f(+ ) f( ) , 0 B. 1.10 f(x) = x2 + 10x + 2 x3+ f(1)=3 . ) f ) f(x) 1. 1.11 f(x) = ln(x + ) ln( x) f(1) = 1 f(6) = 0 . : ) ) f 1.12 f(x) = 2x 3 , x 4 x2 1 , 4 < x 10 . f(3) , f(4) , f(10). . 1.13 f(x) = x + , 6 x < 1 x2 + , 1 x < 7 f(2) = 5 f(5) = 24 . ) f ) ) f(1) f(f(3)) ) f(x) = 3 . 1.14 f(x) = lnx + , x > 0 ex , x 0 f(0) = 1 f(1) = 3 . ) f ) 1.15 f : 3f(x) 2f 1 x = 5x2 , x 0 . . 1.16 f : f(x) + x 2x2 f(x + 1) 3x 1 , x . 1.17 f : f(x) + 2f(3 x) = 2x 1 , x . 1.18 f : f(x) + x x2 f(x + 1) x , x . 1.19 f: f(x)+3 f(2x)=4x , x . : ) f(1) ) f 1.20 f : f(x) + 3x x2 f(x 2) + 7x 10 , x 1.21 f (0, +) f(x y) = f(x) + f(y) , x , y > 0. : ) f(1)=0 ) f(y) = f 1 y , y > 0 ) f x y = f(x) f(y) , x , y > 0 1.22 f : f(x 3) 2f(1 x) = x2 2x , x . 1.23 f : f2(x) = 4ex(f(x) ex) , x . 1.24 f : {0 , 1} f(x) + 3f 1 x = 2x + 1 x 1 , x {0 , 1}
  6. 6. . 6 1.25 : ) f(x) = x2 + 2x 8 . f(x) = |2x 1| 5 . f(x) = ln(x 2) . f(x) = ex + 2 . f(x) = x2 + x 6 x 2 . 1.26 : . f(x) = ln2 x lnx . f(x) = x3 3x2 + 4 . f(x) = e2x 3ex + 2 . f(x) = ex2x2 1 1.27 xx : ) f(x) = 2x2 + 5x + 3 ) f(x) = x3 2x2 5x + 6 ) f(x) = 9x2 9x 4 3x + 1 1.28 Cf xx : ) f(x) = x3 4x ) f(x) = 1 lnx ) f(x) = 2ex 2 1.29 : ) f(x) = x3 + 3x2 2x + 1 g(x) = x2 + x + 1 ) f(x) = x3 g(x) = x2 + x 1 ) f(x) = x lnx 2x g(x) = x ) f(x) = 32x+5 g(x) = 3x+2 + 2 1.30 : ) f(x) = x2 3x + 2 , g(x) = 6 x ) f(x) = x + 1 + 1 x + 1 , g(x) = x2 + x + 2 1.31 f g : ) f(x) = x3 + x g(x) = 3x2 2 ) f(x) = ln2 x g(x) = lnx + 2 ) f(x) = g(x) + x2 1 , x . 1.32 Cf Cg : ) f(x) = x2 g(x) = 6x 8 ) f(x) = x3 3x2 2 g(x) = x2 4x + 1 ) f(x) = x2 ex g(x) = x2 ex 1 1.33 f(x) = x 1 x2 x 12 . : ) f ) Cf ) Cf xx 1.34 f(x) = x2 + x + 3 x 2 g(x) = x2 + 2x . : ) Cf , Cg ) Cg Cf 1.35 : f(x) = 4x 2x+1 g(x) = 2x+2 8 . : ) Cf , Cg ) Cf Cg 1.36 f(x) = x3 x + 2 : 6x y 4 = 0. : ) Cf ) Cf . 1.37 f(x) = ex 1 , x 0 lnx , x > 0 Cf : ) xx ) y = 1 1.38 f(x) = x + 5 x2 + x + 1 ) ) Cf (1 , + 1) 1.39 : f(x) = x3 + 2 g(x) = 2x2 + 5x , , . , Cf Cg x = 1 x = 2. Cf , Cg
  7. 7. . 7 1.40 f(x) = x2 + x + g(x) = x3 3x2 + 6 , , . Cf xx 3 Cg yy 6 : ) ) Cf , Cg 1.41 f(x) = (x )ex + yy 1 x = 1 2 . : ) ) Cf y = 2x 1.42 f(x) = x2 + x + 4 , . Cf (3 , 5) , : ) ) Cf ) Cf g(x) = 4x + 1 1.43 f(x) = ln(x2 2x + ) , . Cf , : ) ) f ) Cf xx ) Cf y = 2ln3 . 1.44 f(x) = x2 + , x 1 |x 2| + + 1 , x > 1 . Cf (3 , 5) , : ) ) Cf 1.45 f: : 2xf(x) x f 1 x = x2 3x 4 , x . : ) f ) Cf ) Cf xx 1.46 f: f x e lnx f(x) 1 , x > 0 : ) f ) Cf 1.47 f , g : f(8 3x) + f(x) = 2g(x) , x . ) Cf , Cg ) 3f(x) 2f(2 x) = 2x x2 , x , f , g . 1.48 f: f3(x) 2f2(x) + 5f(x) = e2x ex , x . Cf xx 1.49 f. ) f ) f ) f(1) ) f(x)=0 ) f(x)>0 f(x) 0 f(x) < 0 . ) 0 . 1.51 f. ) f ) f ) f(x) = 0 , f(x) = 2 f(x) = 2 , ) f(x) > 0 f(x) < 0 , f(x) 2 , f(x) < 2 1.52 f. ) f ) f ) ff(2) ) f(x) = 2 ) f(x) 0 f(x) < 2 1.53 f. ) f ) f ) f(7) , ff(4) ff(6) ) f(x) = 0 , f(x) = 2 ) f(x) < 0
  8. 9. . 9 1.54 f g ) f , g ) fg(0) , gf(0) ) f(x) = g(x) ) f(x) > g(x) ) g(x) 0 1.55 ) f(x) = e3x 2xex xex g(x) = e 2x x 2 ) f(x) = x2 4 x2 + 2|x| , g(x) = 1 2 |x| . 1.56 f(x) = x3 8 x2+ 2x + 4 , g(x) = (x + 3)2 x2 5x 11 1.57 f = g. f g f(x) = g(x) . f(x) = x2 x 6 g(x) = x + 2x 3 . f(x) = x2 + 4x + 3 x2 1 , g(x) = x2 9 x2 4x + 3 . f(x) = ln x2 1 x g(x) = 2lnx ln(1 x) 1.58 f , g : (f(x) + g(x))2 = 4f(x) g(x). f = g 1.59 f , g : : f(x) 2 +g(x) 2 2x = f(x) + g(x) x , x 0 . f = g 1.60 f , g : f2(x) + g2(x) + 8x2 4x(f(x) + g(x)) . f , g . 1.61 , , f(x) = x2(x 1) + x(x 2) + g(x) = x3 + 3x2 8x + 5 . 1.62 f(x) = ( + 1)x 2 1 x 22 + 2 g(x) = [(1 )8+ ]x + ( 3)5 4 x 2 2 f g . 1.63 f(x) = x3+ 3x 4 x2 x + 4 g(x) = x 1 1.64 , , f(x) = x2 x + x + 2 g(x) = x2 ( + 1)x + 2 3 x + 1 . 1.65 , , f(x) = 2x + 5 x2 7x + 10 g(x) = x 2 + x 5
  9. 10. . 10 1.66 f(x) = x 1 g(x) = x2 4 x2 3x . f + g , f g , f g . 1.67 f(x) = x 1 g(x) = 6 x . f + g , f g , f g 1.68 f(x) = x 1 g(x) = 2 x . f + g , f g , f g 1.69 f(x) = x2 9 x + 2 g(x) = x 1 x2 x 6 f + g , f g , f g 1.70 f(x) = x lnx g(x) = 1 2x . f + g , f g , f g 1.71 f(x) = lnx 3 , g(x) = ex 2 . : f g (x) 0
  10. 11. . 11 2.1 f(x) = 2 x g(x) = x2 + 2x 6 . fog . . 2.2 f(x) = x + 1 x + 2 , g(x) = x 3 x 2 , , fog , gof , fof. 2.3 f(x) = 2x 1 x + 1 , g(x) = x 1 x . fog = gof ; 2.4 g(x) = x + 3 x 2 . gog . 2.5 f(x) = 2 x g(x) = lnx . , , fog gof 2.6 f(x) = lnx g(x) = x 1 x . fog . ( 2017 ) 2.7 f(x) = 2x 1 g(x) = ln(9 x2) . gof . 2.8 f(x) = ex ex 1 g(x) = ln(x 1) fog , gof , fof. 2.9 f(x) = x 1 g(x) = x2 2x + 3 . fog gof . 2.10 f(x) = 2x + g(x) = 3x + 2 , . x=1 , fog = gof 2.11 f , g : . : ) f , g , fog . ) f g , fog . 2.12 f(x) = x2 4x + 3 x 3 g(x) = ln(x 1) ) f ) gof ) Cgof xx 2.13 f(x) = x + x + 1 , Cf (2 , 3). ) ) fof ) (fof)(x) g(x) = x 1 x2+ x . 2.14 f(x) = 5 |x| , Cf (3 , 1) . ) . ) Cf ) x Cf g(x) = |x| 4 . ) f h(x) = x2 36 x2 + 6|x| . 2.
  11. 12. . 12 Composite function of f and g , f g 2.15 f (0 , +) 2f(x) f 1 x = lnx3 , x > 0. ) f . ) g(x) = ex + 2 ex 1 gof . 2.16 f , g : (fog)(x) = 3x2 6x + 10 f(x) = 3x + 1 g(x) . . 2.17 f , g : (gof)(x) = 4x2 + 4 f(x) = 2x 1 . g(x) . 2.18 f g : (fog)(x) = 2x + 1 f(x) = lnx , x > 0 . g(x) . 2.19 f , g : (fog)(x) = x + 8 f(x) = ex+1 . g(x) . 2.20 f , g : (fog)(x) = 4x2 14x + 13 g(x) = 2x 3 . f(x) . 2.21 f , g : (fog)(x) = 2x2 11x + 16 g(x) = x 3 . f(x) . 2.22 f , g : (fog)(x) = 2 x 2 + x , x > 0 g(x) = lnx . f(x) . 2.23 g(x), (gof)(x) = x 2x2+ 2x + 1 f(x) = 2x + 1 2.24 f(x), (gof)(x) = 2x 1 x2 x + 1 g(x) = x 2 2.25 f , g : (fog)(x) = 4x2 1 g(x) = 2x + 1 ) f(x) . ) fof 2.26 f(x) f(ex) = 3x2 2x + 4 , x 2.27 f(x) f(2x 1) = 4x2 6x + 3 , x 2.28 f(x) f(lnx) = x2 + 3lnx + 1 , x > 0 2.29 f , g : (gof)(x) = 3x2 6x + 10 g(x) = 3x 2 . : ) f ) x Cf Cg 2.30 f(x) = lnx g(x) = x x + 3 , Cg (5 , 4) : ) ) fog ) fog . 2.31 f : [2 , 1] . f(2x 3) . 2.32 f : [0 , 1] . f(lnx) . 2.33 f : (0 , 1] . : ) f(3x 2) ) f(lnx) ) f(ex) 2.34 f : [1 , 4] . g(x)=f(x2 5)
  12. 13. . 13 2.35 f: (fof)(x) = 3x 2 , x . f(1). . () fof 2.36 f: (fof)(x) = x2 + x , x . f(0). 2.37 f: (0 , +) (fof)(x) = 3x2 + 2x 80 , x . f(5). 2.38 f: (fof)(x) = 3x + 4 , x . . f(3x + 4) = 3f(x) + 4 , x . . f(2) . 2.39 f(x) = x 1 g(x) = 7x , . fog gof . . 2.40 f(x) = x + 1 , g(x) = (3 2)x + 2 1 , . fof = g . 2.41 f(x) = 2x 1 g(x) = 3x + 1 , . fog gof . 2.42 f(x) = 2x + 3, g(x) = x2 + x + , h(x) = 4x2 + x + 2 . , , gof = h 2.43 f(x) = 3 x 2 x . x 2 (fof)(x) = x .