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### Transcript of Maths g lykeiou_raptis

1. 1. . 1.600
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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 .
13. 14. . 14 3.1 ) f(x) = 3x + 7 ) f(x) = x2 + 5x + 2016 ) f(x) = x3 + 4x2 2x + 1 ) f(x) = 3ex + 2lnx + 7x ) f(x) = 3x + 2x + 4 x 2x . 3.2 ) f(x) = xex ) f(x) = xlnx ) f(x) = x2 x ) f(x) = x3 lnx ) f(x) = (x2 2x)ex 3.3 ) f(x) = 2x x + 1 ) f(x) = 3x 1 2x + 5 ) f(x) = x2 3x 2x + 3 ) f(x) = x2 ex ) f(x) = x 3 ex ) f(x) = x lnx ) f(x) = x x 3.4 ) f(x) = ln(x2 3x) ) f(x) = (2x + 3) ) f(x) = (x2 + 5x) ) f(x) = ex2 + 5x 3 ) f(x) = 4x 5 ) f(x) = (3x 2)5 3.5 : ) f(x) = 2x3 + 6x 1 ) f(x) = ex + lnx ) f(x) = x5 x3 lnx . 3.6 : ) f(x) = 1 3x + 1 2x ) f(x) = 4e3x + 2017 ) f(x) = x5 1 2x 3.7 : ) f(x) = x2 + x ) f(x) = 2 x lnx + 1 ) f(x) = 2 3 x 4x 3.8 : ) f(x) = ln(x 1) e2x ) f(x) = 1 x 1 + x ) f(x) = x 1 + 2x 3.9 f , g : . : ) f , g , fog . ) f , g fog . 3.10 f , g : (0, +). f g , h = f g . 3.11 f , g . f + g . 3.12 f . f : ) (2 , 5) (4 , 3) ) (1 , 6) (3 , 8) 3.13 f . Cf xx , yy 2 1 . ) f ) g , gog , fog . 3.
14. 15. . 15 Strictly increasing (decreasing) function () 3.14 f : f3(x) + 2f(x) = 5x + 2 , x . f 3.15 f : f3(x) + ef(x) = 2x 3 , x . f 3.16 f(x) = 1 x lnx ) f ) 1 x2 + 5 1 2x2 + 1 < ln x2 + 5 2x2 + 1 . . 3.17 f(x) = lnx + x . ) f ) ln(x2 + x + 1) + x2 < ln(x + 2) + 1. 3.18 f(x) = x2007 + 2007x . ) f ) 2007 3x 1 2007 x + 3 > (x + 3)2007 (3x 1)2007 . 3.19 f(x) = x3 + 2x . ) f ) (x3 + x2)3 (x + 1)3 > 2(x + 1 x3 x2) 3.20 f(x) = 2 3 x 2x ) f ) 4 9 x 2 3 x < 2x 3.21 f(x) = 2x + 4x . ) f ) 21x + 4 < 4x + 6 3.22 f(x) = 1 x x . ) f ) 1 2x2 + 3 1 x2 + 2x + 6 > 2x2 + 3 x2 + 2x + 6 3.23 f(x) = x + ln(1 + ex) ) f ) (x 1)2 > 1 + e2x 1 + ex2 3.24 e1x < 1 + lnx 3.25 5x3 + lnx < 2 x + 3 3.26 ex + 3x > 1 2 x 3.27 f(x) = x2 + lnx ) f ) x Cf y = 1 ) (3|x| + 1)2 (2|x| + 3)2 > ln 2|x| + 3 3|x| + 1 3.28 f(x) = x + ln(x + 2) ) f ) f(x4 + 1) f(x2 + 1) > 0 ) ln 3x x2+ 2 < 2 3x + 2
15. 16. . 16 3.29 f(x) = lnx + e x ) f ) e3 ex < x 3 x > 3 3.30 f(x) = 8e2 x 2x ) f ) f(x) < 4 ) 8e2 x2 e2 x > 2x(1 x) 3.31 f Cf (2 , 3) , 2 f (x2 3x) + 6 0 . 3.32 f . : (fof)(x2 + x) < (fof)(x + 1) 3.33 f . A A(1 , 5) (2 , 7) : ) f ) f (f(|x| 4) 6) 5 < 0 . 3.34 f(x) = x3 + 2x 3 , . A Cf xx 1 : ) ) f ) f (f(x) + 3x2 + 3) + 3 > 0 . 3.35 f(x) = 1 3 x + x , . A Cf (2 , 13) : ) ) f ) 3x(2x + 5) < 1 . 3.36 f(x) = x5 + x + 2 , . A Cf (1 , 4): ) ) f ) (fof)(x) > 2 . 3.37 f(x) = x lnx , x > 0 , (0 , 1) ) f ) x2+x+4 x2+9 < ln(x2 + x + 4) ln(x2 + 9) 3.38 f(x) = x + x + 1 , > 1 ) f ) x29 x3 < 6 + x x2 3.39 f . A A(4 , 3) (3 ,2 ) : ) f ) fof ) f f (ex1 5) > 2 . 3.40 f . A A(5 , 13) (7 , 11 ) : ) f ) f(f(x) 6) < f(7) + 2 3.41 f(x) = ex + ln(x + 1) 1 ) f ) ex2 + ln(x2 + 1) > 1 ) e x2 e x + 2 > x + 3 x2 + 1 3.42 f(x) = ex + lnx ) f ) x > 0 ln x2+ 1 2x > e2x ex2+ 1 ) x > 0 ln x + 1 x x > (ex ex + 1) ) g(x) = f(x + ) f(x + ) , > > 0 xx 3.43 f : ) g(x) = f(x) x ) f(x2 2x) f(3x 6) > x2 5x + 6
16. 17. . 17 3.44 f : f(2) = 8 ) g(x) = x3 f(x) ) 8x3 < f(2x) 3.45 f(x) = x3 + 5x + 1 ) f ) (3x + 18)3 (7x + 12)3 < 57x+12 53x+18 57x+12+1 + 53x+18+1 3.46 f(x) = lnx e x ) f ) 1 ex 1 ex2+ 1 > ln x x2+ 1 , x > 0 ) x , y > 0 x > y , : 3lnx + ey3 < 3 + ex3 ) x > 0 ln 1 + 1 x > 1 ex+1 1 ex 3.47 f: (0, +) . f(x) + f(3x) < f(2x) + f(7x) 3.48 f(x) = 1 x lnx ) f ) , : f(5) + f(7) > (6) + f(8) ) , : f(2x) + 1 > f(3x) + f(ex) 3.49 f(x) = x3 + 8x ) f ) x > 1 : f(x3) + f(2x) > (x2) + f(2) ) x < 0 : f(3x) + f(5x) > (2x) + f(4x) 3.50 f : f3(x) + f(x) = x , x . ) f ) f(x3) < (3x 2) 3.51 f(x) = e x x3 ) f ) : 1) ex(x3 + 1) < 1 2) ff(x) < 1 e 1 3) e x 1 2 < x3 ln3 2 3.52 f . A A(2 , 1) (5 , 2 ) : ) f ) 2f 2(x) 4 2f(x) 3.53 2x5 + 3ex = 3 . . 3.54 x3 + lnx 1 = 0 . 3.55 2 x = 1 + ln(x 1) . 3.56 e3 x 1 = ln(x 2) 3.57 f(x) = lnx 1 x +1 ) f ) ln(2x + 3) + 1 = 1 2x + 3 ) 2x2 lnx + x2 < 1
17. 18. . 18 4.1 1-1 : ) f(x) = 3ex3 5 ) f(x) = 3 lnx 2 4 ) f(x) = 3 x x + 1 ) f(x) = 1 3 2x ) f(x) = 2 ln(x + 1) 3 ) f(x) = 2 ex 1 ex + 2 . 1-1 4.2 1-1 : ) f(x) = 2x + 3 x 1 ) f(x) = 2 + 3x 4 ) f(x) = 3x2 6x + 1 , x 1 4.3 1-1 : ) f(x) = 3 2 x ) f(x) = ln 1 + x 1 x ) f(x) = 1 2x 3ex ) f(x) = e2x x 4.4 1-1 : ) f(x)= 2 x5 + 7x3 + 3x 5 ) f(x) = 3ex + 2 lnx 1 ) f(x)= 1 2 x 4x3 ) f(x)= 5 x 3 lnx 4.5 1-1 : ) f(x) = x2 1 x2+ 1 ) f(x) = (x 3)(x 4) + 2017 ) f(x) = ln(|x| + 1) 4.6 f : (x 2)f(x 3) (x 3)f(x) = 1 , x ) f(0) , f(2) ) f 1-1 4.7 f : 6f(x2) f2(x) 9 , x . ) f(0) , f(1) ) f 1-1 4.8 f : f2(x) + f(x) + 1 3f(x2 x) x . f 1-1 4.9 f : (fof)(x) + f3(x) = 3x 2 , x . f 1-1 4.10 f : f3(x) + 4f(x) = 2x + 3 , x . f 1-1 4.11 f , g : (gof)(x) = x3 + 3f(x) + 2 , x . f 1-1 4.12 f (0, +) f5(x) + 3f(x) = ln(2x + 1) , x > 0 . f 1-1 4.13 f , g : , gof 1-1 . f 1-1. 4.14 f(x) = x3 3x2 + 4. ) f . ) f 1-1 4.15 f(x) = x + x2 + 1 , (1, 1). : ) . ) Cf yy ) f 1-1. 4.16 f , g : fog 1-1 ) g 1-1 ) x > 0 g( f(lnx) + 1 ) = g(x + 2) f(x) . 4.
18. 19. . 19 4.17 f : : f(x y) = f(x) f(y) , x , y ) f(0) = 0 ) f(x) = f(x) ) f(x) = 0 0 , f 1-1 4.18 f : : |f(x) f(y)| |x y| , x , y f 1-1 4.19 ) g(x) = ex ex 1-1 ) f (0, +) ef(x) ef(x) = elnx 1 x , x > 0 , f 4.20 ) g(x) = x + lnx 1-1 ) f x + ex = f(x) + lnf(x) , f(x) > 0 , f 4.21 : ) ln(x 1)= 2 x ) 3x = 5 2x . 1-1 4.22 f : (fof)(x)+ f3(x) = 2x + 3 , x . ) f 1-1 ) f(2x3 + x) f(4 x) = 0 4.23 f : (fof)(x)+ f3(x) = 2x + 5 , x . ) f 1-1 ) f(2x3 + x 2) = f(2 x) 4.24 f : (fof)(x) = f(x) + ex1 , x . ) f 1-1 ) f(x3 5x) = f(2x 6) 4.25 f : f(x) + f3(x) 2x = 3 , x . ) f 1-1 ) f(et + 1) f(2et) = 0 4.26 f (fof)(x)=(x2) f(x) , x . ) f 1-1 ) f(3) ) fx + 1 f(|x| 1) f(x 2) = 0. 4.27 f : (fof)(x) f(x) = x + 2 , x . ) f 1-1 ) f(2) ) f ) f4 f(|x| 1) = 2 4.28 f : (fof)(x) f(x) = 2x + 2 , x . ) f 1-1 ) f(1) ) f(x2 + x 1) + 2(x + 1) = ff(x) 4.29 f : (fof)(x) f(x) = x , x . ) f 1-1 ) f(0) ) f(x3 + ex) = f(1) 4.30 g(x) = x + 3 e x 2 f : (gof)(x) = 8 3 e x 2 x . ) g 1-1 ) f(2) ) f( f(|x| 3) + ex 1) f(ex + 1)=0. 4.31 f , g : ) fog 1-1 g 1-1 ) x fg(x) = 2x + ex gex2 e2x = g(4x 2x2) 4.32 f (0 , +) g(x) = f(x) 2lnx , x > 0 ) g ) (1 , 2) f , : 1) f(x 1) = 2 + 2ln(x 1) 2) ln(lnx)2 < (lnx) 2
19. 20. . 20 4.33 f : ) g : , g(x) = f(x) x . ) f(x2 3x) f(2x 6) = x2 5x + 6. 4.34 f, g : , (gof)(x)= 2x5 + ef(x) +1 , x . ) f 1-1 ) f(lnx) = f(1 x3). 4.35 f(x) = x x , 0 < < 1 ) f 1-1 ) 2 4 2 = (2 4) ( 2) 4.36 f(x) = x3 + x ) f 1-1 ) ex + x 3 + ex = x + 1 3 + 1 4.37 f(x) = x + x 1 ) f 1-1 ) 2x2 x + 1 x2 + 7 = 6 + x x2 4.38 f(x) = ex + 1 + lnx ) f 1-1 ) : 1) ex2 + 1 + 2lnx = e + 1 2) e3x + 1 + ln3 + lnx = 3 + ln(ln8) 3) x + 1 + ln(lnx) = f e x 4.39 f(x) = ex + lnx + x 1 ) f 1-1 ) ex2+ 1 e2x = ln2x ln(x2 + 1) x2 + 2x 1 4.40 f(x) = x5 + x3 + x 3 . ) f 1-1 ) x5 + x3 + x = 3 ) e5x + e3x + ex < 3 . 4.41 f(x) = ex lnx . ) f 1-1 ) ln = 1 e2 1 e 4.42 ln ex + 1 ex + 1 = 7(ex + 1)3 7(ex + 1)3 . 4.43 f(x) = ex1 + x + 1 . ) f 1-1 ) f(x) = 3 ) ex1 + x 2 > 0 4.44 f(x) = x + ln(x + 1). ) f 1-1 ) f(ex + x 1) = 0 A = [0, +) . 4.45 f(x) = 1 x lnx + 1 ) f 1-1 ) lnxx + x = 1 , x > 0 ) ln x2+ 1 3x2+ 2 = 2x2+ 1 (x2+ 1)(3x2+ 2) 4.46 f(x) = 3x + x3 . ) f 1-1 ) 3x2 4x 3x + 4 = (x2 4x)3 + (x + 4)3 4.47 f(x) = e3 x x + 2 . ) f 1-1 ) : 1) x e3x = 2 2) e3 x2+ 1 x2 + 1 = 2 3) ex2+ 4x + 3 e9 x + 5x = x2 + 6 4.48 f(x) = x3 + e x 1 . ) f 1-1 ) : 1) f(x) = e 2) f(x2 6x + 8) = 0 3) (x + 3)3 (x2 + 1)3 = ex2+1 ex+3 4) ln3 x + x = e1x (x 1)3 4.49 f(x) = e x + x 1 . ) f 1-1 ) : 1) ef(x) + f(x) = 1 2) (x 1)2 = e2x ex2 + 1 3) fex2 4x = f(e4x x2) 4) ex2+2x+2 + (x + 1)2 = e 5) ex23x + x2 3x ex = 1
20. 21. . 21 4.50 f (0, +) f(x) f(y) = f x y . f(x) = 0 : ) f 1-1 ) f(x) + f(x2 + 3) = f(x2 + 1) + f(x + 1) 4.51 f (0, +) f(x y) = f(x) + f(y) , x , y > 0 f(x)=0 : ) f(1) = 0 ) f 1 x = f(x) ) f 1-1 ) f(x) + f(x2 + 1) = f(x + 8) 4.52 f (0, +) f(x) f(y) = f x y , x , y > 0 ) f(1) ) f 1 x = f(x) , x > 0 ) f(x)=0 : 1) f 1-1 2) f(x2) + f(2) = f(12x 16) 4.53 , , : ) f(x) = 2x + 5 ) f(x) = 3x 2 x + 1 . 4.54 , , ) f(x) = x3 2 ) f(x) = x 1 x 2 4.55 , , ) f(x) = 1 + ln(x 3) ) f(x) = 2 + x 1 4.56 , , ) f(x) = 2ex3 1 ) f(x) = 2 + ex 1 4.57 , , ) f(x) = 1 ex1 ) f(x) = 2 3 x 4.58 , , ) f(x) = x2 4x + 5 , x 2 ) f(x) = x2 8x + 10 , x 4 4.59 , , ) f(x) = 8x3 3 ) f(x) = ex + 1 ex 4.60 , , ) f(x) = e x 1 e x + 1 ) f(x) = ln x 1 x 2 4.61 , , ) f(x) = ln x 1 x ( 2017 ) ) f(x) = 1 2ex e x + 1 4.62 f(x) = e x e x 2 4.63 f : 2f3(x) + 4f(x) = x + 4 , x . , , f. 4.64 f : f3(x) 3f2(x) + f(x) + x = 2017 , x . , , f. 4.65 f (1 , +) o f2(x) 2 f(x) = e2x 1. : ) f(x) ) f1 (x) . 4.66 f(x) = 4x + 2 , g(x) = 2f1(x) + 1 . g1 . Inverse Function
21. 22. . 22 4.67 f(x) = x + , 0. , f(x) = f1(x) + 3 . 4.68 f(x) = (2 1)x 3 , , f = f1 4.69 f(x) = + ex 1 , . ) f ) f1(4) = 1 , : 1) 2) 4.70 f(x) = x 1 + x , . Cf (3 , 2) ) ) f ) f = f1 4.71 f(x) = ex + 1 g(x) = ex + 1 ex 1 ) f ) g ) gof1 4.72 f(x) = x3 + 2x . ) f ) f1 (3) ) f1( f (x2 5) + 15 ) = 2 . - 4.73 f(x) = 2 x lnx . ) f ) f(x)=1 ) x+ lnx >1 4.74 f : , , Cf (2 , 6) (4 , 3). ) f ) f ( f1 (x2 5x) + 2 ) = 3 ) f1( f (x2 x) 3 ) < 4 . 4.75 f : , , Cf (1 , 5) (2 , 4). ) f ) f1[3 + f(x2 3x 3)] = 3 ) f 2 + f1 2x + 10 x 1 8 . 4.76 f : , , Cf (1 , 5) (6 , 4). ) f ) f1 1 + f(x2 2x 4) < 6 4.77 f : , , Cf (1 , 5) (3 , 8). ) f ) f ( f1 (x2) 3 ) = 5 4.78 f : , , Cf (2 , 1) (3 , 8). ) f ) f (1 + f1 (x2 + 2x) ) = 1 ) f(f1(lnx) + 1) < 8 4.79 f : , , Cf (3 , 2) (5 , 9). ) f ) f2 + f1(x2 + x) = 9 ) f(f(x2 4x) 6) < 2 4.80 f : , , Cf (2 , 5) (3 , 2). ) f ) f1(5) f1 (2) ) f1 3 + f (x2 + 2x) > 2
22. 23. . 23 4.81 f : f(0) 2 + f(1) 2 + 13 = 6f(0) + 4f(1) ) f ) f(f1(x3 3x + 4) 1) > 3 4.82 f : , , Cf (1 , 4) (2 , 12). ) f ) f1 + f1(3x 17) > 12 ) : K = f1 24 f5f1(12) 8f1(4) 4.83 f : , , Cf (5 , 9) (2 , 3). ) f . ) f3 + f1(x2 + 2x) = 9 ) f1 x ln 2 x + 1 = 2 ) f2(x) 12 f(x) 27 ) f (x + lnx + 4) > 9 4.84 f : (fof)(x) = 3x 5 f(2) = 10 . ) f ) f1 (2) ) f(f1(|x| 2) 5) = 2 4.85 f(x) = e1x x . ) f ) f1(1 x) > . 4.86 f(x) = 2 x3 3x + 1 ) f ) f1( f (x2 4) 22) < 2. 4.87 f(x) = ex + x3 + x + 1 . ) f ) ex2 x + (x2 x)3 + x2 2x = e x + 3 + (x + 3)3 + 3 4.88 f(x) = ex + x . ) f ) f1(x) = x 1 ) f1(x) x 1 4.89 f(x) = 2e2x 3x 2e2 ) f ) f(f1(x 2e2) 1) = 3 ) f1( f (x) 1 2e2) < 0 . 4.90 f(x) = 3x5 + 2x3 1 ) f ) f f1(4x + 2) = 4 ) f1( f (x2 + 2x + 2) 5) > 0 4.91 f : (fof)(x) + f(x) = 3x 4 f(3) = 8 ) f(8) ) f 1-1 ) f1 (3) ) f ( f1 (x2 4x) 3 ) = 3. 4.92 f : f3(x) + f(x) = 27x3 + 8 . ) f 1-1 ) f(x) = 0 ) f( ln2 x) = f(2 lnx + 3). 4.93 f : f(x) = e x 1 + 2x 3 . ) f ) f1(x) = 0 ) f1(lnx) < 1 ) f 1 + f1(x + 1) = 0 4.94 f(x) = ln(x + 1) ex + 2x , f(A) = . ) f . ) f1( ex 2) < 0 . ) f1( x 1) = x (1 , +) 4.95 f : : f (ex + 2) + f (x + 3) = x , x . ) f ) Cf xx ) f 6 f1(x2 4) > 0 .
23. 24. . 24 4.96 f(x) = ex ex 1 . ) f xx , yy ) f ) f1 1 1 e + 2 f (lnx) = 1 . 4.97 f : ef(x) + f(x) = x , x ) f ) f(1) ) e x 4 e 2x + 1 = x + 5 4.98 f : ef(x) + f(x) = x + 2 , x ) f ) f(lnx) = f e x ) f ) (x3 8)(ex 3) < (1) . 4.99 f(x) = ex e x + 1 g(x) = 1 lnx ) f ) (f1 og)(x) . ) 1 < < < e , 1 ln 1 ln > ln ln . 4.100 f(x) = x3 + x + 2 , . fof yy 14 . ) ) f ) Cf , Cf1 ) f ( f(x2 4) + x 1) f(x + 1) = 0 . ) f ( f(|x| 2) 5) < f1(14) . 4.101 f: (0, +) f(1) + f(e) = 2e + 3 , f(x) f(y) = ln x y + 2(x y) ) f(1) , f(e) ) f ) f ) 4(x2 1) < ln x2 + 10 3x2 + 8 4.102 f(x) = e x 2 + x 1 . ) f . ) Cf , Cf1 . , 4.103 f(x) = x3 x + 12 ) f . ) Cf , Cf1 ) f1 ( f (|x| 1) + 8 ) < 1 . 4.104 f(x) = x3 + 2x 2 ) f . ) = f1(1) + f1(10) ) Cf1 y = x 4.105 f(x) = x3 + 4x 4 . ) f . ) Cf , Cf1 ) f1( x2 13) < 2 . 4.106 f(x) = 3x5 + x + 3 . ) f ) Cf , Cf1 ) f1( f (x2 3) 4 ) > 0 . 4.107 f(x) = lnx + x e x ) f ) Cf , Cf1
24. 25. . 25 4.108 f : 2f3(x) + f(x) = x + 16 , x ) f ) f1 ) Cf y = x 4.109 f : f3(x) + f(x) = x 8 , x . ) f ) f ) Cf , Cf1 4.110 f : f3(x) + 3f(x) = x + 3 , x . ) f ) f ) f(x) = f1 (x) 4.111 f : (0 , +) f(x) + lnf(x) = x , x . ) f ) f ) Cf , Cf1 4.112 f(x) = 1 x e x 1 + 1 x > 0 ) f ) Cf , Cf1 . 4.113 f : ef(x) + f(x) = x + 1 , x . ) f ) f1 ) f(x) = f1 (x) 4.114 f(x) = x5 + x3 + x ) f . ) f1(x) = 1 ) Cf , Cf1 ) ef2(x)3f(x) 1 4.115 f(x) = lnx + x 1 ) f . ) Cf1 y = x 4.116 f(x) = 3x + x 9 . ) f . ) o f1(5) ) f(x) = f1 (x) ) f1( f (lnx) 3) > 0 4.117 f(x) = x3 + x 8 . ) f . ) f1(6) f1(2) ) f(x) = f1 (x) ) f (x2 8) = 6 ) f1(logx2) 2 4.118 f(x) = ex + lnx + x e2 x ) f . ) x + xex e2 2x = ex , x > 0 ) f(x) = f1 (x)
25. 26. . 26 5.1 f : lim x2 f(x) = ( 1), lim x2+ f(x) = 5 9 lim x2 f(x) . 5.2 f : lim x 3 f(x) = 2 3 + 2 , lim x3+ f(x) = 2 4 lim x3 f(x) 5.3 f : lim x f(x) = 3 + 3, lim x+ f(x) = 2 22 lim x f(x) 5.4 f . : ) lim x2+ f(x) ) lim x1 f(x) ) lim x1+ f(x) ) lim x1 f(x) ) lim x1+ f(x) ) lim x2 f(x) ) lim x3 f(x) ) f(1) ) f(1) 5.5 f . : ) lim x2 f(x) ) lim x1 f(x) ) lim x0 f(x) ) lim x1 f(x) ) lim x2 f(x) , . Karl Weierstrass (1815-1897) lim xx0 5.
26. 27. . 27 The limit of f as x approaches f x x0 5.6 f . : ) lim x2 f(x) ) lim x0 f(x) ) lim x2 f(x) ) lim x3 f(x) ) lim x4 f(x) 5.7 f . : ) lim x3+ f(x) ) lim x2 f(x) ) lim x0 f(x) ) lim x1 f(x) ) lim x2 f(x) ) lim x3 f(x) ) lim x4 f(x) ) lim x5 f(x) ) lim x6 f(x) 5.8 f lim h0 f(1+h) 1 h = 3 . lim h0 f(1+2h) f(12h) h 5.9 f lim h0 f(1+h) 2 h = 5 . lim h0 f(1+3h) f(12h) h 5.10 lim x2 f(x) 2 x 1 = 3 , lim h0 f(2+h) f(2h) h 5.11 : ) lim x2 x2 4 x 2 ) lim x 3 x2 9 x2 + 3x ) lim x1 x2 + 4x 5 x2 1 ) lim x1 1 x 2x2 7x + 5 ) lim x 2 x3 + x2 2x x4 16 2. (0/0) 5.12 : ) lim x1 2x2 3x 5 x3+ 1 ) lim x 2 2x2 5x + 2 x2 5x + 6 ) lim x1 x3 7x 6 x2 1 ) lim x1 3x4 2x 1 4x2+ x 5 ) lim x 1 2 8x3 1 2x2+ 7x 4 5.13 ) lim x2 1 x 2 4 x3 2x2 ) lim x1 1 x + 1 + 2 x2 1 ) lim x3 1 x 3 2 x2 4x + 3 ) lim x1 1 x2+ x 2 x x3 1 ) lim x1 1 x x x 3 + 2 x
27. 28. . 28 5.14 f : f3(x) 3f2(x) + 3f(x) = x + 9 . : lim x3 f1(x) x2 5x + 6 5.15 f : lim x f(x) x = 3 , lim x (f(x) 3x) = 5 . : lim x xf(x) + f2(x) + 5x2 x2f(x) 3x3+ x2 5.16 : ) lim x2 3 x 1 2 x ) lim x 1 x + 5 2 x2 + x ) lim x1 x 1 x2 + 3 2 . (0/0) 5.17 : ) lim x2 3 x 1 x+7 3 ) lim x 2 5 2x 3 2x + 3x2 + 4 5.18 f : lim x2 f(x) = 2 . lim x2 f3(x) 8 4 f2(x) + 12 5.19 f : lim x3 f(x) = 2. lim x3 f2(x) 4 f(x) + 7 3 5.20 0 < x < 1 f(x) = x 1 x + 3 2 , x > 1 3x2 5x + 2 x 2 x , 0 < x < 1 . lim x1 f(x) . 5.21 f(x) = x2 6x + 5 x 1 , x > 1 x 1 x 1 , 0 < x < 1 . lim x1 f(x) 5.22 f(x) = x2+ 2x x + 4 2 , 4 x < 0 x3+ 2x2 8x 2x2 4x , 0 < x < 2 x2 x 2 x2 3x + 2 , x > 2 . , lim x0 f(x) , lim x2 f(x) 5.23 f(x) = 3x , x 1 x2 x + , 1 x < 1 x3 x2 + , x 1 . , , lim x1 f(x) lim x1 f(x) = 1 . 5.24 f(x) = x2 + 4x + 1 , x < 0 x+1 1 x 1 2 , x > 0 lim x8 f(x) = . lim x0 f(x) 5.25 , , : ) lim x2 |x 3| |x 1| x 2 2x ) lim x 1 x3 3x 1+ x |x3 + 5x + 4| 2 ) lim x 2 x2 4 x2 + 5x + 6 |x2 + 3x| + x . 5.26 , , : ) lim x3 x2 x 6 |x 4| 1 ) lim x 1 |x 2| 1 |x| 1
28. 29. . 29 5.27 lim x1 f(x) = 2 , : ) lim x1 |f(x)+1|+|f(x)3x|4 x 1 ) lim x1 f(x)3x2+|f(x)+1|4 x 1 5.28 f (1 , +) lim x2 f(x) = 3 . : lim x2 |f(x) 2| f2(x) 5 f(x) + 5 f(x) + 1 2 5.29 f x f(x) 2 f(x) x2 5x + 6 , x lim x2 f(x) . lim x2 f(x) . 5.30 f (x 2) f(x) x2 7x + 10 , x lim x2 f(x) . lim x2 f(x) 5.31 f (x 1)f(x) x2 + x 2 , x lim x1 f(x) . lim x1 f(x) 5.32 f lim x3 f(x) 2x + 4 x 3 = 10 . : ) lim x3 f(x) ) lim x3 f(x) 2 x2 3x . 5.33 f lim x2 [ f(x) 3x2 + x 2] = 4 lim x2 f(x) 5.34 lim x2 f(x) : ) lim x 2 [ 2 f(x) + 1 x] = 3 ) lim x 2 f(x) 1 x + 2 = 3 5.35 lim x0 f(x) 2 x = 3 , : ) lim x0 f(x) ) lim x0 f(x)+ 2x 2 x 5.36 lim x2 f(x) lim x2 5f(x) 1 f(x) + 3 = 8 5.37 f lim x0 f(x) 4 f(x) + 2 = 1 , lim x0 f(x) 5.38 lim x1 f(x ) x3 x2 1 = 2 , lim x1 f(x) x x 1 5.39 f lim x2 f(x) x x2+ 5 3 = 4 : ) lim x2 f(x) ) lim x2 f(x) + x 4 |x 3| 1 5.40 f lim x2 [ x f(x) + x2 8 ] = 6 . : ) lim x2 f(x) ) lim x2 f2(x) 5 f(x) f(x) 1 2 5.41 lim x 1 [ f(x) g(x)] , lim x 1 f(x) x + 1 = 2 lim x 1 [ g(x)(x2 x 2)] = 3 5.42 f lim x1 f(x) 4 x 1 = 3 . lim x1 f2(x) f(x) 12 x2 + x 2 5.43 f lim x1 f(x) 2 + (x1)2 x 1 = 100 . : ) lim x1 f(x) ) lim x1 f(x) 2 x 1 ) lim x1 |f(x) 3| 1 x 1
29. 30. . 30 squeeze theorem sandwich rule/theorem 5.44 f lim x 0 f(x) 2 x = 5 , lim x 1 f(x) 1 x 1 = 3 . : ) lim x0 f(x) lim x1 f(x) ) lim x 0 (f(x)2)(f(2x+1)1) 2x2 5.45 f(x) = x2+ x 2 x 1 . , lim x1 f(x) . 5.46 , lim x 1 2 x2+ x + x2 + 3x + 2 = 5 . 5.47 , lim x 1 x2 + x 6 x2 1 = 4 . 5.48 , lim x 1 x2+3 x x 1 = 5.49 , lim x 1 x3 + x2+ x + 1 = 5 . 5.50 , lim x 1 x3 + x x 1 = 2 . 5.51 , lim x 4 |x + 3|+|x5| 3 x2 5x + 4 = 7 . 5.52 f(x)= x2 1 x + 3 2 , 3 x < 1 x2+ x + x2 3x + 2 , 1 < x < 2 , lim x1 f(x) . . 5.53 f 2x2 7x + 5 f(x) x2 x 4 , x (2 , 6) . : ) lim x3 f(x) ) lim x3 f(x) 2 x 3 5.54 f x2 + x f(x) 12x + 3 22 . : ) lim x1 f(x) ) lim x1 f(x) f(1) x 1 5.55 f |f(x) x + 2| x2 2x + 1 x . lim x1 f(x) 5.56 f |xf(x) 2f(x) x2 + 4| x2 4x + 4 , x . lim x2 f(x) 5.57 f f2(x) 4f(x) x2 4 , x . lim x0 f(x) 5.58 f f2(x) 2x2 f(x) , x . lim x0 f(x)
30. 31. . 31 1595 Bartholomaeus Pitiscus . 5.59 f f2(x) + 2 x 2 f(x) x . lim x0 f(x) 5.60 f f2(x) 6 f(x) x + 3 x 3 x > 3 . lim x0 f(x) 5.61 2x + 2 f(x) x + 3 , x 2 , : ) lim x1 f(x) ) lim x1 f(x) 2 x + 1 ) lim x1 2f2(x) 8 x2+ 3x + 2 5.62 f , g : lim x1 f(x) x 1 = 2 , |g(x) 1| |f(x)| x . : ) lim x1 f(x) ) lim x1 g(x) ) lim x1 fg(x) g(x) 1 . 5.63 : ) lim x0 x x2 + x ) lim x0 x x + 4 2 ) lim x0 3x x 5.64 : ) lim x0 x + x 2x + 3 x ) lim x0 x + x 2x + 5x ) lim x0 3x 7x + 93 5.65 ) lim x0 x 1 x ) lim x0 x 1 x ) lim x0 x2 1 x 5.66 f 2 x f(x) + 2x x x2 x . : ) lim x0 f(x) ) lim x0 f(x) + 2x x2 5.67 f f2(x) 2xf(x) 2 x 2x x , x . : ) lim x0 f(x) ) lim x0 f(x) x 5.68 f |x2 f(x) 2 x| x4 . lim x0 f(x) 5.69 f f2(x) 2x x 2x f(x) + 2 x . lim x0 f(x) 5.70 f lim x0 f(x) x = f3(x) + f(x)2 x = 2x2 x x . . 5.71 f lim x0 f(x) x = f3(x) + f2(x)x + x3 f(x) = 2x2 x , x . ) ) lim x0 f2(x) + xf(x) + x x f2(x) + x2+ 2x . 5.72 lim x0 f(x) 3 x = 4 , lim x0 f(x) lim x0 f2(x) 5f(x) + 6 2x + x
31. 32. . 32 5.73 lim x0 f(x) x = 2 , : ) lim x0 f(x) + f(2x) x ) lim x0 x2f(x) + xf2(x) f3(x) + x3 ) lim x0 f(x) + x 1 f(x) + x 5.74 lim x0 f(x) x = 1 , : ) lim x0 f(2x) + f(3x) x ) lim x0 xf(x) + x2 f2(x) + 2x ) lim x0 f(x) + x + 1 1 f(x) + x 5.75 lim x0 f(x) x = 2 , 0, lim x0 xf(2x) + f(x) x 2x2 2x = 8 5.76 f f4(x) + 2006f(x) = 2x + 6 , x . ) f ) f ) lim x0 2f1(x) + x + 6 x 5.77 f lim h0 f(1+h) h = 2 . : ) lim x1 f(x) ) lim x1 f2(x)1 + f(x) 1 f(x) ) lim x1 f(x) f(x) f(x) + f(x)