Επιλεγμένες ασκήσεις από το ανανεωμένο βιβλίο του φίλου Γιώργου Μαυρίδη, για την Γ΄ Λυκείου. Το βιβλίο ξεχωρίζει για τις πολλές λυμένες ασκήσεις που περιέχει, στις οποίες ο μαθητής θα βρει πολλά και εξόχως ενδιαφέροντα θέματα.

1. 9 786188 021495
ISBN: 978-618-80214-9-5

2. .»

3. “
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4. 2013

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7. 243
f :
x :
i)
h 0
lim f x 2h 2f x h f x 0
ii)
h 0
f x 2h f x h
lim f x
h
iii) 2h 0
f x 2h 2f x h f x
h
i) H f
h 0
ii)
h 0 h 0
f x 2h f x h f x 2h f x f x h f x
lim lim
h h h
h 0 h 0
f x 2h f x f x h f x
2lim lim
2h h
2f x f x f x
iii) E De l’ Hospital
0
0
2h 0 h 0 2
f x 2h 2f x h f xf x 2h 2f x h f x
lim lim
h
h
h 0
2f x 2h 2f x h
lim
2h
h 0
f x 2h f x h
lim
h
ii)
f x

8. 244
f :
g :
g x f x 2x
y 2x fC
x
lim f x
i) g
ii)
x
lim f x
x
lim g x 0
iii) g x 0 x
iv) fC
i)
g x f x 2
f
f
1 2 1 2
1 2
1 2
1 2
f x f x
f x 2 f x 2
g x g x
g g
ii) y 2x fC
x
x x
lim f x lim f x 2x 2x
x
g
g

9. x
f x
lim
x
x
De
L’ Hospital
x x x
f x f x
lim lim lim f x
x x
x
2 lim f x
x x x x
iii) g
x x x
g x 0 x
iv) iii) g
fC
x x
g
x x x
g x 0
f x 2x 0 x
f x 2x x
fC

10. 246
1
1 xf x x , x 0, 1 1,
g x
x
g x g 1 0
i) N
ii) N g
iii) N f
iv) f x
i)
g 1 0
1
g x g 1
g
2
g x
x
x x
Fermat
2
g 1 0 0
1 1
g 1 0
g
x 1
Fermat

11. 247
ii) g
i
1
x
2
2
1
x
1 1
x x
x 1
x
2
x 1
x
2
x 1
x
2
x 1
x
g
g
x 1
iii)
1
1 x
f x x 0
1
1 x
1 x 1 x
f x e e e
x 0 1
g
g
0

12. 248
1 x 1 x
1
1 x
1
1 x
1
1 x
1
1 x
2
2
2
2
f x e e
1 x
x
1 x
1
xx
1 x
1
xx
1 x
g x
x
1 x
i) g
x 1
g x g 1 0
1
1 x
2
g x
f x x 0
1 x
f
iv) iii)
f
x 0 xx 1 x 1
x 0 x 01 x 1 x
x 0
1
lim 1
1 x x 0
f

13. 249
u
1 x
u1 x
x 0 x 0 u
0
0
1
xlim lim lim lim 1
1 x 1 x1 x
u
1 x
u 11 x
x 1 x 1 u 1
1
e
1
xlim lim lim lim 0
1 x 1 x1 x
u
1 x
u 01 x
x x u 0
x 0 x 1
1
e
xx 1
1
e
f x
1 1
e e
1
e
x
y
y 1
y
O 1
1
e

14. 3 4 3 2
i) f
ii)
iii) N f
i)
3 4 3 2
2 3 3 2
2 3 2
1
x
2 3 2
2 2
2
1
x
2
1
x
x 0
f
f
f

15. ii) f
x x 0
2
x x
2
x
lim x 72 36
x x
2
x
lim x
x x x
x
lim 72 36 72 lim 36 lim
x x x x
72 lim 36 0
x
1
72 lim 36
x
2
x 0 x 0
2
x 0 x 0
2
x 0
x 0
2
x 0
72lim 36 0 2
1
x
72lim 0 2
1
x
1
x72lim 2
1
x
f
x 0
limf x
x
x

16. iii) ii)
f
0
0
f
0f x f x 0 0
0f x f x 0 0
f 0
0 f 0x
0 0
f
f
x 0 0x
f
f
0

17. 2
i) f
ii) N f
iii) N f
iv)
x
i)
2 1
x
x 0
2 2x 2
2
x x
2x 2
x
x 02x 2
x
x 02x 2
x
f
f
x 0 1
f
f
0

18. f
fC
ii) i) f
f
x 1
f x f 1 0
f
ii) f
2
x 0 x 0
2 2
x 0 x 0 x 0
2
1
x
1
2
xlim lim lim 2x 0
1
1
x
x
2
x x
2
x
lim x 1 2
x
2
x
lim x
f
f
f
f
f
f

19. x x
x
lim 1 2 1 2 lim
x x
1 2 lim
x
1
1 2 lim
x
x 0 x
iv) H
x
x 0
2
2
2
f x (1)
f 1 1
(1)
f x
fC
y
fC
x
y
fC
y
O 1
1

20. f : 1,
f e 1
2
f xf x
x 4 4x
f xf x
x 1
i)
2
ii) N fC
iii) f
iv) f
f
i)
2
x f x f x 4 f x f x 4xf x f x x 1
222
x f x 4 f x 4xf x f x 0 x 1
22
xf x 2 f x 2 xf x 2 f x 0 x 1
2
xf x 2 f x 0 x 1
xf x 2 f x 0 x 1
xf x 2 f x x 1
f x 1
x2 f x
x 1
x 1
x 1

21. x e
x 1
2
ii)
2
x x
0 0
0 0 0
0 0 00 f x f x 0 x 0 0 0x f x f x 0
20
0 0
0
2
0 0
0 0
0 0
0
2
0
x
22 2 2
0
2
2
0 2 2 2
e e e
2
2
4
y 4 x e
e
2 2
4 4
e e
0 0

22. iii)
x
f
fC
x 1 x 1 x 1
fC fC
1
2
x
x 1 xx
y 0 x x
fC fC
iv)
2
2 2
f x f x 2 2 2
x x x
1
x2 2
x x
x
x 1
x
x 1
x
f
fC
x 1 e
f
f
0