-
1
1.
1.1
a
f(x) a, L
x a
lim f(x) L
f(x) x a L
x a
lim f(x) L
f(x) x a L
x alim f(x) L
f(x) x a L
(1)x alim f(x) L
x alim f(x) L
x alim f(x) L
(2) L
x a
lim f(x)
,x a
lim f(x)
,x alim f(x)
f x=a
1.2
a,c,A,B n
f,gR x alim f(x) A
x alim g(x) B
(1)x alim c c
(2)x a x alim cf(x) c lim f(x)
(3) x a x a x alim f(x) g(x) lim f(x) lim g(x)
(4) x a x a x alim f(x)g(x) lim f(x) lim g(x)
(5) x ax a
x a
lim f(x)f(x)lim
g(x) lim g(x)
(6) n nx a x alim f(x) lim f(x)
(7) n n 1 n n 1n n 1 1 0 n n 1 1 0x alim(c x c x ... c x c ) c a
c a ... c a c
(8) fog(x) x alim g(x) A
(Limitofcompositefunctions)
x Alim f(x) f(A)
x a x alim(fog)(x) f(lim g(x))
=f(A)
x Alim f(x) f(A)
x a g(x) A x Alim(fog)(x) lim f(g(x)) lim f(x)
-
2
(1) x a x a
lim f(x), lim f(x)
(2) 00
,
,
(InderterminateForm: IF)
(3) f(x) x=a
1.2
2x 2
4 xlim
3 x 5
1.3 2.6 3.9 4.
f(a) IF
(1) f(a)
x alim f(x) f(a)
(2) f(a) A
0
x alim f(x)
f(a) IF
00
,
,
()
x = a f(x)
f(a)
f(a)
-
3
2.2
2 2x 4
| 4 x | x 2x 8lim
12 x x (x 4)
1.1
7 2.
4
7 3.
9
2 4.
3. f g
f(x)= 210 ; x 4
x 16; x 4
x 4
g(x)=7x3 x 1lim(fog)(x)
1.10 2.8 3.6 4.4
4. 2
2
2x 1( )x 1
22
2x 3xsin 1 ; x 0
x 2
2 ; 0 x 2f(x)
log (x x 2) ; x 2
x 1 x 1lim f(x 1) lim f(2x)
1.0 2.2 3.4 4.
-
4
5. 2
3 3x 2
2x x 6lim
6 x 2 3x 7
1. 175
2.
17
5 3.
84
25
4.
84
25
6. f(x)=2x 1
1 x
| 1 x |g(x)
1 | x |
1.f x=1 2.g x=1
3. f(1)=4 f x=1 4. g(1)=1 g x=1
7.x 1
x 2 2x 3lim
3x 7 2x 6
1.2 2. 12
3.1
2 4.2
-
5
8. f(x)=
| x | 1; x 1
1 x| 1 x |
; x 11 x
1.x 1lim f(x)
x 1lim f(x)
2.
x 1lim f(x)
>0
x 1lim f(x)
-
6
1.3
A f
xlim f(x)
=A x f(x) A
xlim f(x)
=A x f(x) A
xlim f(x) A
y=A (Horizontal asymtote)
(1) c R c0 xlim c c
xlim c c
(2) xlim f(x) A
xlim g(x) B
A,B
x xlim cf(x) c lim f(x)
cR
xlim[f(x) g(x)]
=x
f(x)lim
x
g(x)lim
xlim[f(x) g(x)]
=x
f(x)lim
x
g(x)lim
x
f(x)lim
g(x)
= x
x
lim f(x)
lim g(x)
x
(3) c R,c0 f
xlim f(x)
xlim f(x)
x
clim 0
f(x)
xlim f(x)
xlim f(x)
x
clim 0
f(x)
(4) k R+ x>0 k
xlimx
x
-
7
1.1/xx 0
1lim
3 2 +
1/x
1/xx 0
1 2lim
3 2
1.0 2. 13
3.1
2 4.1
2.4
2 2x
4x 1 x 3lim[ ]
6 x 3x x 5x 6
1.0 2. 23
3.2
3 4.
3
2
3. 3
2
x 0 x
x 1 1lim lim ( x 1 x)
x
1.0 2. 13
3.1 4.
4. 2xlim(x x 1)
+2x
2x 3lim
3x 2
-
8
1.4
f a
f x = a
(1) f(a)
(2) x alim f(x)
(x xa x xa
lim f(x) lim f(x)
)
(3)x a
f(a) lim f(x)
(1) f(x)=anxn+an1xn1++a1x+a0
x=c c
(2) f(x)= p(x)q(x)
p(x),q(x) q(x)0
x=a a q(a)0
(3) f g x=a c
fg,fg, fg
cf x=a
fg
g(a) 0
(4) ()
f x=a g x=f(a)
gof x=a
x = a
(1) f()x=a
x a x a x a
lim f(x) lim f(x) lim f(x)
(2)
x a
f(x)lim
g(x) =L ( )
x alim g(x) 0
x alim f(x) 0
(3) fx=a
x a
x a x a
f(a) lim f(x)
f(a) lim f(x) lim f(x)
-
9
1. f(x)= 3g(x) : x 2
x A; x 2
x 2
f x=2 2
x 3
x 1lim (x A) g( )
x 1
2. a,b x 1
3x 1 alim b
x 1
a+b
-
10
3. x 3 ; x 3
2x 10 x 13f(x)
a ; x 3
a
f x=3 a(PAT1:5 2554)
4. f
2
x a , x 2
2x b , 2 x 3f(x)
5
x 6x 11 , x 3
a,b f x=2 x 3lim f(x)
| a 5b | (PAT1:1.. 2557)
-
11
5. k f g x=4
2
x 4f(x) ; x 4
x 2g(x)
4 kx ; x 4
f y=x+1 x=4 k
1.(3,1) 2.(2,0) 3.(1,1) 4.(0,2)
6. f(x)=22x x 6
x 7 3
g(x)= x 7 f x=2
gof(2)
1.7 2.8 3.9 4.10
-
12
2.
2.1
y=f(x)
y x x x+x
y f(x x) f(x)
x x
y x x
x 0 x 0
y f(x x) f(x)lim lim
x x
y = f(x) x x
x 0 x 0
y f(x x) f(x)lim lim
x x
f x f(x) dydx
x h
f(x)=h 0
f(x h) f(x)lim
h
f(a) =h 0 x a
f(a h) f(a) f(x) f(a)lim lim
h x a
y=f(x)
2
2 h 0
d y f (x h) f (x)f (x) lim
hdx
3
3 h 0
d y f (x h) f (x)f (x) lim
hdx
n (n 1) (n 1)
(n)
n h 0
d y f (x h) f (x)f (x) lim , n 1,2,3,...
hdx
f x=c f x=c
f x=c f x=c
-
13
2.2
1. f(x)=a a df (x) (a) 0dx
2. f(x)= nx n n n 1d
f (x) (x ) nxdx
4.d
(sin x)dx
=cosx
5.d
(cos x)dx
=sinx
6. x xd
(e ) edx
7.d
(ln x)dx
= 1x
8. f g x c
d d(cf(x)) c (f(x))dx dx
d d d(f(x) g(x)) (f(x)) (g(x))dx dx dx
d d d
(f(x)g(x)) f(x) (g(x)) g(x) (f(x))dx dx dx
2
d dg(x) (f(x)) f(x) (g(x))
f(x)d dx dxdx g(x) (g(x))
g(x)0
9.
( : Chain Rule) y=g(u) u=f(x) g(u) f(x)
dy dy dudx du dx
(gof) (x) g (f(x)) f (x)
10.
f x 1f f (x) 0
1f x
1
1
1(x)f
f f (x)
dy 1
dx dx
dy
11. f(x,y)=C C
y=f(x) dydx
x y dydx
dydx
-
14
1. f '(x) dy
dx
1.1f(x)=5x3+2x2+6x+8 1.2.f(x)=4 3 1
3 2 2x 3x 2x 1
1.3.y=4x6+7x-5+3x-2+9 1.4.y=3 1 3
2 4 56x 2x x 8
1.5.f(x)=(6x2+4)(3x3+5) 1.6.f(x)=2
2
4x 7x 1
3x 8
1.7f(x)=(4x2+8x+1)5 1.8f(x)= 2x 1
2. f(x)= 3x 1 x 0
f(1 x) f(1)lim
x
1. 14
2.1
2 3.
3
4 4.1
-
15
3.
7 5 4
3 3 3
2
3x 12x 24xf(x)
x
h 0
f(x h) f(x)lim
h
x=8
1.0 2.1 3.2 4.3
4. y=f(x)=4+ x 1 a y x
x=1 x=5 b y x x=2
1.a=b 2.|a+b|=1 3.|ab|=1 4.a=2b+1
5. y=f(x)=x+ 4x
x[1,4] c(1,4)
f(c) y x x=1 x=4
f(c)
1.1 2. 12
3.1 4.1
2
-
16
6. y= x +x x f(x)3 [a,b]
y x x=a x=b
1. 14
2.4 3.1
5 4.5
7. 2x a
f(x)x b
a b
f(0)=4 f(0)=8 f(0)
1.2 2.2 3.1 4.1
8. g(x) f(x)=(x1)2g(x)
x2 f(x) 3 x2 f(x) 4 g(2)
1.3 2.2 3.1 4.4
-
17
9. f(x)= 3x 1 g (fog)(x) =x2+1 x
f(1)+g(1)
1. 4112
2.35
12 3.
33
4 4.
39
4
10. f(x)=x26x+c c
a b f(x)=0 3a+2b=20 f(c)
1.38 2.26 3.26 4.38
11. y=f(x)= | x |x
x{0}
. f(0)=0 x{0}
. f(0) f(0)
1.. . 2.. . 3.. . 4.. .
-
18
12. f(x+1)= 2x 1 x
f(2)+f(2)
1.0 2.2 3.4 4.6
13. f f(0)= f (0) =2
2x 0
xlim
(f(x)) f(x) 2
14. f(x) f(x) x=2 2x 2
f(x) 5lim 3
x 4
f(x) x=2
-
19
15. 3
1 1 1f (x)2 x x
h 0
f(1 h) f(1)lim
f(4 h) f(4)
(PAT 1 .. 2552)
16. f(5)=0 h 0
f(5 h)lim 10
h
g(x)= f(x)
x g (5)
17. R f : R R f(3)=111
x 3
xf(x) 333lim 2013
x 3
f(x) x x=3
(PAT1...2557)
-
20
18. f f 2x+y=4 x=1
f (1) 2 x 1
f(x) 2lim
x 1
19. F(x)=f(g(x)) g(2)=4, g (2) 5, f (2) 6, f (4) 9 F (2)
20. 3f(x) 2x 5 g f g(21)
21. f f(x) x
3(f(x)) xf(x) 8 f(0)
-
21
22. g f(x)=xg(x)
f(x)= 3 24x 9x f(0)=0 d f(x)dx g(x 1)
x=2
23. f(x)=|x| 2x g(x)= 2| x x | 4 (fog)(2)
24. f
2
2
16 x; x 4
x 4f(x)
kx x 2 ; x 4
f x=4 (fof)(5)
-
22
25. f,g h f(1) g(1) h(1) 1 f (1) g (1) h (1) 2
(fg h) (1) (PAT 1: 11 .. 2552)
26. R f : R R g : R R
23f(x) 3x ,g(1)=8 2g (1)
3 (fog) (1) (PAT1)
27. 2 2
2 2
x y1
a b P(x0,y0)
1. 0 02 2x x y y
1a b
2. 0 02 2
x x y y1
a b
3. 0 02 2xx yy
1a b
4. 2 20 0
1 x 1 y( ) ( ) 1x ya b
-
23
2.3
y=f(x)
(1) f P 1 1(x , y ) f( 1x )
1 1 1y y f (x )(x x )
(2) f P(x,y) P(x,y)
(3) f P 1 1(x , y ) 1
1
f (x )
1 11
1y y (x x )
f (x )
1
(1) y=f(x) 24 x=4.
(2) y=f(x) (1,2) 3...
(3) y=f(x) x=7 12.......
(4) y=f(x) (2,5) 7........
(5) y=f(x) x=2
2x+y=9...............................................
(6) y=f(x) (2,1) y=5x+3
.....
(7) y=4x5 y=f(x) x=5
.....
(8) 2x+y=5 y=f(x) y= 1
.....
y
x
y=f(x)
P(x1,y1)
0
-
24
2 3 1y 2xx
x=1
(1) (2) (PAT1)
3. y=f(x)= 23x 4x 2
f x=1
4. 1L 4x3y+10=0
2L 2 8 7y x x
3 3
2L 1L 1L 2L (PAT1)
-
25
5. y=x|x|+ x (1,2)
L P L
1.7x+3y+13=0 2.7x3y+1=0 3.3x7y11=0 4.3x+7y+17=0
6. L 2 y=x2+2
(a,b) L a+b
1. 15
2.1
5 3.
2
5 4.
2
5
7. (a,b) 2 2x y 2x 6y 30 0 4
yx+1=0 a+b
1.1 2.1 3.2 4.2
-
26
2.4
(1)
AR f A R BA
(1)f B x1,x2B x1
-
27
(2) (Maximum and Minimum Value)
c f
(1)f(c) f (a,b) c(a,b)
f(c)f(x),x(a,b) (c,f(c))
(2)f(c) f (a,b) c(a,b)
f(c)f(x),x(a,b) (c,f(c))
c f f
f(c)=0 f(c)
c f (c,f(c)) f
1 c f(c)=0 f(c)
2 c 1 f(c)
1 ( )
(1)
(2)
2 ( f(x))
f(x) c
(1) f(x)0
(3) f (x) = 0 *** 1
2 c f(c)=0
-
28
(3) (Absolute Maxima and Minima)
f D cD
(1)f c f(x)f(c) xD
(2) f c f(x)f(c) xD
f [a,b]
(1) f
(2) f(a) f(b)
(3) f(a),f(b)
(4) f(a),f(b)
1. 3 2f(x) x 3x 9x 10
2. y=f(x)= 3 21 1
x x 2x3 2
x[3,4]
-
29
3. y=f(x) x
(1) x=2 y=f(x)..
(2) y=f(x) x=1, x=1..
(3) y=f(x) x=3..
(4) y=f(x) x=4
(5) y=f(x) x=5 x=2
(6) (2,4) y=f(x)..
(7) (1,1) y=f(x)
(8) y=f(x) 5 x=1..
(9) y=f(x) 1 x=5..
(10) y=f(x) 9 x=3 (2,3)
.
4. f(x)=3x10 F(x)=(f g)(x) =ax2+bx+c
F(0) =1 F x=2 5 g(1)
-
30
5. y=f(x) 3 x=2
3x+y7=0 (1,4)
g(x)= 2x f(x) g (2) g (1)
6. f f(0)=2
f x=1 x=1
f(4)
7. A,B,B0 2
Ax 9 ; x 1f(x)
Bx Ax 5 ; x 1
f x x=2 A+B
-
31
8. f
(1) f ....................... (2) f........................
(3)f x
..................................................................
(4) f
x....................................................................
(5)f..................................................................
(6) f
X......................................................................
(7)
x.......................................................................................
(8)
x.........................................................................
9. f(x)=a 3x +bx a b f
2 x=1 g(x)= 3x +f(x) g
1.(0,2) 2.(3,1) 3.(1,1) 4.(2,0)
10. y=f(x)=detx 1
x 1 x
x[2,2]
F=(fof)f F
1. 2 2. 12
3.1
2 4.
1
4
Y
X0
2
2
1 2 3 4 5 6
y f (x)
(4,2)
(2, 2)
-
32
11. f(x)=3x10 h(x)=(f g)(x) = 2ax bx c
h(0)=1 h x=2 5 g(1)
1.2 2.3 3.5 4.6
12. y= 4 2x 2x kx 4 k x=2
24
1.7 2.8 3.9 4.10
13. n nf =nx2n2x g(x)=
10
nn 1
f (x)
x g x
1.2.5 2.2.7 3.3.2 4.3.5
-
33
3.
3.1
f
F(x) F(x)=f(x) x f
f(x) f(x)dx
f(x)dx F(x) C C
k C
kdu = ku+C
nu du =n 1u
Cn 1
,n1
kf(u)du= k f(u)du
[f(u)g(u)]du= f(u)dug(u)du sinudu = cosu+C
cosudu = sinu+C
u
e du =u
e +C
1
udu = ln|u|+C;u0
(Integration by Substitution)
f(g(x)) g '(x)dx
f(x) f '(x) dx
f(g(x)) g '(x)dx =f(u)du
u=g(x) du=g(x)dx
-
34
1.
1.1(3x2+x2)dx 1.2(5 2
2 52x x )dx
1.3 4 2
4
6 7x x
x
dx 1.4x2(3x+4)dx
1.5 2(1 )x
x
dx 1.6 x (x3)2(2x+1)dx
1.7x2 3 1x dx 1.84x 21 2x dx
-
35
2. 3 2f (x) x 4x x 6 f(x) f(1)=3
3. 2
2 3f (x) 2x 4x 1 g(x)= 21 x (fog)(1)=1 f(x)
4. 2f (x) x 4x 3 f (1,7)
f
5. f f(x)=2x+1
f 12
x=1 f
-
36
6. f(x)= 2(3x 2) dx g(x)=4x 2x
f '(x)
g(2)
7. y=f(x) x2
(2,4
3) 3 f (a,b) 9 a+b
8. y=f(x) (x,y) 3kx 10x 6 k
(1,3) x f(1)
-
37
9. R f : R R
f (x) 6x 4 x y=f(x)
(2,19) 19 f(1) [PAT1:6.. 2553]
10. y x. (x,y)
3
2
2x 3
x
x+
(1) (1,1) (2, 52
)
(2) x=2
4x4y+11=0
1. (1) 2. (2)
3. (1) (2) 4. (1) (2)
11. y=f(x) y x (x,y)
4 x
y 2
f (4,1)
(1) f .. (4,2) (4,1)
(2) f (4,3) y
1. (1) 2. (2)
3. (1) (2) 4. (1) (2)
-
38
3.2
(Fundamental Theorem of Calculus)
y=f(x) [a,b]
F(x) [a,b] F(x)=f(x)
b
af(x)dx =F(b)F(a)
b
F(x)a
=F(b)F(a)
f [a,b]
1. b
a
f(x) dx
2. f(x)0 x[a,b]b
a
f(x) dx 0
3. f(x)0 x[a,b]b
a
f(x) dx 0
4. b
a
f(x) dx =a
b
f(x) dx
5. b
a
kf(x) dx =kb
a
f(x) dx k
6. a
a
f(x) dx=0
7. b
a
f(x) dx =c
a
f(x) dx +b
c
f(x) dx
8. c[a,b] b
af(x)dx f(c)(b a)
-
39
1.
(1)1 3 2
0(8x 6x 1) dx
(2)1 2
2(3x 4)
dx
(3)1 2 3
13x x 1 dx
(4)3
1
0 4
xdx
x 9
-
40
2. 2f (x) 3x x 5 f(0)=1 1
1
f(x)dx
[PAT1:11.. 2552]
3. 2f (x) x 1 1
0
f(x)dx 0 | f(1) |
[PAT1:10.. 2552]
4. f(x) y=f(x)
(1,2) 4 2
1
f(x)dx 12
f( 1) f ( 1)
[PAT1:3.. 2553]
-
41
4. 2
n 2n0
1a dx
x n n
n 1
(1 2n)a
5. y=f(x) XY f x=1
2h 0
f(1 h) f(1)lim 3
h 2h
y=g(x) (1,2)
3
2
g(x)dx
6. f(x)=2kx ; x 2
2x k ; x 2
f x=2 5
2
f(x)dx
1.23 2.25 3.27 4.29
-
42
3.3
f [a,b]
f x = a x = b
f x x=a x=b
2
1. f(x)0 x[a,b]
A X A=b
a
f(x) dx
2. f(x)0 x[a,b]
A X A=b
a
f(x) dx
4 f g [a,b]
f g x = a x = b
f g x=ax=b
f g
f g [a,b] f(x)g(x) x
[a,b]
A f g x=a x=b
A=b
a[f(x) g(x)] dx
y=f(x)
Y
Xa b0
A
y=f(x)Y
Xa b0A
Y
Xa
b0
A y=f(x)
y=g(x)
-
43
1. f(x)=(x1) 3 x=1 x=2
2. y=x 3 y=x
3. A y=1 2x x
B 2xy4
x x=c x=c
c A=B [PAT17.. 2552]
-
44
B-PAT1 : 2551
1. f(x)= 2x d 0
f(x d) f(x d)lim
d
1.x 2.2x
3.4x
4.
2. f g
. f '(a) 0 f x=a
. f '(x) g '(x) f(x)g(x)
1.. .
2.. .
3.. .
4.. .
3. a (a,b) C y= 2x
P C C P
(0,0) (a,b) P
1. 1 1( , )2 4
2.2a a
( , )2 4
3.2a a
( , )4 16
4.2a a
( , )3 9
-
45
4. f(a)=2a
1 2
t 1dt
t
a 1 f(2551)
1.2550
2. 2(2550)
3.2(2550)
2551
4.2
2550
2551
PAT 1 : 2552
5.
A 2y 1 x X
B 2x
y4
X x=c x=c
c A=B
1. 2 2.2
3.2 2
4.4
6. f(x)= 4 2x 3x 7 f
1.(3,2)(2,3) 2.(3,2)(1,2)
3.(1,0)(2,3)
4.(1,0)(1,2)
7. 3
1 1 1f '(x)
2 x x
h 0
f(1 h) f(1)lim
f(4 h) f(4)
1.1
2.16
5
3.7
5
4.1
5
-
46
PAT 1 : 2552
8. 2f (x) 3x x 5 f(0)=1 1
1
f(x)dx
1. 53
2.7
3
3.2
3
4.1
3
9. f,g h f(1)=g(1)=h(1)=1 f (1) g (1) h(1) 2
(fg h) (1)
1.1 2.2
3.4
4.6
10. 3 1y 2xx
x=1
1.13x2y11=0 2.13x+2y15=0
3.2x13y+11=0
4.2x+13y15=0
PAT 1 : 2552
11. 2f (x) x 1 1
0
f(x)dx 0 | f(1) |
12. 2f(x) ax b x a b b0
2f (1) f(1) f(4)
f (9)
-
47
13. y=f(x) x=1 f (x) 4 x
f(1)+f(3)=0 f
PAT 1 : 2553
18. R f : R R g : R R
23f(x) 3x ,g(1)=8 2g (1)
3 (fog) (1)
1. 13
2. 23
3.1
4. 43
19. a b f
3
2
x 3x 2 , x 2x 2
f(x) a b , x 2
x ax 1 , x 2
f 2 2a b
20. R f : R R
f (x) 3 x 5 x f(1) 5 2
x 4
f(x ) 2lim
f(x)
-
48
21. R f : R R
f (x) 6x 4 x y=f(x) (2,19)
19 f(1)
PAT 1 : 2553
22. a b f
3| x 1 |, 1 x 1
x 1
f(x) ax b , 1 x 5
5 , x 5
f (1,) ab
1. 54
2.7
4
3. 15
4. 10
23. x 3 2x 450x 60,200x 10,000 200
()
24. f(x)
y=f(x) (1,2) 4 2
1
f(x)dx 12
f(1)+ f ( 1)
-
49
25. h(x)=f(x)g(x) y=f(x) (x,y)
22x y=f(x) 5
g g(2)= g (2) 5 h (2)
PAT 1 : 2553
26. R f : R R x=1
g
g x=1
(fog)(1)
1.2 3
2.2
3.2 7
4. 7 2
27. a b f 4 3 2f(x) x 2x x ax b
Q(x) Q(x)= 2(Q(x)) 10
f(x)dx
1. 7130
2. 3130
3. 1130
4. 130
x>1
x1
x 3 2
x 1g(x)
f(x)
| x | 7
-
50
PAT 1 : 2554
28. 3 2
2x 0
x x xlimx
1. 12
2. 12
3.1
4.1
29. f f (x) ax b a b
f(0)=2 f (1,5) 2a+3b
1.12 2.20
3.42
4.48
30. R
g : R R 1g(x)2x 3
x 32
f : R R (f g)(x) x x
1f ( )2
1. 12
2. 12
3.8
4.8
31. R
f : R R g : R R xR
2 6 4 3 2g(x) x 2x 5, (f g)(x) x 2x 2x x 2x 5 f(0)=0
(f g )(1) (g f )(0)
-
51
32. y=f(x) 2xy+3=0 (0,3) 2
0f (x)dx 3 g(x) x 2 f(x) g (2) 0 f(2)
34. x 3
2x 10 x 13f(x)
a
a
f x=3 a
PAT 1 : 2554
35. f : R R 2
3f(x) x
L f(x) (a,f(a)),a>0
L y 52
L
1.(2,7) 2.(1,4)
3.(2,4)
4.(3,5)
36. A(0,0),B(1,0) C 1 3( , )2 2
ABC
f(x)= 2ax bx c A B AC BC f
A B f AB
1. 36
2. 33
3. 32
4. 2 33
x3
x=3
-
52
37. f,g,h x 6(f g)(x) 3x 14, f x 2, h(2x 1) 6g(x) 123
h (0)
38. f : R R f ''(x) 0
f(0)=23f(1)=103 1
0
f(x)dx
39. L (0,10) 1 0
L X x=0 x=6
51
L X x=0 x=3
40. 3 3x 0
xlim
x 8 x 8
-
53
PAT 1 : 2555
41. R
f:RR f (x) 2x 1 f (2) 2
y=f(x) (1,3)
1. 1y x 22
2.1 5
y x2 2
3.1 5
y x2 2
4.1
y x 22
42. R
f:RR,g:RR h:RR
2
ax 1f(x)
x 1
a
2g(x) (x 1)f (x)
f(x) ; x 2
h(x)g(x) ; x 2
h x=2 2h(2)h(2)
1.0.6 2.0.8
3.1
4.3
43. R
f:RR,g:RR h:RR
h(x)= 2x 4 , g(x) h(f(x) 1) f (1) g(1) 1
f(1)
1.2 2.1.5
3.1
4.0.5
-
54
44. R f:RR g:RR
f(x)=2x+3 (gof)(x)= 3 28x 44x 80x 48 x
60
f(g(x))dx
45. f(x)= 3x ax b a b
1
L 2
L x=a x=b
1
L 2
L h 0
9hlim 1
f(1 h) f(1)
20
f(x)dx
46. 3 2
2x
4
(cot x 1)cos ec xlim
1 cos2x 2sin x
PAT 1 : 2555
47. 3 2f(x) x 26x bx 216 b
1 2 3a , a , a f(x)=0
f (1)
1.211 2.107
3.101
4.85
-
55
48. f(x)
f(0)=1 f(x+1)=f(x1)+x+1 x
1
2
f(x)dx
1.3 2.2
3. 23
4. 13
49. 2
x 1
1 x 2xlim
x 3 2
1.12
2.0
3.12
4.
50. R f : R R g : R R
1.(fg)(x)=2x+3 x
2. f g x
3. f 2 x=1
4. g (x) 2 x
g
51. P(x)
2 4 2P(x 3) 3x 24x 40
x
0
f(x) P(t)dt
x 2lim P(x) f(x)
-
56
52. P(x) P(0)=1
h 0
3xh 2hlim 1
P(x h 2) P(h 2) P(x 2) P(2)
P(12)
53. L (0,1) x+2y=6
(quadrant) 1 x y L
x+2y=6
PAT 1 : 2556
54. xlim x(x 1) x 2
1.0
2. 12
3.1
4. 32
55. C 4
3
3x 2y
x
x>0 L C
(1,1) L x(x1)=y1 A B
A B
1. 4 82 2. 8 82 3. 4 41 4. 8 41
-
57
56. 2
2x 8, x 4
2x 4x 3x 12f(x)kx
, x 43
k
f x=4 f(k+1)
57. f f(x)
x 3ax bx a b 3g(x) (x 2x)f(x)
f (1) 18, f (0) 6 f(2) f(1) f(0) g ( 1)
58. f(x)
x+1 f(x),5+2i f(x)=0 f(0)=53
2
0
[f(x) f( x)]dx
PAT 1 : 2556
59. a b
2x ax b , x 2
f(x) x 1 , 2 x 5
ax b , x 5
f ab
1.5 2.8 3.11
4.12
-
58
60. 2
2
2
ax 7x 6 dx
b a b b0
... a b 1 a+b
1.33 2.69
3.102
4.104
61. 3
6 3
4xf(x)
x 3x 64
x
() f (0,3)
() f 413
1.() ()
2.() ()
3.() ()
4.() ()
62. 2f(x) x ax b a b
f(1)=2 (f f)(0) 10 2
1
f(x)dx
63. R
f : R R f (x) 3 6x x
y=f(x) (2,22) 20
x 4lim f(x)
-
59
PAT 1 : 2557
64. f
2
x a , x 2
2x b , 2 x 3f(x)
5
x 6x 11 , x 3
a,b
f x=2 x 3lim f(x)
| a 5b |
1.8 2.18
3.88
5
4.102
5
65. b>1 b
1
x 1dx 4
x x
21 b b
1.21 2.31
3.91
4.111
66. R a,b
f : R R 3f(x) a bx x x
5xy+13=0 f x=1
2
0
f(x)dx
-
60
67. R f : R R f(3)=111
x 3
xf(x) 333lim 2013
x 3
f(x) x x=3
68. 2f(x) ax bx c a,b,c a0
f(1)=0 f 1x3
F( , ) f(x)dx
F(0, t) F(1, t) 1 t>1
() F(1,2) F(2,3) 10
() 2
f(x)
x
2
3
3x 2x 2
x
1.() ()
2.() ()
3.() ()
4.() ()
-
61
: 2552
1. 2xlim x 6x x
: 2554
2. f(x)=
4
2
ax x 1 , x 1
1 , x 1
x 2a , x 1
x 1lim f(x)
f(1)x 1lim f(x)
3. 62x 10
x=0 x=3
Clearinghouse : 2555
4.2
0
6x | x 2 |dx
-
62
Clearinghouse : 2556
5. 3 2f(x) x 3x 9x 1 [-1,2]
6. f(x) 2x+5
y=g(x) (x,y) 23x
f g (1,2)
f (1)g
1.5
2.2
3.1
4.2
5.5
7. g(x)
2
x 1 ; x 11 x
f(x) g(x) ; 1 x 2
2x 3 ; x 2
f 2
1
g (x) dx
1. 32
2. 12
3.0
4. 12
5. 32
-
63
8. A={1,2,3,4,5,6} B= 2{ p(x) | p(x) ax bx c a,b,cA}
p(x) S p(x) 1
0
p(x) dx
1. 112
2. 212
3. 312
4. 412
5. 512
9. f
f
. f(x)=x 2
