2010 ________________________________________PAT1 (1)

(Function) f f Df f f Rf f y = f(x) f x 1. f A B f A B f : A B 2. f A B f A B f : A B 3. f A B f A B y Rf x Df (x, y) f x1 x2 A f(x1) = f(x2) x1 = x2 f : A 1 - 1 B 4. f A B (one - to - one function from A onto B one - to - one correspondence) f f : A onto1 - 1 B f A 1. f (increasing function) A x1 x2 A x1 < x2 f(x1) < f(x2) 2. f (decreasing function) A x1 x2 A x1 < x2 f(x1) > f(x2) (Polynomial function) f(x) = anxn + an-1xn-1 + + a1x + a0 an, an-1, ..., a1, a0 n

PAT1 (2) _______________________________________ 2010

1. (Linear function) f(x) = ax + b a, b f(x) = -3x + 5 a = 0 f(x) = b (Constant function) f(x) = 5 a = 1 b = 0 f(x) = x (Identity function) 2. (Quadratic function) f(x) = ax2 + bx + c a, b c a 0 f(x) = 3x2 + 1 (Composite function)

gof

B Cf g

A

fD gRgf D ,R

(gof)(x) = g(f(x)) Dgof = {xDf| f(x) Dg} gof RfI Dg (Algebra of function) (f + g)(x) = f(x) + g(x) ; Df + g = DfI Dg (f g)(x) = f(x) g(x) ; Df g = DfI Dg (f g)(x) = f(x) g(x) ; Df g = DfI Dg

gf (x) = g(x)f(x) ; Dr/g = DfI Dg - {x | g(x) = 0}

(Inverse function) f f-1 f 1 - 1 1. (gof)-1(x) = (f-1og-1)(x) 2. f() = f-1() =

2010 ________________________________________PAT1 (3)

1 f(x) = x2 - 1 x (-, -1]U [0, 1] g(x) = 2x x (-, 0] (PAT 152) 1) Rg Df 2) Rf Dg 3) f 1 - 1 4) g 1 - 1

2 f(x) = 3x - 1 g-1(x) = x2 , x 0

-x2 , x < 0 f-1(g(2) + g(-8)) (PAT 152) 1) 3

2 1- 2) 32 1+ 3) 3

2 1-

- 4) 32 1

-+

3 f(x) = 10x g(x) = 23x 100- Rgof (ENT47) 4 f(x + 1) = 3x + 2 + f(x) g(3x - 1) = 2x + 8 f(0) = 1 g-1(f(2)) (ENT 44) 1) -1 2) 0 3) 1 4) 2 5 f(2x - 1) = x2 + 5x - 1 g 2 1 x

2 - = 3x - 2 f-1(-5) + g-1(-2) 1) - 27 2) - 25 3) 25 4) 27 6 g-1(x) = 2 x 5- (gof)(x + 1) = 2 - 6x - 3x2 f(x - 1) 7 f(x) = 2 sin 2x g(x) = 1 x2 - (RfI Dg) - Rgof (ENT 44) 1) {-1, 1} 2) {-2, 2} 3) [-2, - 3 ]U [1, 2] 4) [-2, -1] U ( 3 , 2]

8 f(x) = 1 - x , x [0, 1] 1 + 1 x- , x (1, ) . f-1(x) f(x) x (1, ) . a 0 2 f-1(a) = a (ENT47) 1) . . 2) . . 3) . . 4) . .

PAT1 (4) _______________________________________ 2010

1. f(x) = x + 1 , x 0 x - 1 , x < 0 g gof (ENT47) 1) g(x) = 1 x (-, -1)U [1, ) 2) g(x) = f-1(x) x (-, -1)U [1, ) 3) g(x) = (x - 1)

2 , x 1 (x + 1)2 , x < -1 4) g(x) = x

3 x (-, -1)U [1, ) 2. A = {1, 2, 3, 4} B = {1, 2, 3, 4, 5} f A B f(1) = 2 f(2) = m m

f (ENT48) 1) 75 2) 150 3) 425 4) 500 3. f(x) = ax2 + b g(x - 1) = 6x + c a, b, c f(x) = g(x) x = 1, 2 (f + g)(1) = 8

(fog-1)(16) (ENT47) 1) 931 2 961 3) 10 4) 20 4. f(x) = 2 x 1

x- x (-1, 1)

. f-1(x) = 2x 4x 1 12+-- , x 0

0 , x = 0 . f (-1, 1) (ENT47) 1) . . 2) . . 3) . . 4) . . 5. f, g Df = [0, ) f-1(x) = x2, x 0 g-1(x) = (f(x))2 + 1, x 0 a > 0 f(a) + g(a) = 19 f-1(a) + g-1(a) (ENT46) 1) 273 2) 274 3) 513 4) 514 6. a > 0 f(x) = ax2 x 0, g(x) = x3 (f-1og)(4) = 2 (64)g

(64)f11

--

(ENT46) 7. f(x) = -(x - 1)2 x 1 g(x) = x 1- x 1 . f-1 = 1 - |x| x 1 . (g-1of-1) 41- = 43 (ENT46) 1) . . 2) . . 3) . . 4) . . 8. I f, g I I f(x) = 2x

g(x) = 2x x

x x gof - f I I (ENT45) 1) 2) 3) 4)

2010 ________________________________________PAT1 (5)

1. 4) g(x) = x3 x (-, -1)U [1, ) f(x) = x + 1 , x 0 x - 1 , x < 0 1) g(x) = 1 gof(x) = g(f(x)) = 1 a R

axlim

(gof)(x) = 1

gof 2) g(x) = f-1(x) f-1(x) = x - 1 , x 1 x + 1 , x < -1 gof(x) = g(f(x)) = f-1(f(x)) = f

-1(x + 1) , x 0 f-1(x - 1) , x < 0

x 0 x + 1 1 f-1(f(x)) = (x + 1) 1 = x x < 0 x - 1 < -1 f-1(f(x)) = (x 1) + 1 = x gof(x) = x a R

axlim

(gof)(x) = gof(a) = a gof 3) g(x) = (x - 1)

2 , x 1 (x + 1)2 , x < -1

gof(x) = g(f(x)) = g(x + 1) , x 0 g(x - 1) , x < 0 x 0 x + 1 1 g(f(x)) = ((x + 1) - 1)2 = x2 x < 0 x - 1 < -1 g(f(x)) = ((x - 1) + 1)2 = x2 gof(x) = x2 a R

axlim

(gof)(x) = gof(a) = a2 gof 4) g(x) = x3 gof(x) = g(f(x)) = g(x + 1) , x 0 g(x - 1) , x < 0 =

(x + 1)3 , x 0 (x - 1)2 , x < 0

-0x lim (gof)(x) = (0 - 1)2 = -1 +0x lim (gof)(x) = (0 + 1)

2 = 1

-0x lim (gof)(x) +0x lim (gof)(x) 0xlim (gof)(x) gof

PAT1 (6) _______________________________________ 2010

2. 3) 425 A B f(1) = 2 f(2) = m m P f(1) = 2 Q f(2) = m m n(PU Q) 1 f(1) = 2 1 f(2) = 1, 2, 3, 4 5 5 f(3) = 1, 2, 3, 4 5 5 f(4) = 1, 2, 3, 4 5 5 n(P) = 1 5 5 5 = 125 2 f(2) = m m f(2) = 1, 3 5 3 f(1) = 1, 2, 3, 4 5 5 f(3) = 1, 2, 3, 4 5 5 f(4) = 1, 2, 3, 4 5 5 n(Q) = 3 5 5 5 = 375 3 f(1) = 2 f(2) = m m f(1) = 2 1 f(2) = 1, 3 5 3 f(3) = 1, 2, 3, 4 5 5 f(4) = 1, 2, 3, 4 5 5 n(PI Q) = 1 3 5 5 = 75 n(PU Q) = n(P) + n(Q) - n(PI Q) = 125 + 375 - 75 = 425 3. 4) 20 f(x) = ax2 + b g(x - 1) = 6x + c a, b, c A = x - 1 x = A + 1 g(A) = 6(A + 1) + c = 6A + 6 + c g(x) = 6x + 6 + c x = 1 f(1) = g(1) a + b = 6 + 6 + c a + b - c = 12 ...(1) x = 2 f(2) = g(2) 4a + b = 12 + 6 + c 4a + b - c = 18 ...(2) (2) - (1) ; 3a = 6 a = 2 b - c = 10 ...(3)

2010 ________________________________________PAT1 (7)

(f + g)(1) = 8 f(1) + g(1) = 8 2 + b + 12 + c = 8 b + c = -6 ...(4) (3) + (4) ; 2b = 4 b = 2 c = -8 f(x) = 2x2 + 2 g(x) = 6x - 2 g-1(6x - 2) = x 6x - 2 = A x = 6 2 A + g-1(A) = 6 2 A + g-1(x) = 6 2 x + (fog-1)(16) = f(g-1(16)) = f

+6 2 16 = f(3) = 2(3)2 + 2 = 20

4. 3) . . . f(x) = 2 x 1

x- ; x (-1, 1)

y = 2 x 1x

- ; x (-1, 1) x = 2y 1

y- ; y (-1, 1)

x(1 - y2) = y x - xy2 - y = 0 xy2 + y - x = 0

y = 2xx)4(x)( 1 1 ---

= 2x4x 1 1 2+- x 0

-1 < y < 1 f-1(x) = 2x 4x 1 12++- x 0 .

. f(x) = 2 x 1

x- x (-1, 1)

f(x) = 222

) x (12x) x( ) x (1

----

= 2222

) x (12x ) x (1

-- +

= 2222

) x (12x ) x (1

-- +

= 222) x (1

x 1-+

x (-1, 1) 1 + x2 > 0 (1 - x2)2 > 0 f(x) > 0 f (-1, 1) .

PAT1 (8) _______________________________________ 2010

5. 1) 273 f-1(x) = x2, x 0 y = x2, x 0 y 0 x = y2 , y 0 x 0 y = x , x 0 f(x) = x , x 0 g-1(x) = (f(x))2 + 1 , x 0 g-1(x) = ( x )2 + 1 = x + 1 , x 0 y = x + 1 , x 0 y 1 x = y + 1 , y 0 x 1 y = x - 1 , x 1 g(x) = x - 1 , x 1 f(a) + g(a) = 19 a > 0 a + a - 1 = 19 a + a - 20 = 0 ( a + 5)( a - 4) = 0 a = -5 () a = 4 () a = 16 f-1(a) + g-1(a) = a2 + a + 1 = 162 + 16 + 1 = 256 + 16 + 1 = 273 6. 0.5 g(x) = x3 y = x3 x = y3 y = 3 x g-1(x) = 3 x (f-1og)(4) = 2 f-1(g(4)) = 2 f-1(64) = 2 g(x) = x3 g-1(x3) = x x3 = 64 x = 4 g-1(64) = 4 (64)g

(64)f11

--

= 42 = 21 = 0.5

2010 ________________________________________PAT1 (9)

7. 1) . . . f(x) = -(x - 1)2 x 1 y = -(x - 1)2 , x 1 , y 0 x = -(y - 1)2 , y 1 , x 0 -x = (y - 1)2 , y 1 , x 0 y - 1 = x- , y 1 , x 0 y = x- + 1 , y 1 , x 0 y 1 y = f-1(x) = 1 - x- |x| = -x x 0 f-1(x) = 1 - |x| . . g(x) = x 1- x 1 y = x 1- , x 1 , y 0 x = y 1- , y 1 , x 0 x2 = 1 - y , y 1 , x 0 y = 1 - x2 , y 1 , x 0 g-1(x) = 1 - x2 , x 0 (g-1of-1)

41- = g-1

.

41f 1 --

= g-1

41 1 ---

= g-1

21 1-

= g-1

21

= 1 - 41 = 43 . 8. 1) f(x) = 2x g(x) = 2

x x x x

(gof)(x) = g(f(x)) = g(2x) = 2

2x = x 2x (gof - f)(x) = (gof)(x) - f(x) = x - 2x = -x y = -x gof - f

Y

X-5 5

2

4

-2

PAT1 (10) ______________________________________ 2010

2 (Distance between two points) P1(x1, y1) P2(x2, y2) | 21PP | = 221221 )y (y ) x (x -- + y1 = y2 P1 P2 X | 21PP | = |x1 - x2| x1 = x2 P1 P2 Y | 21PP | = |y1 - y2| A(x1, y1), B(x2, y2), C(x3, y3)

ABC = 21 = 21 [(x1y2 + x2y3 + x3y1) - (x1y2 + x3y2 + x2y1)]

3 2 P(x, y) P1(x1, y1) P2(x2, y2) x = 2

x x 21 + y = 2y y 21 + (Point of Division) P(x, y) P1(x1, y1) P2(x2, y2) r1 : r2 x = 12

2112 r rxr xr

++ y = 12 2112 r r

yr yr++

(The Centriod of the triangle) A(x1, y1), B(x2, y2) C(x3, y3) AD, BE CF P(x, y) x = 3 x x x 321 ++ y = 3 y y y 321 ++ L P1(x1, y1) P2(x2, y2) x1 x2 L = 21 21 x x

y y--

= 12 12 x xy y

--

1. L1 L2 m1 m2 L1 L2 m1 = m2 2. L1 L2 m1 m2 L1 L2 m1m2 = -1

2010 _______________________________________PAT1 (11)

L P1(x1, y1) m y - y1 = m(x - x1) 1. y = mx + b b y - intercept Y m 2. (First degree Equation) (Linear Equation) (General form Equation) Ax + By + C = 0 A, B C A 0 B 0 - BA L Ax + By + C = 0 P(x1, y1) L d d = 22 11 B A

|C By Ax|+

++ d Ax + By + C1 = 0 Ax + By + C2 = 0 d = 22 21 B A

|C C|+

- (Circle) (Center) (Radius) 1. (h, k) r (x - h)2 + (y - k)2 = r2 2. x2 + y2 + Dx + Ey + F = 0

2E ,2D - r = 2

4F E D 22 -+ (Parabola) (Directrix) (Focus) 1. (h, k) () Y (x - h)2 = 4c(y - k) (h, k + c) y = k - c c > 0 c < 0 2. (h, k) () X (y - k)2 = 4c(x - h) (h + c, k) x = h - c c > 0 c < 0 3. |4c|

PAT1 (12) ______________________________________ 2010

(Ellipse) (focus) 1. (h, k) X 2

2

ah) (x- + 2

2

bk) (y - = 1 a > b

() (h a, k), (h c, k), (h, k b), |2a| , |2b| x - h = ca

2

2. (h, k) Y 2

2

bh) (x- + 2

2

ak) (y - = 1 a > b

() (h, k a), (h, k c), (h b, k), |2a| , |2b| y - k = ca

2

3. a, b c a2 = b2 + c2 4. a2b2 (Hyperbola) 1. (h, k) X 2

2

ah) (x- - 2

2

bk) (y - = 1 () (h a, k), (h c, k),

(h, k b), |2a| , |2b| (asymptote) y - k = ab (x - h) 2. (h, k) Y 2

2

ak) (y - - 2

2

bh) (x- = 1 () (h, k a), (h, k c),

(h b, k), |2a| , |2b| (asymptote) y - k = ba (x h) 3. a, b c c2 = a2 + b2 4. a2b2 5. (rectangular hyperbola) (b = a)

2010 _______________________________________PAT1 (13)

9 y2 - 4y + 4x = 0 (a, b) a + b (PAT 152) 1) 4 2) 5 3) 6 4) 7 10 (2, 1) x = 1 3

1

(PAT 152) 1) (0, 1) 2) (0, 2) 3) (1, 0) 4) (3, 0)

11 (3, 0) (2, 221 ) (PAT 152) 1) (-4, 0) 2) (0, 2

25 ) 3) (6, 0) 4) (0, -3 2 )

12 x = y 91) (x2- + 4

1) (y 2- = 1 A B F1 F2 AF1 + AF2 + BF1 + BF2 (ENT48) 13 y2 - 4y - 16x - 12 = 0 L 3x - 2y + 5 = 0 Y L (ENT48) 14 A C 3 C 2y2 - 12y - 3x2 + 6x + 9 = 0 A (ENT47) 1) 5 2) 18 3) 20 4) 24 15 L (2, 1) L 1 L y = ax2 - 4x + 1 (2, 1) a (ENT47) 1) - 163 2) - 161 3) - 83 4) - 81 16 E (-3, 2) (5, 2) 12 A B E E Y C D ABCD (ENT46)

PAT1 (14) ______________________________________ 2010

1. (ENT48) 1) y = 3x + 2 3x - y - 4 = 0 2) y + 5x + 8 = 0 5y = x + 3 3) (0, 0) 3x + 4y - 10 2 4) x - 2y + 5 = 0 x - 2y - 5 = 0 2 2. F1 F2 x2 + 3y2 - 2x - 23 = 0 P(4, 5 ) cos (F1 P F2)

(ENT47) 1) - 91 2) - 71 3) 43 4) 53 3. A 91) (x

2- - 162) (y 2- = 1 A

3 A (ENT47) 4. P x2 + 8y + 2x + a = 0 a < 0 y = 4 P X

A A P (ENT47) 5. C y = 1 - 8(x - 2)2 3x - 4y + 5 = 0

C C (ENT46) 1) (0, 1 + 5 ) 2) (1, -2 2 ) 3) (-1, -1) 4) (-2, 2) 6. H 12y2 - 4x2 + 72y + 16x + 44 = 0 F1 F2 E

H F1 F2 Y E X A B AB (ENT46) 1) 8 2) 7 3) 6 4) 5 7. L1 (-2, 0) (-1, 2) L2 L1

X L1 L2 (ENT41) 8. (0, -1) 3x2 + 4y2 - 16y + 4 = 0

(ENT42) 1)

1 ,32 2)

1 ,2

3 3)

0 ,31 4)

0 ,21

2010 _______________________________________PAT1 (15)

1. 4) x - 2y + 5 = 0 x - 2y - 5 = 0 2 1) y = 3x + 2 3 3x - y - 4 = 0 13-- = 3 2) y + 5x + 8 15- = -5 5y = x + 3 51 -5 51 = -1 3) (0, 0) 3x + 4y - 10 = 0 22 4 3

|10 4(0) 3(0)|+

+ - = 16 9|10|

+- = 25

10 = 510 = 2

4) x - 2y + 5 = 0 x - 2y - 5 = 0 22)( 1

|5)( 5|---

+ = 4 1

|10|+ = 5

10 = 5510 = 2 5

2. 1) - 91 x2 + 3y2 - 2x - 23 = 0 (x2 - 2x + 1) + 3y2 = 23 + 1 (x - 1)2 + 3y2 = 24 24

1) (x 2- + 8y2 = 1

c2 = a2 - b2 = 24 - 8 = 16 c = 4 F1 F2 (1 - 4, 0) (1 + 4, 0) F1(-3, 0) F2(5, 0) P(4, 5 ) PF1 + PF2 = 2a PF1 + PF2 = 2 24 PF2 = 22 )5( 5) (4 +- = 6 PF1 + 6 = 2 24 = 4 6 PF1 = 3 6 F1PF2 (F1F2)2 = (PF1)2 + (PF2)2 - 2(PF1)(PF2) cos(F1 P F2) 82 = (3 6 )2 + ( 6 )2 - 2(3 6 )( 6 ) cos(F1 P F2) 64 = 54 + 6 - 36 cos(F1 P F2) 36 cos(F1 P F2) = -4 cos(F1 P F2) = - 36

4 = - 91

PAT1 (16) ______________________________________ 2010

3. 9 91) (x

2- - 162) (y 2- = 1 a2 = 9 b2 = 16

a = 3 b = 4 A A

3 A k |k - 3| = 2a = 2(3) = 6 k - 3 = 6, -6 k = 9, -3 k k = 9 4. 0.5 x2 + 8y + 2x + a = 0 x2 + 2x + 1 = -8y - a + 1 (x + 1)2 = -8

8a 1 y -- (x - h)2 = 4c(y - k) 4c = -8 c = -2 P 8a 1 1, -- y - k = -c y - 8a 1- = -(-2) = 2 a < 0 y = 4 4 - 8a 1- = 2 8

a 1- = 2 1 - a = 16 a = 1 - 16 = -15 P x2 + 8y + 2x - 15 = 0 (-1, 2) X ; y = 0 x2 + 2x - 15 = 0 (x + 5)(x - 3) = 0 X = -5, 3 P X A A (-5, 0) A(-5, 0) (-1, 2) = 1)( 5 2 0 --- - = 42-- = 21 = 0.5

2010 _______________________________________PAT1 (17)

5. y = 1 - 8(x - 2)2 y - 1 = -8(x - 2)2 (x - 2)2 = - 81 (y - 1) 4c = - 81 c = - 321 3231 2, C 3231 2, 3x - 4y + 5 = 0

r = 22 4 35 32314 3(2)

+

+

- = 16 9

5 831 6++-

= 258

57 = 4057

C (x - 2)2 + 23231 y - = 2

4057

(x - 2)2 + 2

3231 y

- = 16003249

x = 0, y = 1 + 5 4 + 2

32532 1

+ = 4 + 1024

564 5121+ = 1024564 9217 + 16003249

(0, 1 + 5 ) C x = 1, y = -2 2 1 +

232

31 264

-- = 1 + 1024

23968 9153 + = 1024564 10177 + 16003249

(1, -2 2 ) C x = -1, y = -1 9 + 23263 - = 9 + 10243969 = 102413185 16003249 (-1, -1) C x = -2, y = 2 16 + 23233 = 16 + 10241089 = 102417473 16003249 (-2, 2) C C 6. 2) 7 12y2 - 4x2 + 72y + 16x + 44 = 0 12(y2 + 6y + 9) - 4(x2 - 4x + 4) = -44 + 108 - 16 12(y + 3)2 - 4(x - 2)2 = 48 4

3) (y 2+ - 122) (x 2- = 1

a2 = 4, b2 = 12 c2 = 4 + 12 = 16 a = 2, b = 12 c = 4 (2, -3) (2, 1) (2, -7) E (2, -3) (2, 1) (2, -7) Y a = 4 E Y b = 2 E 163) (y

2+ - 42) (x 2- = 1

E X A B

PAT1 (18) ______________________________________ 2010

y = 0 169 + 42) (x2- = 1

42) (x 2- = 167

(x - 2)2 = 47

x - 2 = 27 , - 2

7

x = 27 4 + , 2

7 4-

E X A

+ 0 ,2

7 4 B

0 ,2

7 4-

AB = 2 7 4 2 7 4 --+ = 7 7. 0.8 L1 (-2, 0) (-1, 2) L1 1)( 2 2 0 --- - = 12-- = 2 y = mx + b L1 y = 2x + b (-2, 0) 0 = 2(-2) + b b = 4 L1 y = 2x + 4 ...(1) L2 L1 L2 - 21 (0, 0) L2 y = - 21 x + b (0, 0) 0 = - 21 (0) + b b = 0 L2 y = - 21 x ...(2) (1) (2) 2x + 4 = - 21 x 4x + 8 = -x 5x = -8 x = - 58 = -1.6 x = -1.6 (2) y = - 21 (-1.6) = 0.8 L1 L2 (-1.6, 0.8)

2010 _______________________________________PAT1 (19)

Y

X-5 5

2

4

-2

-4

6

(-1.6, 0.8)

(-1, 2)

(-2, 0)

1L

2L

X L1 L2 21 2 0.8 = 0.8

8. 1)

1 ,32

3x2 + 4y2 - 16y + 4 = 0 3x2 + 4(y2 - 4y + 4) = -4 + 16 3x2 + 4(y - 4)2 = 12 4x

2 + 3

2) (y 2- = 1 X, a = 2, b = 3 c = 3 4- = 1 (1, 2) (-1, 2) (0, -1) x2 = 4c(y + 1) 12 = 4c(2 + 1) 1 = 12c c = 121 x2 = 4 121 (y + 1) x2 = 31 (y + 1)

1) x = 32 , y = 1 2

32

= 31 (1 + 1) 32 = 32

1 ,32

2) x = 23 , y = 1 2

23

= 31 (1 + 1) 23 = 32

1 ,23 3) x = 31 , y = 0

231

= 31 (0 + 1) 91 = 31 0 ,31 4) x = 21 , y = 0

221

= 31 (0 + 1) 41 = 31 0 ,21

PAT1 (20) ______________________________________ 2010

f(x) L1 x a (x < a) L1 f a -ax lim f(x) = L1 f(x) L2 x a (x > a) L2 f a +ax lim f(x) = L2 f L L f a

axlim

f(x) = L f f a a f g a 1.

axlim

f(x) = L -ax

lim

f(x) = +ax

lim f(x) = L 2.

axlim

c = c c 3.

axlim

x = a 4.

axlim

xn = an 5.

axlim

cf(x) = cax

lim

f(x) c 6.

axlim

(f(x) g(x)) = ax

lim

f(x) ax

lim

g(x) 7.

axlim

(f(x) g(x)) = ax

lim

f(x) ax

lim

g(x)

8.

g(x)f(x)lim

ax = g(x)lim

f(x)lim

ax

ax

axlim

g(x) 0 9.

axlim

(f(x))n = (ax

lim

f(x))n n = 1, 2, 3, ... 10.

axlim

n f(x) = n ax f(x)lim n = 1, 2, 3, ... 11.

axlim

(f(x))n/m = (ax

lim

f(x))n/m n, m = 1, 2, 3, ... ax

lim

f(x) 0 12. f f(x) = cnxn + cn-1xn-1 + ... + c1x + c0 c0, c1, ..., cn c0 0 axlim f(x) = f(a)

2010 _______________________________________PAT1 (21)

a f a f 1. f(a) 2.

axlim

f(x) 3. f(a) =

axlim

f(x) y = f(x) R R x y f(x) x x x + h y f(x) f(x + h) x h y f(x + h) f(x) y x = h f(x) h) f(x -+ y x x h f(x) h) f(xlim0x

-+

y = f(x) h f(x) h) f(xlim0x

-+

h f(x) h) f(xlim0x

-+

f x f x f(x) dxdy () y dxd f(x) 1. dxdc = 0 c 2. dxdx = 1 3. dxd xn = nxn-1 n 4. dxd [f(x) g(x)] = dxd f(x) dxd g(x) 5. dxd cf(x) = c dxd f(x) c 6. dxd [f(x) g(x)] = f(x) dxd g(x) + g(x) dxd f(x)

7. dxd

g(x)f(x) = 2(g(x))

g(x)dxd f(x) f(x)dx

dg(x) - g(x) 0

8. dxd [f(x)]n = n[f(x)]n-1 dxd f(x) 9. dxd (gof(x)) = dyd g(y) dxd f(x) y = f(x) (gof)(x) = g(f(x))f(x) a. (Chain rule) s = f(t) t t + h h f(t) h) f(t -+ t t + h h 0 t t + h t t h f(t) h) f(t lim0h

-+

s f(t)

PAT1 (22) ______________________________________ 2010

y = f(x) 1. P(x, y) P h f(x) h) f(xlim0h

-+

y f (x) 2. P(x, y) f(x) f(x) 2 f f(x),

22

dxyd , 2

2dxd f(x)

3, 4, ... n 3 fn x fn x n + 1 f x fn+1(x) f s 1. f(x) < 0 x s f s 2. f(x) > 0 x s f s 1. f x = c (a, b) Df c (a, b) f(c) > f(x) x (a, b) x c 2. f x = c (a, b) Df c (a, b) f(c) < f(x) x (a, b) x c f s c f f(c) = 0 1. f(c) > 0 f(c) 2. f(c) < 0 f(c) f x = c f(c) > f(x) x f x c f x = c f(c) < f(x) x f x c 1. y 2. x y x 3. y x 4. dxdy y y x 5. dxdy = 0 x 6. 5 y 7. x 6 y

2010 _______________________________________PAT1 (23)

1. F f F(x) = f(x) x f 2. f F(x) = f(x) x f f f(x)dx f(x)dx = F(x) + c c 1. kdx = kx + c k c 2. xndx = 1 nx

1n++

+ c n -1 3. kf(x)dx = kf(x)dx k 4. [f(x) + g(x)]dx = f(x)dx + g(x)dx 5. [f(x) - g(x)]dx = f(x)dx - g(x)dx f [a, b] F [a, b] F(x) = f(x) =

b

a-F(a) F(b) f(x)dx

F(x) ab

F(b) - F(a)

F(x) = f(x) =b

aF(x) f(x)dx a

b = F(b) - F(a)

f [a, b] A f x = a x = b 1. f(x) 0 x [a, b] A X A =

b

af(x)dx

2. f(x) 0 x [a, b] A X A = -

b

af(x)dx

PAT1 (24) ______________________________________ 2010

17 f(x) = x4 - 3x2 + 7 f (PAT 152) 1) (-3, -2)U (2, 3) 2) (-3, -2)U (1, 2) 3) (-1, 0)U (2, 3) 4) (-1, 0)U (1, 2) 18

2

22 x 4

-- (ENT48)

1) (3.1, 3.2) 2) (3.2, 3.3) 3) (6.1, 6.2) 4) (6.2, 6.3) 19 f f(2) = -1, f(1) = -3 f(x) = 3 x f(0) (ENT42) 1) 5 2) 6 3) 12 4) 15 20 f(x) g(x) f(g(x)) = (x)g

1 f(g(0)) = 5 f(g(2)) (ENT47) 1) 1 2) 3 3) 5 4) 7

21 f(x) = 21

+ 3x

1 x1 f(4) h) f(4 f(1) h) f(1lim0h -

-++

(PAT 152)

1) 1 2) 516 3) 57 4) 51

22 f(x) = 1 , x 0 0 , x > 0

. -0x

lim

(fof)(x) = 0 . +0x lim (fof)(x) = 1 (ENT48) 1) . . 2) . . 3) . . 4) . . 23 y = f(x) (x, y) 2x - 4 f 10 y = f(x) X x = 0 x = 3 (ENT48) 1) 33 2) 36 3) 39 4) 42 24 A y = 1 - x2 X B y = 4x

2 X x = -c x = c c A = B (PAT 152) 1) 2 2) 2 3) 2 2 4) 4

2010 _______________________________________PAT1 (25)

1. . f g h f(x) h) f(xlim0h

-++

= h f(x) h) f(xlim

0h-

-+

= g(x) g(x) = f(x)

. f f(x) > 0 x f(a) 0 y = f(x)1 a (a)f

1 (ENT48) 1) . . 2) . . 3) . . 4) . . 2. f(x) = x - n , 2n x 2n + 1

n + 1 , 2n + 1 x 2n + 2 n = 0, 1, 2, ..., 9

20

0f(x)dx (ENT47)

1) 105 2) 115 3) 125 4) 135 3. y = f(x) 3 x = 2 3x + y - 7 = 0

(1, 4) g(x) = x2f(x) 21

(x)dxg (ENT47) 1) 5 2) 7 3) 8 4) 10 4. f F(x) = 15 (f(x))3 + F(1) = f(1) = 4 F(1) (ENT41) 1) 21 2) 23 3) 8 4) 24 5. y = x2 - 3x + 2 x = 0 x = 2 x (ENT41) 1) 61 2) 32 3) 65 4) 23 6. a, b, c, d f(x) = ax3 + bx2 + cx + d f 2 x = 1 f(1) = -4

f(0) = 1 f (ENT42) 1) x = -3 2) x = - 31 3) x = 31 4) x = 3 7. f (0, 2) f(x) = 3x2 - 12x + 9 f (ENT42) 1) 2 2) 3 3) 6 4) 8 8. f(x) = x3 + cx2 - 9x c f 1 f

(ENT43) 1) (-3, 1) 2) (-, -3)U (1, ) 3. (-1, 4) 4. (-, -1)U (4, )

PAT1 (26) ______________________________________ 2010

1. 2) . . . h f(x) h) f(xlim0h

-++

= h f(x) h) f(xlim

0h-

-+

= g(x)

h f(x) h) f(xlim0h-+

g(x)

f(x) g(x) g(x) = f(x) . . y a dxdy y x = a y = f(x)1 y = f-1(x) y = -f-2(x)f(x) = 2(f(x))

(x)f- x = a y 2(f(a))

(a)f- (a)f 1 . 2. 1) 105 f(x) = x - n , 2n x 2n + 1

n + 1 , 2n + 1 x 2n + 2 n = 0, 1, 2, ..., 9

n = 0 f(x) = x , 0 x 1 1 , 1 x 2

n = 1 f(x) = x - 1 , 2 x 3 2 , 3 x 4

n = 2 f(x) = x - 2 , 4 x 5 3 , 5 x 6

M

n = 9 f(x) = x - 9 , 18 x 19 10 , 19 x 20

2010 _______________________________________PAT1 (27)

20

0f(x)dx =

1

0f(x)dx +

2

1f(x)dx +

3

2f(x)dx +

4

3f(x)dx +

6

5f(x)dx + ... +

19

18f(x)dx +

20

19f(x)dx

=

102

x2 + x21 +

32

x 2x2 - + 2x

43 +

54

2x 2x2 - + 3x

65 + ... +

1918

9x 2x2 - + 10x

1918

=

0 21 - + (2 - 1) +

2 24 3 29..

--- + (8 - 6) +

8 216 10 225..

--- + (18 - 15) + ... +

162 2256 171 2361..

--- + (190 - 180) = 21 + 1 + 23 + 2 + 25 + 3 + ... + 219 + 10 = 105 3. 2) 7 y = f(x) 3 x = 2 f(2) = 0 f(2) = 3 3x + y - 7 = 0 -3 y = f(x) (1, 4) (1, 4) y = f(x) x = 1 -3 f(1) = 4 f(1) = -3 g(x) = x2f(x) g(x) = x2f(x) + f(x)(2x) 2

1(x)dxg = g(x)

21

= [x2f(x) + f(x)(2x)]21

= [4f(2) + f(2)(4)] - [1f(1) + f(1)(2)] = [4(0) + (3)(4)] - [1(-3) + (4)(2)] = [0 + 12] - [-3 + 8] = 12 - 5 = 7 4. 2) 23 F(x) = 15 (f(x))3 + F(x) = [(f(x))3 + 15]1/2 F(x) = 21 [(f(x))3 + 15]-1/2[3(f(x))2f(x)] = 15 (f(x))2

(x)f3(f(x))3

2

+ = 2F(x) (x)f3(f(x))

2

F(1) = 2F(1) (1)f3(f(1))2 = 2(4) (4)3(f(1))

2 = 2

3(f(1))2

F(1) = 15 (f(1))3 + = 4 (f(1))2 + 15 = 16 (f(1))2 = 16 1 = 1 F(1) = 23(1) = 23

PAT1 (28) ______________________________________ 2010

5. 3) 65 y = x2 - 3x + 2 = (x - 1)(x - 2) Y

X5

2

4

21 y = x2 - 3x + 2 x = 0 x = 2 x +

1

02 2)dx 3x (x - =

10

+ 2x 23x 3x

23 -

= 31 - 23 + 2 = 6 12 9 2 +- = 65 6. 2) x = - 31 f(x) = ax3 + bx2 + cx + d f(x) = 3ax2 + 2bx + c f(x) = 6ax + 2b f 2 x = 1 2 = f(1) = a + b + c + d ...(1) 0 = f(2) = 3a + 2b + c ...(2) f(0) = 1 f(1) = -4 d = 1 6a + 2b = -4 ...(3) d = 1 (1) 1 = a + b + c ...(4) (2) - (4) ; 2a + b = -1 ...(5) (5) 2 ; 4a + 2b = -2 ...(6) (3) - (5) ; 2a = -2 a = -1 b = 1 c = 1 f(x) = -x3 + x2 + x + 1, f(x) = -3x2 + 2x + 1 f(x) = -6x + 2 f(x) = -3x2 + 2x + 1 = 0 3x2 - 2x - 1 = 0 (3x + 1)(x - 1) = 0 x = 1, - 31 x = - 31 f 31 - = -6 31 - + 2 = 4 > 0 f x = - 31

2010 _______________________________________PAT1 (29)

7. 3) 6 f(x) = 3x2 - 12x + 9 f(x) = 6x - 12x f(x) = x3 - 6x2 + 9x + c f(x) = 3x2 - 12x + 9 = 0 x2 - 4x + 3 = 0 (x - 3)(x - 1) = 0 x = 3, 1 x = 3 f(3) = 6(3) - 12(3) = 18 - 12 = 6 > 0 f x = 3 x = 1 f(1) = 6(1) - 12(1) = 6 - 12 = -6 < 0 f x = 1 f(x) = x3 - 6x2 + 9x + c (0, 2) f(0) = c = 2 f(x) = x3 - 6x2 + 9x + 2 f f(1) = 1 - 6 + 9 + 2 = 6 8. 1) (-3, 1) f(x) = x3 + cx2 - 9x f(x) = 3x2 + 2cx - 9 f(1) = 0 3 + 2c - 9 = 0 c = 3 f(x) = x3 + 3x2 - 9x f(x) = 3x2 + 6x - 9 f f(x) < 0 3x2 + 6x - 9 < 0 x2 + 2x - 3 < 0 (x + 3)(x - 1) < 0 x (-3, 1) f (-3, 1)