y = ex · e-y
R2 P = (-1, 2)
f(x) = ex, x " (-", +") g(y) = e-y, y " (-", +")
g(y) = 0 y " (-", +") F (x) = ex, x " (-", +") H(y) = ey, y " (-", +")

f 1/g F H
A = infx"(-",+") F (x) = 0 B = supx"(-",+") F (x) = +" C = infy"(-",+") H(y) = 0 D =
supy"(-",+") H(y) = +" H H-1(t) = ln t, t " (0, +")
Å‚ " (C - B, D - A) = (-", +") "Å‚
(-", +") Å‚ 0
"Å‚ = {x " (-", +") : C < F (x) + Å‚ < D} = ;
(ln(-Å‚), +") Å‚ < 0
ÕÅ‚I = ln(ex + Å‚), x " I,
Å‚ " (-", +") I "Å‚
P = (-1, 2)
Õ(x) = ln(ex + e2 - e-1), x " (-", = "),
Å‚ = H(2) - F (-1) = e2 - e-1 > 0
Ć(¾, ·, x) = ln(ex + e· - e¾), (¾, ·, x) " V,
V = {(¾, ·, x) " R3 : (¾, ·) " R2, · ¾, x " (-", +")}*"
{(¾, ·, x) " R3 : (¾, ·) " R2, · < ¾, x " (ln(-e· + e¾), +")}.
P = (-1, 2)
 (¾, ·)
y = x · (1 + y2)
R2 P = (0, 1)
f(x) = x, x " (-", +") g(y) = 1 + y2, y " (-", +")
g(y) = 0 y " (-", +") F (x) = 1/2x2, x " (-", +") H(y) = arctan y, y " (-", +")

f 1/g F H
A = infx"(-",+") F (x) = 0 B = supx"(-",+") F (x) = +" C = infy"(-",+") H(y) = -Ä„/2 D =
supy"(-",+") H(y) = Ä„/2 H H-1(t) = tan t, t " (-Ä„/2, Ä„/2)
Å‚ " (-", Ä„/2) "Å‚
" " " "
(- Ä„ - 2Å‚, - -Ä„ - 2Å‚) *" ( -Ä„ - 2Å‚, Ä„ - 2Å‚), Å‚ " (-", -Ä„/2]
"Å‚ = ;
" "
(- Ä„ - 2Å‚, Ä„ - 2Å‚), Å‚ " (-Ä„/2, Ä„/2)
ÕÅ‚I = tan(1/2x2 + Å‚), x " I,
Å‚ " (-", Ä„/2) I "Å‚
P = (0, 1)
Õ(x) = tan(1/2x2 + Ä„/4), x " (- Ä„/2, Ä„/2),
Å‚ = H(1) - F (0) = Ä„/4 " (-Ä„/2, Ä„/2)
P = (0, 1)
y = 4y(1 - y)
R2 P = (1, 1/2)
Q = (1, 1)
f(x) = 1, x " (-", +") F (x) = x, x " (-", +")
f A = -" B = +" g(y) = 4y(1 - y), y " (-", +")
g(y) = 0 y = 0 y = 1
Õ1(x) = 0, Õ2(x) = 1, x " (-", +");
T1 = R × (-", 0)
y
1
T2 = R × (0, 1) T3 = R × (1, +") H1(y) = ln , y " (-", 0)
4 y-1
-1
1/g|(-",0) C1 = -", D1 = 0 H1 H1 (t) =
1
, t " (-", 0) Å‚ " (-", +") "1Å‚ "1Å‚ = {x " (-", +") :
1-e-4t
-" < x + Å‚ < 0} = (-", -Å‚) T1
1
Õ1Å‚(x) = , x " (-", -Å‚)
1 - e-4(x+Å‚)
Å‚ " (-", +") Õ1Å‚ Õ1
limy0- H1(y) = -"
T2
1
Õ2Å‚(x) = , x " (-", +")
1 + e-4(x+Å‚)
Å‚ " (-", +") T3
1
Õ3Å‚(x) = , x " (-Å‚, +")
1 - e-4(x+Å‚)
Å‚ " (-", +") T2 T3
Õ1 Õ2 R2
T1, T2, T3 Õ1, Õ2
P = (1, 1/2) " T2
1
Õ(x) = , x " (-", +")
1 + e-4(x-1)
Å‚ = -1
Q = (1, 1)
Õ2
P = (1, 1/2)
Q = (1, 1)
"
3
y = ex · y
R2 P = (0, 1)
Q = (0, -1) S = (0, 0)
F (x) = ex, x " (-", +") f(x) = ex, x " (-", +")
"
3
A = 0 B = +" g(y) = y, y " (-", +") g(y) = 0
y = 0
Õ0(x) = 0, x " (-", +");
T1 = R × (-", 0)
3 3
T2 = R × (0, +") H1(y) = y2, y " (-", 0) 1/g|(-",0)
2
-1 8
C1 = 0, D1 = +" H1 H1 (t) = - t3
27
t " (0, +") Å‚ " (-", +") "1Å‚
(-", +") Å‚ 0
"1Å‚ = {x " (-", +") : C1 < F (x) + Å‚ < D1} = ;
(ln(-Å‚), +") Å‚ < 0
T1
8
Õ1Å‚I(x) = - (ex + Å‚)3, x " I
27
Å‚ " (-", +") I "1Å‚ Õ1Å‚I
Å‚ < 0 Õ0 limy0- H1(y) = 0 " R
R × (-", 0] Õ1Å‚I Å‚ 0 I = (-", +")
0, x " (-", ln(-Å‚)]
Õ" (x) = ,
1Å‚
8
- (ex + Å‚)3, x " (ln(-Å‚), +")
27
Å‚ < 0
T2
8
Õ2Å‚I(x) = (ex + Å‚)3, x " I
27
Å‚ " (-", +") I "2Å‚ = "1Å‚
R × [0, +") Õ2Å‚I Å‚ 0 I = (-", +")
0, x " (-", ln(-Å‚)]
Õ" (x) = ;
2Å‚
8
(ex + Å‚)3, x " (ln(-Å‚), +")
27
Å‚ < 0 T1 T2
T1 T2
R2 Õ0
R × (-", 0] R × [0, +")
P = (0, 1) " T2
8
ÕP (x) = (ex + 1/2)3, x " (-", +")
27
Å‚ = 1/2 > 0
Q = (0, -1) " T1
8
ÕQ(x) = - (ex + 1/2)3, x " (-", +")
27
Å‚ = 1/2 > 0 S = (0, 0)
Õ" Õ" Å‚ -1 ln(-Å‚) 0 Õ0
1Å‚ 2Å‚
P = (0, 1)
 Q = (0, -1)
S = (0, 0)
3
y = y2
R2 P = (-1, 1)
F (x) = x, x " (-", +") f(x) = 1, x " (-", +")
3
A = -" B = +" g(y) = y2, y " (-", +")
g(y) = 0 y = 0
Õ0(x) = 0, x " (-", +");
T1 = R × (-", 0)
"
3
T2 = R × (0, +") H1(y) = 3 y, y " (-", 0) 1/g|(-",0)
-1 1
C1 = -", D1 = 0 H1 H1 (t) = t3 t " (-", 0)
27
Å‚ " (-", +") "1Å‚
"1Å‚ = {x " (-", +") : C1 < F (x) + Å‚ < D1} = (-", -Å‚);
T1
1
Õ1Å‚(x) = (x + Å‚)3, x " (-", -Å‚)
27
Å‚ " (-", +") Õ1Å‚ Õ0 limy0- H1(y) =
0 " R R×(-", 0]
Õ0
1
(x + Å‚)3, x " (-", -Å‚)
27
Õ" (x) = ,
1Å‚
0, x " [-Å‚, +")
Å‚ " (-", +")
T2
1
Õ2Å‚(x) = (x + Å‚)3, x " (-Å‚, +")
27
Å‚ " (-", +") R × [0, +")
Õ0
0, x " (-", -Å‚]
Õ" (x) = ,
2Å‚
1
(x + Å‚)3, x " (-Å‚, +")
27
Å‚ " (-", +") T1 T2
T1 T2
R2 Õ0 R ×
(-", 0] R × [0, +")
Å„Å‚
1
ôÅ‚27(x + Å‚1)3, x " (-", -Å‚1)
òÅ‚
Õ" (x) = 0, x " [-Å‚1, -Å‚2] ,
Å‚1Å‚2
ôÅ‚
ół
1
(x + Å‚2)3, x " (Å‚2, +")
27
Å‚1, Å‚2 " (-", +") Å‚1 Å‚2
P = (-1, -1) " T1
1
(x - 2)3, x " (-", 2)
27
ÕP (x) =
0, x " [2, +")
Å„Å‚
1
ôÅ‚27(x - 2)3, x " (-", 2)
òÅ‚
ÕP Å‚(x) = ,
0, x " [2, -Å‚]
ôÅ‚
ół
1
(x + Å‚)3, x " (Å‚, +")
27
Å‚ -2
P = (-1, -1)
y y
y = + 1 - ( )2
x x
D- *" D+
D- = {(x, y) " R2 : x " (-", 0), x y -x},
D+ = {(x, y) " R2 : x " (0, +"), 0), x y -x}.
1
z = 1 - z2
x
- +
T = (-", 0) × [-1, 1], T = (0, +") × [-1, 1]
-
T
- -
È1 (x) = -1, È2 (x) = 1 x " (-", 0)
Å„Å‚
ôÅ‚1 x " (-", -eÄ„/2-Å‚]
òÅ‚
-
È1Å‚(x) = sin(ln(-x) + Å‚), x " (-eÄ„/2-Å‚, -e-Ä„/2-Å‚) ;
ôÅ‚
ół
-1 x " [-e-Ä„/2-Å‚, 0)
+
T
+ +
È1 (x) = -1, È2 (x) = 1 x " (0, +")
Å„Å‚
ôÅ‚-1 x " (0, e-Ä„/2-Å‚]
òÅ‚
+
È1Å‚(x) = sin(ln(x) + Å‚), x " (e-Ä„/2-Å‚, eÄ„/2-Å‚) ;
ôÅ‚
ół
1 x " [eĄ/2-ł, +")
D-
Õ-(x) = -x, Õ-(x) = x x " (-", 0),
1 2
Å„Å‚
ôÅ‚x x " (-", -eÄ„/2-Å‚]
òÅ‚
Õ- (x) = ;
x sin(ln(-x) + ł), x " (-eĄ/2-ł, -e-Ą/2-ł)
1Å‚
ôÅ‚
ół
-x x " [-e-Ä„/2-Å‚, 0)
D+
Õ+(x) = -x, Õ+(x) = x x " (0, +")
1 2
Å„Å‚
ôÅ‚-x x " (0, e-Ä„/2-Å‚]
òÅ‚
Õ+ (x) = ;
x sin(ln(x) + ł), x " (e-Ą/2-ł, eĄ/2-ł)
1Å‚
ôÅ‚
ół
x x " [eĄ/2-ł, +")
y y
y = + (cos )2
x x
D- *" D+
D- = {(x, y) " R2 : x " (-", 0), y " R}, D+ = {(x, y) " R2 : x " (0, +"), y " R}.
P = (1, Ä„/4)
Q = (-1, -3Ä„/4)
1
z = (cos z)2
x
- +
T = (-", 0) × R, T = (0, +") × R
-
T
Ä„
-
Èk (x) = + kÄ„, x " (-", 0), k " Z
2
- Ä„
Tk = (-", 0) × (-Ä„ + kÄ„, + kÄ„) k " Z
2 2
-
ÈkÅ‚(x) = kÄ„ + arctan(ln(-x) + Å‚), x " (-", 0),
Å‚ " (-", +")
+
T
Ä„
+
Èk (x) = + kÄ„ x " (0, +") k " Z
2
+ Ä„
Tk = (0, +") × (-Ä„ + kÄ„, + kÄ„) k " Z
2 2
+
ÈkÅ‚(x) = kÄ„ + arctan(ln(x) + Å‚), x " (0, +"),
Å‚ " (-", +")
D-
Ä„
Õ-(x) = x( + kÄ„), x " (-", 0), k " Z,
k
2
Õ- (x) = x(kÄ„ + arctan(ln(-x) + Å‚)), x " (-", 0), k " Z,
kł
Å‚ " (-", +")
D+
Ä„
Õ+(x) = x( + kÄ„), x " (0, +"), k " Z,
k
2
Õ+ (x) = x(kÄ„ + arctan(ln(x) + Å‚)), x " (0, +"), k " Z,
kł
Å‚ " (-", +")
P = (1, Ä„/4)
Õ+(x) = x arctan(ln(x) + 1), x " (0, +")
P
k = 0, Å‚ = 1
Q = (-1, -3Ä„/4)
Õ-(x) = x(Ä„ + arctan(ln(-x) + 1)), x " (-", 0)
Q
k = 1, Å‚ = 1
P = (1, Ä„/4)
Q = (-1, -3Ä„/4)
(2x - y)y = 5x - 2y
R2
y
5 - 2x
y =
y
2 -
x
D1 *" D2 *" D3 *" D4
D1 = {(x, y) " R2 : x < 0, y < 2x}, D2 = {(x, y) " R2 : x < 0, y > 2x},
D3 = {(x, y) " R2 : x > 0, y > 2x}, D4 = {(x, y) " R2 : x > 0, y < 2x}.
1 (z - 2)2 + 1
z = ·
x -(z - 2)
T1 *" T2 *" T3 *" T4
T1 = (-", 0) × (2, +"), T2 = (-", 0) × (-", 2)
T3 = (0, +") × (2, +"), T4 = (0, +") × (-", 2),
T1
1
È1Å‚(x) = 2 + e-2Å‚ - 1, x " (-e-Å‚, 0), Å‚ " (-", +")
x2
T2
1
È2Å‚(x) = 2 - e-2Å‚ - 1, x " (-e-Å‚, 0), Å‚ " (-", +")
x2
T3
1
È3Å‚(x) = 2 + e-2Å‚ - 1, x " (0, e-Å‚), Å‚ " (-", +")
x2
T4
1
È4Å‚(x) = 2 - e-2Å‚ - 1, x " (0, e-Å‚), Å‚ " (-", +").
x2
D1
Õ1Å‚(x) = 2x - e-2Å‚ - x2, x " (-e-Å‚, 0), Å‚ " (-", +")
D2
Õ2Å‚(x) = 2x + e-2Å‚ - x2, x " (-e-Å‚, 0), Å‚ " (-", +")
D3
Õ3Å‚(x) = 2x + e-2Å‚ - x2, x " (0, e-Å‚), Å‚ " (-", +")
D4
Õ4Å‚(x) = 2x - e-2Å‚ - x2, x " (0, e-Å‚), Å‚ " (-", +").
Õ1Å‚, Õ2Å‚, Õ3Å‚, Õ4Å‚
x = 0
Õ1´ Õ4Å‚
Õ2´ Õ3Å‚ ´, Å‚ " (-", +") ´ = Å‚
R
Õ"(x) = 2x - e-2Å‚ - x2, x " (-e-Å‚, e-Å‚), Å‚ " (-", +")
Å‚
Õ""(x) = 2x + e-2Å‚ - x2, x " (-e-Å‚, e-Å‚), Å‚ " (-", +").
Å‚
2x - y = 0
y y
y = ln
x x
D- *" D+
D- = (-", 0) × (-", 0), D+ = (0, +") × (0, +").
P = (-1, -1)
1
z = z(ln z - 1)
x
- +
T = (-", 0) × (-", 0), T = (0, +") × (0, +").
È-(x) = e, x " (-", 0), È+(x) = e, x " (0, +")
T1 = (-", 0) × (0, e)
Å‚
È1Å‚(x) = e1+xe , x " (-", 0), Å‚ " (-", +")
T2 = (-", 0) × (e, +")
Å‚
È2Å‚(x) = e1-xe , x " (-", 0), Å‚ " (-", +")
T3 = (0, +") × (e, +")
Å‚
È3Å‚(x) = e1+xe , x " (0, +"), Å‚ " (-", +")
T4 = (0, +") × (0, e)
Å‚
È4Å‚(x) = e1-xe , x " (0, +"), Å‚ " (-", +").
Õ-(x) = ex, x " (-", 0), Õ+(x) = ex, x " (0, +")
D1 = {(x, y) " R2 : x " (-", 0), ex < y < 0}
Å‚
Õ1Å‚(x) = xe1+xe , x " (-", 0), Å‚ " (-", +")
D2 = {(x, y) " R2 : x " (-", 0), y < ex}
Å‚
Õ2Å‚(x) = xe1-xe , x " (-", 0), Å‚ " (-", +")
D3 = {(x, y) " R2 : x " (0, +"), y > ex}
Å‚
Õ3Å‚(x) = xe1+xe , x " (0, +"), Å‚ " (-", +")
D4 = {(x, y) " R2 : x " (0, +"), 0 < y < ex}
Å‚
Õ4Å‚(x) = xe1-xe , x " (0, +"), Å‚ " (-", +").
P = (-1, -1) " D1
Õ(x) = xe1+x x " (-", 0).
 P = (-1, -1)
y2 - x2
y =
2xy
D1 *" D2 *" D3 *" D4
D1 = (-", 0) × (-", 0), D2 = (-", 0) × (0, +"),
D3 = (0, +") × (0, +"), D4 = (0, +") × (-", 0).
1 y x
y = ( - )
2 x y
D1 *" D2 *" D3 *" D4
1 z2 + 1
z =
x -2z
T1 *" T2 *" T3 *" T4
T1 = (-", 0) × (0, +"), T2 = (-", 0) × (-", 0),
T3 = (0, +") × (0, +"), T4 = (0, +") × (-", 0).
T1
1
È1Å‚(x) = - e-Å‚ - 1, x " (-e-Å‚, 0), Å‚ " (-", +"),
x
T2
1
È2Å‚(x) = - - e-Å‚ - 1, x " (-e-Å‚, 0), Å‚ " (-", +"),
x
T3
1
È3Å‚(x) = e-Å‚ - 1, x " (0, e-Å‚), Å‚ " (-", +"),
x
T4
1
È4Å‚(x) = - e-Å‚ - 1, x " (0, e-Å‚), Å‚ " (-", +").
x
D1
Õ1Å‚(x) = - -xe-Å‚ - x2, x " (-e-Å‚, 0), Å‚ " (-", +"),
D2
Õ2Å‚(x) = -xe-Å‚ - x2, x " (-e-Å‚, 0), Å‚ " (-", +"),
D3
Õ3Å‚(x) = xe-Å‚ - x2, x " (0, e-Å‚), Å‚ " (-", +"),
D4
Õ4Å‚(x) = - xe-Å‚ - x2, x " (0, e-Å‚), Å‚ " (-", +").
x2
2x arctan y + y = 0
1 + y2
R2
R2
F (x, y) = x2 arctan y, (x, y) " R2
Å‚ " (-", +") F (0, 0) = 0 Å‚ = 0

|Å‚| Å‚
F (2 , ) = Å‚)
Ä„ |Å‚|
Õ(x) = 0, x " (-", +")
Å‚ = 0
Å‚
ÕÅ‚I(x) = tan , x " I,
x2
Å‚ = 0 I

2|Å‚| 2|Å‚|
(-", - ) *" ( , +").
Ä„ Ä„
" ex
x + ex y + y = 0
"
2 y
T = R × (0, +")
 T
1 "
F (x, y) = x2 + ex y, (x, y) " T
2
"
"
1
Å‚ " (0, +") F ( Å‚, Å‚2e-2 y) = Å‚
4
1
ÕÅ‚(x) = e-2x(Å‚ - x2)2, x " (- 2Å‚, 2Å‚),
2
Å‚ " (0, +")
1 1 y
"
+ - y = 0
2 x - y2 2 x x - y2
G = {(x, y) " R2 : x > 0, x > y2}
P = (1, 0)
G
"
F (x, y) = x - y2 + x, (x, y) " G
G1 = {(x, y) " R2 : x > 0, x > y2, y < 0}, G2 = {(x, y) " R2 : x > 0, x > y2, y > 0};
Å‚ " (0, +")
9 Å‚ 9 Å‚
"
(F ( Å‚2, -" ) = Å‚, F ( Å‚2, ) = Å‚).
16 16
2 2
G1
"
Å‚2
Õ1Å‚(x) = - x - (Å‚ - x)2, x " ( , Å‚2),
4
G2
"
Å‚2
Õ2Å‚(x) = x - (Å‚ - x)2, x " ( , Å‚2),
4
Å‚ " (0, +")
Õ0(x) = 0, x " (a, b)
P
P = (1, 0)
-y x
+ y = 0
x2 + y2 x2 + y2
G = R2 \ {(0, 0)}.
T1 = R × (-", 0), T2 = (0, +") × R, T3 = R × (0. + "), T4 = (-", 0) × R,
x y
F1(x, y) = - arctan , (x, y) " T1 F2(x, y) = arctan , (x, y) " T2,
y x
x y
F3(x, y) = - arctan , (x, y) " T3 F4(x, y) = arctan , (x, y) " T4
y x
F G C1, C2, C3, C4
F = F1 + C1 T1 F = F2 + C2 T2 F = F3 + C3 T3 F = F4 + C4 T4
T1 )" T2 F2 - F1 = -C2 + C1 = -Ä„
2
T2 )" T3 F3 - F2 = -C3 + C2 = -Ä„
2
T3 )" T4 F4 - F3 = C3 - C4 = -Ä„
2
T4 )" T1 F1 - F4 = C4 - C1 = -Ä„
2
0 = -2Ä„
x
ln y + x2 - y = 0
y
T = (0, +") × (0, +")
1
µ(x) = , x " (0, +").
x2
ln y 1
+ 1 - y = 0
x2 xy
T
ln y
F (x, y) = x - , (x, y) " T
x
Å‚ " (-", +") F (1, e1-Å‚) = Å‚
T
ÕÅ‚(x) = ex(x-Å‚), x " (0, +"),
Å‚ " (-", +").
ex + (ex + ey)y = 0
T = R2
½(y) = ey, y " (-", +").
exey + (exey + e2y)y = 0
T
1
F (x, y) = exey + e2y, (x, y) " T
2
" "
Å‚ " (0, +") F (ln(1 Å‚, ln Å‚) = Å‚
2
T
ÕÅ‚(x) = ln( 2Å‚ + e2x - ex), x " (-", +"),
Å‚ " (0, +").
y = y tan x - y2 cos x
Ä„
T = (-Ä„ , ) × R
2 2
Ä„ Ä„
Õ0(x) = 0, x " (- , ).
2 2
Ä„
T1 = (-Ä„ , ) × (0, +") T2 =
2 2
Ä„
(-Ä„ , ) × (-", 0)
2 2
T1
z = -z tan x + cos x.
T
Ä„ Ä„
ÈÅ‚(x) = (x + Å‚) cos x, x " (- , ),
2 2
Å‚ ÈÅ‚ T1 x " "1Å‚
Ä„ Ä„
(-Ä„ , ) Å‚
2 2 2
"1Å‚ = ;
Ä„ Ä„
(-Å‚, ) Å‚ " (-Ä„ , )
2 2 2
 T1
1
Õ1Å‚I(x) = , x " I,
(x + Å‚) cos x
Å‚ " (-Ä„ , +") I "1Å‚
2
Ä„
Õ1Å‚I Å‚ " (-Ä„ , ) Õ0
2 2
T2
z = -z tan x + cos x,
ÈÅ‚ T2
x " "2Å‚
Ä„
(-Ä„ , ) Å‚ -Ä„
2 2 2
"2Å‚ = ;
Ä„
(-Ä„ , -Å‚) Å‚ " (-Ä„ , )
2 2 2
T2
1
Õ2Å‚I(x) = , x " I,
(x + Å‚) cos x
Ä„
Å‚ " (-", ) I "2Å‚
2
Ä„
Õ2Å‚I Å‚ " (-Ä„ , ) Õ0
2 2
T1 T2 Õ0
3 "
3
y = y - y ln x
2x
T = (0, +")×R P = (1, 0)
Õ0(x) = 0, x " (0, +").
T1 = (0, +") × (0, +") T2 =
(0, +") × (-", 0)
T1
1 2
z = - z - ln x.
x 3
T
1
ÈÅ‚(x) = (- ln2 x + Å‚)x, x " (0, +"),
3
Å‚ ÈÅ‚ Å‚ " (0, +")
" "
T1 x " (e- 3Å‚, e- 3Å‚)
T1
" "
1
3Å‚
Õ1Å‚(x) = ((- ln2 x + Å‚)x)3, x " (e- 3Å‚, e ),
3
Å‚ (0, +") Õ1Å‚
Õ1Å‚ Å‚ " (0, +") Õ0
"
T1 = (0, +") × [0, +")
Õ0
Å„Å‚
"
ôÅ‚0,
x " (0, e- 3Å‚]
ôÅ‚
òÅ‚
" "
3Å‚
Õ" (x) = .
((-1 ln2 x + Å‚)x)3, x " (e- 3Å‚, e )
1Å‚
3
ôÅ‚
"
ôÅ‚
ół0,
3Å‚
x " [e , +")
T2
1 2
z = - z - ln x,
x 3
" "
3Å‚
(e- 3Å‚, e ) T2
" "
1
3Å‚
Õ2Å‚(x) = - ((- ln2 x + Å‚)x)3, x " (e- 3Å‚, e ),
3
Å‚ (0, +")
"
T2 = (0, +") × (-", 0] Õ0
Å„Å‚
"
ôÅ‚0,
x " (0, e- 3Å‚]
ôÅ‚
òÅ‚
" "
3Å‚ 3Å‚
Õ" (x) = ,
2Å‚
3
ôÅ‚- ((-1 ln2 x + Å‚)x)3, x " (e- , e )
"
ôÅ‚
ół0,
3Å‚
x " [e , +")
Å‚ (0, +")
T1 T2
" "
T1 T2 Õ0
" "
3Å‚
P = (1, 0) Õ0 1 " (e- 3Å‚, e ))
P = (1, 0)
3
3
y = y + x y2
x
T = (-", 0) × R P =
1
(-1, )
27
Õ0(x) = 0, x " (-", 0).
T1 = (-", 0) × (0, +") T2 =
(-", 0) × (-", 0)
T1
1 1
z = z + x;
x 3
 T1
1
È1Å‚I(x) = (- x + Å‚)(-x) x " I,
3
Å‚ I
(-", 0) Å‚ 0
"1Å‚ = ;
(-", 3Å‚) Å‚ < 0
T1
1
Õ1Å‚I(x) = ((- x + Å‚)(-x))3 x " I,
3
Å‚ I "1Å‚
"
T1 = (-", 0) × [0, +")
Õ0 Õ1Å‚I Å‚ 0
((-1x + Å‚)(-x))3, x " (-", 3Å‚)
3
Õ" (x) =
1Å‚
0, x " [3Å‚, 0)
Å‚ < 0
"
T2 = (-", 0)×
(-", 0] Õ0
0, x " (-", 3Å‚]
Õ" (x) =
2Å‚
((-1x + Å‚)(-x))3, x " (3Å‚, 0)
3
Å‚ < 0
T
" "
T1 T2
Å„Å‚
ôÅ‚((-1x + ´)(-x))3, x " (-", 3´)
òÅ‚
3
Õ´Å‚(x) = ,
0, x " [3´, 3Å‚]
ôÅ‚
ół
((-1x + Å‚)(-x))3, x " (3Å‚, 0)
3
´, Å‚ ´ Å‚ < 0
1
P = (-1, ) " T1
27
1
Õ(x) = x6, x " (-", 0)
27
Å‚ = 0
1
P = (-1, )
27
y1 = y1 - e-xy2 + x
y2 = exy1 + xex,
1
Åš1(x) = , x " (-"; +")
ex
y1 = y1 - e-xy2
y2 = exy1.
z2 = z2
Õ2
z2 = (a22 - a12)z2
Õ-1
È2(x) = ex, x " (-", +")
É2(x) = -x, x " (-", +")
-1, x " (-", +")
1
É2 = a12È2
Õ1
0 -x
Åš2(x) = É2(x)Åš1(x) + = , x " (-", +")
È2(x) ex(1 - x)
Åš1, Åš2
1 -x
W (x) = = ex;
ex ex(1 - x)
x -x 1 x
W1(x) = = xex, W2(x) = = 0
xex ex(1 - x) ex xex
W1 W2
g1, g2 ,
W W
x2
g1(x) = , g2(x) = 0, x " (-", +");
2
x2
2
Åš0(x) = g1(x)Åš1(x) + g2(x)Åš2(x) = , x " (-", +")
ex x2
2
x2
1 -x
2
Śł1ł2(x) = ł1 + ł2 + , x " (-", +"),
ex ex(1 - x)
ex x2
2
Å‚1, Å‚2
 Åš1
y1 = xy1 + (1 - x2)y2 + x
y2 = y1 - xy2 + x2,
1 + x2
Åš1(x) = , x " (-"; +")
x
y1 = xy1 + (1 - x2)y2
y2 = y1 - xy2.
2x
z2 = - z2;
1 + x2
1
È2(x) = , x " (-", +")
1 + x2
x
É2(x) = , x " (-", +")
1 + x2
1 - x2
, x " (-", +");
(1 + x2)2
0 x
Åš2(x) = É2(x)Åš1(x) + = , x " (-", +")
È2(x) 1
Åš1, Åš2
1 + x2 x
W (x) = = ex;
x 1
x x 1 + x2 x
W1(x) = = x - x3, W2(x) = = 0
x2 1 x x2
W1 W2
g1, g2 ,
W W
1 1 1
g1(x) = x2 - x4, g2(x) = , x " (-", +");
2 4 5
1 1 1
x2 + x4 - x6
2 4 20
Åš0(x) = g1(x)Åš1(x) + g2(x)Åš2(x) = , x " (-", +")
1 1
x3 - x5
2 20
1 1 1
1 + x2 x x2 + x4 - x6
2 4 20
Śł1ł2(x) = ł1 + ł2 + , x " (-", +"),
1 1
x 1 x3 - x5
2 20
Å‚1, Å‚2
Åš1
y1 = y1 + 2y2 + ex
.
y2 = 3y1 + 2y2
y1 = y1 + 2y2
.
y2 = 3y1 + 2y2
1 -  2
= ( + 1)( - 4)
3 2 - 
1 = -1, 2 = 4
1 - 1 2 w1 1 - 1 2 w1
= 0, = 0
3 2 - 1 w2 3 2 - 1 w2
1 2
“1 = , “2 = .
-1 3
1 2
Åš1(x) = e-x , Åš2(x) = e4x , x " (-", +")
-1 3
e-x 2e4x
W (x) = = 5e3x;
-e-x 3e4x
ex 2e4x e-x ex
W1(x) = = 3e5x, W2(x) = = 1
0 3e4x -e-x 0
W1 W2
g1, g2 ,
W W
3 1
g1(x) = e2x, g2(x) = - e-3x, x " (-", +");
10 15
1
Åš0(x) = g1(x)Åš1(x) + g2(x)Åš2(x) = ex 6 , x " (-", +")
-1
2
1
1 2
Śł1ł2(x) = ł1e-x + ł2e4x + ex 6 , x " (-", +"),
-1 3 -1
2
Å‚1, Å‚2
Å„Å‚
ôÅ‚y1 = y1 + 6y2 + y3
òÅ‚
.
y2 = y1 + 2y2 + y3
ôÅ‚
ół
y3 = 4y3
1 -  6 1
1 2 -  1 = -( - 4)2( + 1)
0 0 4 - 
1 = 4 2 = -1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 - 1 6 1 w1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
1 2 - 1 1 w2 = 0
0 0 4 - 1 w3
îÅ‚ Å‚Å‚
2
ðÅ‚ ûÅ‚
“ = 1 ;
0
îÅ‚ Å‚Å‚
2
ðÅ‚ ûÅ‚
Åš1(x) = e4x 1
0
Åš1
1
z2 = -z2 + z3
2
z3 = 4z3
Õ2 Õ2
z2 = (a22 - a12)z2 + (a23 - a13)z3
Õ1 Õ1
( ).
Õ3 Õ3
z3 = (a32 - a12)z2 + (a33 - a13)z3
Õ1 Õ1
1 = 4 2 = -1
-1 - 1 1 w1 -1 - 2 1 w1
2 2
= 0, = 0
0 4 - 1 w2 0 4 - 2 w2
1 1
"2 = , "3 = .
10 0
1 1
¨2(x) = e4x , ¨3(x) = e-x , x " (-", +")
10 0
3
É2(x) = 8x, É3(x) = - e-5x, x " (-", +")
5
8, 3e-5x, x " (-", +")
3
1
Ék = a1½È½k, k = 2, 3
Õ1 ½=2
îÅ‚ Å‚Å‚
16x
0
ðÅ‚ ûÅ‚
Åš2(x) = É2(x)Åš1(x) + = e4x 8x + 1 , x " (-", +")
¨2(x)
10
îÅ‚ Å‚Å‚
-6
5
0
ðÅ‚ ûÅ‚
Åš3(x) = É3(x)Åš1(x) + = e-x 2 , x " (-", +")
5
¨3(x)
0
Åš1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
2 16x -6
5
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
Śł1ł2ł3(x) = ł1e4x 1 + ł2e4x 8x + 1 + ł3e-x 2 x " (-", +").
5
0 10 0
y1 = y1 - y2
.
y2 = 4y1 - 2y2
" "
-1 + 7i -1 - 7i
1 -  -1
= ( - )( - )
4 -2 - 
2 2
" "
-1 + 7i -1 - 7i
1 = , 2 = .
2 2
1 - 1 2 w1
= 0,
3 2 - 1 w2
1
"
“ = .
3- 7
i
2
"
-1+ 7i 1
"
2
Åš(x) = e x , x " (-", +")
3- 7
i
2
" "
7 7
1 1
cos x sin x
2 2
" " " "
Śł1ł2(x) = ł1e- 2 x " + ł2e- 2 x " ,
3 7 1 7 3 7 1 7
cos x + 7 sin x sin x - 7 cos x
2 2 2 2 2 2 2 2
x " (-", +"), Å‚1, Å‚2
12
y = y
x2
(0, +")
Õ1(x) = x4, x " (0, +")
8
z = - z;
x
1
È2(x) = , x " (0, +")
x8
1
É2(x) = - , x " (0, +")
7x7
È2
1
Õ2(x) = Õ1(x)É2(x) = - , x " (0, +")
7x3
Õ2 Õ1
1
ÕÅ‚1Å‚2(x) = Å‚1x4 + Å‚2 , x " (0, +"),
x3
Å‚1, Å‚2
 Õ1
2x 2
y = y - y - 1
x2 + 1 x2 + 1
R
Õ1(x) = x, x " R
2x 2
y = y - y
x2 + 1 x2 + 1
x > 0
2
z = - z;
x(x2 + 1)
1
È2(x) = 1 + , x " (0, +")
x2
1
É2(x) = x - , x " (0, +")
x
È2
Õ2(x) = Õ1(x)É2(x) = x2 - 1, x " (0, +")
(0, +")
Õ"(x) = x2 - 1, x " (-", +")
2
(-", +") Õ1
(-", +") Õ1, Õ" (-", +")
2
x2 + 1 = 0

Õ0(x) = g1(x)Õ1(x) + g2(x)Õ"(x) x " (-", +"),
2
g1, g2 : (-", +") (-", +")
x2-1 -x
,
x2+1 x2+1
1
g1(x) = x - 2 arctan x, g2(x) = - ln(x2 + 1), x " (-", +").
2
1
ÕÅ‚1Å‚2(x) = Å‚1x + Å‚2(x2 - 1) + (x - arctan x)x - (x2 - 1) ln(x2 + 1), x " (-", +"),
2
Å‚1, Å‚2
 Õ1
y = 3y.
" " W () = 2 - 3 1 =
3, 2 = - 3
" "
3x
ÕÅ‚1Å‚2(x) = Å‚1e + Å‚2e- 3x, x " (-", +")
Å‚1, Å‚2
y(5) - 2y(4) + y(3) = 0.
W () = 5-24 +3 = 3(-1)2
1 = 0 2 = 1
ÕÅ‚1Å‚2Å‚3Å‚4Å‚5(x) = Å‚1 + Å‚2x + Å‚3x2 + Å‚4ex + Å‚5xex, x " (-", +"),
Å‚1, Å‚2, Å‚3, Å‚4, Å‚5
y = -4y.
W () = 2 + 4 1 =
2i, 2 = -2i
ÕÅ‚1Å‚2(x) = Å‚1 cos 2x + Å‚2 sin 2x, x " (-", +")
Å‚1, Å‚2