Äîìàøíåå çàäàíèå 2 äëÿ ïëàòíûõ ãðóïï
ÈÁÌ.
Çàäà÷à 1.
Íàéòè àñèìïòîòû ãðàôèêà ôóíêöèè.
Çàäà÷à 2.
Ïðîâåñòè ïîëíîå èññëåäîâàíèå è ïîñòðîèòü ãðàôèê ôóíêöèè.
Çàäà÷à 4.
Íàéòè ïðîèçâîäíóþ
Çàäà÷à 5.
Íàéòè ïðîèçâîäíóþ
yx0
yx0
íåÿâíîé ôóíêöèè
F (x, y) = 0.
ôóíêöèè, çàäàííîé ïàðàìåòðè÷åñêè.
 âàðèàíòà
Çàäà÷à 1
Çàäà÷à 2
1
5x
x−1
ln x
x
x2 e2y − y 2 e2x = 0
2
x2 + 5
+ 2x
x2 − 1
2x2 + 4x + 3
x2 + x + 1
y sin x − cos (x − y) = 0
3
x
+x
2x − 1
exp(−x)
2x
x + y = ex−y
4
3
x−2
−x2 + 5x − 6
x2 − 3x + 3
y tg(xy) − y 3 e2+x = 1
5
x−2
x+4
ln(x − 2)
x−4
(x − y)2 + yex = 0
6
x
1 + x2
3x2 + x + 2
x2 + x + 1
sin yex + x2 e2y = 0
7
x2 − 1
x4
ln(1 − x)
x
cos(x2 + y) + sin (x + y 2 ) = 0
8
2
x+3
2x2 − 4x + 3
x2 − 3x + 1
sin y(x2 − 1) + x2 y + 3e2 = 0
ln(1 − x)
−3 − x
2 − sin yxex + e2y = 0
9
x2
6
− 16
10
x2
x+4
x2 + 5x + 3
x2 + x + 1
11
x2 + 8x − 6
x
exp (−x)
x2
12
2x2 + x + 3
x+6
−x2 + 7x + 9
x2 − 3x + 3
13
2
2
x −4
ln2 x
x
Çàäà÷à 3
ln
p
y
x2 + y 2 = arctg
x
y + sin x2 y − x = 0
ey
2 −1
− xy + 5 = 0
ln y − arctg
2x
=0
y
Çàäà÷à 4
(
√
x = 1 − t2
√
y = tg 1 + t2

3t2 + 1

x =
3t2 3
t

y = sin
+t
3

√
x = 2t − t2
1
y = p
3
(1 − t)2
(
x = arcsin (cos t)
y = arccos (sin t)
(
√
x = ln (t + t2 + 1)
√
y = t t2 + 1
(
√
x = 2t − t2
y = arcsin (t − 1)
(
x = ctg(2 exp (t))
y = ln (tg(exp (t)))

x = ln (ctgt)
1
y =
cos2 t
(
x = arctg(exp (t/2))
p
y = exp (t) + 1
r

1−t

x = ln
√ 1+t

y = 1 − t2

1

x = ln √
1 − t4
2

y = arcsin 1 − t
1 + t2

x = arctg1 − t2
t
y = √
1 − t2
(
√
x = arcsin ( 1 − t2 )
y = (arccos t)2
14
x2 + 1
2x + 3
x2 + 3x + 2
x2 + x + 1
15
x2 + 1
x
x
exp(1/x)
16
x+5
x2 − 1
−x2 − x + 3
x2 − 3x + 3
3x2 y 2 + sin y = 3y
17
x2
x2 − 4
ln(x − 2)
2x + 6
sin yex + x2 e2y = 0
18
1
3−x
−x2 + 3x + 1
x2 + x + 1
19
x2 − 5x + 6
x
20
x2 − 4
x2 − 9
3x2 + x + 2
x2 + x + 1
21
1
1 − x2
ln2 x
x2
22
1 − x3
x2
3x3
3x2 + 4x + 4
23
x2
1
−4
x2 exp
1
x
exp x
x3
sin yex + x2 e2y = 0
exy +
2
y
= cos 3x
x
sin x
=x+y
y2
x2 − sin xy + xy = 0
√
x + y cos (xy) = 0
x2 + y 2 = 17
y 2 − x − ln
tgy −
x2
y
=0
x
x+1
=0
+ 4x + 5
24
1
4 − x2
3x3 + x + 2
x2 + 2x + 3
exy
= (x + 1)y
x2 + y
25
1−x
x2
ln x
x3 − 1
2 cos(xy) = y 2

t

x = √
2
1 − t√
2

y = ln 1 + 1 − t
t

x = (1 + cos2 t)2
cos t
y =
sin2 t
(
x = ln 1 − t2 1 + t
√
y = 1 − t2


x = arccos 1
t
√
1

y = t2 − 1 + arcsin
t

1

x =
ln t √
2

y = ln 1 + 1 − t
t
(
√
x = arcsin t
p
√
y = 1+ t

x = arcsin2 t
t
y = √
1 − t2

√
x = t t2 + 1
√
2
y = ln 1 + 1 − t
t

x = arctgt
2
y = ln 1 + t
t+1
(
x = ln(1 − t2 )
√
y = arcsin 1 − t2

x = arctg t + 1
t−1
y = arcsin √1 − t2
r


x = ln 1 − sin t
1 + sin t

y = 1 tg2 t + ln cos t
2