Grzegorz Mrzygłocki, WILi , sem. III, gr.2
1
Zad.1
(
)
dl
y
x
L
+
2
2
( )
1
,
1
A
;
( )
4
,
4
B
+
=
+
=
b
a
b
a
4
4
1
=
=
0
1
b
a
x
y
=
1
'
=
y
(
)
3
2
126
2
2
1
1
4
1
2
4
1
2
2
=
=
+
+
dx
x
dx
x
x
Zad.2
(
)
−
L
dl
y
x
2
( )
2
,
2
A
;
(
)
4
,
2
−
B
+
−
=
+
=
b
a
b
a
2
4
2
2
=
−
=
3
2
1
b
a
3
2
1 +
−
=
x
y
2
1
'
−
=
y
5
6
3
4
5
2
5
3
2
5
2
5
2
1
1
3
2
1
2
2
2
2
2
2
2
2
2
−
=
−
=
−
=
+
−
+
−
−
−
x
x
dx
x
dx
x
x
Zad.3
( )
L
dl
y
x
y
4
2
=
( )
0
,
0
O
;
( )
2
,
1
P
( )
x
x
y
1
1
'
2
2
=
−
=
(
)
1
8
3
4
3
4
2
1
1
2
1
2
2
1
2
3
2
1
1
0
1
0
−
=
=
=
=
=
+
=
+
=
+
t
dx
t
dt
dx
t
x
dx
x
dx
x
x
x
Zad.4
( )
L
dl
y
3
x
y
=
( )
1
,
1
A
;
( )
8
,
2
B
( )
( )
4
2
2
2
9
3
'
x
x
y
=
=
(
)
3
3
145
10
3
145
10
2
3
2
4
2
1
4
3
10
145
54
1
54
1
18
1
18
1
9
1
9
1
−
=
=
=
=
=
+
=
+
t
dt
t
tdt
dx
x
t
x
dx
x
x
Grzegorz Mrzygłocki, WILi , sem. III, gr.2
2
Zad.5
L
dl
y
x
4
x
y
1
=
2
1
≤
≤ x
( )
4
2
2
2
1
1
'
x
x
y
=
−
=
(
)
2
2
17
17
6
1
6
1
2
1
2
1
1
1
1
17
2
3
17
2
2
3
2
4
4
2
1
3
4
4
2
1
5
−
=
=
=
=
=
+
=
+
=
+
t
dt
t
tdt
dx
x
t
x
dx
x
x
dx
x
x
x
Zad.6
( )
dl
x
y
L
cos
2
( )
x
y sin
=
2
0
π
≤
≤ x
( )
( )
x
y
2
2
cos
'
=
( ) ( )
( )
( )
( ) ( )
(
)
1
2
2
3
2
3
2
2
sin
cos
cos
1
cos
1
cos
sin
2
1
2
3
1
2
2
2
2
2
2
0
−
=
−
=
−
=
−
=
=
+
=
+
t
dt
t
tdt
dx
x
x
t
x
dx
x
x
x
π
Zad.7
+
L
dl
x
4
9
1
x
x
y
=
4
0
≤
≤ x
( )
x
x
y
4
9
2
3
'
2
2
=
=
22
18
4
8
9
4
9
1
4
9
1
4
9
1
4
0
2
4
0
4
0
=
+
=
+
=
+
=
+
+
x
x
dx
x
dx
x
x
Zad.8
(
)
+
L
dl
y
x
( )
0
,
0
O
;
( )
0
,
1
A
;
( )
1
,
0
B
≤
≤
=
=
≤
≤
=
=
≤
≤
+
−
=
1
0
0
:
1
0
0
:
1
0
1
:
3
2
1
t
t
x
y
l
t
t
y
x
l
x
x
y
l
(
)
(
)
=
=
+
+
−
=
−
1
0
1
0
2
2
1
1
1
1
dx
dx
x
x
dl
y
x
l
(
)
=
+
=
−
1
0
2
1
1
0
2
dt
t
dl
y
x
l
(
)
=
+
=
−
1
0
2
1
0
1
3
dt
t
dl
y
x
l
(
)
2
1
2
1
2
1
2
+
=
+
+
=
+
L
dl
y
x
Grzegorz Mrzygłocki, WILi , sem. III, gr.2
3
Zad.9
dl
xy
L
2
6
( )
t
x
cos
3
1
=
( )
t
y
sin
3
1
=
4
0
π
≤
≤ t
( )
( )
( )
t
t
x
2
2
2
sin
9
1
sin
3
1
'
=
−
=
( )
( )
( )
t
t
y
2
2
2
cos
9
1
cos
3
1
'
=
=
( )
( )
( )
( )
( )
( )
162
2
648
2
81
2
27
2
cos
sin
sin
cos
27
2
9
1
sin
9
1
cos
3
1
6
2
5
2
2
0
3
2
2
0
2
2
4
0
2
4
0
=
=
=
=
=
=
=
=
u
du
u
du
dt
t
u
t
dt
t
t
dt
t
t
π
π
Zad.11
+
L
dl
y
xz
2
1
t
x
= ;
2
t
y
= ;
3
3
2
t
z
=
1
0
≤
≤ t
( )
1
'
2
=
x
( )
2
2
4
'
t
y
=
( )
4
2
4
'
t
z
=
(
)
15
2
3
2
2
1
2
1
3
2
4
4
1
2
1
3
2
1
0
4
2
2
1
0
2
4
4
2
1
0
2
4
=
=
+
+
=
+
+
+
dt
t
dt
t
t
t
dt
t
t
t
t
Zad.12
( )
L
dl
xy
t
e
x
= ;
t
e
y
−
=
;
t
z
2
=
1
0
≤
≤ t
( )
t
e
x
2
2
'
=
( )
t
e
y
2
2
'
−
=
( )
2
'
2
=
z
( )
(
)
( )
=
+
=
+
+
=
+
+
=
+
+
⋅
⋅
−
−
−
dt
e
e
dt
e
e
e
dt
e
e
dt
e
e
e
e
t
t
t
t
t
t
t
t
t
t
t
1
0
2
2
2
1
0
2
2
2
2
1
0
2
2
2
2
1
0
1
1
2
2
2
(
) (
) (
)
1
1
0
1
0
1
1
0
2
1
1
1
−
−
−
−
−
=
+
−
−
=
−
=
+
=
+
=
e
e
e
e
e
e
dt
e
e
dt
e
e
t
t
t
t
t
t
Zad.13
(
)
+
L
dl
z
y
x
( )
t
x cos
=
( )
t
y sin
=
t
z
4
3
=
π
2
0
≤
≤ t
( )
( )
t
x
2
2
sin
'
=
( )
( )
t
y
2
2
cos
'
=
( )
16
9
'
2
=
z
Grzegorz Mrzygłocki, WILi , sem. III, gr.2
4
( ) ( )
( ) ( )
( )
( )
( )
=
=
=
=
+
=
+
+
du
dt
t
u
t
dt
t
t
t
t
dt
t
t
t
cos
sin
cos
4
3
sin
cos
4
5
16
9
1
4
3
sin
cos
2
0
2
0
π
π
( )
( )
( )
( )
( )
(
)
( )
( )
0
cos
16
15
sin
sin
16
15
sin
1
'
cos
'
cos
16
15
4
5
2
0
2
0
2
0
2
0
0
0
=
=
−
⋅
=
=
=
=
=
=
+
π
π
π
π
t
dt
t
t
t
t
q
p
t
q
t
p
dt
t
t
du
u
Zad.14
( )
L
dl
z
( )
t
t
x
cos
=
( )
t
t
y
sin
=
t
z
=
1
0
≤
≤ t
( )
( )
( )
(
)
( )
( ) ( )
( )
t
t
t
t
t
t
t
t
t
x
2
2
2
2
2
sin
sin
cos
2
cos
sin
cos
'
+
−
=
−
=
( )
( )
( )
(
)
( )
( ) ( )
( )
t
t
t
t
t
t
t
t
t
y
2
2
2
2
2
cos
sin
cos
2
sin
cos
sin
'
+
+
=
+
=
( )
1
'
2
=
z
(
)
2
2
3
3
3
1
3
1
2
2
1
1
3
2
3
3
2
2
2
2
1
0
2
1
0
2
−
=
=
=
=
=
+
=
+
=
+
+
u
du
u
udu
tdt
u
t
dt
t
t
dt
t
t
Zad.15
=
L
dl
L
( )
x
y ln
=
5
2
≤
≤ x
( )
2
2
1
'
x
y
=
( )
(
)
( )
=
+
−
+
=
=
+
=
=
+
=
=
+
=
+
=
5
2
2
5
2
2
2
2
1
2
5
2
2
5
2
2
1
1
1
1
'
ln
'
1
1
ln
1
1
x
dx
x
x
x
v
x
x
u
x
v
x
u
dx
x
x
dx
x
L
(
)
(
) (
)
+
+
+
−
=
+
+
+
−
−
=
+
+
−
−
=
26
5
5
2
ln
2
5
5
26
5
2
ln
26
5
ln
2
5
5
26
1
ln
2
5
5
26
5
2
2
x
x
Zad.16
=
L
dl
L
( )
t
x
cos
7
=
( )
t
y
sin
7
=
π
4
3
0
≤
≤
t
( )
( )
t
x
2
2
sin
49
'
=
( )
( )
t
y
2
2
cos
49
'
=
4
21
7
49
4
3
0
4
3
0
π
π
π
=
=
=
dt
dt
L
Zad.17
( )
γ
γ
cos
1
+
=
r
π
γ
≤
≤
0
(
)
(
)
+
=
+
=
γ
γ
γ
γ
sin
cos
1
cos
cos
1
:
y
x
L
Grzegorz Mrzygłocki, WILi , sem. III, gr.2
5
(
)
(
)
γ
γ
γ
γ
γ
γ
γ
cos
cos
1
sin
'
sin
cos
1
cos
sin
'
2
+
+
−
=
+
−
−
=
y
x
( )
(
) (
)
( )
(
)
(
)
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
2
2
2
4
2
2
2
2
2
2
2
cos
cos
1
cos
sin
cos
1
2
sin
'
sin
cos
1
cos
1
cos
sin
2
cos
sin
'
+
+
+
−
=
+
+
+
+
=
y
x
( )
γ
γ
sin
'
−
=
r
( ) ( )
( )
[
]
( )
[ ]
γ
γ
2
2
2
2
'
'
'
r
r
y
x
+
=
+
( ) ( )
(
)
(
)
γ
γ
γ
γ
γ
γ
cos
1
2
cos
cos
2
1
sin
cos
1
sin
'
'
2
2
2
2
2
2
2
+
=
+
+
+
=
+
+
=
+
y
x
( )
(
)
4
2
sin
4
2
cos
2
2
cos
2
2
cos
1
2
,
0
0
0
2
0
=
=
=
=
+
=
=
π
π
π
π
γ
γ
γ
γ
γ
γ
γ
d
d
d
dl
y
x
f
L
L
Zad.18
( )
=
γ
γ
3
1
sin
3
3
r
( )
( )
=
=
γ
γ
γ
γ
sin
3
1
sin
3
cos
3
1
sin
3
:
3
3
y
x
L
( )
( )
γ
γ
γ
γ
γ
sin
3
1
sin
3
cos
3
1
cos
3
1
sin
3
'
3
2
−
=
x
( )
( )
γ
γ
γ
γ
γ
cos
3
1
sin
3
sin
3
1
cos
3
1
sin
3
'
3
2
+
=
y
( )
( )
γ
γ
2
2
2
2
'
'
'
r
r
y
x
+
=
+
( )
=
γ
γ
γ
3
1
cos
3
1
sin
3
'
2
r
=
+
=
+
=
+
γ
γ
γ
γ
γ
γ
3
1
sin
3
1
cos
3
1
sin
9
3
1
sin
3
3
1
cos
3
1
sin
3
'
'
2
2
4
2
3
2
2
2
2
y
x
=
=
γ
γ
3
1
sin
3
3
1
sin
9
2
4
( )
( )
π
α
α
α
γ
α
γ
γ
γ
π
π
2
9
sin
9
3
3
1
3
1
sin
3
,
0
2
3
0
2
∗
=
=
=
=
=
=
d
d
d
dl
y
x
f
L
L
( )
( ) ( )
( )
( )
( )
( )
( ) ( )
[
]
( )
=
+
−
=
−
=
=
=
=
=
=
∗
π
π
π
π
α
α
α
α
α
α
α
α
α
α
α
α
α
0
2
0
0
0
2
cos
cos
sin
cos
cos
'
sin
'
sin
sin
sin
sin
)
d
v
u
v
u
d
d
( )
(
)
( )
( )
2
sin
sin
sin
1
0
0
2
0
2
0
2
π
α
α
α
α
π
α
α
π
π
π
=
−
=
−
+
=
d
d
d
Grzegorz Mrzygłocki, WILi , sem. III, gr.2
6
Zad.19
2
,
1
A
−
11
,
11
B
( )
30
,
2
2
y
x
y
x
+
=
ρ
+
−
=
+
=
b
a
b
a
11
11
2
=
−
=
4
11
4
3
b
a
4
11
4
3
+
−
=
x
y
( )
16
9
4
3
'
2
2
=
−
=
y
( )
−
−
=
+
−
=
+
−
+
=
=
1
11
2
1
11
2
2
16
121
8
33
16
25
24
1
16
9
1
4
3
4
11
30
1
,
dx
x
x
dx
x
x
dl
y
x
M
L
ρ
43
1152
49536
1152
16236
33300
384
1452
3960
1152
33300
384
121
384
33
1152
25
1
11
2
3
=
=
+
=
+
+
=
+
−
=
−
x
x
x
Zad.20
x
y
=
2
0
≤
≤ x
( )
x
x
y
x
=
,
ρ
( )
4
2
2
2
1
1
'
x
x
y
=
−
=
( )
=
+
=
+
⋅
=
+
⋅
=
=
dx
x
x
dx
x
x
x
x
dx
x
x
x
dl
y
x
M
L
2
0
4
4
2
0
2
4
2
0
1
1
1
1
,
ρ
Zad.21
x
e
y
2
−
=
1
0
≤
≤ x
( )
2
,
y
y
x
=
ρ
( )
(
)
x
x
e
e
y
4
2
2
2
4
2
'
−
−
=
−
=
( )
=
−
=
−
=
=
+
=
+
=
=
−
+
−
−
−
−
4
4
1
5
4
4
4
1
0
4
16
1
16
1
4
1
4
1
,
e
x
x
x
x
L
dt
t
dt
dx
e
t
e
dx
e
e
dl
y
x
M
ρ
(
)
+
−
=
− 3
4
3
4
1
5
24
1
e
Zad.22
( )
( )
=
=
=
t
z
t
y
t
x
3
2
sin
4
2
cos
4
π
2
0
≤
≤ t
(
)
π
ρ
2
1
,
,
t
y
x
+
=
( )
( ) ( )
(
)
( )
( )
( )
( ) ( )
(
)
( )
( )
( )
=
=
=
=
−
=
9
'
2
sin
2
cos
64
2
sin
2
cos
8
'
2
sin
2
cos
64
2
sin
2
cos
8
'
2
2
2
2
2
2
2
2
2
z
t
t
t
t
y
t
t
t
t
x
(
)
( ) ( )
=
+
+
=
=
dt
t
t
t
dl
z
y
x
M
L
π
π
ρ
2
0
2
2
2
sin
2
cos
128
9
2
1
,
,
Grzegorz Mrzygłocki, WILi , sem. III, gr.2
7
Zad.23
2
x
y
=
1
0
≤
≤ x
( ) ( )
2
2
2
4
2
'
x
x
y
=
=
( )
12
1
5
4
1
4
1
4
1
4
1
,
3
0
5
1
2
0
2
2
1
0
2
0
0
−
=
=
=
=
+
=
+
=
=
=
ρ
ρ
ρ
ρ
ρ
dt
t
tdt
xdx
t
x
dx
x
x
xdl
dl
y
x
x
M
L
L
Y
Zad.24
( )
t
t
x
sin
−
=
( )
t
y
cos
1
−
=
π
2
0
≤
≤ t
( )
( )
(
)
( )
( )
( )
( )
=
+
−
=
−
=
t
y
t
t
t
x
2
2
2
2
2
sin
'
cos
cos
2
1
cos
1
'
( )
( )
( )
( )
( )
(
)
( )
dt
t
dt
t
dt
t
t
t
dl
y
x
M
L
−
=
−
=
+
+
−
=
=
π
π
π
ρ
ρ
ρ
ρ
2
0
0
2
0
0
2
0
2
2
0
cos
1
2
cos
1
2
sin
cos
cos
2
1
,
( )
( )
(
)
( )
dt
t
t
t
M
dl
y
x
x
M
M
M
x
L
Y
C
cos
1
sin
2
,
1
2
0
0
−
−
=
=
=
π
ρ
ρ
( )
( )
(
)
( )
dt
t
t
M
dl
y
x
y
M
M
M
y
L
X
C
cos
1
cos
1
2
,
1
2
0
0
−
−
=
=
=
π
ρ
ρ
Zad.25
( )
( )
=
=
=
t
z
t
y
t
x
sin
cos
2
0
π
≤
≤ t
( )
( )
( )
( )
( )
=
=
=
1
'
cos
'
sin
'
2
2
2
2
2
z
t
y
t
x
(
)
( )
( )
2
2
2
1
cos
sin
,
,
0
2
0
0
2
0
2
2
0
π
ρ
ρ
ρ
ρ
π
π
=
=
+
+
=
=
dt
dt
t
t
dl
z
y
x
M
L
(
)
( )
π
ρ
ρ
ρ
π
2
2
cos
2
,
,
1
0
2
0
0
−
=
−
=
=
=
=
M
dt
t
M
dl
z
y
x
x
M
M
M
x
L
YZ
C
(
)
( )
π
ρ
ρ
ρ
π
2
2
sin
2
,
,
1
0
2
0
0
=
=
=
=
=
M
dt
t
M
dl
z
y
x
x
M
M
M
y
L
YZ
C
(
)
( )
8
2
2
2
8
8
2
,
,
1
0
0
2
0
2
2
0
0
π
πρ
ρ
π
ρ
π
ρ
ρ
π
=
⋅
=
=
=
=
=
M
dt
t
M
dl
z
y
x
z
M
M
M
z
L
XY
C
−
8
2
,
2
,
2
π
π
π
C
Grzegorz Mrzygłocki, WILi , sem. III, gr.2
8
Zad.26
x
e
y
=
1
0
≤
≤ x
(
)
=
+
=
=
dx
e
e
dl
y
z
y
x
I
L
x
x
X
1
0
2
0
2
1
,
,
ρ
ρ
Zad.27
=
=
γ
γ
sin
cos
y
x
+
=
=
x
z
z
2
0
( )
( )
γ
γ
2
2
2
2
cos
'
sin
'
=
=
y
x
1
sin
(cos
'
'
2
2
2
2
=
+
=
+
γ
γ
y
x
(
)
=
+
=
+
=
π
π
π
γ
γ
2
0
4
0
4
cos
2
d
Zad.28
=
=
γ
γ
sin
2
cos
2
y
x
+
+
=
=
2
2
1
0
y
x
z
z
( )
( )
γ
γ
2
2
2
2
cos
4
'
sin
4
'
=
=
y
x
2
sin
(cos
4
'
'
2
2
2
2
=
+
=
+
γ
γ
y
x
(
)
=
=
+
=
π
π
π
γ
γ
2
0
2
0
20
10
2
4
1
d
d
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