"f "f
A 1. " f (x0 + "x, y0 + "y) H" f (x0, y0) + (x0, y0) � "x + (x0, y0) � "y
"x "y
"
1.2
" f (x0 + "x, y0 + "y) = 0.9 ln
0.9
"
y
" f(x, y) = x ln
x
" x0 = 1, "x = -0.1, y0 = 1, "y = 0.2
"
"f 1 y 1 "f x
"
" = ln - " , = ,
"x 2 x x x "y y
"f "f
" (1, 1) = -1, (1, 1) = 1, f(1, 1) = 0
"x "y
"
1.2
" 0.9 ln H" 0 + (-1) � (-0.1) + 1 � 0.2 = 0.3
0.9
2. " x + y + z = 27, z = 27 - x - y, x, y, z > 0
" I(x, y) = xy(27 - x - y), x > 0, y > 0, x + y < 27
"I "I
" = y(27 - 2x - y), = x(27 - x - 2y)
"x "y
ńł
"I
�ł ńł ńł
�ł
�ł = 0,
�ł �ł �ł
y(27 - 2x - y) = 0, x = 9,
"x
" �!�! �!�!
�ł ół ół
"I
�ł
x(27 - x - 2y) = 0 y = 9
�ł
= 0
ół
"y
�ł łł
-2y 27-2x-2y
"2I "2I "2I
�ł �ł
" = -2y, = -2x, = 27 - 2x - 2y, W = det
"x2 "y2 "x"y
27-2x-2y -2x
�ł łł
-18 -9
"2I
�ł �ł
" W (9, 9) = = 243 > 0, (9, 9) = -18 < 0  (9, 9) maksimum lokalne
"x2
-9 -18
właściwe
" uzasadnienie, że jest to maksimum globalne
" I = 729
x2 x2

3. " f(x) = =
3
3x2 - 2
-2 1 - x2
2
"

3n
" f(x) = - x2n+2
2n+1
n=0



3 2 2

" x2 < 1 �!�! |x| < , R =

2 3 3
" c17 = 0 : f(17)(0) = 0
38 38
" c18 = - : f(18)(0) = - 18!.
29 29
1
y
1
x
2
-1
4. "
1 x=2-y2

" xy dxdy = dy xy dx
D -1
x=y2
1 x=2-y2
x2
" xy dxdy = ... = y dy
2
x=y2
D -1
1

" xy dxdy = ... = 2 y 1 - y2 dy
D -1

" xy dxdy = ... = 0
D
5. " rysunek
2 2

"z "z
" |Ł| = + + 1 dxdy
"x "y
D
"z -x "z -y
" = , =
"x "y
R2 - x2 - y2 R2 - x2 - y2

R dxdy

" |Ł| = , D : x2 + y2 r2
R2 - x2 - y2
D

R� d�d�
" |Ł| = , " = [0, 2Ą] � [0, r]
R2 - �2
"
2Ą
r
R� d�
" |Ł| = d�
R2 - �2
0
0

" |Ł| = 2ĄR R - R2 - r2
6. " 2 + 1 = 0, 1 = i, 2 = -i
" y(t) = C1 cos t + C2 sin t + �(t)
" �(t) = C1(t) cos t + C2(t) sin t
�ł łł
�ł łł �ł łł
2
0
cos t sin t C1(t)
�ł śł
" �ł �ł �ł �ł
=
�ł �ł
1
2
- sin t cos t C2(t)
sin t
cos t
2 2
" C1(t) = -1, C2(t) =
sin t
" C1(t) = -t, C2(t) = ln sin t
" �(t) = -t cos t + sin t ln sin t
" y(t) = C1 cos t + C2 sin t + t cos t + sin t ln sin t
2
"f "f
B 1. " f (x0 + "x, y0 + "y) H" f (x0, y0) + (x0, y0) � "x + (x0, y0) � "y
"x "y
"
" f (x0 + "x, y0 + "y) = (1.1)0.9 0.9
"
" f(x, y) = xy y
" x0 = 1, "x = 0.1, y0 = 1, "y = -0.1

"f " "f " 1
" = yxy-1 y, = xy ln x y +
"
"x "y 2 y
"f "f 1
" (1, 1) = 1, (1, 1) = , f(1, 1) = 1
"x "y 2
"
" (1.1)0.9 0.9 H" 1 + 1 � 0.1 + 0.5 � (-0.1) = 1.05

2. " C = (x, y, 2), |CA| = (x - 1)2 + y2 + 4, |CB| = x2 + (y + 2)2 + 4,
" d(x, y) = |CA|2 + |CB|2 = 2x2 - 2x + 2y2 + 4y + 13, x, y " R
"d "d
" = 4x - 2, = 4y + 4
"x "y
ńł
ńł
"d
�ł ńł
�ł
1
�ł = 0, �ł
�ł �ł �ł
4x - 2 = 0, x = ,
"x
2
" �!�! �!�!
�ł ół �ł
"d
�ł ół
4y + 4 = 0
�ł y = -1
= 0
ół
"y
�ł łł
4 0
"2d "2d "2d
�ł �ł
" = 4, = 4, = 0, W = det
"x2 "y2 "x"y
0 4

1 "2d 1 1
" W , -1 = 16 > 0, , -1 = 4 > 0  , -1 minimum lokalne właściwe
2 "x2 2 2
" uzasadnienie, że jest to minimum globalne

1
" C = , -1, 2
2
x x

3. " f(x) = =
2
2x2 + 3
3 1 - - x2
3
"

2n
" f(x) = (-1)n x2n+1
3n+1
n=0



2 3 3

" - x2 < 1 �!�! |x| < , R =

3 2 2
28 28
" c17 = : f(17)(0) = 17!.
39 39
" c18 = 0 : f(18)(0) = 0
3
y
"
3
1
x
4. "
"
x=y2+1
3

1 1 1 1
" + dxdy = dy + dx
x2 y2 x2 y2
x=y
D 1
"
3
x=y2
+1
1 1 1 x
" + dxdy = ... = y - + dy
x2 y2 x y2
x=y
D 1
"
3

1 1 1 1
" + dxdy = ... = 1 + - dy
x2 y2 y2 y2 + 1
D 1

1 1 2 Ą
"
" + dxdy = ... = -
x2 y2 12
3
D
5. " rysunek
2 2

"z "z
" |Ł| = + + 1 dxdy
"x "y
D
"z -x "z -y

" = , =
"x - x2 - y2 4
"y - x2 - y2
4

2 dxdy
" |Ł| = , D : x2 + y2 3
4 - x2 - y2
D


"
2� d�d�

" |Ł| = , " = [0, 2Ą] � 0, 3
4 - �2
"
"
2Ą 3

2� d�

" |Ł| = d�
4 - �2
0 0
" |Ł| = 4Ą
6. " 2 + 1 = 0, 1 = i, 2 = -i
" y(t) = C1 cos t + C2 sin t + �(t)
" �(t) = C1(t) cos t + C2(t) sin t
�ł łł �ł łł �ł łł
2
cos t sin t C1(t) 0
" �ł �ł �ł �ł �ł �ł
=
2
- sin t cos t C2(t) cos2 t
2 2
" C1(t) = - cos2 t sin t, C2(t) = cos3 t

1 1
" C1(t) = cos3 t, C2(t) = sin t 1 - sin2 t
3 3


1 1 1
" �(t) = cos4 t + 1 - sin2 t sin2 t = 1 + sin2 t
3 3 3

1
" y(t) = C1 cos t + C2 sin t + 1 + sin2 t
3
4
"f "f
C 1. " f (x0 + "x, y0 + "y) H" f (x0, y0) + (x0, y0) � "x + (x0, y0) � "y
"x "y
"
0.3
" f (x0 + "x, y0 + "y) = 1.1 arc tg
1.1
"
y
" f(x, y) = x arc tg
x
" x0 = 1, "x = 0.1, y0 = 0, "y = 0.3

" "
"f 1 y 1 y "f 1 1
"
" = arc tg + x
2 - , = x
2
"x 2 x x y x2 "y y x
1 + 1 +
x x
"f "f
" (1, 0) = 0, (1, 0) = 1, f(1, 0) = 0
"x "y
"
0.3
" 1.1 arc tg H" 0 + 0 � 0.1 + 1 � 0.3 = 0.3
1.1

1
2. " C = x, y, ,
xy
1
" d(x, y) = x2 + y2 + , xy = 0

x2y2
"d 2 "d 2
" = 2x - , = 2y - , x, y " R
"x x3y2 "y x2y3
ńł ńł
"d 2
�ł �ł ńł ńł ńł ńł
�ł �ł
�ł = 0, �ł2x- = 0,
�ł �ł �ł �ł �ł �ł
x1 = 1, x2 = 1, x3 = -1, x4 = -1,
"x x3y2
" �!�! �!�! (" (" ("
�ł �ł 2 ół ół ół ół
"d
�ł �ł
y1 = -1 y2 = 1 y3 = 1 y4 = -1
�ł �ł
= 0 2y- = 0
ół ół
"y x2y3
�ł łł
6 4
2+
�ł
"2d 6 "2d 6 "2d 4 x4y2 x3y3 śł
�ł śł
" = 2 + , = 2 + , = , W = det �ł śł
�ł 4 6 �ł
"x2 x4y2 "y2 x2y4 "x"y x3y3
2+
x3y3 x2y4
"2d "2d
" W (1, -1) = W (1, 1) = W (-1, -1) = W (-1, 1) = 48 > 0, (1, -1) = (1, 1) =
"x2 "x2
"2d "2d
(-1, -1) = (-1, 1) = 8 > 0  (1, -1), (1, 1), (-1, -1), (-1, 1) minima lokalne
"x2 "x2
właściwe
" uzasadnienie, że są to minima globalne
" C1 = (1, -1, -1), C2 = (1, 1, 1), C3 = (-1, -1, 1), C4 = (-1, 1, -1)
x2 x2

3. " f(x) = =
4 - 2x2
x2
4 1 -
2
"

x2n+2
" f(x) =
2n+2
n=0


" "
x2

" < 1 �!�! |x| < 2, R = 2

2
" c17 = 0 : f(17)(0) = 0
1 18!
" c18 = : f(18)(0) = .
210 210
5
y
2
1
x
4. "
2 x=y

x2 x2
" dxdy = dy dx
y2 y2
1
D 1
x=
y
2 x=y
x2 x3
" dxdy = ... = dy
y2 3y2
1
x=
D 1
y
2
x2 1 1
" dxdy = ... = y - dy
y2 3 y5
D 1

x2 27
" dxdy = ... =
y2 64
D
5. " rysunek
2 2

"z "z
" |Ł| = + + 1 dxdy
"x "y
D
"z "z
" = x, = y
"x "y


" |Ł| = x2 + y2 + 1, D : r2 x2 + y2 R2
D


" |Ł| = �2 + 1� d�d�, " = [0, 2Ą] � [r, R]
"
2Ą
R

" |Ł| = d� �2 + 1� d�
r
0


2
" |Ł| = Ą 1 + R2 1 + R2 - 1 + r2 1 + r2
3
6. " 2 + 1 = 0, 1 = i, 2 = -i
" y(t) = C1 cos t + C2 sin t + �(t)
" �(t) = C1(t) cos t + C2(t) sin t
�ł łł
�ł łł �ł łł
2
0
cos t sin t C1(t)
�ł śł
" �ł �ł �ł �ł
=
�ł �ł
1
2
- sin t cos t C2(t)
cos t
sin t
2 2
" C1(t) = - , C2(t) = 1
cos t
" C1(t) = ln cos t, C2(t) = t
" �(t) = cos t ln cos t + t sin t
" y(t) = C1 cos t + C2 sin t + cos t ln cos t + t sin t
6
"f "f
D 1. " f (x0 + "x, y0 + "y) H" f (x0, y0) + (x0, y0) � "x + (x0, y0) � "y
"x "y

" "
4
" f (x0 + "x, y0 + "y) = ln 1.1 + 0.8 - 1
"
"
4
" f(x, y) = ln x + y - 1
" x0 = 1, "x = 0.1, y0 = 1, "y = -0.2
"f 1 1 "f 1 1
" = " " , = "
" "
4
4 4
"x x + y - 1 2 x "y x + y - 1
4 y3
"f 1 "f 1
" (1, 1) = , (1, 1) = , f(1, 1) = 0
"x 2 "y 4

" "
4
" ln 1.1 + 0.8 - 1 H" 0 + 0.5 � 0.1 + 0.25 � (-0.2) = 0

2. " C = (x, y, -x - y), |CA| = (x - 1)2 + (y - 2)2 + (x + y + 3)2
" d(x, y) = |CA|2 = (x - 1)2 + (y - 2)2 + (x + y + 3)2, x, y " R
"d "d
" = 2(x - 1) + 2(x + y + 3), = 2(y - 2) + 2(x + y + 3)
"x "y
ńł
"d
�ł ńł ńł
�ł
�ł = 0,
�ł �ł �ł
2(x - 1) + 2(x + y + 3) = 0, x = -1,
"x
" �!�! �!�!
�ł ół ół
"d
�ł
2(y - 2) + 2(x + y + 3) = 0 y = 0
�ł
= 0
ół
"y
�ł łł
4 2
"2d "2d "2d
�ł �ł
" = 4, = 4, = 2, W = det
"x2 "y2 "x"y
2 4
"2d
" W (-1, 0) = 12 > 0, (-1, 0) = 4 > 0  (-1, 0) minimum lokalne właściwe
"x2
" uzasadnienie, że jest to minimum globalne
" C = (-1, 0, 1)
x x

3. " f(x) = =
4
3 + 4x2
3 1 - - x2
3
"

4n
" f(x) = (-1)n x2n+1
3n+1
n=0
" "


4 3 3

" - x2 < 1 �!�! |x| < , R =

3 2 2
48 48
" c17 = : f(17)(0) = 17!.
39 39
" c18 = 0 : f(18)(0) = 0
7
y
1
x
e
1
e2
4. "
2 x=e2

y y
" dxdy = dy dx
x x
D 1 x=ey
2
y 2
" dxdy = ... = [y ln |x|]x=e dy
x=ey
x
D 1
2
y
" dxdy = ... = y (2 - y) dy
x
D 1

y 2
" dxdy = ... =
x 3
D
5. " rysunek
2 2

"z "z
" |Ł| = + + 1 dxdy
"x "y
D
"z "z
" = 2x, = 2y
"x "y


" |Ł| = 4x2 + 4y2 + 1, D : x2 + y2 1
D


" |Ł| = 4�2 + 1� d�d�, " = [0, 2Ą] � [0, 1]
"
2Ą
1

" |Ł| = d� 4�2 + 1� d�
0 0

"
Ą
" |Ł| = 5 5 - 1
6
6. " 2 + 1 = 0, 1 = i, 2 = -i
" y(t) = C1 cos t + C2 sin t + �(t)
" �(t) = C1(t) cos t + C2(t) sin t
�ł łł �ł łł �ł łł
2
cos t sin t C1(t) 0
" �ł �ł �ł �ł �ł �ł
=
2
- sin t cos t C2(t) sin2 t
2 2
" C1(t) = - sin3 t, C2(t) = cos t sin2 t

1 1
" C1(t) = cos t 1 - cos2 t , C2(t) = sin3 t,
3 3


1 1 1
" �(t) = 1 - cos2 t cos2 t + sin4 t = 1 + cos2 t
3 3 3

1
" y(t) = C1 cos t + C2 sin t + 1 + cos2 t
3
8