math 3
Universiti Tun Hussein Onn Malaysia
Pusat Pengajian Sains
Semester 1 2008/2009
BSM 2913
TEST 1
DURATION : 60 minutes
ANSWER ALL QUESTIONS.
1. Given the function f ( x, y ) = x 2 + 2 y 2 − 1
a. Find the domain and range of the function.
(2 marks)
b. Sketch the contour map of the function f ( x, y ) using three level curves, c = 1, 2, 3 .
(4 marks)
c. Use 3D-contour map to sketch roughly the surface of f ( x, y ) .
(2 marks)
2
4 x2 − y 2
,
x2 + 2 y 2 f ( x, y ) along x- axis and y-axis,
Given the function f ( x, y ) =
a. find the lim ( x , y )→(0,0)
(4 marks)
b. does the lim ( x , y )→(0,0) f ( x, y ) exist?
(1 marks)
c. is the function f ( x, y ) continuous?
(2 marks)
3
Given the function f ( x, y ) = cos( xy ) + e x y . Find f x , f y , f xy and f yy .
2
(8 marks)
4
Use chain rule to find
∂z
∂z
and if z = xy + xy 2 , x = u sin v, y = v sin u .
∂v
∂u
(10 marks)
5
6
Suppose that a particle moving along a metal plate in the xy-plane with the rate 1 cm/s along x-direction and - 4 cm/s along y-direction at the point (3, 2). Given that the temperature of the plate at points in the xy-plane is T ( x, y ) = y 2 ln x, x ≥ 1 in degree
Celcius, at what rate is the temperature changing at the point (3, 2)?
(7 marks)
∂z
∂z and using implicit differentiation
Given F ( x, y , z ) = sin( y + z ) − x = 0 , find
∂x
∂y formula. (5 marks)
9th August 2008
[Total marks : 45]
Questions
Q1 (a)
Answer
D = {( x, y ) : x ∈ , y ∈
R = { z : z ≥ −1, z ∈
Q1 (b)
x2
( 2)
2
x2
( )
3
x2
+
22
2
}
}
y2
+ 2 =1
1
+
y2
⎛ 3⎞
⎜
⎟
⎝ 2⎠
y2
( 2)
2
2
=1
=1
Ellipse shape , correct vertices
2a
(4 marks)
2b
2c
3
(8 marks)
Along x-axis
4 x2 − y 2 lim( x , y )→(0,0) 2 x + 2 y2
4x2 − 0
= lim( x )→(0) 2 x +0
4x2
= lim( x )→(0) 2 = 4 x = lim( y )→(0)
0 − y2
0 + 2 y2
= lim( y )→(0,0)
− y2
1
=−
2
2y
2
Does not
Pusat Pengajian Sains
Semester 1 2008/2009
BSM 2913
TEST 1
DURATION : 60 minutes
ANSWER ALL QUESTIONS.
1. Given the function f ( x, y ) = x 2 + 2 y 2 − 1
a. Find the domain and range of the function.
(2 marks)
b. Sketch the contour map of the function f ( x, y ) using three level curves, c = 1, 2, 3 .
(4 marks)
c. Use 3D-contour map to sketch roughly the surface of f ( x, y ) .
(2 marks)
2
4 x2 − y 2
,
x2 + 2 y 2 f ( x, y ) along x- axis and y-axis,
Given the function f ( x, y ) =
a. find the lim ( x , y )→(0,0)
(4 marks)
b. does the lim ( x , y )→(0,0) f ( x, y ) exist?
(1 marks)
c. is the function f ( x, y ) continuous?
(2 marks)
3
Given the function f ( x, y ) = cos( xy ) + e x y . Find f x , f y , f xy and f yy .
2
(8 marks)
4
Use chain rule to find
∂z
∂z
and if z = xy + xy 2 , x = u sin v, y = v sin u .
∂v
∂u
(10 marks)
5
6
Suppose that a particle moving along a metal plate in the xy-plane with the rate 1 cm/s along x-direction and - 4 cm/s along y-direction at the point (3, 2). Given that the temperature of the plate at points in the xy-plane is T ( x, y ) = y 2 ln x, x ≥ 1 in degree
Celcius, at what rate is the temperature changing at the point (3, 2)?
(7 marks)
∂z
∂z and using implicit differentiation
Given F ( x, y , z ) = sin( y + z ) − x = 0 , find
∂x
∂y formula. (5 marks)
9th August 2008
[Total marks : 45]
Questions
Q1 (a)
Answer
D = {( x, y ) : x ∈ , y ∈
R = { z : z ≥ −1, z ∈
Q1 (b)
x2
( 2)
2
x2
( )
3
x2
+
22
2
}
}
y2
+ 2 =1
1
+
y2
⎛ 3⎞
⎜
⎟
⎝ 2⎠
y2
( 2)
2
2
=1
=1
Ellipse shape , correct vertices
2a
(4 marks)
2b
2c
3
(8 marks)
Along x-axis
4 x2 − y 2 lim( x , y )→(0,0) 2 x + 2 y2
4x2 − 0
= lim( x )→(0) 2 x +0
4x2
= lim( x )→(0) 2 = 4 x = lim( y )→(0)
0 − y2
0 + 2 y2
= lim( y )→(0,0)
− y2
1
=−
2
2y
2
Does not