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