‫الـمـمـلكـة الـعـربـيـة الـسـعـوديـة‬
‫وزارة الـتـعـلـيـم الـعـالـي‬
‫جـامـعـة الـمـجـمـعـة‬
‫كلية العلوم بالزلفي‬
Kingdom of Saudi Arabia
Ministry of Higher Education
Majmaah University
College Of Sciences in Alzulfi
Questions bank for math 203
(Differential and Integral Calculus in Several Variables)
‫ تطبيق‬+ ‫ تذكر‬: )‫*المستوى األول (معرفي‬
[1] evaluate
lim
( x , y ) (0,0)
xy
x  y2
2
[2] change from cylindrical to rectangular coordinates
(1,
[3] find all second partial derivatives f xx , f yy , f xy , f yx of
3
, 2)
2
f ( x, y)  x3 y 4  3xy 2  2 xy
[4] find an equation of the tangent plane to the surface at the given point
z  4x2  y 2  2 y
(-1,2,4)
[5] for the function f ( x, y, z )  cos(4 x  3 y  2 z ) find f xy and f zz .
[6] find
lim
( x , y )(6,3)
xy cos( x  2 y) .
Find all of the first order partial derivatives for the following functions.
[7]
[8]
[9]
[10] Determine if the following limits exist or not. If they do exist give the value
of the limit.
(a)
(b)
1
(c)
(d)
[11] show that f ( x, y ) 
x
is differentiable at (6,3)
y
[12] change from rectangular to spherical coordinates
[13] if
f ( x, y, z )  cos(4 x  3 y  2 z )
find f xyz
(0, 3,1)
and
f yzz
-------------------------- ---------------------------------------------------------------------‫ مهارة عملية‬+ ‫ مهارة ذهنية‬: )‫ (مهاري‬: ‫*المستوى الثاني‬
w  x2  y 2  z 2
[14] find dw if
[15] find surface area of z  x 2  y 2 lies above the disk x 2  y 2  9 .
[16] find the local maximum and local minimum values and saddle points of :
f ( x, y)  x 4  y 4  4 xy  2
2

[17] evaluate :
0

[18] evaluate
D
[19] find dz if

2
0
x sin ydydx
x2  y 2  9
xydA , where D is the disk
z  x3 ln( y 2 )
[20] use the chain rule to find
z  x 2  xy  y 2
and x  s  t
,
y  st
[21] find the local maximum and minimum values and saddle point(s) if any of the
function:
f ( x, y )  x sin y
[22] find the absolute maximum and minimum values of f on the set D:
,
D  ( x, y )  0  x  3, 0  y  2 f ( x, y)  x 4  y 4  4 xy  2
2
[23] use Lagrange multipliers to find the maximum and minimum values of :
2
2
f ( x, y )  x 2 y On x  y  1
[24] find the differential dz of the function z  y cos( xy ) .
evaluate the following integrals:

2 2
[25]
  x cos ydydx
0 0
[26]
 ( x  y)dA
where D is bounded by y  x and y  x 2
D
 cos( x
[27]
2
 y 2 )dA
x2  y 2  9
where R is the disk
(use polar coordinates).
R
[28] use the chain rule to find
dz
if z  x 2 y  xy 2
dt
, x  2  t4 ,
y  1 t3 .
[29] find the area of the part of the paraboloid z  4  x 2  y 2 that lies above the xypalne.
evaluate the following integrals:
[30]

[31]

E
E
xydv , where E  ( x, y, z )  0  x  3, 0  y  x, 0  z  x  y
x 2  y 2 dv
where E is the region that lies inside the cylinder x 2  y 2  16 and
between the planes z  5 and z  4 (use cylindrical coordinates).
1 x2
[32] evaluate :
  ( x  2 y)dydx
0 0
[33] You decide to build a box that has the shape of a rectangular prism
with a volume of 1000 cubic centimeters. Find the dimensions x, y and z of
the box so that the total surface area of all 6 faces of the box is minimum.
[34] Find the dimensions of a six-faced box that has the shape of a
rectangular prism with the largest possible volume that you can make with
12 squared meters of cardboard.
3
[35] Find the distance from the point (1,2,-1) to the plane given by the
equation x - y + z = 3.
[36] Evaluate the following integral.
[37] Evaluate
where E is the region under the plane
that lies in the first octant.
[38] Evaluate
where E is the solid bounded by
and the plane
Compute
for each of the following.
[39]
,
[40]
,
,
Use a tree diagram to write down the chain rule for the given derivatives.
[41]
for
,
, and
[42]
,
for
,
, and
4
,
[43] Compute
for
if
and
Compute the differentials for each of the following functions.
[44]
[45]
--------------------------------------------------------------------------------------------------------:‫ مهارة انتقالية‬: ‫*المستوى الثالث‬
[46] find and sketch the domain of
f ( x, y ) 
1
x  y2 1
[47] find and sketch the domain of
f ( x, y ) 
3x  5 y
x  y2  4
2
2
Determine the domain of each of the following , and graph it using a graphing
software.
[48]
[49]
[50]
5
6
7
8