Multivariable Calculus (MATS3101)
BSc, Chemistry - CHMS
Semester: Resit
Level: 300
Year: 2017
REPUBLIC OF CAMEROON REPUBLIQUE DU
CAMEROUN
Peace-Work-Fatherland Paix-Travail-Patrie
THE UNIVERSITY OF BAMENDA UNIVERSITE DE
BAMENDA
P.O. Box 39 Bambili
Faculty of Science Department of Chemistry Lecturer: Shadrach/KWalar
Course Code: MATS3101 Course Title: Multivariable Calculus
Date: 18/09/2017
hall:Asanji time: 10:00-12:30 (2.5 hours)
Instructions: Please answer all questions
INSTRUCTIONS: CAREFULLY DO ALL THE QUESTIONS I - IV, FOLLOWING STRICTLY THE
INSTRUCTIONS FOR EACH. USE SYMBOLS OF THE COURSE AS MUCH AS POSSIBLE
I. Carefully find:
1. The first and second partials of f, if f is given by f(x, y)=Arctan (
) (10 marks)
2.
and
, if the univariate function.y is given by
= Arctan (
(12 marks)
3. The constants Arctan
+ Arctan
, given that the function f, such that f(x,y) = Arctan
+ Arctan
, is a constant function on
a) (R
*+
)
2
b) (R
*-
)
2
c) R
*+
X R
*-
(6 marks)
4. The set of continuity of f, given by f(x,y) = (x + y)sin
sin
for
0 and f(0,0)=0
(7 marks)
5.
(4 marks)
6.
π
(6 marks)
II. In each statement that follows, state briefly, in point form (using math symbols, where
possible),
why the statement is true
A. f is continous at (0, a) and g is continous at (0, 0), where f is given by
f (x, y) =
!"#
for x 0, f (0, a) = a; and g is given by
g (x, y) = (x + y) sin
sin
for
0 and g (0,0) = 0. (4
marks)
B. Both of the functions in A are continous functions. (4
marks)
C. The function f, given by f (x, y) = (
$
, for (x, y) (0, 0) and f (0, 0) = 0, is a continous
function. (5 marks)
D.
www.schoolfaqs.net
1. If the function is given by f (x, y) = (xy
%
), for if (x, y) (0, 0) and f (0, 0)
=0, it is a bounded function on the rectangle
&
' (
)
X
&
' (
)
*
2. It is not every subinterval of a compact interval, in the set of real numbers, that is
compact
3. The range of the function f, given by f (x, y) =,
+
is the compact interval
[0,3].
4. The bivariate function u, given by u= Arctan (2xy/x
2
— y
2
), satisfies the equation ,
+
,
= 0. ( 8 marks)
III. In each case below, just write T, if the statement is true or F, if it is false and then explain
why.
1. If the function f is given by f (x, y) = lnj(x
2
+ xy + y
2
), then xf
x
(x
,
y) + yf
y
(x, y) = 2 for all (x, y)
in Dom f.
2. If u = (x - y)(y - z)(z - x)
f
then U
x
+ U
y
+ U
z
= 0
3. A bivariate function f, that has first partials, is continuous.
4. A univariate differentiable function is continuous.
5. If the function f is given by f (x, y) = ln(x +
then F
x
is continous at (0, 0). (20
marks)
IV. In each case that follows, give a justified example.
1. A discontinuous bivariate function f, which is such that f
2
is continuous
2. A nondifferentiable bivariate function
3. A bivariate function f, which is discontinuous at (0, 0), but has first partials at (0, 0)
4. A subset of the real line that is neither open nor closed
5. A noncompact subset of a compact set, in the Euclidean plane. (20 marks)
WE WISH YOU THE BEST
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