Multivariable Calculus (MATS3101)
BSc, Chemistry - CHMS
Semester: Resit
Level: 300
Year: 2016
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: kwalar
Course code: MATS3101 Course title: Multivariable Calculus
Date: 20/09/2016 hall:PBA08 time: 15:00-17:00 (2 hours)
Instructions: Please answer all questions.
Instructions: Follow the instructions for Questions I-III. Use mathematical symbols as much as possible
and avoid the use of too many words
I. Define any five of the following concepts and in each case, give a justified example. (30 marks)
1.A real multivariable function
2.A bounded bivariate real function
3.A partial derivative
4.A continuous bivariable function
5.A global extremum of a trivariate real function
6.A critical value of a bivariate real function
7.An open set in R
2
II. In your answer booklet, write down just the most appropriate word(s), phrase or symbol(s) that fit the
blank in each case (24 marks)
1.New closed sets are formed from given closed sets by taking or
2. In R
3
the boundary of the open ball B((x
0
,y
0
,Z
0
); r) is given by the equation .
3. A chemical equation containing carbon, hydrogen and oxygen atoms can be verified with the help of
equations in __ space.
4. By inspection, write, down the global maximum and the point where it occurs for the function given by
f(x,y)=7-( − 2)
– ( + 3)
. _____and____
5. Without using the sigma notation, the general form of a bivariate polynomial F, of degree 3 is given by
F(x,y)=______, where ___ are ____ with the condition
6. The centre and radius of the circle x
2
+y
2
-6x+4y+12=0 are and
7. Point set topology is the study of such sets as , and
8. A trivariate function has ________ second order partials
9. The theorem on mixed second partials is called ___ theorem
10. If f is a bivariate function, then f
xy
and f
yx
stand for ____and ____respectively, in other symbols.
11. To move in a direction in which the bivariate functional decreases the most, one should proceed in the
direction of ___, in symbols
III. Find, in simplified form (where applicable), any five of the following: (46 marks)
1. The domain of definition of the following function, f, given by f(x,y)= √[(x
2
+y
2
-l)(9-x
2
-y
2
)] and a sketch
of its domain (7mks)
2. limf in each of
(x,y)→ ( x
0
, y
0
)
a)
f(x,y)=xy(x
2
-y
2
)/(x
3
-y
3
), (x
0
,y
0
)=(l,l); (2mks)
b)
f(x,y)=x
2
y/(x
2
+y
2
), (x
0
,y
o
)=(0,0); (2mks)
c)
f(x,y)=(y
4
-81x
4
)/(y-3x), (x
0
,y
0
)=(3,I); (2mks)
d) f(x,y)=x
2
y
2
/[√[l+xysin(xy) ]- √ cos(xy)]. (2mks)
3. 2
nd
partial derivatives of f, given by f(x,y)=Arctan(x/y)+Arctan(y/x)+sin(x
2
+y
2
) (10mks)
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4
a) The minimum value of f, given by f(x,y)=3x
?
+y
2
-4x-6y+2, when y=l (3mks)
b) All local extrema and saddle points, if any, of f, given by f(x,y)=x
3
+y
3
-6xy (9mks)
c) Both first partials, at the point (1 ,ln2), of f, given by f(x,y)=x
exp(x
y)
(4mks)
5. Set of discontinuity of the function f, given by f(x,y)=ln(1-x
2
-y
2
) (3mks)
END
WE WISH YOU THE BEST!!
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