North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH-211 (Calculus of Several Variables) - university paper

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NORTH MAHARASHTRA UNIVERSITY, JALGAON

(New Syllabus w.e.f. June 2008)
Class- S. Y.B.Sc.
Subject : Mathematics
Paper MTH-211
(Calculus of Several Variables)

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I) Objective kind ques. (2 Marks each)
1) describe neighbourhood of a point in a plane .
2) describe simultaneous limit of a function f(x,y) as (x,y) ? (a,b)
3) describe Continuity of a function f(x,y) at a point (a,b)
4) obtain 2 repeated limits of f(x,y) = two 2
2 2
x y
x y
+
-
, (x,y) ? (0.0).
As x ? 0, y ?0
5) Evaluate the limit,
( x,y ) ( , )
lim
? 0 0 two 6
3
x y
xy
+
if exists
6) describe x f (a,b) and y f (a,b)
7) describe xx f (a,b) and yy f (a,b)
8) describe xy f (a,b) and yx f (a,b)
9) If u= tan-1
x
y , obtain
x
u
?
?
and
y
u
?
?
10) obtain x f (x,y) and y f (x,y) if f(x,y)= ex sinxy
11) If u(x,y) = y2
x – x2
y obtain ux(1,1) and uy(1,1).
12) describe differentiability of a function f(x,y) at a point (a,b) of its domain.
13) State the necessary condition for differentiability of a function f(x,y) at a point (a,b).
14) State sufficient condition for differentiability of a function f(x,y) at a point (a,b)
15) State Young’s theorem for the equality of xy f and yx f
16) State Schwarz’s theorem for the equality of xy f and yx f
II) Multipal option ques. (1 Marks each)
select the accurate choice .
1) If a function f(x,y) is discontinuous at a point (a,b) then
a)
( x,y ) ( a,b )
lim
?
f(x,y) exist and equal to f(a,b)
b) f(x,y) is not differentiable at (a,b)
c) f(x,y) is differentiable at (a,b)
d) None of these.
2) If the simultaneous limit
exists and has the identical value along any 3 various paths then
a)
( x,y ) ( a,b )
lim
?
f(x,y) exists.
b)
( x,y ) ( a,b )
lim
?
f(x,y) may or may not exist
c)
( x,y ) ( a,b )
lim
?
f(x,y) does not exist
2
d) None of these.
3) If a function f(x,y) is differentiable at a point (a,b) then,
a) x f (a,b)& y f (a,b) may or maynot exist.
b) x f (a,b) & y f (a,b) both exist .
c) only 1 of x f (a,b) and y f (a,b) exists.
d) x f (a,b) and y f (a,b) both does not exist.
4) If f(x,y) = x3 + y3 - 2x2 y2 then
f(x,x) (1,1) = --------
a) one b) -1
c) 0 d) None of these.
III) Theory and Examples (4- Marks each)

Earning:   Approval pending.