North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH-211 (Calculus of Several Variables) - Question 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 ques. ( two marks each)
1) Write the simplified mathematical Taylor’s expansion for f(x,y)
2) Write the simplified mathematical Maclaurin’s expansion for f(x,y).
3) describe absolute maximum of the function f(x,y) at (a,b)
4) describe absolute minimum of the function f(x,y) at (a,b)
9
5) State the necessary condition for the extreme value.
6) Write the condition for critical point (a,b) to become a function f(x,y) maximum.
7)Write the condition for critical point (a,b) to become function f(x,y) minimum.
8) Under what condition the critical point (a,b) will be saddle point ?
9) prove that, ( )
2!
1 ( )
ex y x y x + y2
+ = + + + + - - - - - - - - -
10)State the working rule to determine extreme value of the function f(x,y)
11) prove that
sin (x+y) = (x+y) - -
+
3!
(x y)3 ------------.
12) prove that cos (x + y) = one - ( ) -
+
+
+
!
x y
!
( x y )
2 3
2 3
----------------
13) obtain the Critical point for f(x,y) = xy +
x y
50 + 20
14) obtain the Stationary points for f(x,y) = x3 y2 (1- x - y)
15) obtain the Stationary points for f(x,y) = x3 + y3 - 3axy
16) obtain the minimum value of f(x,y) = 1+ x2 + y2
II ) Multiple option ques. (1 mark every )
select the accurate choice from the provided 4 choices :
1) If f(x,y) = x2 - y2 + four then f has extreme value at - - - - -
a) (1,1) b) (0,0)
b) (1,0) c) None of these.
2) To expand x2 + 2xy + y3 in powers of x -4 and y+2 by Taylor’s theorem the values of h
& k are - - - - - .respectively.
a) four , two b) -4 , 2
c) four , -2 d) None of these.
3) If (a,b) is the Stationary point of the function f(x,y) and
f ( a,b )>0, f ( a,b ) f ( a,b ) -[f ( a,b )]2 <0 xx xx yy xy then f(a,b) is,
a) Minimum b) Saddle point
c) Maximum d) None of these
4) If (a,b) is the critical point of the function f(x,y) and f (a,b)>0, f (a,b) f (a,b) - [f (a,b)]2 > 0 xx xx yy xy
then f(a,b) is,
a) Maximum b) Saddle point
c) Minimum d) None of these
III) Theory and Examples.(4 marks every )
1) State and prove Taylor’s theorem for f(x,y)
2) State Taylor’s theorem and hence find Macluarin’s expansion in simplified form
3) State and prove necessary condition for extreme values of the function f (x,y)

Earning:   Approval pending.