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

4) State and prove the sufficient condition for the maximum value of the function f (x,y)
5) State and prove the sufficient condition for the minimum value of the function f (x,y)
6) discuss the Lagrange’s method of undetermined multipliers to obtain extreme values of the
function f (x,y,z).
7) Prove that e by by abxy e[(a x ab xy ) u ( a bx y b y ) v]
u
ax three sin three cos
6
sin = + + three 3 - two 2 + two 2 - three 3
10
Where u = a?x,v = b?y
8) Expand sinxy in powers of (x-1) and ( y -
2
?
) upto and including 2nd degree term.
9) Prove that ex+ y = + x + y + x + y two + (x + y)3 e?x+?y
6
( ) 1
2
1 ( ) 1
10) Show that for 0 sinx siny = xy - [(x 3xy )cos?x sin?y (y 3xy )sin?x cos?y]
6
1 three + two + three + 2
11) Expand x three + y3 + xy2 in powers of (x-1) and (y-2)
12) Expand x two y as polynomial in (x-1) and (y+2) by using Taylor’s theorem
13) Expand the function f(x,y) = x2 + xy - y2by Taylor’s theorem in powers of (x-1) and (y+2) .
14) Expand x2 y + 3y -2 in powers of (x-1) and (y-2)
15) Expand e2x cos y as a Taylor’s series about (0,0) up to 1st 3 terms.
16) Write down the Taylor’s expansion of eax cosby about (0,0) up to and including terms of the
second degree.
17) Expand eax log(1+ y ) in powers of x and y up to terms of 3rd degree.
18) Expand ex sin y in powers of x and y as for as terms of 3rd degree.
19) obtain extreme values of f ( x, y ) = x3 y2 (12 - 3x - 4y )
20) obtain extreme values of f(x,y) =
y
a
x
xy a
3 3
+ +
21) obtain the Stationary points and determine the nature of the function
f(x,y) = x3 + y3 - 3x -12y + 20
22)Find the lowest value of the function, f(x,y) = xy +
x y
50 + 50
23) Investigate the maximum and minimum values of
f(x,y) = 3x2 y - 3x2 - 3y2 + y3 + 2
24) explain the maximum and minimum of the function u =
x y
x2 + y2 + two + 2
25) obtain the rectangle of perimeter 12cm which has maximum area.
26) obtain the points on the surface z two = xy +1nearest to the origin.
27) Determine the minimum distance from origin to the plane
3x + 2y + z -12 = 0
28) obtain the minimum and maximum distance from the origin to the curve

Earning:   Approval pending.