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

1) If a function f(x,y) is differentiable at a point (a,b) of its domain then show that
i) f(x,y) is continuous at (a,b)
ii) x f and y f exist at (a,b)
2) State and prove sufficient condition for differentiability of the function f(x,y).
3) State and prove Schwarz’s theorem for equality of xy f and yx f at a point (a,b)
4) State and prove Young’s theorem for equality of xy f and yx f at a point (a,b)
5) Evaluate
Lim two 2
2
x y
x ( x y )
+
+
(x,y) ? (0,0)
6) Show that the limit
Lim
x y
tan( x y )
+
2 +
does not exist.
(x,y) ? (0,0)
7) Evaluate
Lim two four 2
2 4
( x y )
x y
+
(x,y) ? (0,0)
8) explain the continuity of the function f described by
f(x,y) =
x y
x y
-
3 + 3
when x ? y and
= 0 for x = y
at (0,0)
9) Show that the function f(x,y)is continuous at (0,0)
where f(x,y) =
( x y )
xy( x y )
2 2
2 2
+
-
for (x,y) ? (0,0)
= 0 for (x,y)=(0,0)
3
10) Examine the continuity of the function f(x,y) at (0,0)
Where
f(x,y) = two 2
2 2
x y
xy sin x y
+
+
for (x,y) ? (0,0)
= 0 for x =y
at (0,0)
11) Show that the function f(x,y) is continuous at (0,0)
Where
f(x,y) =
y
y + x sin one when y ? 0
x may or may not be 0.
= 0 when y = 0
12) explain the continuity of the function f(x,y) at (0,0)
Where
f(x,y) = two 2 2
2 2
x y ( x y )
x y
+ -
for (x,y) ? (0,0)
= 0 for (x,y) =(0,0).
13) If u = log ( tanx +tany+ tanz)
Prove that
Sin2x
x
u
?
?
+ sin2y
y
u
?
?
+sin2z
z
u
?
?
= 2
14) If u =
y
y tan x
x
x2 tan-1 y - -1
obtain xy u (x,y)
15) If u =
2 two 2
1
x + y + z
, show that ?2u = 0 where ?2 = 2
2
2
2
2
2
x y ?z
?
+
?
?
+
?
?
16) Let f be a function described by
f(x,y) = two 2
2 2
x y
x y
+
for (x,y) ? (0,0)
and f(0,0) = 0
show that xy f (x,y) and yx f (x,y) are equal for (x,y) ? (0,0)
and also show that they are not continuous at (0,0).
17)For the example 16 prove that xy f (0,0)= yx f (0,0)
18) If f(x,y)= two 2
2 two 2 2
x y
x y ( x y )
+
-
for x2 + y2 ? 0 and f(0,0)= 0
4
Show thtat f ( , ) f ( , ) xy yx 0 0 = 0 0
19) Let a function f be difined as,
f(x,y) =
y
y tan x
x
x2 tan-1 y - two -1 when xy ? 0
and f(x,y) =0 when xy = 0
show that , f ( , ) xy 0 0 =1 and f ( , ) yx 0 0 = -1
20)Let f be function described by ,
f(x,y) =
x2 y2
xy
+
if (x,y) ? ( 0,0 )
f(0,0) = 0
show that f(x,y) is not differentiable at (0,0).
21) Show that the function f described by,
f(x,y) = two 2
4 4
x y
x y
+
+
when x2 + y2 ? 0
= 0 when x2 + y2 = 0
is differentiable at (0,0).
22) Show that the function f(x,y) = x + y ? x,y?R
Is continuous at the origin but not differentiable there.
23) explain the differentiability of a function f , where
f(x,y) = x2 y2
xy
+
when (x,y) ? (0,0)
f(0,0) = 0 at the origin.
24) Let a function f be described by ,
f(x,y) = two 2
2 2
x y
xy( x y )
+
-
when (x,y) ? (0,0)
= 0 when x= 0 =y
Show that , f ( , ) f ( , ) x y 0 0 = 0 = 0 0
Also show that ,
f ( x, y ) x = two 2 2
4 four two three 5
( x y )
x y x y y
+
+ -
for (x,y) ? (0,0)
25) Prove that the function described in example 24 is differentiable at (0,0).
26) Using differentials obtain an approximate value of ( 2.01) ( 3.02 )2
27) Estimate f(1.02,1.97), where ,f(x,y)= x2 + y2
28) obtain approximate value of ( 3.9 )2 ( 2.05 ) + ( 2.05 )3
29) obtain approximately the value of ( 5.12 )2 ( 6.85 ) - 3( 6.85 )
30) obtain approximate value of, ( 3.012 )2 + ( 3.977 )2

Earning:   Approval pending.