2008

2007

1) Show that the series ∑(−1)n[√n2+1−n] Is conditionally convergent.

[10M]

2) Applying Cauchy’s criterion for convergence, show that the sequence (sn) defined by sn=1+12+13+… is not convergent.

[10M]

3) Show that ∬D(x−y)(x+y)3dxdy does not exist, where D={(x,y)∈R2∣0≤x≤1,0≤y≤1}

[10M]

4) If f:R2→R such that f(x,y)={xy(x2−y2)x2+y2,(x,y)≠(0,0)0,(x,y)=(0,0)
Then show that fxy≠fyx

[10M]

2006

1) Evaluate the double integral ∬Rx2dxdy where R is the region bounded by the line y=x and tha curve y=x2z

[10M]

2) Show that the function f(x) defined by f(x)=1x,x∈[1,∞) is uniformly continuous on [1,∞)

[10M]

3) Show that the series x4+x41+x4+x4(1+x4)2+… is not uniformly convergent on [0,1).

[10M]

4) Examine the following function for extrema: f(x1,x2)=x31−6x1x2+3x22−24x1+4.

[10M]

5) Show that (i) h(x)=√x+√x,x≥0 is continuous on [0,∞) (ii) h(x)=esinx is continuous on R

[10M]

6) If f(x,y)={x2−y2x2+y2, where ,x,y≠00, where x=y=0 show that at (0,0)∂2f∂x∂y∂2f∂v∂y

[10M]

2005

1) Evaluate the double integral ∬Rx2dxdy where R is the region bounded by the line y=x and tha curve y=x2

[10M]

2) Show that the function f(x) defined by f(x)=1x,x∈[1,∞) is uniformly continuous on [1,∞)

[10M]

3) Show that (i) h(x)=√x+√x,x≥0 is continuous on [0,∞) (ii) h(x)=esinx is continuous on R.

[10M]

4) If f(x,y)=xyx2−y2x2+y2, when (x,y)≠(0,0) and f(0,0)=0, show that at (0,0) ∂2f∂x∂y≠∂2f∂y∂x

[10M]