2008

2007

1) Show that the series $\sum (-1{)}^n[\sqrt{n^2+1}-n]$ Is conditionally convergent.

[10M]

2) Applying Cauchy’s criterion for convergence, show that the sequence $(s_n)$ defined by $s_n=1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}+\dots$ is not convergent.

[10M]

3) Show that ${\iint }_D{\displaystyle \frac{(x-y)}{(x+y{)}^3}}dxdy$ does not exist, where $D=\{(x,y)\in R^2\mid 0\le x\le 1,0\le y\le 1\}$

[10M]

4) If $f:R^2\to R$ such that $f(x,y)=\{\begin{array}{c}{\displaystyle \frac{xy(x^2-y^2)}{x^2+y^2}},(x,y)\ne (0,0)\\ 0,(x,y)=(0,0)\end{array}$
Then show that $f_{xy}\ne f_{yx}$

[10M]

2006

1) Evaluate the double integral ${\iint }_R{x}^{2}dxdy$ where $R$ is the region bounded by the line $y=x$ and tha curve $y=x^2$z

[10M]

2) Show that the function $f(x)$ defined by $f(x)={\displaystyle \frac{1}{x}},x\in [1,\mathrm{\infty })$ is uniformly continuous on $[1,\mathrm{\infty })$

[10M]

3) Show that the series $x^4+{\displaystyle \frac{x^4}{1+x^4}}+{\displaystyle \frac{x^4}{{(1+x^4)}^2}}+\dots$ is not uniformly convergent on [0,1).

[10M]

4) Examine the following function for extrema: $f(x_1,x_2)=x_1^3-6x_1{x}_2+3x_2^2-24x_1+4$.

[10M]

5) Show that (i) $h(x)=\sqrt{x+\sqrt{x}},x\ge 0$ is continuous on $[0,\mathrm{\infty })$ (ii) $h(x)=e^{\sin x}$ is continuous on $R$

[10M]

6) If $f(x,y)=\{\begin{array}{cc}{\displaystyle \frac{x^2-y^2}{x^2+y^2}},& \text{ where },x,y\ne 0\\ 0,& \text{ where }x=y=0\end{array}$ show that at (0,0)${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y}}{\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }v\mathrm{\partial }y}}$

[10M]

2005

1) Evaluate the double integral ${\iint }_R{x}^{2}dxdy$ where $R$ is the region bounded by the line $y=x$ and tha curve $y=x^2$

[10M]

2) Show that the function $f(x)$ defined by $f(x)={\displaystyle \frac{1}{x}},x\in [1,\mathrm{\infty })$ is uniformly continuous on $[1,\mathrm{\infty })$

[10M]

3) Show that (i) $h(x)=\sqrt{x+\sqrt{x}},x\ge 0$ is continuous on $[0,\mathrm{\infty })$ (ii) $h(x)=e^{\sin x}$ is continuous on $R$.

[10M]

4) If $f(x,y)=xy{\displaystyle \frac{x^2-y^2}{x^2+y^2}},$ when $(x,y)\ne (0,0)$ and $f(0,0)=0,$ show that at (0,0) ${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y}}\ne {\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }y\mathrm{\partial }x}}$

[10M]