2004

1) Show that the sequence $\left\{f_n\right\},$ where $f_n(x)=nx{e}^{-n{x}^2}$ is pointwise, but not uniformly convergent in $\left[0{,}^{\circ }\right)$.

3) Show that $f(x,y)=x^4+x^{2}y+y^2$ has a minimum at (0,0).

2003

1) Evaluate $I=\iint {(a^2-x^2-y^2)}^{1/2}dxdy$ over the positive quadrant of the circle $x^2+y^2=a^2$.

4) Let $(a,b)$ be any open interval, $f$ a function defined and differentiable on $(a,b)$ such that its derivative is bounded on $(a,b).$ Show that $f$ is uniformly continuous on $(a,b)$

5) If $f$ is a continuous function on $[a,b]$ and if ${\int }_0^b{f}^2(x)dx=0$ then show that $f(x)=0$ for all $x$ in $[a,b].$ Is this true if $f$ is not continuous?

6) Find the maximum and minimum distances of the point $(3,4,12)$ from the sphere $x^2+y^2+z^2=1$.

2002

1) Evaluate $\iiint \sqrt{1-z}dxdydz$ through the volume bounded by the surfaces $x=0,y=0,z=0$ and $x+y+z=1$

2) Define a compact set. Prove that the range of a continuous function defined on a compact set is compact.

3) A function $f$ is defined in [0,1] as $f(x)=(-1{)}^{r-1}$ ; ${\displaystyle \frac{1}{r+1}}<x<{\displaystyle \frac{1}{r}},$ where $r$ is a positive integer show that $f(x)$ is Riemann integrable in [0,1]$\mathrm{\&}$ find its Riemann integral.

4) A function $f$ is defined as $f(x,y)={\displaystyle \frac{x^2{y}^2}{x^2+y^2}},(x,y)\uparrow (0,0)=0,$ otherwise. Prove that $f_{xy}(0,0)=f_{yx}(0,0)$ but neither $f_{yx}$ nor $f_{xy}$ is continuous at $(0,0)$.

2001

1) Change the order of integration in the integral ${\int }_0^{4a}{\int }_{\frac{s^2}{4a}}^{2\sqrt{ax}}dydx$ and evaluate it.

2) If $a_n=\log (1+{\displaystyle \frac{1}{n^2}})+\log (1+{\displaystyle \frac{2}{n^2}})+\dots .+(1+{\displaystyle \frac{n}{n^2}})$ find lim $\underset{x\to \mathrm{\infty }}{lim}{a}_n$.

3) State the weierstrass M-test for uniform convergence of an infinite series of functions. Prove that the series $\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{x}{n(1+n{x}^2)}}$ with $\alpha <0$ is uniformly convergent on $(-\mathrm{\infty },\mathrm{\infty })$.

2000

1) Change the order of integration in ${\int }_0^{2a}{\int }_{\frac{s^2}{4a}}^{3ax}\varphi (x,y)dxdy$

2) Test the convergence of the integral ${\int }_0^1{\displaystyle \frac{\sin {\displaystyle \frac{1}{x}}}{\sqrt{x}}}dx$

3) Test for uniform convergence the series $\sum _{n=1}^{\mathrm{\infty }}{2}^n{\displaystyle \frac{x(2^n-1)}{1+x^{2n}}}$

4) Prove that the function $f(x,y)=x^2-2xy+y^2-x^2-y^3+x^5$ has neither a maximum nor a minimum at the origin.

5) Evaluate ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\log }_e(x^2+1)}{x^2+1}}dx$ by using method of residues.