2004

1) Show that the sequence \(\left\{f_{n}\right\},\) where \(f_{n}(x)=n x e^{-n x^{2}}\) 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 \(I=\iint\left(a^{2}-x^{2}-y^{2}\right)^{1/2} d x d y\) 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 \(\int_{0}^{b} f^{2}(x)d x=0\) 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 \(\iiint \sqrt{1-z} d x d y d z\) 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 \(f(x)=(-1)^{r-1}\) ; \(\dfrac{1}{r+1} < x<\dfrac{1}{r},\) where \(r\) is a positive integer show that \(f(x)\) is Riemann integrable in [0,1]\(\&\) find its Riemann integral.

4) A function \(f\) is defined as \(f(x, y)=\dfrac{x^{2} y^{2}}{x^{2}+y^{2}},(x, y) \uparrow(0,0)=0,\) 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 \(\int_{0}^{4 a} \int_{\frac{s^2}{4a}}^{2 \sqrt{a x}} d y d x\) and evaluate it.

2) If \(a_{n}=\log \left(1+\dfrac{1}{n^{2}}\right)+\log \left(1+\dfrac{2}{n^{2}}\right)+\ldots .+\left(1+\dfrac{n}{n^{2}}\right)\) find lim \(\lim _{x \rightarrow\infty} a_{n}\).

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

2000

1) Change the order of integration in \(\int_{0}^{2 a} \int_{\frac{s^2}{4a}}^{3 a x} \phi(x, y) d x d y\)

2) Test the convergence of the integral \(\int_{0}^{1} \dfrac{\sin \dfrac{1}{x}}{\sqrt{x}} d x\)

3) Test for uniform convergence the series \(\sum_{n=1}^{\infty} 2^{n} \dfrac{x\left(2^{n}-1\right)}{1+x^{2 n}}\)

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

5) Evaluate \(\int_{0}^{\infty} \dfrac{\log _{e}\left(x^{2}+1\right)}{x^{2}+1} d x\) by using method of residues.