IFoS PYQs 5
2004
1) Show that the sequence {fn}, where fn(x)=nxe−nx2 is pointwise, but not uniformly convergent in [0,∘).
| 2) Evaluate ∫1−1f(x)dx, where $$f(x)= | x | ,$$ by Riemann integration. |
3) Show that f(x,y)=x4+x2y+y2 has a minimum at (0,0).
2003
1) Evaluate I=∬ over the positive quadrant of the circle x^{2}+y^{2}=a^{2}.
| 2) Investigate the continuity of the function $$f(x)=\dfrac{ | x | }{x}forx \neq 0andf(0)=-1$$. |
| 3) Let $$f(x)= | x | , x \in[0,3]where[x]denotes the greatest integer not greater thanx,Prove thatfis Riemann integrable on [0,3] and evaluate\int_{0}^{3} f(x) d x$$. |
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.