This problem from TIFR 2013, Problem 19 discusses the example of a non-uniformly continuous function. Question: TIFR 2013 problem 19 True/False? Every differentiable function $f:(0,1) \to [0,1]$ is uniformly continuous. Hint: $\sin(1/x)$ is not uniformly continuous. However, range is not $[0,1]$. But its a simple matter of scaling. Discussion: Let $f(x)=\sin(\frac{1}{x})$ for all $x\in(0,1)$. For […]
Try this problem, useful for Physics Olympiad based on Constructing Parallel Plate Capacitor. The Problem: Constructing Parallel Plate Capacitor Suppose you are to construct a parallel plate capacitor of (1\mu F) by using paper sheets of thickness (0.05mm) as dielectric and a number of circular parallel metal foils connected alternately. If the dielectric constant of […]
Try this problem, useful for Physics Olympiad, based on Capacitance of a Spherical Capacitor. The Problem: A spherical capacitor has inner radius (a) and outer radius (b). It is filled with an inhomogeneous dielectric with permittivity $$ \epsilon=\epsilon_0K/r^2$$ for (a<r<b). The outer sphere is grounded and a charge is placed on the inner sphere. Find […]
Try this problem from TIFR 2013 problem 18 based on limit. Question: TIFR 2013 problem 18 True/False? Let \(P(x)= 1+x+ \frac{x^2}{2!} +... \frac{x^n}{n!} \) where n is a large positive integer. Then \(\lim_{x\to\infty} \frac{e^x}{P(x)} = 1 \) Hint: n really doesn't matter! Discussion: As \(x\) approaches 'infinity', both \(e^x\) and \(P(x)\) tends to 'infinity' (i.e, […]
TIFR 2013 Problem 17 Solution has been written for TIFR entrance preparation series. A problem on convergence of improper integral from real analysis.
TIFR 2013 Problem 16 Solution has been written for TIFR entrance preparation series. A problem on bounded sequence from real analysis.
Try this problem based on convergence from TIFR 2013, Problem 15. Question: TIFR 2013 problem 15 True/False? Let \(x_1\in(0,1)\). For \(n>1\) define \(x_{n+1}=x_n-x_n^{n+1} \) Then \(lim_{n\to \infty} x_n \) exists. Hint: A bounded monotone sequence is convergent. Discussion: By induction, we can prove that the sequence is between 0 and 1 always. Suppose, \(x_n\in(0,1)\). Since […]
TIFR 2013 Problem 14 Solution has been written for TIFR entrance preparation series. A problem on uniform continuity from mathematical analysis.
Try this true/false problem from TIFR 2013 Problem 13 based on composite of Riemann integrable functions. Question: TIFR 2013 problem 13 True/False? Let \(f:[a,b]\to[c,d]\) and \(g:[c,d]\to\mathbb{R}\) be Riemann integrable functions. Then the composite \(gof\) is also Riemann integrable. Hint: Can the Thomae function(which is known to be integrable) be composed with some function to give […]
Try this problem, useful for Physics Olympiad, based on Mass over a Smooth Pulley. The Problem: One end of a string is attached to a rigid wall at point O, passes over a smooth pulley and carries a hanger S of mass M at its other end. Another object P of mass M is suspended […]