Some problems for next week

Tuesday we will continue with limits, due to my blunder wasting a third of Friday.

Thursday or Friday, we will cover at least some of the following problems, subject to time and demand:

1. Through this problem, let f be a convex strictly increasing function; part (a) does not utilize all these properties, but in part (b) we shall specialize to an exponential function. 
   (a) Explain why it is so that (edit: "for all x" was missing here!) if \(f(x)>x\) for all x, then \(f(x)>f^{-1}(x)\) for all x.
   (b) There is precisely one base number a such that the function a^x and its inverse function log_ax are equal at exactly one point. The topic of this question is to find this a:
   - Explain why it suffices to find an y such that the equation \(\mathrm e^{xy}=x\) has precisely one solution.
   - Explain why it suffices then to find an y such that the function \(g(x)=\mathrm e^{xy}-x\) has a minimum value of zero.
   - Solve the equation system
   \(\mathrm e^{xy}=x\)
   \(y\mathrm e^{xy}=1\)
   (Hint: why is \(y=1/x\)?)
   - Find a.
2. Find the derivative of the function \(f(x)=u(x)^{v(x)^{w(x)}}\)
3. Throughout this problem, r and q are related through (1+r)^t = e^qt for all t; the problem is about finding the one-year effective rate from a given continuously compounded interest rate, and vice versa.
   - Suppose the one-year interest rate is r = 0.02.  What is the equivalent continuously compounded rate q?
   - Suppose that q = 0.2.  What is the effective annual interest rate r?
    
4. In this problem, you shall show that more frequent accumulation of interest, leads to more rapid growth: that g(n) = (1+r/n)ⁿ increases wrt. n, when r>0.  I said it was not easy, so you may need the following hints:
   - Put z = 1+r/n, and notice that (since r>0) z decreases when n increases.  Then g(n)=z^r/(z-1)
   - Let h(z)=z^1/(z-1) . Why does it suffice to show that ln h is decreasing for z>1?
   - Show that ln h is decreasing as long as u(z)=1-1/z - ln z <0. 
   -  u(1)=0. What is the derivative of u? 
   - Complete the proof :-)
    
5. Find the limit \(\displaystyle\lim_{x\to0}\frac{(2x-1)x^2}{e^x-1-x-x^2}\)