Problems for last seminar

Again, exam problem sets plus some smaller bits.

• Elasticities: Problem 23 from the compendium, try to do it by manipulating logarithmic differentials without writing ? y' ?.
  Extra problem for drilling functions: for certain positive values of the constant, the y of Problem 23 has precisely one stationary point; find a necessary and sufficient condition for this.
• Approximations: It is sometimes convenient to calculate with the approximating polynomials (or indeed the full infinite power series!) e.g. as follows: suppose the kth order approximation of f(x), g(x) resp. h(x) around zero, are the respective kth order polynomials p, q and r.
  (a) Explain why the function F = f + g + h has kth order approximation equal to p + q + r. 
  (b) It is also a fact that the function G(x) = f(x) g(x) h(x) has kth order approximation equal to the kth order approximation of the product p(x) q(x) r(x); that is, from this product one simply deletes all terms of higher order exponent.
  Use this to find the quadratic variation around zero for G when f(x) = ln(1+x) and g(x) = ln(1+x²) and h(x) = ln(1+2e^x) (recall last week's problems).

Exam problem sets: Spring 2010 and Autumn 2011. The latter will be given priority.