For next seminar: full exam sets

For next week I intend to assign two entire fairly recent exam problem sets. I recommend that you simulate an exam situation for each:

• Allocate 3 hours, and do not open the problem sets until that time.
• Arm yourself with notes and books and paper and pencil.
• Start your stopwatch. Open the problem set. Work three hours.
• Only afterwards, see what you would be able to improve with more time: crack the remaining nuts.

I have not completely decided which ones are most appropriate to assign (I have three on mind, one is likely a bit odd at this stage and could serve as a "practice set" I think). Most likely you should drill a little bit into this week's curriculum before starting. I may assign some such problems by tomorrow, but here are a few: 

• Make sure you got exam Autumn 2012 Problem 4 (d) right.
• Let a > 0, b and c be constants, and consider the minimum value V = V(a,b,c) of the quadratic ax² + bx + c. (That is, minimise wrt. x only.)
  Find the partial derivatives of V using the envelope theorem, and verify by calculating V first and differentiating.
• Same as the previous question, except instead for the function
  V(p,q) = min_x {e^qx - px}, where p > 0, q > 0.
  (Again: min. wrt. x only.)
• Let V(p,q) = min_x {e^qx - px + qx²}, where  p > 0, q > 0.
  Find as simple expressions as possible for the first-order partial derivatives.