ECON4140 - Spring 2017

Seminar problems

This is a live document to be updated weekly. 

If you wish to start doing problems before I have announced them, then you can have a look at the 2016 problems, from which I will recycle a bit. Most of them are also listed below, but not in order; before deciding on the problem set, I will however need to know that the lectures have progressed sufficiently far.

Most problems will be taken from old exam problems or the compendium of exam-type problems, all of which are - hopefully - to be found in the "Problems for seminars etc." heading in the left margin on the course site (or just click the link from here). 

In 2016, the following compendium and old exam problems were assigned (in addition to a few problems typed).

Compendium problems:

  • (1-01,) 1-03, 1-04, 1-05, 1-06, 1-07, 1-08, 1-12(a), 1-13, 1-14
  • 2-01, 2-02, 2-03, 2-06, 2-07
  • 3-03, 3-05, 3-07, 3-12, 3-13, 3-14, 3-21
  • 4-01, 4-02, 4-05, 4-08, 4-09, 4-10
  • 5-06, 5-14
  • 6-10, 6-12, 6-15 (and generally, start from the beginning of section 6 to drill 2nd order diff.eq's)
  • 7-01, 7-02, 7-04, 7-05
  • 8-01, 8-06
  • 9-01, 9-03, 9-02, 9-05, 9-07, 9-12, 9-17
  • 10-03, 10-04, 10-05
  • 11-01, 11-04, 11-07

Exam problems

  • Exam 2008 problem 1
  • Exam 2011 problems 1d and 3
  • Exam 2014

 

The upcoming seminar: #1, January 31st:

First, the "pre-seminar" problems given the first week - you seemed not to want more than these, but I'll throw in a couple at the end. 

  • Let f(x) = g(||x||). What is the gradient of f, in matrix form?
  • General differentiation/integration drill for trig. functions: 
    • Differentiate f(t) = tan t = sin t / cos t, and write the answer in two ways: one which has only "cos" (no sin nor tan) and one which has only "tan" (no sin nor cos).
    • From the previous point, you should be able to show that \(\frac{\mathrm d}{\mathrm d t}\arctan t = \frac1{1+t^2}\qquad. \ \) (The last vertical bar is a bug in the publishing system.)
    • Here is a one which you should not spend too much time on, but which shows the method of substitution in a way that might be a bit less familiar: 
      Find \(\displaystyle \int(1-x^2)^{-1/2}\:d x.\qquad\)Hint: The substitution x = sin t (yes, it is x = sin t, not u = sin x). 
  • The Leibniz rule, all from the compendium:
    • 4-01
    • 4-02
  • Double integrals, from the compendium:
    • 4-05
    • 4-08. 

Then the following (including "optionals" from the previous list):

  • 4-04
  • 4-10
  • Let A be the domain bounded by the lines y=0, x=1 and y=x. Phrase and compute the double integral (over A) of \(\displaystyle x e^y\quad \)in "both the dx dy and the dy dx order".
  • Let  p>1>q>0 and let f be a nonnegative function defined on the bounded domain between the functions \(y=x^p\quad\text{and}\quad y=x^q\quad.\)
    Formulate in terms of iterated integrals - both the dx dy order and the dy dx order - the volume under the graph of f. 

 

 

The non-seminar "Aktualitetsuka":

A few proofs. The point is not the proofs themselves, but getting the manipulation of functions of several variables at your fingertips. 

  • Let f be a function of n variables defined as f(x) = g(Ax) where g is a given twice continuously differentiable function of m variables with Hessian H = H(y), and A is a given (constant) mxn matrix.
    Express the Hessian of f in matrix form by means of H and A. (This could be a tedious problem, and you may try two variables to see what happens.)
  • Try to show without using the quasiconcavity/homogeneity argument, that \(\|\mathbf x\| = \sqrt{\sum_ix_i^2}\qquad\)is a convex function. Approach this way:
    • Because both sides are nonnegative, then the inequality \(\|\lambda \mathbf u+(1-\lambda) \mathbf v \| \leq \lambda \|\mathbf u\|+(1-\lambda)\|\mathbf v \|\qquad \)is equivalent to (by squaring both sides)
      \(\|\lambda \mathbf u+(1-\lambda) \mathbf v \|^2 \leq \left(\lambda \|\mathbf u\|+(1-\lambda)\|\mathbf v \|\right )^2\qquad\).
    • Write this out in terms of dot products (using \(\|\mathbf x\|^2=\mathbf x\cdot\mathbf x\qquad)\), and see that you will be done when you can show that \(\quad |\ \mathbf u\cdot \mathbf v\ | \leq \|\mathbf u\|\ \ \|\mathbf v\|\)
    • To show this, let \(\mathbf w = \|\mathbf u\|\ \mathbf v\pm\|\mathbf v\|\ \mathbf u\qquad\)and calculate \(\mathbf w\cdot\mathbf w,\qquad\)which is nonnegative.
  • Show that the closed unit ball \(\qquad\{\mathbf x\in\mathbb R^n;\ \|\mathbf x\|\leq 1\}\qquad\)is a a convex set. 
    • What about the open unit ball \(\qquad \{\mathbf x\in\mathbb R^n;\ \|\mathbf x\|< 1\}\qquad\text{?}\)
  • Suppose g is a concave function. Fix two points u and v. Show that f(t) = g(u+tv), is a concave function (of a single variable t).
  • Do compendium problems 2-01, 2-02, 2-03 and 2-07. In particular: 
    • How much of these can you do without differentiating?
    • In 2-01 (b) and (d): write down the associated matrix, and work with that directly.
    • (Do the rest as well.)
  • Do exam 2011 problem 3 (a).

 

 

Seminar #2, February 14th:

The previous week's problems: if you want any of those covered, then let me know in the Monday lecture, and I will forward to Eric.

  • Exam 2011 problem 1(d)
  • Exam 2008 problem 1(c) (can be done without anything from (a) and (b)).
  • Compendium problem 1-13
  • Prove from the definition of positive semidefiniteness that the odd powers \(\mathbf A^{2p+1}\) (where p is a natural number) of a positive semidefinite matrix A is positive semidefinite. Hint:  Putting y=Ax should resolve case p=1 and give a hint for higher p. 
    Also, answer the following:
    • If A is also positive definite, must then the odd powers also be? What about the odd negative powers?
    • If A is positive semidefinite but not positive definite: could then any odd positive power be positive definite? 
    • What about odd powers of negative definite matrices?
    • What about even powers?

If that is not enough, do Exam 2011 problem 3(b) - we have not gotten as far as to Kuhn-Tucker yet, but this is something you should be able to do anyway. You should even be able to do the first question of 3(c), but the last part takes one of the nitpickeries that you do not see in Mathematics 2.

 

 

Seminar #3, February 28th:

This is a mix of the ones assigned for this seminar-free week (that you hadn't touched), and some concerning linear independence and rank.

  • Compendium problems 1-03 through 1-07, and 1-12.
  • Also, have a look at 1-11, although it should not be given priority for this seminar. 

Expect that at least some of the other that I posted in messages, will be assigned for later. That goes for 1-11 as well. And probably 1-14.

 

 

Seminar #4, March 07th:

 

 

Seminar #5, March 14th:

We only got to scratch the surface of second-order differential equations, but you should be able to do 6-08 and (by the hints given) 6-10 - and if you managed 6-10, you can do 6-13 as extra. Also, the information problem was left over and you can take up that should the following not be enough.

  • Exam 2015 problems 1 and 2.
  • Compendium 5-14
  • 6-08 
  • 6-10 (for (b): guess a not-too-complicated function of x.  Easier than exponentials ... )

 

 

Seminar #6, March 21st:

Many, but some are quickly done. If this is not enough, then exam 2015 problem 3 will be assigned later.

 

 

Seminar #7, March 28th:

Is this maybe too much?  Maybe drawing phase diagrams will take too much time?

  • 7-02
  • 7-04
  • 7-05
  • Classify the origin as equilibrium point for the system x' = 1 - exp(x-y),  y' = -y.
    (Classify only locally.  The example is from the book, found under Olech's theorem for global asymptotic stability.  That test is not hard in this case (try it!) but it is not curriculum - therefore, the seminar problem is to classify locally.)
  • 8-01
  • 8-06

     

     

    Seminar #8, April 4th:

    Some optimal control theory, and then a full exam problem set. You may want to allocate three full hours and see how much you can do in that timeframe / how long time it takes to do it all. 

    • 9-03 (if you need an even easier one, look at 9-01 first)
    • 9-05
    • Exam 2011 problems 4 
    • Exam 2015 problems 3 and 4 (one detail requires Monday's lecture)
    • The entire 2009 exam. 

     

     

    Seminar #9, April 18th:

    I know it is Easter, but still ... another full exam problem set. The induction problems do not take that much time.

    • It is a fact that a product of two odd integers, is an odd integer. Formulate as proof by induction the fact that the product of any (natural) number of odd integers, is an odd integer.
    • Show by induction that 1 + 2 + ... + n = (n+1)n/2.
    • Show by induction that for every natural number n, the number \(9^n-1\)is divisible by 8. (That is, (9n-1)/8 is an integer.)
      Hint: Write \(9^{n+1}=9^n\cdot(8+1)\)
    • 9-12; also: can we know that this is indeed the solution?
    • 9-17
    • The entire 2014 exam

     

     

    Seminar #10, April 25th:

    • 10-03
    • 10-04
    • 10-05
    • Look at the difference equation given in problem 3 (a) in this exam in ECON5150. Here, the unknown function is v, while  p∈(0,1) and K are constants. (Do not be confused by the running index being called i rather than t - it is not supposed to be time in that application. And it has nothing to do with the imaginary unit.)
      • Is there any p∈(0,1) such that the equation is (update:) globally asymptotically table? (Updated: do not worry so much over the stable-but-not-asymptotically-stable case. Also, this question requires Monday's lecture.)
      • Find the general solution. Update: a constant solution will be a particular solution to the homogeneous, so for a particular solution to the inhomogeneous, try Li except when p=1/2; then try a quadratic Qi2
      • Find the particular solution which satisfies \(v_0=v_N=0\).
        (The Department has discontinued Mathematics 4. If you think this application looks interesting, then try STK2130 - be aware though, that for the time being 2*** courses may not be useful towards your degree even though the content is at a level we used to assign to PhD candidates.)
    • The entire 2012 exam.

     

     

    The final seminar #11, May 2nd

    If you need one more dynamic programming problem: Exam 2008 problem 2.

    Published Jan. 26, 2017 12:32 AM - Last modified Feb. 7, 2020 4:19 PM
    E-mail this page