Seminar problems

General notes:

For seminars, we assign exam problems and old exam-type problems. If you need "lighter" problems first, then use the textbook.
We expect you to have made your best shot at the problems before the seminars. (You learn much more from doing problems than from seeing someone else solve them - that makes them look all too easy.)

Review problems (not for seminars):

If you need to review something, have a look at the following:

• This problem set (recycling one assigned by Arne Str?m in 2011). 
• Knut Syds?ter and Arne Str?m gave this review note for Mathematics 3 (yes "3") back in the time when that was compulsory in one of the Master programmes' first semester. Even Mathematics 2 students should know already nearly everything in items 1 through 17 (and then 18 and 19 early in the course); except the possibly mysterious vector notation (boldface) you should know the content of items 25 through 30 and 32, 37 and 38. I also assume you have somehow during your studies picked up item 21.

Problems for seminar #1 (February 2nd/5th):

• Exam spring 2004 problem #1
• Exam spring 2005 problem #1 and  #2 part (a)
• Problems 18 and 117 from the compendium (they are short) 
• Problem 107 from the compendium (somewhat more involved).

Problems for seminar #2 (February 9th/12th):

All from the compendium except noted:

• 80 (Functions, review; but, touching a topic new to Math2)
• Exam spring 2015 problem 3 (a). (Limits.)
• 49 (Integration.)
• 100 (a), (b) (ditto)
• 119. Also think of the following: how close can you get to the answer before you have to actually integrate?
  (Modification: find the derivative of \(\displaystyle \int_4^x \left(g(u) + \frac x{\sqrt u} \right) du\), where g is arbitrary.)
• Why is \(\displaystyle\int_{-\pi}^\pi x^{123456789}e^{-x^{2016}}dx=0\)? 

Problems for seminar #3 (February 16th/19th)

All from the compendium

• 88
• 100 (c)
• 120
• 127
• 134

Remember the teaching-free week. Seminar #4 (March 1st/4th)

You have two weeks on this set, which is why it is long. Unfortunately, we have not gotten far enough in linear algebra to assign so many interesting problems.
Instead, there will be a healthy mix of problems of diverse levels of difficulty - some which could be quite a bit demanding even when they do not need sophisticated tools.

Linear algebra:

• Do these problems from the English OR the Norwegian textbook (as you should know, you do not need both English and Norwegian books):
  English EMEA: 15.7.8, 15.8.4, 15.4.6 
  Norsk: LA: 2.2.4, 2.3.3, 3.3.6 
• You are soon to learn matrix products. The following covers a special case, where one of them is a vector. 
  Let A be a matrix of three columns, and v be the column vector \(\begin{pmatrix}x\\ y\\ z \end{pmatrix}\).
  • Explain why the dot product between v and any row of A, is well-defined.
  • Is it so that the dot product between v and any column of A, is well-defined? Always? Never? Could be, but we need more information?
  • The product Av is a column vector where
    the top element is the dot product between v and the first row of A,
    the middle element is the dot product between v and the second row of A,
    and so forth. 
    Question: If A is such that Av does not depend on x (recall that x is the top element of v), what do we then know about A?

Functions etc.

• 28 (quite involved, start early. Should be covered in the seminar.)
• 32 (easier than 28, I believe - you can do this as a warm-up)
• 43 (likely not to be prioritized)

Integration and differential eq's (and other things yes):

• 71 (make sure you can at least "translate the problem into formulas"; the seminar leader can choose to stop there and not do the calculations)
• 84
• 111 (one differential eq. problem should be covered in class. Likely this one.)

Seminar #5 (March 15th/19th)

The three last bullet points - 46//72/108 - are added because you have an extra week. The rest of problems 72 and 108 will be assigned for the seminar after Easter, and if short on time, Eivind may choose to postpone them entirely until then.

• For numbers, we have (a+b)² = a² + 2ab + b² and (a-b)(a+b) = a² - b².  
  Do the same hold true when a and b are replaced by nxn matrices A and B? (Always? Never? Sometimes?)
• There are four 2x2 diagonal matrices A for which the two main diagonal entries are 1 or -1.  (Hard to remember what this means? Here they are.)
  For each of those, compute the square A²=AA. How does this relate to squares of real numbers?
• Compendium problem 5
• Compendium problems 37 and 59.  They have many of the same features, and 37 is likely the easiest; therefore, do 37 first, but expect the seminar leader to prioritize 59.
• Compendium problem 139. 
• 46
• 72 (a) and (d)
• 108 (a) and (b)

Seminar #6 (March 29th/April 1st)

Priorities, as there are quite a few problems this time.
Problem 58abc is fairly similar to 72, and will likely not be prioritized except if requested; it is given because it there are many such given in exams, so if you have fiddled around a bit with 72, you can next try to see if you can do 58 without struggling that much. 
Part "2" of the 139 modification could take some time. The core of it is whether you after solving Ax = (1,0,0)' by Cram¨¦r's rule, understand the connection asked for in the end. And if you don't understand it, it is hardly anything one can write an exam question on.
Also, the last text problem is a "puzzle" thing that could take you much more time to figure out, than it will take Eivind to explain. So you may want to be light on it.

• The rest of problem 108
• The rest of problem 72.
• Problem 58
  (Problems like parts (a) and (b) of 58 and 72 do, by the way, fairly often show up in exams.)
• Problem 69
• Problem 139, modified as follows: Find the inverse using both the following methods:
  1. Solve the matrix equation A X = I using Gaussian elimination.
  2. Solve the three equation systems Ax = (1,0,0)',  Ay = (0,1,0)' and Az = (0,0,1) using Cram¨¦r's rule. Explain why the inverse then is the matrix with x, y and z as columns.
  • Did you notice any connection between any of these methods, and the "third" method in this course?
• Consider an equation system Ax = b with n equations in n-1 unknowns. Then the augmented coefficient matrix M is square. It is a fact that if this is invertible, then the equation system is inconsistent (i.e., it has no solution). You shall fill in the missing parts of each of the following two arguments for this claim.
  1. Form the n-vector y by "putting -1 at the bottom of x": That is, y_i = x_i for i<n, and the last coordinate is y_n=-1.
     Why is My = Ax - b?
     Since Ax shall equal b, then My must equal 0. What do we know about y?
     Why is this impossible?
  2. What happens if you do Gaussian elimination?

Seminar #7 (April 5th/8th)

• Compendium problem 38
• Compendium problem 67
• Exam autumn 2010 problem 1
• Exam autumn 2010 problem 2
• Exam autumn 2010 problem 3 (looks a bit odd, but that semester there were more problems of that kind)

Seminar #8 (April 12th/15th)

For next week, the autumn 2011 exam set is assigned, though with a few reservations for the Lagrange problem 3, which requires knowledge we have not yet covered - but which most of you should have seen in ECON2200.  So here is the assignment:

• Exam autumn 2011 problems 1, 2 and 4.
• For problem 3 of exam 2011, do as much as you can do, and take a stand on the following:
  • What part(s) of problem 3 can be done using your math background from before this course?  (ECON2200, or whatever course you had from outside this Department.)
  • Is there any question that the lectures up to and including Wednesday April 6th should enable you to answer?
  • Is there any question that the April 11th lecture - and for those on the Friday seminar, the April 13th lecture - should enable you to answer?
• Compendium problem 39
• Compendium problem 63

Seminar #9 (April 19th/22nd)

Do not confuse the weeks - this is posted early because I (Nils) will be away 14th/15th.
For this week, I assign an entire exam problem set; I suggest that you allocate three hours to it, and see how far you get. 
I suggest that you practice Lagrange problems first, though; if you need more than the ones below, they are easily found in the compendium, e.g. 78 and 73.

• Compendium problem 44
• Compendium problem 86
• Compendium problem 138 (a) - two constraints, that could take some time.
• The entire exam autumn 2004.

Seminar #10 (April 26th/29th)

If you want to simulate an exam for yourself by using the sets below, I suggest that you practice a bit on a nonlinear programming problem. Compendium problem 24 could be useful (focus on (a)-(c)). For those who like applications, I once made a somewhat lengthy problem of an information economics model - a lot of work (and the below is already a full day!) and not easy, but for practice, it will even be useful to just formulate the conditions. 

Seminar problems:

• In the lecture, exam autumn 2007 problem 4 was sketched; do this problem in detail. Eivind may choose to skip this.
• The entire spring 2013 exam
• The entire spring 2005 exam

Seminar #11 (May 3rd/6th)

• The entire spring 2011 exam
• The entire spring 2012 exam