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.)

List of compendium & exam problems already assigned 

For completeness, the following lists compendium problems and exam problems that have been assigned this far in the semester. (Other seminar problems are not listed.) "Extras" to the end.

• Compendium: 5, 9, 18, 19, 28 (sort of), 30, 32, 35, 37 (modified), 38, 39, 49(a), 51, 59, 69, 70, 71, 77, 78, 84, 86, 88, 97, 100, 107, 109, 117, 127, 128, 134, 131, 138, 139
• 2004-spring #1
• 2005-spring #1, #2(a)
• 2007-spring #3
• 2010-spring, the entire set
• 2011-spring #2
• 2012-spring, the entire set
• 2012-autumn #2, #4
• 2013-spring, the entire set
• 2014-spring #1, #2, #4
• 2015-autumn, the entire set
• 2016-spring, the entire set
• (2016-autumn for the review lecture)
• "Extras" assigned: Compendium 21, 44(a-c), 94, 104, 108 (modified), 111.

Review problems (pre-seminar, see messages):

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 (January 30th to February 2nd):

• Exam spring 2004 problem #1. Part (c) subject to covering the relevant piece of theory in class - watch this document for changes.
• 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 the seminar-free week between seminars #1 and #2:

• Compendium #97 (general problem on functions)
• Compendium #9 (ln and elasticity)
• Compendium #32 (may be a bit of work)
• Compendium problems #49(a) and #100(a) (integrals)

"extras", not to be covered: Compendium #44 (a)-(c), and #21.

Problems for seminar #2: 

• Do the above problems, Matteo will poll you for requests - although, do not prioritize Compendium 32, we leave that until after the teaching-free week. 
• Compendium problems 71 and 77

If this is not enough: Do problem 94 and 104. 

Problems for seminar #3: 

For the seminar after the teaching-free week. You have already gotten a sneak peak at the term paper set, so even if there is more time, I should not overload you. All from the compendium

• 32 (still a bit of work).
• 88.  Solve part (b) using both methods.
• 100 (b) and (c)
• 109
• 127

If you need extras: 111, 134. 

Problems for seminar #4: 

A bit of this, a bit of that. Both diff.eq.'s, Lagrange (p)review, a single variable function, ...

• 134 (was "extra" for seminar #3)
• 84
• 39
• 78; Lagrange has not yet been reviewed in class, consider this a review problem from 2200
• 28, a long and not-too-easy one. Matteo may choose to cover only the first part of this.  If so, the rest could be assigned for later. In part (e), you need to differentiate implicitly.

Problems for seminar #5: 

This might be a bit of work, since some have three variables and two constraints.

• Spring 2004 problem 5. How "easy" can you make this? 
• Spring 2005 problem 4. Also, show that a minimum indeed exists.
• Compendium 86. Note there is a max and min problem, so in (b) you will have a bit of work to do if you want to use Kuhn--Tucker, but as you have already done the Lagrange in part (a) ...
• Compendium 51. Also: answer the following: can any of the tools in the course help us conclude that (1,1,1) indeed maximizes?
• Compendium 138 (a)

Problems for seminar #6: 

I realize that problem 51 for the seminar #5 should have been "51 parts (a) and (b)", leaving part part (c) for seminar #6. ... 
If you are short on time (term paper deadline approaching), then I suggest that you give priority to 2012 problem 4, as it tests multiple elements of the theory.

• Compendium 138 (b) (you have done (a), this should be short).
• Compendium 19
• and 70
• and 30. Do part (b) in two ways: (I) By calculating the value function and differentiating, and (II) by pointing out how you use the envelope theorem.  (That makes "at least two", I'd say.)
• Exam autumn 2012 problem 4; you can replace "= 900" in the (N) problem by less than or equal to 900, I do not know if I will cover mixed problems this year.
• Exam autumn 2012 problem 2 (reviewing integration).

Problems for seminar #7: 

Revised after Friday's lecture:

• Exam spring 2014 problem 2
• Exam spring 2014 problem 4
• The following "preliminary" is in Norwegian but should be doable with Google translate (if not, send a mail to Nils). Consider this midterm exam in MAT1001. For each of the problems 1 through 5 on page 2, take a stand on whether we by Ola's Friday lecture have covered enough to be able to solve it - and, if applicable, whether you should be able to solve it even before starting on Mathematics 2.
• English EMEA: 15.7.8, 15.8.4, 15.4.6 
  Norsk: LA: 2.2.4, 2.3.3, 3.3.6 
• Compendium problem 37 modified as follows:
  • In (a), do not do the determinant |A|
  • in place of (b), show that A(A-I)² = I and explain then why it also holds true that (A-I)²A = I  (why is that not completely obvious?)
• These two problems will appear easier after a while ... or possibly you will have forgotten what the issue is?
  • 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.  (Not sure what that means? Here they are.)
    For each of those, compute the square A²=AA. How does this relate to squares of real numbers?

If you need more, the following will probably be assigned for later:

• Compendium problem 5; as of now, you can only do part (a) and not the "determinant" question
• Compendium problem 59 (a) and (c).

Problems for seminar #8: 

(May be revised after Friday's lecture - if so, it will be announced in Messages.) All from the compendium.

• 5
• 59 (if (b) is possible using what you know by Friday, otherwise just (a) and (c))
• 69 (if (a) is possible using what you know by Friday, otherwise just (b))
• The following modification of 139:
  • First (b): find the inverse by solving AX = I  by Gaussian elimination.
  • Then (a): when does that inverse exist - i.e., when does your method from (b) actually yield an X?
  • If you by Friday can compute the determinant of A: verify that the determinant is nonzero precisely when the inverse exists.
  • (c) it is a fact that a matrix has an inverse if and only if it is square with nonzero determinant. In part (c), the point of assuming nonzero determinants is precisely to ensure existence of inverse. Use this to answer part (c). 
• 128

If this is not enough (or maybe, if we have to revise the problem set): 131. Before doing part (b): is it obvious up-front that a solution will exist - or do you need to start solving to see that?

Problems for seminar #9, right after Easter - note it is Wednesday and Thursday: 

This time you get too much work to do in a "proper" way if you are going away for a week and a half (next week is half a working week though!), but you will "save time" if you do as follows. 
First, do exam spring 2014 number 1 (you should drill a bit more linear algebra before starting on exams). 
I have assigned full exam sets. I suggest that you simulate an exam situation: allocate three hours, leave the problem set unseen until you can start - and then work out as much as you can in three hours, and take note on how much you are missing. Then take note on how much time you need to do the rest. (Do not panic if you miss a lot, there will be more full-exam drills the next weeks.)
After that, see how much time you have left. 

• You did problem 128 for seminar #8. What could a = |A| and b = |B| possibly be?
• Exam spring 2014 number 1
• The entire spring 2010 exam (suggested to be given priority for Wednesday's seminar) 
• The entire spring 2013 exam (suggested to be given priority for Thursday's seminar)

Problems for seminar #10: 

Again I assign a full exam set, but first there are some problems with differentiation in equation system, which you should probably try first to speed yourselves up. 
If you need more of equation systems than the ones assigned, then you can do problems 63, 67bc and 130.
If on the other hand this topic is something you find easy and you rather spend your time on something else, then the sets for spring 2008 and spring 2016 are recommended (at least one will be assigned for the final seminar), or autumn 2010. For more specific review: Autumn 2008 #1 (linear algebra), #2 (function of two var's), #3 (diff.eq) or spring 2014 #3 (Kuhn-Tucker).

The problems: 

• Consider exam spring 2007 problem 3 (This was on the screen at the end of Wednesday's lecture.)
  Do it first "the full cookbook way" and then take a stand on how much you could have shortened it down by taking note of what the question asks.
• Compendium problem 35. 
• Compendium problem 38.
• The full exam spring 2012. 
• If time permits, problem 108 where you can for part (c) also do the following (a tool from Mathematics 3, was in Math 2 in old days): 
  Let M be the augmented coefficient matrix (i.e. the coefficient matrix but with the RHS stacked up as third column; in this problem, it the augmented matrix is a square matrix). Then the system says M (x, y, -1)' = 0; that is, it says M (x, y, z)' =0 AND z=-1.
  Since M is square in this problem the determinant |M| is well-defined, which leads to the question: 
  Explain why |M| = 0 is necessary for a solution to exist.

Problems for the final seminar #11: 

Exam spring 2014 #4 has in fact been assigned for seminar #7, due to a typo - I indended to assign #3. As of then, we had not covered it and you were not at all supposed to be able to solve it by then (but after this Wednesday's lecture, you are). None of us got any question. 

• Exam spring 2014 #4 - do this first, as now you should be able to catch it without looking up your notes. Do not expect Matteo to spend much time on it though. 
• The entire exam spring 2016 (top priority). 
• The entire exam autumn 2015.
• Exam spring 2011 #2 (requires Friday's lecture. If you need extra problems, you can do the rest of that set as well.)

For exam preparations, do full exam problem sets, and ...

• The review lecture routinely covers the most recent ordinary exam (to be uploaded soon). This semester I may or may not deviate, as we will already have covered the term paper set in class. Nevertheless, you should do autumn 2016.
• Other sets you could do: Spring sets of 2010 through 2013 are either assigned or recommended already, do them if you have not; also recommended as "extras" were spring 2008 and autumn 2010. You could very well also do autumn 2009, autumn 2011 and autumn 2014. 
  Autumn 2012 and autumn 2013 contains particular bits that could be said to have had slightly less focus this semester (that may admittedly go for some others too), but nearly all should be doable; also, the oldest sets may have had a somewhat different focus. Spring 2015 was a too hard set for the usual grading scale, which was adjusted to accommodate it.