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 - merely seeing the solution being done, often makes it look easier than it is.) 

More review problems:

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 (30th/2nd/4th):

Optimization, except the first, which is just drilling differentiation. 

• Compendium problem 43
• Compendium 39
• Compendium 138
• Exam spring 2005 problem 4
• Exam spring 2010 problem 3 parts (a), (b) and (c). 
  Also: The maximum value V depends on (a,b,C).  Find the partial derivatives in two ways: (i) by inserting t* and then differentiating; (ii) by differentiating and then inserting t*. (What is the latter method called?)
• Concerning the Lagrange problem covered in class (last two pages in the notes):
  • For each of the two points (the max and the min), calculate the multipliers.
  • Inserting the multipliers for the max. point: is the Lagrangian then a concave function?
  • Inserting the multipliers for the min. point: is the Lagrangian then a convex function?
  • What implications would the above have, if you were asked to solve the problem without using the extreme value theorem?

Problems for seminar #2 (September 6th/9th/11th):

Term paper no. 1, and any leftovers that couldn't be covered in seminar #1.

If you need more: Have a look at Compendium problem 28 (mid-1980s problem, not easy ...). In part (b) take for granted that the limit is positive infinity. Part (e) is implicit differentiation (cf. marginal rate of substitution formula!). You cannot do part (f) yet. 

Problems for seminar #3 (September 13th/16th/18th):

• Compendium 31; also, decide if the functions are homothetic.
• Compendium 117 (limits)
• Find the limit (note: it is Q that tends to 0):
  \(\displaystyle \lim_{Q\to 0}\left(\frac{aK^{-Q}+bL^{-Q}}{a+b}\right)^{-m/Q}\) 
  (This is a CES function. The book has the same problem except with m=1 and b=1-a and Greek letter ? for Q.)
• Compendium 32 (inflection point = point where the second derivative changes sign).
• Compendium 19, solved by way of Kuhn-Tucker. 
• Find the elasticity of substitution for the Cobb-Douglas function x^ay^b using both methods (manipulating differentials and formula). Is the answer any surprising, given the previous bullet item? 
• True or false claim? "The elasticity of substitution is unaffected by a strictly increasing transformation."
  More precisely, suppose g'>0 everywhere, and let G(x,y)=g(F(x,y)): is it so the case that the elasticity of substitution of F (at level curve C), is the same as the one for G (at level curve D=g(C))?
  • Suggestion:
    First, argue as economists - ought it to be the case? (Presuming that the elasticity of substitution indeed measures what you think it is supposed to do.) 
    Then check your argument using the definition.

Problems for seminar #4 (September 20th/23rd/25th):

Correction made September 19th: removed problem 43 which was already assigned. Instead you can do 53(a) which is assigned for seminar #6. Unless you request otherwise, it will have low priority at least on Friday's seminar, due to the late call.

There are many bullet items, but several of the integration problems are short - and some of those will require the lecture on the 18th. Hence the longer problem at the end, which could be suited for discussion in the workshop.

• Compendium 18 (limits, might require some thought ...)
• Suppose that the integral \(\int x^2e^{2x}dx\) has been calculated. How would you calculate \(\int 4x^3e^{2x}\:dx\ \)?
• Compendium 40 (in (b), what happens when the alpha is actually two?)
• Compendium 49 (a)
• Compendium 53 (a). This problem replaces 43; due to the short notice, it might have low priority.
• [Hint: "More conceptual" questions, don't try to antidifferentiate! A (rough!) sketch could be helpful though.]
  Explain why
  •  \(\int_{-e}^e\sqrt[2019]t\:e^{-t^2}\:dt = 0\)
  • \(\int_{-e}^e\sqrt[2020]{|t|}\:e^{-t^2}\:dt = 2\cdot \int_{0}^e\sqrt[2020]t\:e^{-t^2}\:dt\) and is >0
  • The function \(F(T)= 2\cdot \int_{0}^T\sqrt[2020]{|t|}\:e^{-t^2}\:dt\) is strictly increasing (be careful to also address the case T<0)
• And an optimization problem: Compendium 60.
  Also: The optimal value is a function V(a, b, alpha, A). Find expressions for the first-order partial derivatives.
• Optimization formulated as application [this might take just as long time to read, as to actually do - could possibly be postponed. The "application" part of this is not exam material, but the mathematics could surely be - indeed, this is a "lots of talk" version of an earlier exam problem.]
  Consider an Edgeworth box where two agents share 1 unit of each of two goods. The agents have the same utility function u, strictly increasing in the amount of each good. (You can assume that no marginal utility ever hits zero, and you can assume whatever degree of continuous differentiability you need.)
  Consider two versions of agent number 1's problem:
  • Whatever utility level w the other agent gets, I want to maximize my utility. 
  • Impose a minimum utility level w for the other agent. Whatever w is, I want to maximize my utility.
    
    Now, here are the tasks for you:
    • Formulate either of these as a constrained maximization problem; remember that each good is available in fixed supply 1!
    • There is an "obvious" candidate for a Pareto efficient allocation, namely a fifty-fifty split: each agent takes half of each good. Let w be the particular number that matches the the agent's utility (= yours, because utility functions are the same!) from getting half of each.
      Check whether this allocation satisfies the Kuhn-Tucker conditions associated to one of the above problem versions (you pick one).
    • Under what conditions can we (using Math2) know that this allocation actually solves agent 1's problem?
    • If w were increased only slightly, how much would agent 1's value change?

HAND-IN #2: Problems for seminar #5 (September 27th/30th & Oct 2nd):

Term paper hand-in #2 is posted in Canvas. (Nils will give all three seminars, schedule may or may not be updated.)

Problems for seminar #6 (October 4th/7th/9th):

Revised September 27th: 100(c) out (due to progress), 81(a) in. Nils will give the seminars.
Edit Oct. 4th: Errors galore, sorry for very late corrections:
81: I had a nonsensical sentence. Struckout: the right thing is "only get this with the hint" (if at all). And, there is no "81 (a)", I meant the first integral. A bit late for corrections I know, sorry. 
100: (b) is actually not curriculum without the following hint: rewrite 1/(u(u-1)) as 1/(u-1)  - 1/u.
Exam 2015 but which? Spring 2015 #3. At this stage it is better to give it least priority?
Edit again, argh ... in any case, skip part (b)!

• Compendium 81 (a).  first integral. (The second could be covered if time if you actually did this before this correction.)
  Knowing how to rewrite the integrand is no longer curriculum, so rest assured that you cannot get only get this with the hint on the exam.
• Compendium 53 (b)
• Compendium 49 (b)
• Compendium 71
• Compendium 100 (b). And (a), though it should not be hard after having done 40 and 53. 
• Compendium 109; part (b) requires the October 2nd lecture and maybe even the October 3rd lecture.
• Exam spring 2015 problem 3 parts (a) and (c).
• Exam autumn 2017 problems 1 and 2.

HAND-IN #3: Problems for seminar #7 (October 11th/14th/16th):

Problem set to be posted in Canvas. Tuomas will give the seminars. 

Problems for seminar #8 (October 18th/21st/23rd):

18. Oct.: another annoying error: the last diagram, corrected from -t to +t.

First exam set, yay!

Exam spring 2006 is one where you "can in principle do without" linear algebra theory - although it would be helpful! So do that, and for the equation system: rewrite it using linear algebra language when we have done that. 

• For numbers it is true that \(\smash{a^2-b^2=(a-b)(a+b)}\).
  If we replace a and b by matrices A and B, is the formula then still true? Alternatives
  [_] yes, always!
  [_] only iff both A and B are n x n, 
  [_] only iff A and B are proportional (meaning, one is a scaling of the other) and n x n
  [_] typically false, but there are cases where the formula holds true even if the matrices are not proportional
  [_] never except when both matrices are n x n AND furthermore either n=1 or one matrix is zero.
  (If you want to rule out some wrong alternatives, then there should be quite a lot of info in checking 2x2 cases.)
• Compendium 69 (b), solved in any way you like. 
• Compendium 42 (b) and (c).
• Exam spring 2006: the entire set!
• In the spring 2006 exam, problem 2:
  • Write the equation system on matrix form \(\smash[b]{\mathbf A_t\left(\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right)=\mathbf b}\), where \(\mathbf A_t\) means that t enters somewhere in the coefficient matrix.
  • After the lecture(s) on Gaussian elimination, take note how you could solve this and 69(b) with linear algebra.
• Compendium 37 except the determinant |A|. (b) requires the very end of the Wednesday 16th lecture (that will be reviewed Thursday 17th). 
• Here is part of a directional diagram for a differential equation of the form 
  \(\dot x = m(x) + t\). (Now with correct form "+t". The form is essential to the question!)
  
  Indicate the directions for t = 2, x in {-2, -1, 0, 1, 2}.

Mock exam October 24th; no seminar October 25th/28th/30th.

There was one too many in the schedule, and then this week is the one which will be removed.

HAND-IN #4: Problems for seminar #9 (November 1st / 4th / 6th):

Problem set to be posted in Canvas with deadline likely November 1st. The "mock exam" the 24th will have the same problem set. 

• Important: If you need Hand-in #4 approved, then read the instructions on precisely what to hand in.
  (If you hand in too much, you won't be penalized in any way.)

Seongbong will give the seminars.

Seminar #10: November 8th/11th/13th

No change from draft. Seongbong will give the seminars.

• Compendium 42 (a) and 69 (a). (The other parts were assigned for seminar #8.)
• Compendium 139
• Exam autumn 2017 problem 3
• Exam spring 2015 problem 2: 
  • In (b): what difference does it make, whether you are asked for a "general expression" or for \(\smash{v'_p(1,1)}\)?
• The autumn 2009 exam.

Final seminar #11: November 15th/18th/20th.

Seminar leader: as originally scheduled, Seongbong / Vegard / Vegard.

• One problem on the new topic of polynomial approximation:
  Problem 3(b) of exam spring 2015
• Spring 2012: the entire set
• Autumn 2018: the entire set
  • An actual exam paper that got an "A" is posted in Canvas.

(Note that the draft had a different exam, which I left out: autumn 2011.) 

Full exam sets recommended between last seminar and the exam.

Many of you might want to do old exam problems under time pressure, and I do get questions on which are more and less relevant. Generally speaking, the focus on differential equations and Kuhn--Tucker started around 2006. I suggest you go to https://folk.uio.no/ncf/4120_oldexams/ , as those have indicated when corrections were necessary afterwards.

I am a bit worried that specific recommendations will be taken to mean way more than intended - generally, and in particular for this year's new format. However, there could be specific reasons to do particular ones. Among those not touched for seminar problems: 

• Spring 2016: one of the problems is solved in a video in Canvas, if that is what you want. (Which? If you do not want a spoiler, do not click here.)
• Autumn 2015, spring 2017 and spring 2018 (the last spring version), if you think recent past is more relevant. Also the 2014 exams, although those years maybe had a slightly higher focus on a particular piece of theory. 
• ... and if you want a handwritten solution to compare with what you write yourself: there are such for spring 2014 and autumn 2011. 
  (Also autumn 2011 has a "rare" question though - and beware that the handwritten note is very terse.)

You will probably also ask yourselves "is this anything we should expect?" at some exam sets. I do not wish to go at lengths elaborating on that, but again: some problems reflect what was the focus at that time.
Just to give one general example and two specific ones: Many of the exams 2006 to 2013 have questions about tangents (which isn't too hard, but we haven't stressed this geometric application so much this semester).  Part of autumn 2010 problem 3 was a copy of two seminar problems assigned that semester. If you want to do autumn 2012, there is a problem "(N)" where might want to make an "obvious" tweak to make the question more familiar, and you shouldn't spend time debugging a ... well, you will see which problem, be satisfied when you have gotten the method right, that was sufficient for "A"-score.     

And one reason to assign only part of spring 2015 for seminar, was that this was a too hard problem set overall - so I chopped it apart to leave other sets as full sets. Don't worry so much over spring 2015, the grading thresholds were altered significantly that semester.

- Nils