Messages

The last exponent missed a negative sign, it should be 
¨C (t⁹ ¨C 1) / 27.

Marcus and myself had an improvised meeting with the Department's administrative resources right now.

• You can contact him on the matter, and
• the Department has agreed to forward communication from him, most likely early next week - thus, keep an eye on the inbox of your UiO e-mail (the one that the University uses in formal communication, e.g. "student.*.uio.no" type). 

Edit: his e-mail is marcusbr@student.sv.uio.no

The questions during the exam took more time to answer than usual, and those minutes may certainly have been less efficient.

I will make sure that the grading guidelines include information about the circumstances. The committee will apply their best judgment.

- Nils

Nobody showed up for today's consultation hours, so I doubt there is much demand for the following. Nevertheless, given that next Wednesday I have reserved half the "consultation hour" slot for Mathematics 3 (1215-14 to be more precise), here is a suggestion: I will be available next Tuesday, "lecture time" (1415-16). Open agenda. If enough people will leave me an e-mail that they want to show up, then we can find a room. If only a couple ... then my office is just as convenient.

- Nils

As far as I can see, there is nothing in particular that has been requested for review? Or have I forgotten something?

I will not be as much in office as one could hope - I've got an issue with my back, and have been ordered to spend less time in the chair and more walking. 
If you send an e-mail I can easily be in office by appointment, but the chance of finding me by just knocking my door are somewhat reduced. 

- Nils

I posted my notes for seminar 11, but there is an error (and there may be more; let me know!):

The last expression on page 8 should have bBy^b, not bBy, in the denominator, and aAx^a, not aAx, in the numerator.

Good luck on the exam!

Eivind

This is the one I will cover in the review lecture.

Exam problem sets for next week: Spring 2011 and Spring 2014.

Before you do those, you should likely do a couple of quadratic approximation problems, and maybe also a couple involving homogeneity. 

• For homogeneity: Give a simple argument why a homogeneous f(x ,y) can be written as x^k u(y/x) as long as x>0, where k is the degree of homogeneity.
  (Hint: f(x,xy/x) = ... ?)
• Do Problem 22 part (d), which concerns quadratic approximation.  Parts abc will likely not be given priority.
• Find the quadratic approximation of ln(1+x²) around x = 0. Compare it to the first-order approximation of ln(1+x). Comment?
• Find the quadratic approximation of ln(1+ e^x) around ln 2.

I got an e-mail that there is no specific reference in the schedule. You already know that the Lagrange multipliers do have an interpretation, and I will make that more precise, in the context of the envelope theorem. 

I do not have the books in front of me right now, and when I am back at Blindern you will most likely have looked up those parts by yourselves. So, I will cover:

• the envelope theorem ("omhyllingssetningen" in Norwegian);
• interpretations of the Lagrange multipliers;
• remaining bits and pieces on optimization;
• a few examples
• if I start on homogeneous and homothetic functions tomorrow, we will be very light on it.

• Seminar problems posted
• You have a couple of weeks, and not all of you are on holiday.
  There are a couple of old exams which by now you should be able to solve. My suggestion is: simulate an exam. Allocate three hours, leave it unseen until you start the clock, see how far you get. (You will likely not get through it in three hours - you might need more practice yet!)
  The sets: Autumn 2004 and Autumn 2005.

I've posted solutions to seminar 5 along with a note on cofactors and inverse matrices.

In 92(b) we could also have used the inverse of the coefficient matrix to find solutions for x' and y' (the inverse of a 2x2 matrix is easy).

Eivind

Updated under schedule as well. All from the compendium: 

• For linear algebra: 72 and 108.  For 108 part (c), also do the following (a tool from Mathematics 3): 
  Let M be the augmented coefficient matrix (i.e. the coefficient matrix but with the RHS stacked up as third column); then the system says M (x, y, -1)' = 0; that is, it says M (x, y, z)' = 0 AND z=-1. Explain why |M| = 0 is necessary for a solution to exist.

• For implicit functions: 35, 38, 63, 67 (b) and (c), 105.

In problem 69 (a) you are not asked to calculate the inverse (as I by mistake wrote), only the determinant - now you can do that by cofactor expansion.

For 139 (b) you shall calculate the inverse though. 

(Of course you can also calculate the inverse in 69 (a) as well, by using cofactors, but do not expect that to be given priority in the seminar.)

Eivind

The last two are problems for last week, but now you shall use a different method or possibly two. Meaning four matrix inversions - there might not be time in the seminar.

Trykkfeil korrigert.

I was made aware that parts of the curriculum was not in the lecture schedule - indeed, curriculum for next week was not filled in. Fixed now; in addition I have retro-inserted the sections concerning the basic properties of the inverse back when it was actually covered.

A 'thank you' to the student who alerted me. - Nils

* Review of differential equations: problems 84, 111 from the compendium.
* Linear algebra problems:
52 (note that the 1 vector is a row here - I would have denoted it 1').
55, the rest.
59, the rest
69 - use row operations (like in Gaussian elimination) to calculate the determinant.
and - this is a difficult one at this stage, do not worry if you have to spend a lot of time on it: problem 139, where you in parts (a) and (b) solve for the inverse using Gaussian elimination.