Last-minute "remember this": to review

I will be generally available on e-mail except tomorrow Friday (Department seminar), in office Mon/Tue from not-too-early til whenever needed I - can go to the university at short notice as well.  

A few bits and pieces I would have highlighted if there was another hour:

General: 

•     Do not divide by anything unless checking whether it could be zero.
  •     Exception: If you are asked for, say, "an expression for" an implicit derivative. You divide, you get an expression if the derivative exists (... up to a detail you should not worry about).
•     Generally, read the problem text. For example, you can be asked about whether one can find, not where to find. 
  • This course has two "nonconstructive" existence results in analysis, that do not say anything about where: The extreme value theorem, and the intermediate value theorem. (For linear algebra, there is also a "non-constructive" existence and uniqueness result: zero determinant for 

Implicit differentiation:

•     Remember what "differentiate the equation system" means.  After having done so, identify the linear equation system.  The rest is then dirty work.

Max/min. Unconstrained: 

•     Stationary points are found by solving equations.  Do not divide by anything variable on both sides, until you have checked whether it could be zero.  (2y = xy is equivalent to x=2 or y=0, do not cancel the y.)
•     You are expected to recognize (if they are not too complicated) concave/convex functions of one/two variables even if the second derivative / the Hessian determinant should vanish at isolated points.  For example the fourth power.
•     You should know that a sum of concaves is concave (and the sum of convexes is convex).
•     You are expected to classify a global extremum of a concave or a convex. (0 for x⁴).
•     Once you have to resort to classifying a stationary point for local extremum, you are allowed to write "no conclusion" if the Hessian determinant vanishes.
•     If you are asked whether a function of two variables has a global extremum, and you suspect not: try to see what happens "at infinity" (say, let y=ax and send x to positive or negative infinity); that is not fool-proof, but if you can establish that f takes all values, then no global extrema exist.
•     You are allowed to know the envelope theorem without deducing it.

Max/min. Constrained:

•     Read the problem text! If it says "find all the points that satisfy the Lagrange conditions", then do that, even though you know what the solution to the max/min problem is. If it says "solve the problem", you have free choice of tool.
•     Kuhn--Tucker: it is possible that the multiplier is zero AND SIMULTANEOUSLY the constraint is active.  (All admissible stationary points satisfy Kuhn--Tucker, even those at the boundary.)
•     If you have a boundary point and a multiplier (that would suffice for Lagrange), then for Kuhn--Tucker: check that the multiplier is nonnegative.
•     You are allowed to know the shadow price interpretation without deducing it from the envelope theorem.
•     Existence of solution?  Possibly the extreme value theorem can help you.  If the admissible set is unbounded, you may try to see what happens "at infinity" (that again is often best used to disprove existence).  Of course, if you have found a point where sufficient conditions apply, you have existence.

Integration:

•     Remember the "+C" in indefinite integrals.
•     If you substitute in definite integrals substitute the limits too; if you are not sure about that, do the indefinite integral first and insert the limits after having substituted back.
•     We basically know two methods of integration, and basically two types of differential equations where the respective method applies.
•     Linear differential equations: a formula in the book, if you want it: if you want to use the formula, remember which side of the equality sign you find the "a" and "b" terms.
•     Separable differential equations: remember to check for a constant solution (that is: do not divide by zero, have I said that before?)

Other analysis items:

•     In quadratic/higher order approximations, remember the 1/2, resp. the 1/n! (n! = n*(n-1)*...*2*1 - I did not mean "!" as in "remember!")
•     The elasticity of substitution: Formula in the book (in the problems part).
•     Homogeneous functions: Recognize them and apply the variants of the Euler theorem.
•     To recognize a homothetic: try one-to-one transformations until you end up with a homogeneous.

Linear algebra:

•     Do not divide by a vector or a matrix.  Check whether a matrix has inverse before applying it.
•     AB is not the same as BA.
•     For linear equation systems: Gaussian elimination irrespective of whether the coefficient matrix is square.
•     For square coefficient matrix: the connection between invertibility (thus nonzero determinant) and unique solution.
•     Basically two methods to calculate a determinant of order 3x3 and above (cofactor expansion and row/column operations). 
•     Basically two methods to calculate the inverse (cofactor formula and is to solve AX = I), but if you can recognize it ... as long as X is square, it suffices to check that AX = I or XA = I, the other will follow.