Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2022
Sporkz
Views
29
Quadratic approximation Quadratic approximation is used to determine the change in a function f of (x,y)when x and y are changed a little bit. It can be expressed as the sum of two terms:the first-order term and the linear term. The first-order term is f sub x times thechange in x, while the linear term is f sub y times the change in y. If the solution is atthe critical point, then the derivative of the solution will be 0 at the critical point. Sothat term will go away, and that term will also be 0 at the critical point, so that termwill also go away. The next term, a quadratic term, is derived from the Taylor expansion of a function ofa single variable. The first derivative times x minus x0 plus 1/2 of the secondderivative times x minus x0 squared equals the nth derivative times xn-1. And see,this side here is Taylor approximation in one variable looking only at x. But of coursewe also have terms involving y and terms involving simultaneously x and y. Andthese terms are f xy times change in x times change in y plus 1/2 f yy times y minusy0 squared. There's no 1/2 in the middle because, in fact, you would have two terms,one for xy, one for yx but they are the same. To make the approximation more accurate, one must consider cubic terms involvingthe third derivative and so on, but we are not actually looking at them. And so nowwhen we do this approximation well, the type of critical point remains the same whenwe replace the function with this approximation. And so we can apply the argumentthat we used to deduce things in the quadratic case. In fact, it still works in generalusing this formula for approximation. We previously referred to the coefficient a as little a and called b capital B and ccapital C. When you substitute these definitions into the various cases we haddiscussed, you end up with the second derivative test. In the degenerate case, wecannot say that a critical point is always a maximum or minimum. The approximationformula is justified only if higher-order terms are negligible.
In the case of non-degenerate functions, the shape of the graph is determined by thesecond and third derivatives, not by higher order derivatives. In the degenerate case, where the function is extremely flat or has a very smallderivative in a certain region, the sign of the function's second derivative can changefrom plus to minus or minus to plus. This changes the sign of the value of thefunction at that point, which can either make it a minimum (where it stops goingdown) or a saddle point (where it doesn't go down at all). In real life, you have to be extremely lucky for this quantity to end up being exactly 0.If it happens, then what you should do is maybe try by inspection, see if there's agood reason the function should always be positive or always be negative orsomething or plot it on a computer and see what happens.
Quadratic Approximation and Critical Points Analysis
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