MS-A0211_aalto-CUR-165356-3086063: Lecture 7 - 31.1.2023

This course space end date is set to 20.02.2023 Search Courses: MS-A0211

Lecture 7 - 31.1.2023

In this lecture we studied:

Absolute extreme values (min/max). We stated the theorem that continuous functions of two variables defined on closed and bounded regions attain their absolute extrema. How to find the absolute extrema on a closed and bounded domain D? Step 1. Find critical and singular points on the interior of D (as we know from the previous lecture). Step 2. Find any point on the boundary of D where f might have extreme values. Step 3. Evaluate f at all the points found in steps 1 and 2. We found the absolute extrema of f(x,y) = x^2 + 2 y^2 on the disk of radius one centered at (0,0). For that, on the boundary (step 2) we reduced the problem to a one variable problem.
Method of Lagrange multipliers. We discussed the method of Lagrange multipliers and justified it intuitively by looking at a sketch of level curves and using the fact that the gradient vectors are orthogonal to the corresponding level curves. We noticed that the absolute extrema of f(x,y) subject to g(x,y)=c can occur at points where the gradient of f is parallel to the gradient of g (Lagrange multipliers) or at points where the gradient of f is not defined. We re-solved step 2 of the previous example by using the method of Lagrange multipliers. We did another example of Lagrange multipliers: on the curve x^2 + xy+y^2=1 which points are closest and furthest from the origin?

The notions studied in this lecture can be found in sections 13.2 and 13.3 of Adam and Essex’s Calculus book (seventh edition).