MS-A0211 - Differential and Integral Calculus 2, Lecture, 10.1.2023-20.2.2023
Kurssiasetusten perusteella kurssi on päättynyt 20.02.2023 Etsi kursseja: MS-A0211
Lecture 6 - 26.1.2023
Suorituksen vaatimukset
In this lecture we studied:
- Taylor approximations. We reviewed Taylor series and Taylor polynomials for functions of one variable.
We gave the definition of Taylor polynomial of degree k at (a,b) for a function of two variables f(x,y). We paid special attention to the cases k=1 and k=2. These are, the 1st and the 2nd order Taylor polynomials. Noted that the 1st order Taylor polynomial is the equation of tangent plane at (a,b) as expected, and that the 2nd order terms of the 2nd order Taylor polynomial can be written as 1/2[x-a y-b] H(a,b) [x-a y-b]^T (where T denotes transpose and H is the Hessian matrix). We did an example of computing the 1st and the 2nd order Taylor polynomials.
- Local extreme values (min/max). We reviewed local extreme values and the 2nd derivative test to classify critical points for functions of one variable. We studied when a function of two variables can have an extreme value, and we stated the 2nd derivative test for critical points of a function of two variables. For that, we have to consider the Hessian matrix. We did an example of finding and classifying local extrema.
Maple codes of some examples seen during the lecture: