- 3.4: Second-Order Approximations
- In one-variable calculus, Taylor polynomials provide a natural way to extend best affine approximations to higher-order polynomial approximations. It is possible to generalize these ideas to scalar-valued functions of two or more variables, but the theory rapidly becomes involved and technical. In this section we will be content merely to point the way with a discussion of second-degree Taylor polynomials. Even at this level, it is best to leave explanations for courses in advanced calculus.
- 3.5: Extreme Values
- After a few preliminary results and definitions, we will apply our work from the previous sections to the problem of finding maximum and minimum values of scalar-valued functions of several variables. The story here parallels to a great extent the story from one-variable calculus, with the inevitable twists and turns due to the presence of additional variables. We will begin with a definition very similar to the analogous definition for functions of a single variable.
Thumbnail: Real function of two real variables. (Public Domain; Maschen).