1 page double spaces

After this write 2 reply on others discussion

about 100 words for each relpy.

PROMPT:

How did you find our study of linear algebra?

Can you think of other situations where linear programming would be applicable?

Can you think of other situations where Markov chains might be applicable?

Lesson

This module and the next comprise our unit on the mathematics of optimization. Our optimization comes in two flavours – optimizing a function of one variable and optimizing a function of more than one variable. You may have come across optimizing a function of one variable already, so part of this module will be a review. This lesson discusses how to optimize a function of a single variable, that is, find its maximum or minimum, and then introduces some of the mathematical machinery required for optimizing a function that has many variables. The basic ideas are straightforward. Functions that resemble a hill, strictly concave functions, can have a unique (global) maximum, while functions that look like a valley, strictly convex functions, can have a unique (global) minimum. The adjective strictly means that regardless of the x values, there are no wiggles in the function nor any flat surfaces. As noted before, mathematicians use adjectives like strictly to make it clear that some property holds completely. If the function isn’t strictly convex or concave, then when you find a point that looks like a maximum or minimum, it might not be global but local, that is, one of the functions possible maxima or minima over some set of x values. A saddle-shaped function is an example. Sometimes when we restrict the possible values of x we can find a maxima or minima within the interval or at the boundaries of the interval. For example, if we look at the sine function over the interval 0 to pi radians, it has a maximum at pi/2, then repeats the maximum it obtains 2 pi radians later, the 4 pi radians, etc.. If, however, your restrict your attention to the interval 0 to 2 pi, then there is a unique maximum, as well as a unique minimum. If we restrict the values of the sine function to begin at pi/2 to 3pi/2, then the maximum happens at the boundary of the interval. You will see how to determine if a function is concave or convex or neither and strictly so or not. You will learn how to assess whether the necessary conditions for an optimum hold. You will see how to calculate the partial derivatives of a function of several variables.