Maximum, Minimum, and a Warm-Up Example

Experimentation is normally expensive, and sometimes very tedious, and difficult. As an example, advances in nanotechnology make it easier to examine nature at a deep level because today we have advanced microscopes and tools that allow us to better observe nature and that have improved our understanding of the micro and nano worlds, but such equipment is very expensive and the experiments rather difficult. When planning new experiments it is important to determine, first, what experiments to do. Several mathematical techniques can help us, in specific cases, to determine what and how many experiments represent the optimal set. One of these techniques is the D-optimal design that normally significantly reduces the number of experiments and delivers good results. For example, if you have a mathematical model such as the one depicted in Fig. 11.1 (Michaelis-Menten model) and you want to fit the curve with experimental data, then to determine the parameters of the model (Fmax and K,), the question is how many experiments are required and for what values of the S-axis. D-optimal design helps to efficiently fix the set of experiments. 

As shown in Fig. 11.2, x = a and x = b represent a relative maximum and minimum for/(x), respectively. We use the term relative because if we extend the amplitude of x values, we can probably find other maximum and minimum values. Why do we call x = a the maximum Because in the vicinity of x = a,f a) is the largest value for/(x). In the same way, x = b represents the minimum because/(6) is the minimum value of/(x) in the vicinity ofx=b. 

As indirectly mentioned in Sect. 11.3, you are probably not yet familiar with the concept of derivatives, which is essential in handling maximum and minimum problems. Although we will present and solve interesting and applicable maximum and minimum engineering problems, it is not our intention to familiarize you with the concept of derivatives. Soon in your career you will be introduced by experts to this important and key mathematical concept for aU engineers. You are most probably familiar with the use of spreadsheets. In this section we will show you how to solve maximum and minimum problems using spreadsheets. 

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