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Upcoming events (1)
This topic will explore the pattern of partial variation for finding minima and maxima (extrema) in geometry. How can we find extrema by looking at the partial variation, the variation of just one or two variables.
In particular, this group exploration will examine Chapter 8 §1–6 on pages 121–131 (with an in depth focus on §5) and Examples 15–22 on pages 133–134 in George Pólya's "Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics" https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n139/mode/2up .
Prerequisite: high school geometry and algebra.
To guide your preparation & participation, consider:
➀ Our first concern will be an in depth exploration of each others' understanding and interpretation of the text in §5 of Chapter 8 on pages 128–130 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n147/mode/2up). Bring your notes, questions, insights, difficulties, and critiques for these three pages of text.
● In the text, Pólya solves the problem of finding the maximum of the product of n parts into which a line segment of length l has been subdivided. On page 130, Pólya writes, "The reader can learn a great deal in clarifying the foregoing proof. Is it quite satisfactory?" At the event, we will explore your attempts to clarify the proof that Pólya sketches. We will also explore your attempts to assess if the proof is satisfactory.
➁ Our second concern will be an in depth look at the "Examples and Comments on Chapter VIII" #15–22 of Chapter 8 on pages 133–134 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n151/mode/2up).
• These problems invite you to exercise your knowledge and skill to resolve mathematical questions. Bring your partial solutions so we can work together to strengthen everyone's understanding. Through this practice, we will all come to understand the particular situations and the pattern of partial variation better. Even when none of us can solve all the problems, hopefully each of us will understand enough to fill in each others' gaps to give everyone a better idea about how each one works.
Pólya's book is in the public domain, so you can find free copies of it in PDF, EPUB, Kindle, text, and other formats at https://archive.org/details/Induction_And_Analogy_In_Mathematics_1_ (scroll down to the "Download Options"). The PDF version (without text) might give the best printed version.
In addition to the main material described above, the following passages introduce Pólya's approach in the book which will be useful but not strictly necessary for the event:
• The Preface on pages v–x https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n7/mode/2up , pay particular attention to §1–4.
• The very important Hints to the Reader on pages xi–xii https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n13/mode/2up .
• Chapter 1 on pages 3–11 https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n21/mode/2up , pay special attention to Example Problems 9–14 on pages 9–11 https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n27/mode/2up .