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Lecture 2 (Basic Techniques)

Lecture 2 (Basic Techniques). Some Basic Techniques. Drawing a Picture Reformulate the Problem Use Symmetry, Create Symmetry. Symmetry in Calculus. Problem 1:. Mentally (or graphically) calculate (if exists):. Idea:. Need to understand the meaning of the double angle formula:

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Lecture 2 (Basic Techniques)

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  1. Lecture 2 (Basic Techniques)

  2. Some Basic Techniques • Drawing a Picture • Reformulate the Problem • Use Symmetry, Create Symmetry

  3. Symmetry in Calculus

  4. Problem 1: Mentally (or graphically) calculate (if exists):

  5. Idea: Need to understand the meaning of the double angle formula: Note: L’Hopital’s Rule does not apply here. Why?

  6. Symmetry in Geometry

  7. Problem 2: Find the length of the shortest path along the outer surface of a cube between two opposite corners.

  8. Idea: Draw a flattened picture of the cube.

  9. Problem 3: Find the length of the shortest path from the point (3,5) to the point (8,2) that touches both the x-axis and the y-axis.

  10. Idea: Use symmetry about the x- and the y- axes.

  11. Symmetry in Combinatorics (The Art of Counting)

  12. Problem 4 How many subsets of the set X={1,2,3,…,109} have the property that the sum of the elements of the subset is greater than 2997?

  13. Idea: Consider the map sending each subset S  X to its complement Sc = X  S.

  14. Symmetry in Algebra

  15. Problem 5 Show that: (a + b)(b + c)(c + a)  8abc, for all positive numbers a, b, and c, with equality iff a = b = c.

  16. Idea: Use the Arithmetic-Geometric mean inequality.

  17. The Arithmetic/Geometric Mean Inequality: Show that for x, y > 0, Generalize the corresponding inequality for n positive numbers.

  18. Problem 6:Let ai, bi > 0, for i = 1, 2,…, n. Show that:

  19. Idea: Use the Cauchy-Schwarz Inequality.

  20. The Cauchy-Schwarz Inequality In other words: xy  |x||y|. Generalize the corresponding inequality in the nth dimensional space.

  21. Thank You for Coming Wafik Lotfallah

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