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: Note: L’Hopital’s Rule does not apply here. Why?

Symmetry in Geometry

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

Idea: Draw a flattened picture of the cube.

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.

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

Symmetry in Combinatorics (The Art of Counting)

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?

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

Symmetry in Algebra

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.

Idea: Use the Arithmetic-Geometric mean inequality.

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

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

Idea: Use the Cauchy-Schwarz Inequality.

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

Thank You for Coming Wafik Lotfallah