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):

x

xdtt

x 0

2cos1

lim

Idea: Need to understand the meaning of the double

angle formula:

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

)2cos(2

1

2

1cos2 tt

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 4How 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 5Show 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.

2

yxxy

Problem 6:

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

2

2

2

1

1

2

2

1

1 na

b

a

b

a

b

b

a

b

a

b

a

n

n

n

n

Idea: Use the Cauchy-Schwarz Inequality.

The Cauchy-Schwarz Inequality

22

22

22

21

21

21

2212121 zyxzyxzzyyxx

In other words: xy |x||y|.

Generalize the corresponding inequality in the nth dimensional space.

Thank You for ComingWafik Lotfallah