lecture 2 (basic techniques) some basic techniques drawing a picture reformulate the problem use...
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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