Optimization Problems:

1
Optimization Problems

1. Understand the problem what is unknown? what is
   given? conditions?
2. Diagram identify given required quantities.
3. Notation for function to be optimized and other
   unknowns.
4. Express this function in terms of other
   quantities.
5. Function of one variable eliminate others by
   using relations between unknowns. Find domain.
6. Find abs min/max.

Algebra Find two positive numbers whose product
is 100 and sum is minimal.
Geometry Find the area of largest rectangle that
can be inscribed in a semicircle of radius r.
Find point on parabola y22x closest to the point
(1,4).
Economics The manager of 100-units apartment
complex knows from experience that all units will
be occupied if the rent is 800 per month. A
market survey suggest that, on average, one
additional unit will remain vacant for each 10
increase in rent. What rent should the manager
charge to maximize the revenue?
2
Real life During the summer months Terry makes
and sells necklaces on the beach. Last summer he
sold the necklaces for 10 each and his sales
averaged 20 per day. When he increased the price
by 1, he found that he lost 2 sales per day.
Assuming linear relation between the price and
the number of sales per day, what should be the
selling price be to maximie his profile? The
material for each necklace costs Terry 6.
Farmer wants to fence an area of 1.5 million
square feet in a rectangular field and then
divide it in half with a fence parallel to one of
the sides of rectangle. How can he do this so
that the cost is minimized?
The top and bottom margins of a poster are each
6cm and the side margins are each 4cm. If the
area of printed material on the poster is fixed
at 384cm2, find the dimension of the poster with
smallest area.
A painting in art gallery has height h and is
hung s.t. its lower edge is a distance d above
the eye of observer. How far from the wall should
be observer stand to get the best view?
3
df. F antiderivative of f on an interval I, if
F?(x) f(x) ?x? I.
Th. If F antiderivative of f on I, then most
general antiderivative of f on I is F(x)C,
C-const.
Find function given knowledge about its
derivative Equation that involves derivatives
of function is called differential equation.
Geometry of antiderivatives Direction field
each segment indicates direction in which yF(x)
proceeds at specific point.
4
Area Problem
A area of region S
As n increases, Rn and Ln become better and
better approximation to A ? Define
In general case,
where is some sample point in the
interval xi,xi1.
5
A word (or two) about MatLab
Vectors v ones(n,1) generate a row vector
of ones a v a is a column vector of
ones b zeros(n,1) a row vector of zeroes v
110 a vector 1 2 3 .. 10 a 1 2 3 4
5 6 a 2 by 3 matrix Matrix Operations A
B C A B C Av v must be a column
vector Loops for i110 some commands
here end C fix(10rand(3,2)) create a 3
by 2 matrix, of integers 0-gt9 fix round
toward zero plot(x) plot the values in a
vector x bar(x) create a bar graph form a
vector sum(x) hist(x) create a
histogram from the values of x hist(x,n)
histogram with n bins x rand(n,1) a vector
of n random numbers U0,1 ind find(xgt0.5)
ind is an array of indices satisfying . clear
all help lookfor