Download Week 3 - CMU Math

Optimization Problems:
1.
2.
3.
4.
5.
6.
Understand the problem: what is unknown? what is given? conditions?
Diagram: identify given & required quantities.
Notation: for function to be optimized and other unknowns.
Express this function in terms of other quantities.
Function of one variable: eliminate others by using relations between unknowns. Find domain.
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 y2=2x 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?
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?
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 y=F(x) proceeds at specific point.
y
y
y=x2
x=a
Area Problem
y
S
0
1
x
0
y
x
1
0
y
L8
L4
A – area of region S
0
b
1
lim Rn 
n
3
x
1
S
a
0
R8
R4
x=b
y=f(x)
x
1
0
lim Ln 
n
x
1
1
3
As n increases, Rn and Ln become better and better approximation to A  Define
A  lim Rn  lim Ln
n 
In general case, A  lim
n 
where xi
*
n

i 1
f ( xi )x  lim
n 
n

i 1
n
f ( xi 1 )x   f ( xi )x,
is some sample point in the interval [xi,xi+1].
i 1
*
n 
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 = [1:10];
% 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;
A*v;
% v must be a column vector
Loops: for i=1:10
% some commands here
end
C = fix(10*rand(3,2));
%create a 3 by 2 matrix, of integers 0->9
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 U[0,1]
ind = find(x>=0.5);
% ind is an array of indices satisfying ….
clear all;
help
lookfor