Optimization Problems:

1. Optimization Problems: • 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?

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 y=F(x) proceeds at specific point.

4. y=f(x) x=a x=b S a b 0 y y R8 R4 y y=x2 x x 1 0 1 0 1 y y S L8 L4 x 0 x x 0 1 0 1 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,xi+1].

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 = [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