Vectors and Optimization Exercises
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Vectors and Optimization Exercises
1. vectors. Let vector a = (1, 2) and vector b = (1, 0). plot these vectors on (x1 , x2 ) space. calculate vectors c = a + b and d = b 一 a. plot these vectors on the same diagram. Are any of the vectors, a,b,c,d orthogonal to each other? calculate the scalar products of the corresponding vectors to prove your answer.
2. vectors. Let vector a = (1, 2, 3) and vector b = (2, 一1, 3). calculate vectors c = a + b, and d = a 一 b and scalar products of vectors
a . b
a . c
b . c
3. vectors and matrices. consider vector a = (1, 一1) and matrix B = [ 3(2) 1(1) ] .
Note that a' =(-11). This operation is called tTanspose, the notation is a' , (or aT ) turns a string vector into the column vector and a column vector into the string vector. calculate a . B (this should be a vector of size 1 X 2) and B . a' (this should be a vector of size 2 X 1). calculate scalar products a . a' (this should be a number) and a地 . a (this should be a 2 X 2 matrix).
4. Diferentiate the following functions
(a) f (x) = 3x2
(b) f (x) = x2(3)
(c) f (x) = aex
(d) f (x) = 1- -x
(e) f (x) = a ln (x)
(f) f (x) = h (g (x))
(g) f (x, 9) = 3x3 2
5. Implicit functions 1. Take the budget equation
P1x1 + P2x2 = w
Find the slope of the implicit function x1 (x2 ). show your work.
6. optimization 1. A consumer seeks to maximise her utility by choosing how much of commodities A and B to consume. Let xA and xB denote the quantities demanded, and (PA , PB ) the prices. our consumer has utility
u(xA , xB ) = ln(1 + xA ) + ln(1 + xB ),
and she is subject to the budget constraint
xAPA 十 xBPB = M
(a) Find the optimal bundle (xA , xB ).
(b) How does the level of utility u(xA , xB ) change when the consumer,s income M changes?Relate this to the value of the Lagrange multiplier.
7. optimization 2. You need to enclose a rectangular ield with a fence. You have 100 meters of fencing material. Determine the dimensions of the ield that will enclose the laTgest aTea. set this up as a constrained optimization problem and approach this with Lagrangean. Hints: use all the information to determine the objective function and the constraint. It may help to draw. Recall: what is the area of a rectangle?call the short side x and the long one y. what is the perimeter of such rectangle?
8. optimization with inequality constraints. Find
max f (x, g) = xg
s.to. x 十 g2 ≤ 2
x ≥ 0, g ≥ 0
Approach this formally via Lagrangean. write all the karush-kuhn-Tucker condi- tions. Argue that the non-negativity constraints will not bind and that the x十g2 ≤ 2 constraint will hold as equality. solve the resulting system of equations.
2023-08-30