Year 9 Economics and Maths Research Task
In this task you will use the knowledge, understanding and skills that you have
developed in your Math and Economics classes. You will undertake an investigation into
the relationship between price and demand. The
Properties of Real Functions: Boudedness
Let f : X ! R where X R. We say f is bounded above if
is bounded above.
f (X ) = cfw_r 2 R : f (x) = r for some x 2 X
Let f : X ! R where X R. Similarly, we say f is bounded below if
CHAPTER 2 7
This chapter is meant to serve as a review of various facts about opti-
mization. Typically students will have spent several weeks studying these
methods in a mathematics course prior to studying this text.
27.1 Single variable op
September 22, 2015
1 Linear computations in Rn
1.1 Linear systems and Gauss method . . .
1.2 Matrices and vectors of Rn . . . . . . . .
1.3 Product of a matrix by a vector . . . . .
1.4 Gaussian operations ap
A satisfactory discussion of the main concepts of analysis (such as
convergence, continuity, dierentiation, and integration) must be based on an
accurately defined number concept. (Rudin)
What does this mean? All those concepts use some ide
Vector Space Over the Reals
A vector space V over R is a 4-tuple (V , R, +, ) where V is a set, + : V V ! V
is vector addition, : R V ! V is scalar multiplication, satisfying:
Closure under addition: 8x, y 2 V , (x + y) 2 V
P means P is true, P means P is false (or not P).
P ^ Q means P is true and Q is true.
P _ Q means P is true or Q is true (or possibly both).
P ^ Q stands for (P) ^ Q; P _ Q stands for (P) _ Q.
P ) Q means whenever P is true, Q also holds. (i
September 11, 2015
Open and closed sets
Exercise 1. Let B (a, r) be an open ball of center a and radius r in the normed space
(E, k k) . Prove that B (a, r) = a + r B (0, 1).
Exercise 2. Let (xk )k be a sequence on Rn that
Open and Closed Sets
A set is open if at any point we can find a neighborhood of that point
contained in the set.
Let (X , d) be a metric space. A set A X is open if
8x 2 A 9" > 0
B" (x) A
Remember that B" (x) = cfw_y 2 X : d(y , x) <
September 10, 2015
Problem Set 1
(due on Tuesday 15 September 2015)
Exercise 1. (a) Give an example of finite subset of N.
(b) Give an example of a proper subset of N which is infinite.
Exercise 2. What are the subsets of cfw_x R :
December 12, 2015
This exam is worth 50 points (6 + 6 + 9 + 6 + 13 + 10). Answer all problems in as thorough
detail as possible. Partial credit will be given even if the answer is not fully correct. If you believe
September 10, 2015
Let k k1 and k k2 be two norms on the vector space E. Prove that
k k1 + k k2 is a norm on E.
Exercise 2. Let (E, k k1 ) be a normed real vector space. Let c R be such that c > 0
f : Rn R
a) x A is global minimum of f on A if
x A, f (x) < f (x)
b) x A is local minimum of f on A if
rx > 0
x B(x, r), f (x) f (x)
c)f has maxima on A: x A such that x A, f (x) > f (x)
Negate the a
Prerequisites: You must know how to solve unconstrained and constrained optimization problems with one variable. Furthermore, you must know the first order necessary condition (FONC)
for unconstrained multivariable optimization pr
1. According to the problem will have to find the minimum and maximum points of the function:
Then we should make a large variety of steps including the finding of the partial derivatives of
the above function and create the Hess
Maximize f=81 +42 +1 2 -1 2 - 2 2 subject to constraints:
21 +32 24,
-51 +122 24,
Use Kuhn-Tucker conditions for the solution of this problem, since the task is to maximize the
We could rewrite the cons