Center for Economic Research and Graduate Education Charles University Economics Institute Academy of Science of the Czech Republic
A COOK-BOOK OF MATHEMATICS
Viatcheslav VINOGRADOV
June 1999
CERGE-EI LECTURE NOTES
1
A Cook-Book of MAT HEMAT ICS
Viatchesl
Calculus Integration Techniques
Aim To introduce dierent techniques of integration. Learning Outcomes At the end of this section you will be able to: Understand the process of integration by substitution, Understand the process of integration by parts.
In
Formula: Integral Calculus
xn+1 1. xn dx = +c n +1 2. kxdx= k xdx,
where k is a constant.
3.[ f (x) g(x)]dx = f (x)dx g(x)dx 4. ex dx = ex + c 5. f ' (x)e f ( x) dx = e f ( x) + c
1 6. dx = ln x + c x 7. lnudu = u lnu u + c au 8. au du = +c ln a
ECO201
Section 1
Comparative Statics Qualitative analysis Quantitative analysis Rate of change For example, consider the case of Mr. Mamun. He drives from Uttora to Gulshan 1. Average change Marginal change
Measuring Rate of Changey = f ( x)
x0 initial
ECO201
Section 1 Lecture 6
Necessary and Sufficient Condition:
Pre-requisite to a solution but does not ensure that the value is our solution to a problem. Course registration and Scholarship distribution.
Finding the determinant10 4 A= A = 10.5 8.4 = 50
ECO201
Lecture 4 Section 1
Matrix Algebra:
c1 P1 + c 2 P2 = c 0
1 P1 + 2 P2 = 0
c1 1 c 2 P1 c 0 P = 2 2 0
a11 x1 + a12 x 2 + a13 x3 . + a1n x n = d 1 a 21 x1 + a 22 x 2 + a 23 x3 . + a 2 n x n = d 2 . a m1 x1 + a m 2 x 2 + a m 3 x3 . + a mn x n = d m a 2
Lecture 4
ECO201 Section 1
General Market Equilibrium: In a Market products are interrelated. Every commodity has a substitute and a complement. Coke and Pepsi, Tea and Sugar etc. A demand function thus should take account of its own price as well as pric
ECO201
Section 2 Lecture 2
Domain and Range: Problem Types of Function Constant Function: Range consists of only one element y = f ( x) = 10 i.e y = 10 Polynomial Function: Multi term function
y = a 0 + a1 x + a 2 x 2 + a3 x 3 + . + a n x n
Linear Quad
Mathematical Economics- is an approach in economics, where economists use mathematical tools for stating problems and use existing theorems to aid reasoning. Mathematical economics-uses mathematical symbols, tools and equations to state problems instead
ECO201 Homework 1 Group 1 1. Find the equilibrium Price (P*) and Quantity (Q*) from the following modelQd = 10 2 P Qs = 5 + 3P Qd = Q s 2. Find the equilibrium Price (P*) and Quantity (Q*) for both commodities from the following modelQd 1 = 82 3P1 + P2 Qs
ECO201 Homework 2 Group 1 1. Find the equilibrium Price (P*) and Quantity (Q*) for both commodities from the following modelQd 1 = 82 3P1 + P2 Qs1 = 5 + 15 P1 Qd 1 = Qs1 Qd 2 = 92 + 2 P1 4 P2 Qs1 = 6 + 32 P2 Qd 2 = Q s 2 Using a) Cramers Rule b) Matrix in
Assignment-2 1. Given the consumption function C = a + bY (with a > 0; 0 < b < 1): a) Find its marginal function and its average function. b) Find the income elasticity of consumption CY , and determine its sign, assuming Y > 0. 2. The supply function of
Assignment 1 1. The demand and supply functions of a two commodity market model are as follows: Qd1 = 18 3P1 +P2 Qs1 = - 2 + 4P1 Find Pi* and Qi* (i = 1, 2). Qd2 = 12 + P1 - 2P2 Qs2 = - 2 + 3P2 (Use fractions rather than decimals.)
2. Solve the following
PROBLEM SET Problem 1: A two product firm faces the following demand and cost functions: C= Q12+2Q22+10 Q1=40-2P1-P2 & Q2=35-P1-P2 a. Find the output levels that satisfy the first order condition for profit maximisation (Use fractions) b. Check the second
c1 P1 + c 2 P2 = c 0
1 P1 + 2 P2 = 0
c1 c 2 P1 c 0 P = 2 2 1 0 a11 x1 + a12 x 2 + a13 x3 . + a1n x n = d1 a 21 x1 + a 22 x 2 + a 23 x3 . + a 2 n x n = d 2 . a m1 x1 + a m 2 x 2 + a m 3 x3 . + a mn x n = d m a11 a 21 a 21 a 22 a m1 a m 2 ( m n) Ax = d El
Lecture 2 ECO201 Domain and Range: Problem Types of Function: Constant Function: Range consists of only one element y = f ( x) = 10 i.e y = 10 Polynomial Function: Multi term function y = a 0 + a1 x + a 2 x 2 + a3 x 3 + . + a n x n Linear Quadratic Cubic