Solucionario Matematicas Para Economia

A-PDF MERGER DEMO

Answers Pamphlet for

MATHEMATICS FOR ECONOMISTS Carl P Simon . Lawrence Blume

W.W. Norton and Company, Inc.

Table of Contents Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 Appendix 1 Appendix 2 Appendix 3 Appendix 4 Appendix 5 Figures One-Variable Calculus: Foundations One-Variable Calculus: Applications One-Variable Calculus: Chain Rule Exponents and Logarithms Introduction to Linear Algebra Systems of Linear Equations Matrix Algebra Determinants: An Overview Euclidean Spaces Linear IndependenceLimits and Open Sets Functions of Several Variables Calculus of Several Variables Implicit Functions and Their Derivatives Quadratic Forms and Definite Matrices Unconstrained Optimization Constrained Optimization I: First Order Conditions Constrained Optimization II Homogeneous and Homothetic Functions Concave and Quasiconcave Functions Economic Applications Eigenvalues and Eigenvectors OrdinaryDifferential Equations: Scalar Equations Ordinary Differential Equations: Systems of Equations Determinants: The Details Subspaces Attached to a Matrix Applications of Linear Independence Limits and Compact Sets Calculus of Several Variables II Sets, Numbers, and Proofs Trigonometric Functions Complex Numbers Integral Calculus Introduction to Probability 1 5 9 11 13 15 25 41 45 52 55 60 63 68 77 8288 98 106 110 116 125 146 156 166 174 181 182 188 193 195 199 202 203 205

ANSWERS PAMPHLET

1

Chapter 2 2.1 i) y 3x 2 is increasing everywhere, and has no local maxima or minima. See figure.* ii) y 2x is decreasing everywhere, and has no local maxima or minima. See figure. iii) y ( x 2 1 has a global minimum of 1 at x , 0) and increasing on (0, ). See figure. 0. It is decreasing on

iv) yx 3 x is increasing everywhere, and has no local maxima or minima. See figure. v) y x3 x has a local maximum of 2 3 3 at 1 3, and a local minimum of 2 3 3 at 1 3, but no global maxima or minima. It increases on ( , 1 3) and (1 3, ) and decreases in between. See figure. vi) y |x| decreases on ( , 0) and increases on (0, ). It has a global minimum of 0 at x 0. See figure. 2.2 Increasing functionsinclude production and supply functions. Decreasing functions include demand and marginal utility. Functions with global critical points include average cost functions when a fixed cost is present, and profit functions. 2.3 1, 5, 2.4 a) x f) 2.5 a) x 1 2, 0. 1; b) x x 1; c) all x; d) x 0. 1, 2, d) all x. 1; e) 1 x 1;

1, x

1, b) all x, c) x

2.6 The most common functions students come up with allhave the nonnegative real numbers for their domain. 2.8 a) 1, b) 1, c) 0, d) 3.

2.8 a) The general form of a linear function is f (x) mx b, where b is the y-intercept and m is the slope. Here m 2 and b 3, so the formula is f (x) 2x 3. b) Here m 3 and b 0, so the formula is f (x) 3x.

*All figures are included at the back of the pamphlet.

2

MATHEMATICS FOR ECONOMISTS

c) We know m butneed to compute b. Here m 4, so the function is of the form f (x) 4x b. When x 1, f (x) 1, so b has to solve the equation 1 4 1 b. Thus, b 3 and f (x) 4x 3. d) Here m 2, so the function is of the form f (x) x 2, f (x) 2, thus b has to solve the equation 2 b 2 and f (x) 2x 2. 2x b. When 2 2 b, so

e) We need to compute m and b. Recall that given the value of f (x) at two points, m equals thechange in f (x) divided by the change in x. Here m (5 3) (4 2) 1. Now b solves the equation 3 1 2 b, so b 1 and f (x) x 1. f) m [3 ( 4)] (0 f (x) (7 2)x 3. 2) 7 2, and we are given that b 3, so

2.9 a) The slope is the marginal revenue, that is, the rate at which revenue increases with output. b) The slope is the marginal cost, that is, the rate at which the cost of purchasing x units increases...