simon blume_1994_mathematics_for_economists_solution_manual

# Mathematics for Economists

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Answers Pamphlet for MATHEMATICS FOR ECONOMISTS Carl P. Simon Lawrence Blume W.W. Norton and Company, Inc. A-PDF MERGER DEMO

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Table of Contents Chapter 2 One-Variable Calculus: Foundations 1 Chapter 3 One-Variable Calculus: Applications 5 Chapter 4 One-Variable Calculus: Chain Rule 9 Chapter 5 Exponents and Logarithms 11 Chapter 6 Introduction to Linear Algebra 13 Chapter 7 Systems of Linear Equations 15 Chapter 8 Matrix Algebra 25 Chapter 9 Determinants: An Overview 41 Chapter 10 Euclidean Spaces 45 Chapter 11 Linear Independence 52 Chapter 12 Limits and Open Sets 55 Chapter 13 Functions of Several Variables 60 Chapter 14 Calculus of Several Variables 63 Chapter 15 Implicit Functions and Their Derivatives 68 Chapter 16 Quadratic Forms and Definite Matrices 77 Chapter 17 Unconstrained Optimization 82 Chapter 18 Constrained Optimization I: First Order Conditions 88 Chapter 19 Constrained Optimization II 98 Chapter 20 Homogeneous and Homothetic Functions 106 Chapter 21 Concave and Quasiconcave Functions 110 Chapter 22 Economic Applications 116 Chapter 23 Eigenvalues and Eigenvectors 125 Chapter 24 Ordinary Differential Equations: Scalar Equations 146 Chapter 25 Ordinary Differential Equations: Systems of Equations 156 Chapter 26 Determinants: The Details 166 Chapter 27 Subspaces Attached to a Matrix 174 Chapter 28 Applications of Linear Independence 181 Chapter 29 Limits and Compact Sets 182 Chapter 30 Calculus of Several Variables II 188 Appendix 1 Sets, Numbers, and Proofs 193 Appendix 2 Trigonometric Functions 195 Appendix 3 Complex Numbers 199 Appendix 4 Integral Calculus 202 Appendix 5 Introduction to Probability 203 Figures 205

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ANSWERS PAMPHLET 1 Chapter 2 2.1 i ) y 5 3 x 2 2 is increasing everywhere, and has no local maxima or minima. See figure.* ii ) y 5 2 2 x is decreasing everywhere, and has no local maxima or minima. See figure. iii ) y 5 x 2 1 1 has a global minimum of 1 at x 5 0. It is decreasing on ( 2‘ , 0) and increasing on (0 , ). See figure. iv ) y 5 x 3 1 x is increasing everywhere, and has no local maxima or minima. See figure. v ) y 5 x 3 2 x has a local maximum of 2 6 3 p 3 at 2 1 6 p 3, and a local minimum of 2 2 6 3 p 3 at 1 6 p 3, but no global maxima or minima. It increases on ( 2‘ , 2 1 6 p 3) and (1 6 p 3 , ) and decreases in between. See figure. vi ) y 5 | x | decreases on ( 2‘ , 0) and increases on (0 , ). It has a global minimum of 0 at x 5 0. See figure. 2.2 Increasing functions include 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 2, 0. 2.4 a ) x 1; b ) x . 1; c ) all x ; d ) x 6 1; e ) 2 1 # x # 1 1; f ) 2 1 # x # 1 1, x 0. 2.5 a ) x 1, b ) all x , c ) x 2 1 , 2 2, d ) all x . 2.6 The most common functions students come up with all have the nonnegative real numbers for their domain.

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