Calculus with Applications to Economics

1 Descartes’ Analytic Geometry.- 1.1 Lines.- 1.2 Quadratic Equations.- 1.3 Lines and Circles.- 1.4 Hyperbole.- 1.5 Parabola.- 1.6 Long Division of Polynomials and Horner’s Rule.- 1.7 Roots of Polynomials.- 1.8 Tangents to Polynomial Curves.- 1.9 Derivatives of Polynomials.- 1.10 Polynomial and Rational Curves.- 1.11 Descartes’ Rule of Signs.- 1.12 Tangents to the graphs of inverse and implicit functions.- 2 Functions and Graphs.- 2.1 Functions.- 2.2 Graphs.- 2.3 Implicit Functions.- 2.4 An Application of Conic Sections.- 2.5 Exponents and Logarithms.- 2.6 Cobb-Douglas Functions.- 2.7 Inverse Functions.- 2.8 Trigonometric Functions.- 2.9 Inverse Trigonometric Functions.- 3 Limits and Continuity: The є - δ Method of Weierstrass.- 3.1 Instantaneous Rate of Change.- 3.2 Limits and Infinity.- 3.3 The Limit of a Sequence and Real Numbers.- 3.4 Continuous Functions.- 3.5 Demand and Supply Functions.- 3.6 Newton’s Method.- 3.7 Continuity of Implicit Functions.- 3.8 Classification of Points of Discontinuity.- 3.9 Three Theorems on Limits.- 3.10 Financial Mathematics and Euler’s number e = 2:71828.- 3.11 Remarkable Limits.- 4 Newton’s Method of Fluxions.- 4.1 Basic Rules.- 4.2 Derivatives of Inverse Functions.- 4.3 Tangents and Normals.- 4.4 Linearization and Leibniz Differentials.- 4.5 Related Rates.- 4.6 Rolle’s Theorem.- 4.7 Lagrange’s Theorem.- 4.8 Darboux’ Theorem.- 4.9 Critical Points.- 4.10 Concavity and Inflection Points.- 4.11 The Shape of a Graph.- 4.12 The Shapes of Implicit Functions Graphs.- 4.13 Cost Function, Revenue, Profit.- 4.14 Lorenz Curves.- 4.15 Elasticity.- 4.16 L’Hospital’s Rule.- 4.17 Taylor’s Formula.- 5 Arithmetica Infinitorum: Wallis’ Theory.- 5.1 Areas below Graphs of Monotonic Functions.- 5.2 Areas below Parabolas and Hyperbolas.- 5.3 Areas below Exponentials and Logarithms.- 5.4 Riemann’s Theory.- 5.5 Cavalieri’s Principle.- 5.6 The Rectangle Rules.- 5.7 The Trapezoidal Rule.- 5.8 The Newton-Leibniz Formula.- 5.9 Wallis’ Infinite Product.- 5.10 Brouncker’s Continued Fraction.- 5.11 Evaluation of _ and Brouncker’s Continued Fraction.- 5.12 Lorenz Curves: Robin Hood and Gini Indexes.- 5.13 Consumer’s Surplus.- 5.14 Some Problems in Arithmetica Infinitorum.- 6 Antiderivatives and Indefinite Integrals: Newton’s Theory.- 6.1 Integration Rules.- 6.2 Integration by Parts.- 6.3 Partial Fractions.- 6.4 Trigonometric Integrals.- 6.5 Implicit Functions.- 6.6 Substitutions in Definite Intgrals.- 6.7 Areas between Curves.- 6.8 The Disk Method.- 6.9 The Washer Method.- 6.10 The Shell Method.- 6.11 Elasticity.- 6.12 Applications.- 7 Euler’s Theory of Differential Equations.- 7.1 Graphical Solution of Differential Equations.- 7.2 The Isochrone of Leibniz and Perrault’s Tractrix.- 7.3 Analytic Methods.- 7.4 Integrating Factors.- 7.5 Picard’s Iterative Method.- 7.6 Numerical Solutions: Euler’s Method.- 7.7 Exponential Decay.- 7.8 Bounded Growth.- 7.9 Unbounded and Logistic Growth.- 7.10 Subtangent, Logarithmic Convexity, and Elasticity of Demand.- 7.11 The Solow-Swan growth model.- 8 Optimization.- 8.1 Level Curves and Gradients.- 8.2 General Methods of Optimization.- 8.3 Classification of Critical Points.- 8.4 The Hessian’s Method.- 8.5 Classical Surfaces and their Critical Points.- 8.6 Constraint Optimization.- References.- Index.