Calculus with Applications to Economics

This book presents classical Calculus in a novel way by integrating examples from modern Economics. Drawing inspiration from historical algebra textbooks—rich with buy-sell problems that once prepared students for the economic challenges of their times—the book offers a modern counterpart designed for today's Calculus students, many of whom will pursue careers in business and management. Readers will discover, for example, why Descartes could not derive a formula for the tangents to logarithmic curves, why banks employ functions that describe explosive growth, and why production functions are often modeled by the Cobb–Douglas form. The book also explains the contrasting shapes of demand curves—why a product with many substitutes has a demand curve that is convex downward, whereas a monopoly’s demand curve is convex upward—and shows how the elasticity of demand can be used to achieve maximum revenue, among many other intriguing insights. Mathematics enthusiasts will appreciate the captivating account of Brouncker’s continued fractions and their role in approximating π to many digits as early as 1655. Meanwhile, students of Economics will benefit from a comprehensive treatment of Optimization Theory, covering topics from single-variable problems to the application of Lagrange’s multipliers and utility theory. By interweaving historical insights with practical applications, this book not only reinforces fundamental concepts of Calculus but also demonstrates their relevance in solving modern economic problems. Each chapter is structured to present a historical narrative that elucidates the development of key mathematical ideas, followed by modern examples that illustrate their application in Economics. This dual approach enhances the learning experience and encourages both critical thinking and creative problem-solving. Ultimately, the book serves as a bridge between the theoretical elegance of classical mathematics and the dynamic challenges of contemporary economic analysis. It is our hope that this work will inspire students and educators alike to explore the rich interplay between Mathematics and Economics, fostering a deeper appreciation for the enduring relevance of classical ideas in today’s rapidly evolving academic and professional landscapes.

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 " _ 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.

Sergey Khrushchev is a Full Professor at Satbaev University, where he has been serving since August 1, 2018. Previously, he was a Full Professor at the International School of Economics at Kazakh-British Technical University from August 1, 2011, to May 31, 2018. He also held professorship positions at Eastern Mediterranean University in North Cyprus (2008-2010) and Atilim University in Ankara, Turkey (2001-2008).