Mathematics I – Calculus I

This introductory course aims to fill in the background of students who have already had exposure to calculus in high school, deepen their understanding of the material, and develop their ability for abstract reasoning and mathematical modeling. The course starts with the algebra of real and complex numbers. The course continues with vectors and basic vector operations, including dot and cross products (vectors in two, three, and n-dimensional spaces are discussed). The course also covers the elements of linear algebra, including matrices, determinants, inverse matrices, discussions on systems of first order linear equations, transformations, and matrix equations. The course continues with a review of the theory of elementary functions such as exponentials, logarithms, trigonometric functions, inverse trigonometric functions, hyperbolic functions, and their inverses. The topics covered include limits, continuity, derivatives of functions of one variable, mean value theorem, L’Hopital’s rule, application of derivatives to curve sketching and to simple real-life problems involving related rates, and optimization. Linear and Taylor polynomial approximations are discussed. The course includes a discussion of basic numerical methods such as the method of bisections and Newton’s method. It also has a brief discussion of functions of several variables and partial derivatives. To develop students’ ability for abstract reasoning and to reach a deeper understanding of the material, the discussion often includes proofs. The class meets four hours per week in the first semester.