Mathematical Analysis 1 (Engineering Sciences)

Fall 2026/27        Instructor: Prof. Jonathan Ben-Artzi

Syllabus


  1. Basic notions:
    • Sets
    • Elements of mathematical logic
    • Sets of numbers
    • Cartesian product
    • Relations in the Cartesian plane
    • Factorials and binomial coefficients
  2. Sequences:
    • Subsequences
  3. Functions:
    • Definitions and examples
    • Range and pre-image
    • Surjectivity, injectivity, and invertibility
    • Monotone functions
    • Composition of functions
    • Elementary functions
  4. Limits and continuity:
    • Neighborhoods
    • Limits of sequences
    • Limits of functions
  5. Properties and computation of limits:
    • Uniqueness of the limit and local sign of a function
    • Algebra of limits
    • Comparison theorems
    • Indeterminate forms of algebraic type
    • Substitution Theorem
    • Theorems on limits of sequences
    • Fundamental limits and indeterminate forms of exponential type
  6. Series:
    • Series and their partial sums
    • Geometric series
    • The Cauchy Condensation Test and p-series
    • Necessary condition
    • Convergence tests for series with non-negative terms
    • Alternating series
    • Absolute convergence
  7. Local properties of functions:
    • Landau symbols
    • Infinitesimal and infinite functions
    • Order and principal part of infinitesimals and infinites
    • Asymptotes
  8. Global properties:
    • Theorem of Existence of Zeroes (Bolzano's Theorem)
    • Range of a continuous function defined on an interval
    • Invertibility of continuous functions
    • Lipschitz and uniformly continuous functions
  9. Derivatives:
    • The derivative
    • Differentiation rules
    • Where differentiability fails
    • Extrema and critical points
    • Theorems of Rolle, Lagrange (mean value theorem), Cauchy
    • First and second finite increment formulas
    • Monotonicity intervals
    • Higher order derivatives
    • Convexity and inflection points
    • Qualitative study of a function
    • De l'Hôpital's Theorem
  10. Taylor expansions and applications:
    • Taylor formulas
    • Expanding the elementary functions
    • Operations on Taylor expansions
    • Local behavior of a function via its Taylor expansion
  11. Integral calculus:
    • Primitive functions and indefinite integrals
    • Rules of indefinite integration
    • Definite integrals
    • Cauchy integral
    • Riemann integral
    • Properties of definite integrals
    • Integral mean value
    • Fundamental Theorem of Integral Calculus
    • Rules of definite integration
    • Differentiation of integrals with functional limits
  12. Improper integrals:
    • Improper integrals (Type I and Type II)
    • The Integral Test
  13. Complex numbers:
    • Algebraic operations
    • Cartesian coordinates
    • Trigonometric and exponential form
    • Powers and n-th roots
    • Algebraic equations and the Fundamental Theorem of Algebra
  14. Ordinary differential equations (ODEs):
    • First-order ODEs in normal form
    • Second-order ODEs