Course coordinator

Eric Weber

Catalog description

MATH 414: Analysis I

(3-0) Cr. 3. F.S.SS.

Prereq: Minimum of C- in MATH 201
A rigorous development of calculus of functions of one real variable: real number properties and topology, limits, continuity, differentiation, integration, series.


Basic Analysis I, 5th Edition
ISBN: 9781718862401

The textbook is freely available at


We will cover most of the material of the first 5 chapters of the book, at a pace of roughly one chapter per two and a half weeks.

Learning outcomes

  • Define the set of real numbers, and state its basic properties.
  • Understand the Least Upper Bound Property and the Archimedean Property.
  • Define sequences of real numbers, and understand bounded and convergence sequences.
  • Understand the concept of limit superior and inferior.
  • Know the statement of the Bolzano-Weierstrass theorem, and its consequences.
  • Understand the concept of Cauchy sequences and the completeness of R
  • Define series of real numbers, and identify when they converge or diverge.
  • Understand the idea of a limit of a function; in particular, the ε-δ definition.
  • Define continuous functions, and give relevant examples of continuous and discontinuous functions.
  • State and understand the Extreme Value Theorem and the Intermediate Value Theorem.
  • Define the derivative of a function, and understand the relevant derivate rules, i.e. product, quotient, and chain.
  • State and understand the Mean Value Theorem.
  • Define the Riemann integral in terms of Darboux integrals.
  • Understand the properties of integrals, including linearity of the integral.
  • State and understand the Fundamental Theorem of Calculus.

Free expression statement

Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.

Students with disabilities

Iowa State University is committed to assuring that all educational activities are free from discrimination and harassment based on disability status. Students requesting accommodations for a documented disability are required to work directly with staff in Student Accessibility Services (SAS) to establish eligibility and learn about related processes before accommodations will be identified. After eligibility is established, SAS staff will create and issue a Notification Letter for each course listing approved reasonable accommodations. This document will be made available to the student and instructor either electronically or in hard-copy every semester. Students and instructors are encouraged to review contents of the Notification Letters as early in the semester as possible to identify a specific, timely plan to deliver/receive the indicated accommodations. Reasonable accommodations are not retroactive in nature and are not intended to be an unfair advantage. Additional information or assistance is available online at, by contacting SAS staff by email at, or by calling 515-294-7220. Student Accessibility Services is a unit in the Dean of Students Office located at 1076 Student Services Building. More information about disability resources in the Mathematics Department can be found at