II.  Academic Credit:  Three semester hours

 

III.  Course Description:  An introductory course in rigorous analysis, covering real numbers, sequences, series, continuous functions, differentiation, and Riemann integration.  Prerequisite:  Math 243 or consent of the instructor.

 

IV.  Place of the Course in the Curriculum:  This course is required in the mathematics major.  It may be used as an elective in the mathematics minor.

 

V.  Course Objectives:  This course

      A.  Provides the student with a deeper understanding of the topology of the real numbers and the foundations of the calculus, resulting in an ever greater awareness of the complexity, but also the cohesiveness of the structure of the world that God created.

     B.  Allows the student to investigate the theory on which differential and integral calculus is based so that he/she can better understand the limits and power of the associated algorithms.

      C.  Enables the student to use accepted mathematical terminology and symbolism to define and present proofs of  mathematical concepts involving the real numbers, limits and the calculus in a logical and systematic manner.

      D.  Introduces the student to the requirements that must be met if a mathematician must demonstrate that he/she really knows and understands a concept.

      E.  Requires the student to be able to develop, read, and understand mathematical proofs of the calculus using proper definitions and notation as understood by the international mathematical community.

 

 VI.  Student Outcomes:  Upon successful completion of the course, the student should be able to

      A.  Demonstrate that certain real numbers are irrational.

      B.  Give a completeness axiom for the reals and demonstrate some of its consequences.     

      C.  Determine if given sequences converge or diverge and give examples of each.

      D.  Use the definition of a sequence and/or Cauchy criterion to prove that certain sequences converge or diverge.

      E.  Use the definition of a series and/or the Cauchy criterion to prove that certain infinite series converge or diverge.

      F.  Determine when double summations and products of infinite series converge or diverge.

      G.  Determine when sets are open, closed, or neither and prove simple theorems on the properties of these sets.

     H.  Lead an informed discussion on the Cantor set and its properties.

      I.  Define compact, perfect, connected and dense sets and prove simple theorems on their properties.

      J.  Provide examples of both continuous and discontinuous functions.

      K. State and give the importance of the intermediate value theorem to the calculus.

      L.  Give a formal definition of the derivative and prove its properties.

      M.  State and prove Rolle's theorem, the mean value theorem, and the general mean value theorem with special application to L'Hospital's rule in the 0/0 case.

      N.  Give examples of continuous functions which are nowhere differentiable.

      O.  Describe the difference between uniform convergence and point-wise convergence and their implications for differentiation.

       P.  Determine when power series converge with special attention to the Taylor series.

       Q.  Define the Riemann Integral in your own words.

       R.  Determine when a function can be integrated using this definition.

       S.  Identify the Foourier Series and its uses.

 

VII.  Learning Resources:  You will need the text:  Understanding Analysis by Stephen Abbott

 

 VIII.  Evaluation:  Learning will be measured using textbook assignments, in class demonstrations of proofs, and exams.  Assignments will be picked up and graded.  These, plus the demonstrations will constitute 25% of your grade.  The other 75% of your grade will be based on the three exams (including the final) that will be given. It is possible that I may hand out challenge problems which I will ask you to work on your own and hand in at test time for bonus points.

 

IX.  Grading will be on a percent scale with A's in the 90%. B's in the 80%, C's in the 70%, D's in the 60% range.  Below 60% will be failing. 

 

X.  Attendance Policy:  You are expected to be in class.  Absences make it very difficult to do well in class.  Three unexcused absences will drop you from the class.  Notify me prior to the clss if you must be absent.

 

XI.  The instructor reserves the right to modify, amend, or change the syllabus as the curriculum and/or program require(s).

 

 Instructor:  Larry Matthews                              Office:  Meyer Hall 314

 Email:  lmatthews@blc.edu                               Phone:  344-7587