MA 3345 Real Analysis I

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Course Goals: The course will develop a deeper and more rigorous understanding of Calculus including defining terms and proving theorems about functions, sequences, series, limits, continuity, derivatives, the Riemann integrals, and sequences of functions. The course will develop specialized techniques in problem solving.

Course Objectives:

  1. prove basic set theoretic statements and emphasize the proofs' development
  2. prove various statements by induction and emphasize the proofs' development
  3. define the limit of a function at a value, a limit of a sequence, and the Cauchy criterion
  4. prove various theorems about limits of sequences and functions and emphasize the proofs' development
  5. define continuity of a function and uniform continuity of a function
  6. prove various theorems about continuous functions and emphasize the proofs' development
  7. define the derivative of a function
  8. prove various theorems about the derivatives of functions and emphasize the proofs' development
  9. define a cluster point and an accumulation point
  10. prove the Bolzano-Weierstrass theorem, Rolle's theorem, extreme value theorem, and the Mean Value theorem and emphasize the proofs' development
  11. define Riemann integrable and Riemann sums
  12. prove various theorems about Riemann sums and Riemann integrals and emphasize the proofs' development

Course Outcomes: Students will be able to

  1. prove a basic set theoretic statement
  2. prove an appropriate statement by induction
  3. define the limit of a function at a value, a limit of a sequence, and the Cauchy criterion
  4. prove a theorem about limits of sequences and functions
  5. define continuity of a function and uniform continuity of a function
  6. prove a theorem about continuous functions
  7. define the derivative of a function
  8. prove a theorem about the derivatives of functions
  9. define a cluster point and an accumulation point
  10. state the Bolzano-Weierstrass theorem, Rolle's theorem, extreme value theorem, and the Mean Value theorem
  11. define Riemann integrable and Riemann sums
  12. prove a theorem about Riemann sums and Riemann integrals