• Syllabus
• Midterm 1 topic list
• Midterm 2 topic list

Text: Real Analysis with Real Applications, by Kenneth R. Davidson and Allan P. Donsig, Prentice Hall, 2002.

• Set 1 Due Monday Sept. 9:
  • Read Preface, Chapter 1: Sections 1.1-1.6
  • Section 1.2: Af, D, Ea
  • Section 1.4: A, E (For A: Include the definition of what it means to add two "vectors" and multiply by a "scalar"; the book assumes this is obvious.)
  • Section 1.5: Ba, D
  • Section 1.6: Ea

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• Set 2 Due Monday Sept. 23 Change to Wednesday Sept. 25:
  • Read Sections 2.1 - 2.5, 2.8
  • Section 2.1:
  • Section 2.2: F
  • Section 2.3: Ad, B, D, L
  • Section 2.4: A, D, G
  • Section 2.5: C
  • Section 2.8: G. ALSO: Describe an explicit 1-1, onto map between [0,1] and (0,1). Hint: Consider the irrationals and rationals separately.

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• Set 3 Due Monday Sept. 30:
  • Read Sections 2.6, 2.7
  • Section 2.5: Extra Credit J (not due on Monday)
  • Section 2.6: A, G
  • Section 2.7: A, D, E

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• Exam 1 Fri. Oct. 4. Covers Chapters 1 and 2.

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• Set 4 Due Fri. Oct. 18:
  • Read Sections 3.1, 3.2, 3.4
  • Section 2.5: J (extra credit)
  • Section 3.1: A, D
  • Section 3.2: A, B (Hint: use Cauchy Criterion 3.1.5), C, E (OK to use D), H (extra credit), I (first part only), K, M, O (extra credit). Also know how to decide whether the series in S converge or diverge (not to hand in).

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• Set 5 Due Fri. Nov. 1:
  • Read Sections 3.4, 4.1 - 4.4
  • Section 3.4 B, E
  • Section 4.1 E
  • Section 4.2 A, B, Ca
  • Section 4.3 A, E, L, N
  • Section 4.4 A, C, I

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• Set 6 Due Wed. Nov. 13:
  • Read Sections 5.1 - 5.2
  • Section 5.1 B, E, I, Extra Credit: D.
    Hint for B: f(-x)=f(x) so you can restrict to positive x values at first. It might also help to sketch graphs of f(x) and sec(x) together.
    Hint for E: Look how f behaves as you approach the origin along different parabolas.
    
  • Section 5.2 A, C, G, H

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• Exam 2 Wed. Nov. 20. Covers Chapters 3 and 4 and Sections 5.1-5.2.

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• Set 7 Due Friday, Dec. 6:
  • Section 3.2 S Do any 13 of the 26. Indicate the test(s) used and justify their use, but you need not prove the tests.
  • Read the Inventing Numbers article by David Berlinski.

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• Set 8 Due Monday, Dec. 16. Required only for A or A- in course.:
  • Read Sections
  • Section 5.3 J (4 pts), Extra Credit: H (4 pts)
    Possibly useful hint for H: Show g(x,y)=f(x)-y is continuous.
  • Section 5.4: B (2 pts), E (3 pts; forward direction only), Extra Credit: A (4 pts)
  • Section 5.5: A (4 pts) Alternate (to the book's) hint: Control the "worst case choice" for a.
  • Section 7.1: A (3 pts)
  • Section 8.1: A (4 pts)
  • Section 9.1: A (2 pts), B (3 pts), C (3 pts),
    (4 pts): Do one of the following (or both for extra credit:
    1. Show that if f:(X,d_X)->(Y,d_Y) and g:(Y,d_Y)->(Z,d_Z) are continuous maps between metric spaces, then the composition g o f is continuous.
    2. Show the shift map sigma: Sigma -> Sigma is continuous, where Sigma is the set of sequences (s₀, s₁, ...) with each s_i equal to 0 or 1, and sigma defined by
       sigma (s₀, s₁, ...)=(s₁, s₂, ...).
  • Section 11.1: Prove the contraction mapping theorem for metric spaces:
    Let f:(X,d)->(X,d) be a mapping from a complete metric space (X,d) to itself which satisfies the property: d(f(x), f(y)) < c d(x,y), for some c in (0,1). Show that
    1. (3 pts) the sequence {fⁿ(x)} is Cauchy
    2. (2 pts) Show That fⁿ(x) -> p, where p is a fixed point.
    3. (1 pts) Show That for any y in X, fⁿ(y) -> p.
  • Section 12.1 A (4 pts) with the differential equation changed to f'(x)=f(x), f(0)=1. In showing the solution is valid for all real numbers, you must show that your series converges for all reals, and that the series satisfies the differential equation. Recall (from Calc II??) that inside its radius of convergence, the derivative of a series can be obtained by "term by term differentiation."

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• Corrections to HW's and Tests Due Wednesday, Dec. 18 at 5pm. Corrections handed in by Monday Dec. 16 at 5pm will be returned and may be "recorrected."

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• Final Exam Mon. Dec. 16, 4-6PM in Chem. 251. Cumulative. Optional - will only be counted if it raises your test average.

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• Grading scheme. The grading scheme on the original syllabus will be honored. However, if you choose to do so, it may be altered as follows: Weighting: Test 1 22.5%, Test 2 22.5%, Problem Sets 1-7 55%.

  Grading scale

  Grade		Percentage	Extra
  A		90%	37/42 on Problem set 8
  A-		85%	21/42 on Problem set 8
  B+		80%
  B		75%
  B-		70%
  C+		64%
  C		57%
  C-		50%
  D+		45%
  D		40%

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  • Go to the Fall 2001 Math 5201 Syllabus
  • Go to Bruce Peckham's Home Page
  • Go to the Department of Mathematics and Statistics Home Page

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  This page (http://www.d.umn.edu/~bpeckham/www) is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Friday, 20-Dec-2002 16:03:16 CST.

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