Math 125B: Real Analysis, Spring 2013

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Final Exam

Final exam solutions are here.

Important Dates

• Instruction begins: Mon, Apr 1
• Last day to add: Tue, Apr 16
• Last day to drop: Fri, Apr 26
• Last class: Wed, Jun 5
• Academic holiday: Mon, May 27

Exams

There will be two Midterms and a Final

• Midterm 1: Wed, Apr 24 (in class)
• Midterm 2: Wed, May 22 (in class)
• Final: Tue, June 11, 3:30 –5:30 p.m. (Exam Code N: in 66 Roessler)

Grade

Grade will based on exams and homework, weighted

• 10%: Homework
• 25%: Each Midterm
• 40%: Final

Text

Text: Foundations of Analysis, Joseph L. Taylor, 2012.

Syllabus

A syllabus for the class is here.

Notes

Some class notes are here:

• The Riemann Integral
• The Supremum and Infimum

A set of lecture notes for MAT 125B written by Steve Shkoller is here.

Midterm 1

Midterm 1 covers Sections 5.1-5.3 of the text on the definition, existence and properties on the Riemann integral and the fundamental theorem of calculus. The exam will be closed book, so you should know the precise statements of the theorems in the text and homework problems.

Solutions to Midterm 1 are here.

Midterm 2

Solutions to Midterm 2 are here. Midterm 2 covers material from Sections 7.1-7.3, 8.1, and 9.1-9.4 (up to p. 245) on the text:

• 7.1 Euclidean space
• 7.2 Norms and convergence in Rⁿ
• 7.3 Open and closed sets
• 8.1 Continuous functions in Rⁿ
• 9.1 Partial derivatives
• 9.2 The differential
• 9.3 The chain rule
• 9.4 Directional derivatives, gradients, and tangent vectors

The emphasis will be on the differential calculus in 9.1-9.4, with background material from 7.1-7.3 and 8.1 as needed.

The exam won't be closed book. You're allowed to prepare an 8 1/2 x 11 inch "cheat sheet" (front and back) and use it during the midterm, but no other references are permitted.

Some sample Midterm problems are here. You don't need to turn them in with the Homework problems. (I corrected a typo in Problem 5.)

Solutions to the sample Midterm problems are here. I strongly recommend that you try the problems yourself before you look at the solutions!

Some solutions to selected homework problems in Chapter 9 are here.

Final Exam

• The final exam will be Tuesday, June 11, 3:30--5:30 p.m.
• It will be in 66 Roessler Hall, not our regular classroom.
• The final will be comprehensive and include all the material we've covered in class.
• You're allowed to prepare an 8 1/2 x 11 inch "cheat sheet" (front and back) and use it during the final, but no other references are permitted.
• I'll hold final's week office hours on Monday, June 10, 2:30 p.m. -- 4:00 p.m.

  Here's a detailed list of the sections in the text that we've covered.

  • 5.1 Definition of the integral
  • 5.2 Existence and properties of the integral
  • 5.3 The fundamental theorem of calculus
  • 5.4 Improper integrals
  • 7.1 Euclidean space
  • 7.2 Norms and convergence in Rⁿ
  • 7.3 Open and closed sets
  • 8.1 Continuous functions in Rⁿ
  • 9.1 Partial derivatives
  • 9.2 The differential
  • 9.3 The chain rule
  • 9.4 Directional derivatives, gradients, and tangent vectors
  • 9.6 The inverse function theorem
  • 10.1 Integration over a rectangle
  • 10.2--10.3 Integration over Jordan regions
  • 10.4 Iterated integrals
  • 10.5 Change of variables

  For practice, here's a previous Math 125B final, which was taught by another Professor. (There aren't any multi-variable integration questions, and Problems 4 and 6 are a little outside the material we discussed, but you should still be able to do them.)

Homework

Homework will be assigned weekly and a hard copy will be due in class on Wednesdays. Please write clearly or type, give logical easy to follow proofs, and staple your solutions.

If you want to type good-looking mathematics, the standard tool is LaTeX, or one of its many variants. See here to get started.

Problem numbers refer to the exercises in the text.

Set 1 (Wed, Apr 10)
Sec 5.1, p. 107: 2, 3, 8, 11
In case you don't have the text yet, a copy of the Exercise Sets is here.

Set 2 (Wed, Apr 17)
Sec 5.2, p. 113: 1, 2, 6, 8, 9, 11, 14
Sec 5.3, p. 119: 4, 5, 6, 9, 10, 12
In case you don't have the text yet, a copy of Set 5.3 is here and here.

Some solutions are in the lecture notes.

• 5.2.2: Proposition 1.39
• 5.2.8: Theorem 1.31 and Proposition 2.19
• 5.2.9: Theorem 1.26
• 5.2.14: Example 1.40
• 5.3.9: Proposition 1.35

Set 3 (Wed, Apr 24)
Problem set 3 is here.
It includes a number of sample midterm problems which you should turn in as part of the homework.

Some solutions are in the lecture notes.

• 5.2.10: Theorem 1.26
• Sample question 1: Example 1.57

Set 4 (Wed, May 1)
Sec 5.4, p. 127: 9, 10, 12, 15
Two additional problems on uniform convergence, which are: here
A copy of the problems is Sec 5.4 is here

Some solutions are in the lecture notes.

• 5.4.9: Example 1.70
• 5.4.10: Example 1.68
• Additional question 2: Theorem 1.64

Set 5 (Wed, May 8)
Sec 7.1, p.167: 5, 8, 9
Sec 7.2, p.173: 1, 8, 10
Sec 7.3, p.178: 1, 4, 5, 8
Read over Chapters 7 and 8 in the text.
A copy of the problem sets is here

Set 6 (Wed, May 15)
Sec 9.1, p.228: 1, 6, 8, 10
Sec 9.2, p. 235: 1, 2, 3, 4, 8, 9, 10, 11

Set 7 (Wed, May 22)
Sec 9.3, p. 241: 2, 4, 8, 10
Sec 9.4, p. 250: 1, 2, 3, 6

Set 8 (Wed, May 29)
Sec 9.6, p. 265: 1, 2, 4, 10, 11
The additional problem here
A solution to Problem 9.6.11 on representing a parametric curve by Cartesian equations is here.

Set 9 (Wed, June 5)
Sec 10.1, p. 282: 3, 4, 10
Sec 10.2, p. 287: 3, 4
Sec 10.4, p : 2, 4
Sec 10.5, p. : 3, 4, 7