MAT 125B: Real Analysis (Winter 2015)
Course materials
Some sample exams from classes given by other professors:
Midterm 1,
Midterm 2,
another Midterm 2,
Final,
another Final (without solutions).
Discussion Problems 1. These are review problems for Math 25/125A.
Discussion Problems 2. These are problems on definition of Riemann integral.
Discussion Problems 3. Problems on properties of
Riemann integral and integrable functions.
Jan. 20, 2pm: correction on Problem 2: T(1) is defined to be 0.
Jan. 27, 2pm: correction on Problem 2: In the first paragraph, T(x)>0 rather than T(x)=0.
The Lebesgue condition states that a bounded function on [a,b] is Riemann integrable
if any only if it is continuous almost everywhere on [a,b]. For a quick (but far from easy)
proof, see these
notes
by Leon Simon from Stanford. Some problems have simple solutions if we assume this result,
but it is very easy to make a mistake if you do not completely understand it!
It will not be assumed that you know the Lebesgue condition.
Discussion Problems 4. Problems on fundamental theorems.
Discussion Problems 5. Final problems on Riemann integral.
Mar. 13. Correction in the solution to problem 7, correction to statement of problem 1(b).
Arzela's Theorem is a theorem on interchange of integrals and limits that uses a much weaker
hypothesis than uniform convergence. You can read the statement
and proof in this paper by W. A. J. Luxemburg.
See this paper by J. W. Lewin and its follow-up
by the same author for a more modern take.
Midterm 1 will be on Wed., Feb. 4, 2015, in class. It covers Sections 5.1-5.4
from the book, and first four homework assignments. Topics: definition of Riemann integral
and integrable functions, connection to Riemann sums, properties of Riemann integral
and integrable functions (monotonicity, linearity, domain-additivity, composite with
continuous functions), fundamental theorem of calculus and consequences (integration
by parts, change of variables), integral and limits of functions (including
connection with power series), improper integrals (including comparison theorems and
absolute convergence). You may use any result covered in the lecture or in the discussion
without comment. The official formulation of any theorem is the one given in the lecture.
You should understand the theorems we covered in the lecture. I will not ask you
to reproduce the proofs but often understanding the proofs helps in understanding
the theory. You will need to know how to apply the theorems. Examples from the
lecture, homework problems, and Discussion Problems 2-5 offer
excellent practice (though some problems are clearly too
hard for the midterm). For addition practice, consider sample Midterm 1 above.
Solutions to Midterm 1.
Discussion Problems 6. Problems on topology of Rn.
Discussion Problems 7. Partial derivatives.
Discussion Problems 8. Differentiable functions.
Feb. 25. In the file, the title was Discussion Problems 7. Now corrected.
Discussion Problems 9. Chain Rule and Inverse Function Theorem.
Here are notes on the proof of the Inverse Function Theorem and
the Implicit Function Theorem.
Midterm 2 will be on Wed., Mar. 4, 2015, in class. It covers Sections 9.3, 9.4,
11.1, 11.2, 11.4, 11.5 and 11.6
from the book (only the portions covered in the lecture), and homeworks 5-8.
Topics: functions of several variables: limits, continuity, partial derivatives, directional
derivatives
differentiability, C1 functions, chain rule, inverse function theorem.
Again, you may use any result covered in the lecture or in the discussion
without comment. The official formulation of any theorem is the one given in the lecture.
Also again, you should understand the theorems we covered in the lecture. I will not ask you
to reproduce the proofs but often understanding the proofs helps in understanding
the theory. You will need to know how to apply the theorems.
Examples from the
lecture, homework problems, and Discussion Problems 6-9 offer
excellent practice. The level of problems you should expect are
Problems 2, 3, 5, 6 on Disc. 6; Problem 1 on Disc. 7; Problems 1-4 on Disc. 8;
and Problems 1-5 on Disc. 9. For additional practice, consider the two sample second
midterms (but bear in mind that notation is sometimes a bit different than the one we used).
Solutions to Midterm 2.
Here are my lecture notes on multidimensional integration (theory only, examples will be covered in class).
Discussion Problems 10. Multidimensional integration.
Mar. 16, 2:30pm: Corrections in Problem 6 (A and B were switched) and Problem 7
(r12 was missing in the solution) corrected.
Mar. 18, 8 pm: Corrections in Problem 4 solution.
Finals Info
Final Exam solutions