NEWS:

• Final exam will be on Friday, May 18, 8:00-10:00 (in the usual room CHM0128)
  solution
• Practice problems for final exam:
  2017 final (skip problems 5(b) and 10(c)) ,   solution
• Assignment 3, due Wed, May 9, 8pm
• Exam 2 was on Wed, Apr. 16.
  solution
  Practice Exam A ,   Practice Exam B             solutions A ,   solutions B
  Practice Exam C with solutions ,   Practice Exam D with solutions
• Solution of Exam 1
• CHANGED EXAM DATE: Exam 2 will be on Wednesday, April 18.
• Class notes: Here is the whole Chapter 5 .
  Proposition 7.15:
  Here is the case \(n=2\):
  Assume \(f\) is twice differentiable near \(a\), and \(f''(a)\) exists. Let \(p_2(x) := f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2\). Then \[ \lim_{x\to a} \frac{f(x)-p_2(x)}{(x-a)^2}=0. \] Proof: Using L'Hospital's theorem (case \(\frac{0}{0}\)) gives \[ \lim_{x\to a} \frac{f(x)-p_2(x)}{(x-a)^2} = \lim_{x\to a} \frac{f'(x)-f'(a)-f''(a)(x-a)}{2(x-a)} = \tfrac{1}{2} \lim_{x\to a} \left[ \frac{f'(x)-f'(a)}{x-a} -f''(a) \right] \] Note that we do NOT assume that \(f'''\) exists. Obviously, a corresponding result holds for \(n=3,4,5,\ldots\).
• Exam 1 will be on Wed, Feb. 28.
  Practice problems: MATH410 exam from Fall 2017
  solutions
• Note that series are covered in section 9.1 in the textbook.
• Assignment 2: posted on Feb.9, due on Feb. 16 at 9pm. Problems from the textbook:
  "determine whether the statement is true or false and justify your answer": If the statement is true, show how to prove it using the results from class. If the statement is false, give a counterexample.
  Section 2.1 (p.32): 1,3 (use Thm 1.5 in the textbook),10
  Section 2.3 (p.42): 1,2,7
  Section 2.4 (p.46): 1,2
• Assignment 1 was posted on Jan. 31. It is due on Feb. 7 at 9pm.
  Solution
• Please read the explanation for \(b^{1/2}\) below. I carefully spelled out the complete argument, using only the properties of an ordered field and the least upper bound property.
• There are no office hours on Tue Jan. 30. Instead, there will be office hours on Thu, Feb. 1, 10-12.
• Textbook: Advanced Calculus , Second Edition, Patrick M. Fitzpatrick Pure and Applied Undergraduate Texts, Volume 5 American Mathematical Society, Providence RI, 2009
  We will cover sections A, 1, 2, 3, 4, 7.1-7.3, 8.1-8.3, 9. I will often proceed in a different way and cover material which is not in the textbook. Please read the course notes below.
  Note: The book lists for about 87$ new, and only 69.60$ for AMS members (check out how to become a student member of AMS). The bookstore sells it new for about 87$ and used for about 69$. It is the same edition that was published in 2006 by Thomson Brooks/Cole, but originally sold by them for much more!
  Errata

Course information

• SYLLABUS: Information about exams, homeworks, grades

Additional Course Material (Required Reading)

• Using the Newton method to approximate \(b^{1/n}\)
• Defining the square root \(b^{1/2}\) of a positive number \(b\)
• The real number system, sequences, series, sets of real numbers
• Functions, continuity, differentiation
  Riemann integral
• Uniform convergence

  A sequence of functions \(f_n\) on \([a,b]\) converges uniformly to a function \(f\) on \([a,b]\):
  For all \(\epsilon>0\) exists \(N\) such that
      for any \(n\ge N\) , \(x\in[a,b]\) we have \(|f_n(x)-f(x)| < \epsilon\)

  We say in this case: "\(f_n(x)\) converges to \(f(x)\) uniformly for \(x\in[a,b]\)".
  Main results: (will not be on final exam):
  \(f_n\) continuous \(\implies\) \(f\) continuous
  \(f_n\) uniformly continuous on \([a,b]\) \(\implies\) \(f\) uniformly continuous on \([a,b]\)
  \(f_n\) Riemann integrable over \([a,b]\) \(\implies\) \(f\) Riemann integrable over \([a,b]\) and \(\int_a^b f_n \to \int_a^b f\)
  Differentiation: For continuously differentiable functions \(f_n\) on \([a,b]\)
  \(f_n\to f\) pointwise, \(f_n'\to g\) uniformly on \([a,b]\) \(\implies\) \(f\) continuously differentiable over \([a,b]\) and \(f'=g\).
  You have to know the definition of "uniform convergence". But the theorems from this chapter will not be on the final exam.