Homework 2

Due: Feb 3rd, 2021

Exercises

Exercise numbers are from the Lebl textbook.

  1. Read: https://brownmath.com/stfa/read.htm.

    1. What is one interesting or surprising idea you learned from this reading?
    2. What is one idea you will use when you read mathematics?

  2. Let \(f : [0,1] \to [0,1]\) be continuous. Show that \(f\) has a fixed point. In other words, there exists \(y \in [0,1]\) such that \(f(y) = y\).

    (Hint: Use the Intermediate Value Theorem.)

  3. 5.1.3

  4. One of 5.1.8 or 5.1.10

  5. Choose one of:

    1. [TBB, Exercise 8.2.6] Calculate \(\int_0^1 x^p \, dx\) (for whatever values of \(p\) you can manage) by partitioning \([0,1]\) into subintervals of equal length.

    2. [TBB, Exercise 8.2.7] Calculate \(\int_a^b x^p \, dx\) (for whatever values of \(p\) you can manage) by partitioning \([a,b]\) into subintervals \([a,aq]\), \([aq,aq^2]\), \(\dotsc\), \([aq^{n-1},b]\) where \(a q^{n-1} = b\). (Assume \(0 < a < b\).) (Note that the subintervals are not of equal length, but that the lengths form a geometric progression.)

    For these problems find the values of Riemann integrals using the definition in terms of Darboux sums, not the Fundamental Theorem of Calculus.

Instructions

If you don’t have LaTeX on your computer, you can use Overleaf. Upload the PDF to Gradescope. Don’t submit the LaTeX source, just the PDF.

  • When you upload to Gradescope, please mark which page of the PDF has your answer to each question!

Email your instructor (that’s me) if you have questions or need help.

Please include your name and the homework number (this is Homework 1) within the document. Some additional formatting instructions are in the syllabus. To summarize:

  • Use a new page (\newpage) for each problem.
  • State which question you are answering and the actual question. Then, start your answer in a new paragraph.
  • Use environments such as proof and theorem (via \begin{proof}...\end{proof}).
  • Use 12pt option \documentclass[12pt]{amsart} and \linespread{2.4}.

Please find and use the LaTeX template linked on the course website.

You are encouraged to work together on the homework!

Additional reading