Homework 3

Due: Feb 10th, 2021

Exercises

  1. Lebl 5.2.5. Also, show that the hypothesis of continuity is necessary (give a counterexample to the version of the statement without the continuity hypothesis).

  2. Lebl 5.2.14(a)

  3. Lebl 5.3.11

  4. Choose one of:

    1. Lebl 5.4.6
    2. TBB 8.3.10: If \(f\) and \(g\) are continuous on an interval \([a,b]\) show that \[ \Big( \int_a^b f(x)g(x)dx \Big)^2 \leq \Big(\int_a^b [f(x)]^2 dx \Big) \Big(\int_a^b [g(x)]^2 dx \Big) . \]
    3. TBB 8.8.1: For what values of \(p\), \(q\) are the integrals \[ \int_0^1 \frac{\sin x}{x^p} dx \quad \text{and} \quad \int_0^1 \frac{(\sin x)^q}{x} dx \] ordinary Riemann integrals, convergent improper Riemann integrals, or divergent improper Riemann integrals?

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.

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