Week 2: Self-Guided Problems

When you come to class, be ready to discuss these questions, and bring any additional questions you still have after this exercise!

Submitting Your Work

Your work must be submitted to Anchor for degree credit and to Gradescope for grading.

For any work completed outside of GitHub or Gradescope:

  1. Typeset your work in LaTeX.
  • You should use \begin{enumerate} to create a numbered list, and your solutions to each problem should match with the corresponding number on the assignment.
  • For example, if an assignment has 10 numbered problems, your enumerate should have 10 \item commands, with your solution(s) to problem 1 under the first \item, problem 2 under the second and so on.
  • If you skip a problem, just leave the \item blank, otherwise the numbers of your solutions will not match the assignment document.
  1. Compile into a PDF. This can be done easily through an Overleaf account, otherwise you will need to install LaTeX.
  2. Submit the pdf to Gradescope via the appropriate submission link for the course.
  3. Upload the pdf to Anchor using the form below.

Note: Anchor submissions can occur at any time during the term, but it is critical that you upload all of your work to Anchor before the last day of the term. Gradescope submissions must be submitted before the deadline (or the late deadline, if applicable).

It is required that all assignments are submitted as a PDF generated from a LaTeX document.

Assignments submitted in any other form will earn zero credit.

Self-Guided Problems cannot be submitted late for any reason

This is due to the fact we discuss them in the class. Be sure to stay on top of these Self-Guided problems, and remember it is better to turn in an incomplete set than an empty one.

Problems

  1. Consider a sequence where the first two terms are 1 and 2.
  • What would be the next term if it was an arithmetic sequence?
  • What would be the next term if it was a geometric sequence?

  1. Create your own sequence that starts with 1 and 2. What is the rule for each term in the sequence? Is there a closed form solution for the $n$th term of your sequence? What about the sum up to the $n$th term?

  2. Write code in the language of your choice to calculate the $n$th term of the sequence defined by $$a_n = a_{n-1} + 3, a_0 = 0$$

In two different ways:

  • Using a for loop to compute from $a_0$ to $a_n$
  • Using the closed form for the expression.

Try running the code both ways for $n = 1, 100, 10,000,$ and $1,000,000$.

  1. Is there a significant time difference between the two methods? If there is, at what point do you notice a difference?

  2. Does the countability of a set imply anything about the span of the set? That is, the difference between the smallest and the largest values? What about uncountable sets?

  3. Prove f(x) is bijective. $$f(x) = \frac{3}{2}x + 6$$

  4. Show g(x) is not bijective $$g(x) = 7x^4$$

  5. What are some indicators that a function is bijective? Are there any indicators that a function is not bijective?