Due Sunday, October 18, 11:59pm

This week your OpenLab assignment is another group post.

  • Study groups are listed here.
  • You already have each other’s emails from your last group assignment and you can direct message each other on the OpenLab. You may already have other contact information for your group.
  • You may use the same shared Google doc for your rough work if you like.
  • Meet in your group’s room on Blackboard Collaborate Ultra on Thursday at 10am to compare your work.
  • Decide who will serve as secretary to submit the group post (I suggest a different person volunteer to serve this time, but it’s okay if the secretary is the same person as last time).
  • At least one person will need a (free) account on Desmos.com so they can save and share their work. I recommend everyone sign up for a Desmos.com account, since it is extremely helpful and convenient. (You can graph without signing into your account, but then you can’t save your work.)

Instructions

The culmination of your group work will be an OpenLab post that contains a link to a Desmos graph like this one.

Each group has been assigned a function $f(x)$ and a center $x=a$ below. Your group’s Desmos must show:

  1. the graph of the function $f(x)$,
  2. the point $(a, f(a))$,
  3. the graphs of all the Taylor polynomials up to degree 9 of $f(x)$ centered at $x=a$.

This means that there will be graphs of 11 functions in total: one for $f(x)$ and one for $p_n(x)$ where $n = 0, 1, 2, \dots, 9$. Have fun with the colors!

Suggested timeline

By Thursday, October 15

  • Before you meet with your group on Thursday, each group member should have completed all the work by hand so that you can compare your answers and decide which ones are best.
  • You should also experiment with entering your answers in Desmos yourself, especially if you have not used it before. Desmos is pretty intuitive, especially if you’re not trying to do anything too fancy. A user guide is available here.
  • Decide who will serve as secretary. Also determine who is most comfortable with Desmos.

By Saturday, October 17

  • Continue to stay in touch with your group after Thursday. Everyone in the group should try to enter their answers into Desmos and then share their links with everyone else in the group. Hopefully your Desmos graphs all look the same or at least close!
  • Decide whose graph the secretary will post on the OpenLab. Share the link with the secretary by clicking on the upper-right hand side of the screen as in the figure.
Share your Desmos graph

By Sunday, October 18, 11:59pm

  • The secretary should have the group’s Desmos link by now. Title the post “Group n Taylor Polynomials” (where n is your group number) and select the category Week 7 group post.

Your post must include:

  1. group members’ names,
  2. the function $f(x)$ and center $x=a$ that your group was assigned,
  3. a photo of your hand-written work to show how you found your polynomials,
  4. the Desmos link (if you want to make your post pretty, you can also include an image of your Desmos graph; unfortunately, the embed feature doesn’t work on the OpenLab),
  5. a description of what you notice about the sequence of graphs of polynomials near the center.

For completing this assignment, you will earn participation credit.

Group assignments

Study groups are listed here.

Group 1: Function: $f(x) = \sin(x)$, center: $a= \frac{\pi}{3}$

Group 2: Function: $f(x) = e^{-x}$, center: $a= 1$

Group 3: Function: $f(x) = \sqrt{x+1}$, center: $a= 3$

Group 4: Function: $f(x) = \ln(x)$, center: $a= 1$

Group 5: Function: $f(x) = \cos(x)$, center: $a= \frac{\pi}{6}$

Group 6: Function: $f(x) = \frac{1}{x+1}$, center: $a= 2$

Group 7: Function: $f(x) = \frac{1}{x^2}$, center: $a= 1$