Assignment: Taylor polynomial group post

Due Monday, October 24, 11:59pm

• 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

Tuesday, October 18

• Each group member should complete 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.

Thursday, October 20

• 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.

By Monday, October 24, 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 8 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 a screenshot 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 versus far away from the center.

For completing this assignment, you will earn participation credit.

Group assignments

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

Group 2: Function: $f(x)=\sqrt{x+1}$, center: $a=3$

Group 3: Function: $f(x)=\ln (x)$, center: $a=1$

Group 4: Function: $f(x)=\cos (x)$, center: $a=\frac{\pi }{6}$

Group 5: Function: $f(x)=\frac{1}{x+1}$, center: $a=2$

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