Hi everyone,
After Spring Break we will be launching into a new and incredibly useful topic – using Taylor Series to solve differential equations. Of course, using this method requires you to remember something about Taylor Series (uh oh!). You should have studied them in Calculus, probably Calculus II or III. Don’t remember? This is your chance to refresh your memory.
Assignment (Due Tuesday, April 14th, BEFORE CLASS): Watch all of the following videos carefully. As you are watching, make a note of any questions you have. When you are done, complete the following exercise:
Exercise: Find at least the first 10 terms of the Taylor Series for
at the point
.
Finally, post a comment here confirming that you watched the videos, and including the following:
- Confirm that you watched the videos (say something like “I watched all the videos.”)
- Post ONE of the terms you found while solving the exercise above (for example, you might say “One of the terms is
“). CHALLENGE: Don’t use the same term as anyone else.
- Ask a question about Taylor Series, either about one of the videos (tell me which one), or about Taylor Series in general. If you don’t have a question, tell me something about Taylor Series – what do you like about them, what don’t you like, what’s the big deal, etc…
Here are the videos:
Video 1: Taylor/Maclauren Series intuition
Video 2: Taylor Series for Cosine at
Video 3 (OPTIONAL): Taylor Series for Sine at
Video 4: Visualizing Taylor Series Approximation
(for example, use 
instead of 
. Here’s an example.
is:
, which matches the right hand side of the original equation. Because of this, we would make the following guess for a particular solution:
(without the extra “t”), which will allow the left hand side of the equation to equal the 
on the right side.
.
, then the basic solutions would be:
with 
, approximate the value of 
. Use 5 steps.
. Suppose 
. What is 
? Use a step size of 
.
, use Euler’s method with 4 steps to approximate 
for the particular solution satisfying 
with 
, estimate the value of 
using 6 steps.
