## MAT 2680 Differential Equations - Reitz

### "...how it differs from the rocks"

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 II.  Don’t remember? This is your chance to refresh your memory.

Assignment (Due Tuesday, April 25th, 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 15 terms of the Taylor Series for $y=e^{2x}$ at the point $x=0$.

1. Confirm that you watched the videos (say something like “I watched all the videos.”)
2. Post ONE of the terms you found while solving the exercise above (for example, you might say “One of the terms is $\frac{4x^5}{15}$“).  CHALLENGE: Don’t use the same term as anyone else.
3. Ask a question about Taylor Series, either about one of the videos (tell me which one), or about Taylor Series in general. Make it an honest question – something you are confused about, or stuck on, or genuinely interested in.

Extra Credit.  For extra credit, answer someone else’s question.

Here are the videos:

Video 1: Taylor/Maclauren Series intuition

Video 2: Taylor Series for Cosine at $x=0$ (Maclaurin Series)

Video 3 (OPTIONAL): Taylor Series for Sine at $x=0$  (NOTE: This video is OPTIONAL, but it might help you better understand the next video)

Video 4: Visualizing Taylor Series Approximation

1. I have watched all the videos assigned. One of the terms I found while solving the exercise above is (8x^(7)/315). I am a bit confused about the main difference between a Taylor and Maclaurin series. After researching my question all I really found was that Maclaurin was a speacial case of Taylor series expanded at x=0. Still a little confused though.

2. I watched all the videos. One of the terms I found from the exercise above is (2/3)x^4. I am genuinely interested in how taylor polynomials have the ability to approximate arbitrary functions.

3. I watched all the videos except the optional one. (I will watch it at a later date) I’m still a little confused to if I am doing the steps correctly. The term I found was (16x^4)/24. The question I have is, Is there a faster short cut way to write the result to the series instead of taking the time to write it out n amount of time since its a repetition of a pattern?

• I believe you could find a formula that is different for each function that could tell you the exact iteration without doing calculations with the use of algebra. This formula would include n for the number of orders you have went through and x. If I have a n that is 100 and I have the formula specific to e^2x for a maclaurin polynomial I would just plus in the n and get the answer.

There is a link below that can explain this idea better than I can.

• I believe you could find a formula that is different for each function that could tell you the exact iteration without doing calculations with the use of algebra. This formula would include n for the number of orders you have went through and x. If I have a n that is 100 and I have the formula specific to e^2x for a maclaurin polynomial I would just plus in the n and get the answer.

There is a link below that can explain this idea better than I can.

4. I have watched all of the videos. The term I have found was (2/315)x^8. The question I have is similar to Mohammed Uddin, is there a faster way of approaching these types of questions?

5. I have watched all the videos and one term I found from the given exercise is
(64 * x^(4)) / 720. The question i would ask is “Does the Taylor series and Maclaurin series works for rational function which may contain asymptotes?”. I tried following the same step as in the video but the solution im getting seems off for the function f(x) = 1/ x at point x = 0. I know the function at 0 is undefined but it was just a though that i had.

6. I have watched all the videos. My answer to the given exercise is 1. One question I do have is if there is a way to estimate how many iterations of a Taylor series I would have to do to get an accurate estimation of a function from some point x1 to another point x2 without having to go through the calculations? Could I know what order to stop at before doing calculations to get an accurate estimation of a function between two x values?

7. I have watched all of the videos provided by Khan Academy, but there are more useful tutorial videos based on Taylor Series on Youtube upon Research. One of the term i found is (8/6)^3. Finding these terms can be challenging since this is new to learn but one question i have and would like to understand is, under what circumstances is a function equal to the sum or overall solution of it’s Taylor Series? i am interested in learning if there is a way in making it more simpler to find these terms.

Thank You
Ziaur Khan

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