## 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.

• I just noticed someone else used the same term I used. This is another term I found. 2x^16/638512875. Also Thanks Ryan , that video helped.

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?

• I watched the video you suggest, is easy if we just apply the formula. Thanks bro.

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

8. I have watched the given videos for the Taylor Series and Maclaurin Series. From working on the above problem, one of the terms that I have found is (4x^(2)/(2)). One of the questions that I have is from the video, “Visualizing Taylor Series Approximations”. I understand that we have a given function, for example, it could be sin(x) or cos(x). I am interested in understanding how the higher the order of the derivative, the greater it is in phase with the original function.

9. I watched all of the videos. Like others i wondered about the efficiency of this and if it could be improved. I also kinda wondered how it could be applied in a real world situation. The term i found was (4096e^12)/(12!) or (4e^12)/(467775)

• opps, i actually meant (4096x^12)/(12!) or (4x^12)/(467775) sorry!

• I think that one of the best use or where we practically applied the Taylor Series is to learn about the behavior of the Atom, and The usage of Taylor Polynomials where very important for Einstein in the study of Brownian motion. Essentially finding a way to empirically prove that atoms existed. Essentially what he was doing was trying to figure out how particles were being displaced over a period of time. It was modeled as a probability distribution.

10. I have watched all of the videos. One of the terms that I have found while doing this exercise is (32x^5)/5! Or (32x^5)/120. One of the questions that I had was related to the difference between Maclauren and Taylor series. In the videos, it states that they are essentially the same thing. The only difference is that the approximation of the Maclauren is centered around 0. My question is why is the distinction so significant?

11. I have watched all of the videos. One of the terms that I have found while doing this exercise is (4x^9)/2835. One of the questions that I had was related to the first video. Why is it that we use the factorial? I thought we would divide on the number of the grade so it would cancel later with the derivative.

12. I have watched all of the videos from the open lab. One of the terms that I have found while doing this exercise is 4x^(10)/2835. In the videos, the steps were very clear to what we had to solve for and how we had to do it showing us two different method. I am highly interested in understanding how the higher the order of the derivative, can be the greater phase with the original function. Many of my classmates shows and want to understand like myself, whats the easiest way to solve for these terms.

13. I watched all the videos and one of the terms I found from the exercise is (4 * x^10)/14175. My question is how to solve a Taylor series with a more complicated function.

14. I watched all the videos and one of the terms I found from the exercise is (8x^11)/155925. The question i have is how do we solve complicated terms, And i want to learn how we go about solving logarithmic function. I want to explore about taylor series in general.

15. I watched all the videos and one of the term I found from solving the exercise was(16x^15)/638512875. and my question is if we can still solve a Taylor series question with a more complicated function like if the Y= is a polynomial

16. I watched all the videos. One of the terms I found after solving for the Maclaurin series for e^(2x) is “(8(x^14)/42567525). My question is why did the question ask for Taylor when it centers at 0. Isn’t it a Maclaurin series?

17. I watched all of the videos and one of the terms I got was (128/5040)x^7 and a question I have is what is the difference between Taylor Series and Maclaurin series? I am still a bit confused on that

18. I watched all of the videos. One of the terms that I found was ((4/3) * x^(3)). I found the Taylor Series interesting by just simply using the formula f^(n) (0) * (x^(n) / n!) you can approximate any value you wish as long as you find it’s derivative value when x = 0 or another value.

19. 1) I watched all of the videos.
2) One of the terms I found for the exercise above: (64x^6/720)
3) What exactly could the Taylor Series be used for in Mechanical Engineering ?

• I have the same question but about Computer Engineering. Why do we need this for our major?

20. I watched all the videos. One of the terms I found was ((65536/2.092278989*10^13) (x^16)). When i was watching the last video, I was wondering does any of the terms touches the whole sinx wave or it just gets closer and closer but never be the same wave?

• The answer is no, none of the term touches the whole wave of the sin(x), each term just makes the calculated using Taylor series wave closer and closer to the original one. However, in theory, as it stated in the video #1 , with infinite number of terms all of the terms will be the same (as terms of the original wave) and the functions compared will look like each other. So, not one and not even plenty, only infinite number of term may build the wave identical to sin(x). While one term makes identical only one point it touches, several terms makes identical only some parts of the wave its describes and only infinite number of terms combined together may describes the line which never ends

21. I’ve watched all of the videos and one of the terms I got was (4096/6227020800)x^13. A question that I have is how can Taylor series and Maclaurin series be applied in the real world?

22. I have watched all the videos that were provided. One of the terms i found in the Taylor series y=e^(2x) is (8x^14)/42567525. My question is why would we choose Maclaurin series over the Taylor if the Taylor can be used for any given point, x=a, instead we introduce the special case of the Taylor series which is the Maclaurin?

• We do that in case we have the need to center the function at 0, thus we use Maclaurin to get to the answer quicker since we don’t have to deal with (x – c)^n but just x.

23. Prof. Reitz i have seen all the videos, I think i can get an A on the class as of now. The term that i found is 2x^2. What is more accurate Maclaurin Series or Taylor Series when dealing with trig functions?

24. i have watched all the videos, the terms i found is (2/315)x^8 i did not check if anyone else posted this same answer so another term is (8/6081075)x^13 if the first term was already posted, my question is that is there another way of finding the later term a faster way that was not explained in the videos.

25. watched all, 2x^8/315. no question

26. I’ve watched all the videos. One of the terms I found practicing with the exercise is (4x^17)/10854718875. Watching the first video, I noticed that hard to follow the logic built into Maclaurin and Taylor Series intuition, especially after our Spring break. I had to watch some parts of this video several times to follow the steps shown there. Answers for concrete questions arosen during watching these video were found in the video itself after several reviews of some not so clear part of it. Finally, only this general question remains unanswered: Is some real life challenge push ​the matematitians to look into such series in order of finding the consistent patterns or we, current and already graduated students are those who may try to implements these once discovered patterns into real word / life situations for the first time?

27. I watched all the videos. I found the term (2x)^15/15!.
My question would be: What happen if the exponent is more complex like another derivative such as e^y’.

28. After watching all the videos and doing the exercise of trying to find at least 15 terms with y=e^{2x} at the point x=0 using taylor series one of the terms I found was (4x^18)/(97692469875). My question about taylor series is how much more accurate is it than the other methods we’ve used in class ie. Eulers, Improved Eulers, Runge-kutta.

29. I watched all the videos.

the 12th term is n = 11=> (4x^12/467775)

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