Math n' Stuff w/ The Younge Lady: functions

Sunday, November 29, 2020

Weekly Math Problem! #30

De Moivre's Theorem. Sometimes, when I'm thinking about what problem to post for the week, topics just come to mind. Somehow, DeMoivre and his theorem was this week's winner. 🎉 The last time I really encountered this topic was well over five years ago...way back when I took a precalculus course. Now that I think about it, 🤔 I don't recall going over this topic in any tutoring sessions over the years. But then again, I've been tutoring for a long time and may not really remember if I actually did or not. 🤷🏿

I would also like to say the following: When I learned this topic, I was enrolled in a summer school course--about four weeks long. I learned the topic quickly without knowing what the practical applications of the theorem is. It felt like the topic was being learned for the sake of learning it, without any real-world connections being made. Even though I didn't ask the question then, I can ask it now and find an answer somewhere on this internet. Okay, I am done with that small rant.

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️   How to convert complex numbers to polar form
  ✔️   How to apply DeMoivre's Theorem to find powers of complex numbers
        ✔️   How to evaluate trigonometric functions

WMP! #30 wants us to...

Happy solving!

Check back on Friday, December 4th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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Lowkey, I was a little annoyed by the question, after I began solving it. 😒 I didn't realize the numbers were gonna be this crazy. 😅 As soon as I saw 520 ...👀...smh...no words. I realized it was too late to turn back now, so I pushed forward to finish. I will say this, I was getting nervous about whether or not I was correct and needed to verify my numbers. Oh the anxiety I was having.

Before applying De Moivre's Theorem, some preliminary calculations need to be performed.

Notice, I rounded the angle, 𝛳, in degrees to the nearest thousandth. I didn't want use radians because I am more comfortable with degrees. Next up, the application of De Moivre's Theorem.

Do you see what I mean about the numbers being crazy? 🤪 ...in the trillions! I am aware that answer above is not as accurate as it should be, due to my rounding of 𝛳 to the nearest thousandth. So, I have prepared a more accurate calculation below.

My final thought: I didn't find the problem difficult to execute. Anxiety was triggered in me when I saw the first large number in my calculation. The anxiety lead me to begin doubting my mathematical abilities. Even though I am good at math, I still very much understand, what it feels like to be fearful when doing it.

▪️ What kind of emotions do you feel when doing math??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next...WMP! #31

👩🏿‍🏫➡️

Cheers!

The Younge Lady

Sunday, November 8, 2020

Weekly Math Problem! #27

Hyperbolic Functions. So....about hyperbolic functions...👀 They're not things that I have worked with a lot, however, they are similar enough to trigonometric functions that I don't feel too intimated to work with them. I've had the opportunity to work with them more this semester because of tutoring. So I though it would be fun to have them as part of this week's WMP!

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️   Differentiation rules
  ✔️   Pythagorean identities
  ✔️   Factoring techniques

WMP! #27 wants us to...

Happy solving!

Check back on Friday, November 13th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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Differentiation time! The first thing I do is rewrite the function in a form that allows me to take the derivative more comfortably. What do I mean by that? Well, the second term in the function is rational, so one may be inclined to use the quotient rule to take its derivative. However, I am not so inclined. 🤫 I try to avoid the quotient rule whenever I can. 🤫

By rewriting the second term with a negative exponent, I can take the derivative of it using the power rule. Now, that that's complete, let's get to the taking the derivative of the function.

That's the derivative. It has been taken and simplified, ever so slightly, after applying the applicable differentiation rules. ...But I'm not going to leave my answer like this. Nope! There is more simplification that can be done. Here goes:

These are the kind of problems professors like to give. 👀 Let me tell you why. Are you 👂listening? Good. There was more work involved in simplifying the derivative compared to the work needed to find the derivative. In general, points are often lost in the simplification process. In my opinion, the differentiation process in this week's problem wasn't too bad. I or anyone could easily have made an error in the simplification process. How?

By not knowing you can split the second term in the derivative up as a product of two factors, where one of them can be rewritten as a single hyperbolic function--tanh(x).
By not knowing you could factor the derivative.
By not knowing/being familiar with Pythagorean identities which allow for the derivative to be further simplified.

In a case like this, and lots of other problems, the algebra is where students make mistakes...not the calculus.

▪️ What is your relationship with hyperbolic functions like??

▪️ How did you do this week's problem??