Weekly Math Problem! #18

Fourier Series. A couple posts ago, WMP! #16 to be exact, I mentioned that I learned about Fourier Series over the summer. "Just like that?!," you may ask. Well, no. Actually, I was enrolled in a PDE (Partial Differential Equations) course where Fourier Series was one of the topics we learned...in 8 weeks. 😯  Let me tell you... 👏🏿those 👏🏿 eight 👏🏿 weeks 👏🏿 were 👏🏿 no 👏🏿 joke! [Background: The last time I was in an ODE (Ordinary ...) course, I audited it about about 8/9 years ago.)] Definitely, I was reminded of Taylor and Maclaurin series when we got to Fourier, so that's why I did a Taylor series question recently. I felt like I needed to go over it and to make sure that last time I saw it was weeks ago and not years ago. 

I find that the calculation aspect isn't too bad, especially if you have a good integral calculus background. What can be tricky, as the professor I had reminded us, is knowing the theory well enough to move forward in solving a question. 

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

    ✔️  How to deal with piecewise functions

    ✔️  Integration techniques

    ✔️  How to evaluate trigonometric functions

    

WMP! #18 asks to...

Happy solving! 

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

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Are you ready for this?? I hope so, because I'm still not ready even though I have it done. 😅 Here goes...

Here is graph of the function over the interval [-3π, 3π]:

**This plot was created using GeoGebra's Geometry App.

To begin coming up with the Fourier Series for this function, these are the components that are needed:

The first calculation consisted of finding the coefficient ak in two situations--when k=0 and when k≠0. As you see below, both situations yielded zero.

Next up, the calculation for the coefficient bk. Ahhh...this is where things get interesting. Why? Oh....just take a look 👀... 

The calculation for bk is alternating! An even k does nothing for the series, but an odd k does! To see what the Fourier series, S(x), is for f(x), I put in the various calculation results then simplified.

As you can see, cosine disappears from the final series, leaving sine. I switched out the expression for k with an expression for m that generates odd natural numbers.

Below are four series corresponding to the values where m ends at 1, 2, 3, and 7, respectively. Consequently, m matches how many terms each series has. (I hope that makes sense.) 

I graphed the four series along with the original function, so that it can be seen how each series looks in relation to the original unction. The more terms the series has, the more it closely resembles the original function.

**This plot was created using GeoGebra's Geometry App.

Whew...this was a long one, but a good one. 

◾️ Have you ever learned Fourier series or any or topics from PDE?? 

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

WMP! #19 is next❗️

Cheers!

The Younge Lady

Weekly Math Problem! #17

Complex Numbers. As if numbers weren't already complicated, there exist complex numbers. When did that happen and why? 🤷🏿😂🤦🏿‍♀️ I'm going a bit lite in this week's problem. (At least, I think it's lite.)

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

    ✔️  How to deal with imaginary numbers

    ✔️  How to multiply conjugates

    ✔️  How to square fractions

    

Here is WMP! #17...

Happy solving! 

Check back on Friday, August 28th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Did you find the problem annoying? You can tell the truth. This is a no-judgment zone. Listen, a problem like this requires your attention because mistakes can easily be made. Did I find it annoying? Well, I didn't find it the opposite of annoying and that's my answer. 😁

I did the problem in two parts. The first part was just the squaring and simplification of the rational expression:

Once that was completed, I proceeded to the second part which consisted of rationalizing the denominator of the result:

I definitely did not get this on the first try. How did I know that I was incorrect on the first try? I used Mathway - Algebra to check my answer, because I know with all of those negative signs floating around, I could've easily made a mistake...and I did. So, I went back and searched my work to locate the error. These are the kind of problems a teacher would give as extra credit. 😅

◾️ How did you feel about squaring fractions and rationalizing the denominator?? 

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

Are we up to WMP! #18 already?

Cheers!

The Younge Lady

Weekly Math Problem! #16

Taylor Series. How did I get to this topic for this week? Well, it was inspired by my learning of Fourier series over the summer. (Yeah...so, we'll talk about that in another post.) I do remember learning about Taylor and Maclaurin series in college, but I can't say that it's a topic that has come up in calculus tutoring sessions. Nonetheless, I want to remember how to do these types of problems, so here it is as a WMP! 

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

    ✔️  How to find derivatives

    ✔️  How to evaluate functions

    ✔️  How to work with a series

   

WMP! #16 says to...

Happy solving! 

Check back on Friday, August 21st for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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It's been super long since I've solved a problem like this. Since the Taylor series is centered around x = 0, the problem is really a Maclaurin series for the function. First, I went through the process of calculating the derivatives and evaluating the function and its derivatives at x = 0.

These derivatives got out of hand quickly and are annoying to calculate by hand. If you keep going, you will see that the power of the polynomial part of the derivatives just keep increasing. This also means that evaluating these derivatives at x = 0, yield zero. I wasn't really getting anywhere, so I had to do a little research for help. Then I found a technique someone mentioned, which made sense but I didn't realize I could employ it here--substitution.

Substitution allows you to work with a simplified version of the original function. Actually, you're working with the outer function whose series can more easily be found by hand.

Once the series has been found, just undo the substitution that was done in the first place. Then you'll have your answer.

WolframAlpha has a Taylor Series Calculator widget. I discovered this in my research. Check it out and see what you think. Did I use it? Of course I did! I needed reassurance. Don't judge me. 😜 

Here is the graph of the original function with one approximation:

**This plot was created using Geogebra's Graphing Gaclulator.

I did more work than I thought I would, but it was worth it. My 🧠 is getting 💪🏿. 

◾️ Did you know how to solve this problem right away, or did you have to do some research like I did?? 

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

Moving right along to WMP! #17 

Cheers!

The Younge Lady

Weekly Math Problem! #15

Quadrilaterals. Listen, geometry is not necessarily a favorite topic in mathematics for a lot of people. So, now you're thinking, "What about you, Lori? How do you feel about geometry?"  Well, it's not my favorite either 😅; I'm not in love with geometry. 🤷🏿 I will say, I am very visual, so I love the fact images, and the need to draw pictures, are a crucial part of geometry. I have lots of respect for this topic. Geometry is everywhere.

This week's problem was taken from the last administered NYS Regents High School Examination in Geometry before quarantine--January 2020, Part III, question 32. To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Plotting points on the coordinate plane

    ✔️  Slope, distance, midpoint formulas

    ✔️  Properties and theorems for quadrilaterals

    

WMP! #15 says...

Happy solving! 

Check back on Friday, August 14th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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In high school I learned the two-column proof method for geometry proofs. The proofs I remember doing the most were triangle proofs. I rarely, if ever, used the paragraph-style for writing geometric proofs. So, for this problem I'll complete the necessary computations and accompany them with statements that support those computations. 

There are various way to prove that a quadrilateral is a rhombus. This is because there are various ways to prove that a quadrilateral is a parallelogram, of which a rhombus is a special case. I will prove that quadrilateral NATS is a rhombus with the following method:

            I.   Prove that quadrilateral NATS is a parallelogram

            II.  Prove that the parallelogram is a rhombus.

 

But first, here is an image of quadrilateral NATS (the rhombus in question):

**This plot was created using GeoGebra's Geometry App.

I. Prove that quadrilateral NATS is a parallelogram. I will do this by showing that the diagonals of NATS bisect each other. The midpoint formula was used.

Diagonal NT and diagonal AS share a midpoint. NM = TM and AM = SM. Therefore, diagonals NT and AS bisect each other. Therefore, quadrilateral NATS is a parallelogram. ✔️

II. Prove that parallelogram NATS is a rhombus. I will do this by showing that the diagonals are perpendicular. 

The slopes of diagonal NT and diagonal AS are negative reciprocals of one another. Therefore, diagonal NT and diagonal AS are perpendicular. Therefore, parallelogram NATS is a rhombus. ∎

It's been quite a while since I've done one of these. This has me thinking 🤔 back to when I took my high school math Regents. I remember doing logic proofs. Maybe I'll explore that topic in a future WMP!

◾️ How do you feel about geometry?? 

◾️ What method did you use to complete the proof?? 

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

Without further ado, on to WMP! #16 

Cheers!

The Younge Lady

Weekly Math Problem! #14

Matrix Operations. Ahh...matrices. Just when you thought basic math operations couldn't get more complicated...enter numbers in block form. 😅🤦🏿‍♀️ In my years of doing mathematics, I see that every area of mathematics has practical applications. The truth is, most people won't learn math at the levels where such applications exist. Also for the people who do go on learn math at those levels where these applications exist, accompanying software is also learned. Doing that type of math by hand just isn't efficient. Linear Algebra has some cool applications but, before you can learn the math needed for such applications, you have to start with the basics.  

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

    ✔️  How to perform basic matrix operations

    

Here is WMP! #14...

Happy solving! 

Check back on Friday, August 7th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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In colored fashion, here's how the matrix calculations go:

Matrix multiplication is definitely tricky but, with a keen eye, a pattern can be seen. Yeah...don't have much to say this week; it's been a hectic one. In an upcoming WMP, I'll do a problem from the math journey I took this summer. 👀 Wondering what it is? You should stick around to find out. 😃

◾️ Have you ever done linear algebra or learned about matrix theory before?? 

◾️ If so, how long ago and what was your experience like?? 

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

Next up...WMP! #15 😏

Cheers!

The Younge Lady