Weekly Math Problem! #5

Improper Integrals. A calculus thing. There's just something about solving integrals that I find very intriguing and that's probably because it requires more thought and "tricks" to get the solution, as compared to taking derivatives. But that's also the annoying thing--sometimes trying to figure out what sort of "tricks" are needed to help you figure the integral out. 🙄 

Like verifying trigonometric identities (see WMP! #3), these type of problems are also like puzzles to me. To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Methods for solving integrals

    ✔️  Finding limits 

    ✔️  Using logarithmic rules 

It's been a super long while since I've solved an improper integral so...here goes WMP! #5... 

Happy solving! Check back on Friday, June 5th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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I don't know how much calculus you remember. I don't know how many limit rules you remember. I don't how much you remember about exponential functions. But, if you're like me, you didn't remember enough right off the top of your head to get this problem done quickly. 🤷🏿‍♀️ 🥴 

◾️ Was your solution this long?? (...Or is it just me??) 

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

Coming up next...WMP! #6 👀

Cheers!

The Younge Lady

Weekly Math Problem! #4

System of Linear Equations. Typically, when you've encountered this topic in learning math, you've either encountered a system of two linear equations with two unknowns (from an algebra course), or a system of three linear equations with three unknowns (from a college algebra course). I don't recall ever having to solve a system of four linear equations with four unknowns, however, when I saw it I knew that I would be able to solve it. I knew what I needed to do.

Depending on what level of math you've completed, you're aware that you have a few options when it comes to solving a system of four linear equations with four unknowns. To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Methods for solving systems of linear equations

    ✔️  How to row reduce a matrix

Here goes WMP! #4... 

Happy solving! Check back on Friday, May 29th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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So, I thought it would be cool to solve this problem two ways--using the elimination method and by using Guass-Jordan elimination on the corresponding augmented matrix--just to compare how they look and see how many steps each requires to solve in completion.

Elimination Method

Guass-Jordan Elimination Method

The elimination method definitely required less steps as compared to the Guass-Jordan elimination method. As you can see, when done correctly, either method will yield the same solution. 👍🏿 The substitution method can also be used. I must say, there was something strangely sort of calming from row reducing the matrix. But then again, I'm not your typical person. 🤷🏿‍♀️ I definitely plan on having a problem from linear algebra for an upcoming WMP!  

 

◾️ Did my solution match yours?? 

◾️ Which method do you prefer when solving a system of four linear equations with four variables??

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

See you soon for WMP! #5 😉😁

Cheers!

The Younge Lady

Weekly Math Problem! #3

Verifying Trigonometric Identities. This week's problem comes from an Instagram post...imagine that! 😀  I was scrolling through my feed, like I normally do, and saw @mathematicalmodels's post. Of course I was intrigued. I saw a math problem with trig and said to myself, "Oooo...what's this?" She re-posted from Canadian teen star Maitreyi Ramakrishnan who shared this old test question.

I like these kinds of trig problems. They're like puzzles to me. To solve this week's problem in completion, you need to recall the following math skills:

     ✔️  How to use trigonometric formulas/identities 
     ✔️  Adding/subtracting fractions 

Here goes WMP #3! 

Happy proving! Check back on Friday, May 22nd for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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There is no "one way" to prove/verify this problem. In @mathematicalmodels's post, the solution was completed using the angle-addition formulas. I decided to go with the product-to-sum angle formulas just to see how it would look. Take a look for yourself...

◾️ Would you have done the problem differently?? 

◾️ How do feel about trigonometry...love it, hate it, or are you somewhere in the middle??

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

See you soon for WMP #4! 😉😁

Cheers!

The Younge Lady

Weekly Math Problem! #2

Ordinary Differential Equations (ODE). For some reason, differential equations was on my mind, even when this blog was just a thought. I couldn't help thinking about a notebook 📓 that I did some ode work in years ago. I knew where the notebook was and located the specific problem that kept coming to mind. Let me tell you...when I flipped those pages, the dates were from 2010/2011. 🔟 years ago! Whoa! 😲

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

     ✔️  How to integrate polynomial functions 
     ✔️  Completing the square technique 
     ✔️  Solving higher order polynomial equations

Here goes WMP #2! 

Happy solving! Check back on Friday, May 15th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Based on the nature of the derivative in the problem, the separation of variables method was used:

In finding the domain, I used both methods. What I'll do is show how to use the cubic formula to find the domain in a separate blog post. So, keep an 👁 out for that one. I knew the cubic formula existed but never used it. This week's problem was a great opportunity for that.

This solution had quite a few steps--separate the variables, integrate both functions, use the initial condition to find the constant, isolate y to find the explicit solution, then for accuracy, find the domain of that function. 🥴 *wipes perspiration from forehead* Despite the work, I do like this problem because different types of math were needed to solve in completion.

◾️ Did you arrive at the same answer I have above?? 

◾️ When was the last time you solved an ordinary differential equation??

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

Cheers!

The Younge Lady

Weekly Math Problem! #1

It's SPRING! It's been spring for over a month now, so it's about time I get these problems going. (I know..I know...it's been a long wait. However, I have an excuse. It's the same as everybody else....and you know what it is. 👀) For a little more info on these problems, check out my previous post. As promised, The Younge Lady's weekly math problems are here! So, without further ado, here is WMP #1:

Limits Involving Trigonometric Functions. This week's question came from a tutoring session that happened a couple months ago and is typically learned in a Calculus I course. To solve it in completion, you need to recall the following math skills:

     ✔️  How to deal with fractions 
     ✔️  How to deal with trigonometric functions 
     ✔️  Limit properties
     ✔️  Finding derivatives of trigonometric functions 

I will say that there are two ways to do this problem. Ready?

Happy solving! Check back on Friday, May 8th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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As previously mentioned, there are two ways to find this limit. The first method is algebraic in nature; it is not necessary to know derivatives to find this limit. Limits are most often taught before derivatives are, so algebraic methods are explored. Once derivatives and other topics have been taught, limits are re-explored at some point and the method of L'Hôpital's rule is used to evaluate limits that yield indeterminate forms when evaluated at the specified value.

Algebraic Method

L'Hôpital's Rule Method

That's pretty much it. Two different methods, same answer. That's one of my favorite things about mathematics--one problem can be solved multiple ways. 

◾️ Did you arrive at the same answer I have above?? 

◾️ How did you find the limit?? 

◾️ When was the last time you did a limit problem??

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

Cheers!

The Younge Lady