Weekly Math Problem #72

Limits. I found this week's problem in my personal archives. 😁 I'm inspired by students I'll be working with that are learning limits. I can recall learning the concept of what a limit is and how to evaluate the limit of various types of functions. Honestly, I learned limits more after my Calculus 1 course as tutor than when I was enrolled in the course. Repetition really is the key 🗝 for me. Seriously, if I don't use it, I can definitely lose it. **Whispering** "This is why I started my blog." 😊

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

        ✔️     Substitution        
        ✔️     Simplifying rational expressions
        ✔️     Web Resource: Limits (Evaluating)

             

WMP! #72 want us to...

Happy solving!

Check back on Saturday, February 5th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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To begin the process of finding the limit, we'll use the substitution method. 

Unfortunately, the substitution method yields an indeterminate form expression. What does that mean? 🤷🏿‍♀️ This means that the method used doesn't tell us whether or not the limit exists. Another method for evaluating the limit needs to be used to determine whether or not the limit exists.

The numerator of the expression is a cubic polynomial that can be factored. The form is a difference of cubes. When the cubic expression is factored, you will see that one of the factors matches the expression in the denominator. **Yes!** So, we'll simplify the expression. This time, the substitution method will work.

Nice. The limit does exist, and it's equal to 3. 

Another method that can be used to evaluate a limit that yields and indeterminate form is L'Hôpital's Rule. It involves using derivatives and yields the same results. Take a look...

I didn't do it here, but you can graph the original expression and the simplified expression to see what their graphs look like. Then verify that as x approaches 1 from the left and right, the output value is 3.

▪️ Were you able to find the limit?
▪️ Did you do something else to find the limit? If so, please share. (No judgment.)

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. 😊
Up next...WMP! #73. 

Cheers!

The Younge Lady

Weekly Math Problem #71

Eigenvectors. This week's problem is related to last week's problem, WMP #70. Check it out, if you haven't. Last week we found the eigenvalues for the matrix. This week, we're gonna find the eigenvectors associated with the eigenvalues. Feel free to refer to WMP #45 to help you with the process. 😁 

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

        ✔️     Substitution        
        ✔️     How to perform row operations
             

WMP! #71 says...

Happy solving!

Check back on Saturday, January 29th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Here's what I did to find the eigenvectors for each eigenvalue: 

And there you have it! The eigenvectors are not the same. In fact, they are linearly independent. 

▪️ Did you find the eigenvectors?
▪️ Have you ever done matrix algebra before?

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. 😊
Up next...WMP! #72. 

Cheers!

The Younge Lady

It's My 2nd Blogiversary!

 

YAY! It has officially been 2 years since I published my first blog post. If you haven't read it, you can read it here: Who's That Lady?! You're welcome.☺️ Time is going by faster than I thought. From Monday, January 20, 2020....and now we're here...Thursday, January 20, 2022.🤯 After 2 years, I'm still glad that I created this blog. 🙌🏿 

Last year (First Blogiversary), I created a word search puzzle using some words associated with my blog. This year, I decided to stick with the puzzle theme. This year we have a crossword puzzle. 

 2nd Blogiversary Puzzle! PDF

Happy cross-wording!

Your support is still appreciated and that's not going to change! Thank you for...
        ✔️ clicking the link
        ✔️ reading the blog posts
        ✔️ solving with me 
        ✔️ following me on Instagram
        ✔️ liking and commenting 
I'm grateful!

I haven't done any videos yet, but I haven't given up on that; it's still on my mind. I'm going with the ebb and flow of my life, working out the right time to make it happen. Share your thoughts below ⬇️.

Shameless 🔌 Plug:  Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee:  TheYoungeLady ( I'm gonna need it this year. 😆 )

Cheers!

The Younge Lady

Weekly Math Problem #70

Eigenvalues. This week, linear algebra came to mind. The last time I did a linear algebra related problem on this blog was in WMP #45. Check it out...it may help you with this week's problem. 👀 *clears throat*. Anyhow, eigenvalues are cool and have lots of applications. For more info, check out the Applications of Eigenvalues and Eigenvectors page from Interactive Mathematics website. 

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

        ✔️     How to find the determinant of a 2x2 matrix
        ✔️     How to solve a quadratic equation
             

WMP! #70 wants us to...

Happy solving!

Check back on Saturday, January 22nd for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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🎶Let's get it started...🎶

The process to finding the eigenvalues of a 2x2 matrix requires working with a polynomial called the characteristic polynomial. Ours will be a quadratic polynomial generated from finding the determinant of (Y - λI2), where I2 is the 2x2 identity matrix.

Now that we have the characteristic polynomial, we need to find it roots. The roots of the characteristic polynomial are the eigenvalues of the matrix. Set the characteristic polynomial equal to zero and solve. Ours is factorable, but if it wasn't, then you'd have the option to use other methods for solving a quadratic equation.

The eigenvalues have been found!

▪️ Hopefully you haven't found the process to finding the eigenvalues of a 2x2 matrix daunting.

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. 😊
📗 WMP! #71 is up next. 

Cheers!

The Younge Lady

Weekly Math Problem #69

Volume by Integration. ...And we're back! This is the first problem of 2022, and I'm starting the year off with a problem type that's been on my mind for quite some time--finding the volume of an object via integration. As I reflect on being enrolled in Calculus 2 yeeeaaaarrrsss ago, I can recall learning this topic. It has been such a long time since I've done these types of questions. I'm super rusty, but I want to do it. 

You can use a graphing calculator or other software to assist you with obtaining a visual of the solid, however, it is not required. I'm a visual person, so you know I'll be making use of graphing tools. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to find the area of a circle
        ✔️     How to integrate a function
             

WMP! #69 wants us to...

Happy solving!

Check back on Saturday, January 15th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Let's go...

After reading the problem at least twice, I needed a visual to help me along.

**This plot was created using Geogebra's Graphing Calculator. Click image to enlarge.

The grey shaded area is the bounded region. The space to the left of the y-axis shows the outline of where the solid ends up when the shaded region is rotated about the y-axis. The figure we get looks to me like a tornado, except it would be filled in to be a solid. 🤷🏿‍♀️ How would you describe the shape of this generated solid? Leave your response down below.

Now that we know what kind of shape we're dealing with, we need to find it's volume. The cool thing about this solid figure is that it's horizontal cross-section is a circle...and we know how to find the area of a circle. (I did my best to show the circle in orange above in the visual aid.) Well, to find the volume of the figure, we need to integrate the expression that represents the area of the circle.

How do I know that radius is a function of y? The axis of rotation is the y-axis, which would we would see as the center of the circle from an aerial view. The length of the radius is the distance between the y-axis and the function which is the edge of the circle from the aerial view. So, the function needs to be in terms of y. To get the function in terms of y, isolate the variable x.

Cool. Let's move forward with the integration. To do so, we need one more thing...limits of integration. When you look at our figure, you can see that the circles are growing in area from the bottom of the figure to the top. Each circle has a different area at a different value of y, so the limits of integration start from the "smallest" circle (when y = 0) to the "largest" one (when y = 4).

There you go!

▪️ Have you every used calculus to find the volume of a solid before?

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. 😊
Up next is WMP! #70. 

Cheers!

The Younge Lady