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 ⬇️.

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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

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

Weekly Math Problem #66

Quadratic Inequality. I checked my blog and saw that I have not done a quadratic inequality yet...so, here we are. 🙂 In WMP #31, I solved an absolute value inequality. Make sure to check it out. 😉

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

        ✔️     Methods to solve a quadratic equation
        ✔️     Evaluating an expression
        ✔️     How to test an expression in an inequality

     

WMP! #66 wants us to...

Happy solving!

Check back on Saturday, December 4th 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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I got started solving this week's problem by converting the inequality to an equation. I then solved the resulting quadratic equation by using the AC method for factoring. This process requires factoring by grouping.

Now that I have values for x, I continue to solve the original inequality by considering the three regions of the number line that the values create--the region to the left of 2/9, the region between 2/9 and 3, and the region to the right of 3. I then choose a value in each region to test in the original inequality. If the result of the tested value is greater than zero, that region is part of my solution. If the result of the tested value is less than zero, that region is not part of my solution.

According to the tested results, my solution is as follows:

...And that's a wrap! 

▪️ Were you able to solve this inequality??

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

Thanks for solving with me this week!
WMP! #67 is up next. 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem #65

Definition of a Derivative. One of the nice things about math is that rules just work. Without knowing why rules work, we are able to use them for our purposes. 😁 Derivatives are a classic example of this! I'm so glad that I can just use derivative rules to find the derivatives of functions, instead of using the definition of a derivative all the time. I remember learning precalculus/calculus and being introduced to and having to use the definition of the derivative. Once I learned derivative rules, I breathed a sigh of relief. The definition required lots of writing and, sometimes, fancy algebra tricks. 🙄

Despite my laziness, we're going to use the limit definition of the derivative this week. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to use a formula (substitution)
        ✔️     How to find a limit
        ✔️     Simplifying fractions

     

WMP! #65 says...

Happy solving!

Check back on Saturday, November 27th 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 into the solution for this week's problem! 

The solution requires us to use substitution with the limit definition of the derivative. Then, evaluate the limit.

Houston...we have a problem! When evaluating the limit, we end up with an undefined, rational expression. 🤔 This doesn't tell me anything about the derivative of the function. What is does tell me is that another technique is needed to evaluate the limit. 

What we're going to do is rationalize the numerator. How do I know that? ...From experience. Remembering this technique comes from repetition over the years.  Rationalizing the numerator will yield an equivalent expression that allows us to evaluate the limit.

Now, we can replace the equivalent expression and evaluate the limit.

Now we have a function for the first derivative of the square root of x. 👍🏿 

There is a quicker way to get to this result...and that's by using the power rule for derivatives. 

I don't know about you, but I am grateful for rules and shortcuts. 😁

▪️ Were you able to find the first derivative using the limit definition??

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

Thanks for sticking around and solving with me this week
...I truly appreciate it!
Of course, WMP! #66 is up next. 💪🏿

Cheers!

The Younge Lady

Weekly Math Problem #52

Truth Table. Part of the reason I like math is because it's like solving a puzzle. I like puzzles and puzzle-type games. One official definition of puzzle--as a noun--is "a question, problem, or contrivance designed for testing ingenuity" (Merriam-Webster online dictionary). 🤔 Interesting. I mean...math definitely does that. One area of math that definitely gives me that puzzle vibe is logic. So, that's what we're doing this week. Logic was part of the NYS math curriculum when I attended high school. I vividly remember being tested on truth tables and logic proofs. Checkout WMP #26 for a logic proof.

Also, if you're into puzzles and such, check out these Math and Logic Puzzles from MATHisFUN.com.

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

        ✔️     How to use mathematical logical connectives

WMP! #52 wants us to...

Happy solving!

Check back on Saturday, June 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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Here's my truth table. If you found yourself constantly checking the truth tables for the logical connectives to make sure you weren't mixing up the T's and F's, then you're normal. It's easy to get them mixed up. I went over them multiple times.

▪️ Did you truth table have the same results??

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

Thanks for solving with me this week!

Up next, WMP! #52. 👍🏿

Cheers!

The Younge Lady