Weekly Math Problem #50

Limits. For the most part, I understand limits. It reminds me of integration...where I get the idea but most of the work is knowing the techniques and when to implement them. Such is limits. There are different techniques and knowing when to implement them is important...well...it's important in math. If you're not into math, you most likely don't care. 🤷🏿‍♀️

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

        ✔️     How to find limits involving radical expressions 

Here is WMP! #50:

Happy solving!

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

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✏️📓 Solution Time! 📓✏️

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Just when you think you have it all figured out...you don't, sometimes. 😩 Let's get into this solution. 

The first thing I did was substitute -1 into the expression to see if a value pops out...but I was met with an indeterminate form.

 

The indeterminate form doesn't help me so, I tried another method--rationalizing the numerator--to see if that would help me with substitution.

With the rationalized form of the expression, I tried substitution again.

Lo and behold...indeterminate form again. 😩 Now, I need another method. This one, I know, won't fail me--L'Hôpital's Rule.

Ahhh...an answer, finally! All of that work to come up with 1/6. 😅🤷🏿‍♀️

▪️ How do you solve this problem??

▪️ Did you go straight to L'Hôpital's Rule??

▪️ 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! #51. 👍🏿

Cheers!

The Younge Lady

Weekly Math Problem #48

Radical Equation. I was in a pinch and decided to look to the topic of solving radical equations for this week's problem. Based on my observations from working with others, mistakes tend to happen after the radical disappears--meaning students seem to understand the mechanism by which the radical should "disappear" and stumble to solve the resulting equation. Let's see how solving this one goes.

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

        ✔️     How to use exponent rules
        ✔️     How to factor a trinomial

WMP! #48 says to:

Happy solving!

Check back on Friday, May 7th 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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Solving this equation is straight forward. The radical expression is already isolated so, you can make the radical "disappear" and solve the resulting cubic equation. The factor by grouping method was used to solve the equation, after it was fixed to be equal to zero. 

Checking the potential solutions above wasn't too bad so, it was done below.  

Here is a graph of the two functions involved in the original equation, with the intersection points in black.

If you stuck around to this point, you're a star!

▪️ Let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next, WMP! #49. 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem! #46

Integration. I'll always find my way back to Calculus. 😉 I came across this problem on an exam review sheet. There was a note on the sheet for the students to skip this problem because, it wouldn't appear on the exam. I was intrigued so, it's appearing on the blog this week. 👩🏿‍🏫 I don't really recall solving an integral like this when I first learned integration in college. Since that was too long ago, my recollection is probably off. 

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

        ✔️     How to perform a substitution technique
        ✔️     How to perform completing the square method

(Do you see 👀 the hint I gave above? You're welcome. ☺️) Here is WMP! #46:

Happy solving!

Check back on Friday, April 16th 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 is my solution. I hope you can follow it. My goal is for it to be clear. I got started by dealing with the denominator.

Next, I rewrote the original problem with the completed square and saw that I can now perform u-substitution to simplify the integrand.

After completing the u-substitution, I wasn't able to integrate the function. Upon inspection, (hopefully) it can be seen that trigonometric substitution needs to be used to complete the integration.

Ahhh...integration is complete...but the process isn't complete. 🤨 Conversions need to occur so that the answer can in term of x to match the original problem. Plus, I need to figure out what cos 𝜃 is equal to.

Now, the process is complete! 🥳 Let me know what you did differently, or if you didn't do anything differently.

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

Thanks for solving with me this week!

Up next, WMP! #47. 💪🏿

Cheers!

The Younge Lady

Weekly Math Problem! #23

Radical Equation. The last time I had a radical equation on here was back in February, when I showed a sample WMP! It's time for another one, so here we go.

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

        ✔️ How to deal with radical equations
        ✔️ How to solve quadratic equations
        ✔️ How to solve linear equations
        ✔️ How to substitute values for checking potential solutions

WMP! #23 says:

Happy solving!

Check back on Friday, October 9th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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I did a similar problem type when I posted my WMP! Sample Edition problem back in February. This time, the radical equation involved a square root within a square root. Here's my solution:

As you can see, this problem yielded one extraneous solution and one valid solution.  

 

▪️ How do you feel when you saw this problem?? 

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

Thanks for solving with me this week!

See you next week for WMP! #24

👍🏿

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! #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