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! #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