Math n' Stuff w/ The Younge Lady: Pythagorean

Sunday, March 7, 2021

Weekly Math Problem! #41

Arc Length. Nothing strange going on here this week. I'm just getting back to a calculus problem I wanted to do last week, this week. A little application situation in the form of finding the arc length. 👀 There are times when it is needed to find the length of a curve and there aren't any regular geometric techniques that'll help with that, so calculus steps in to save the day. 😁

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

        ✔️   How to find the first derivative
        ✔️   How to work with hyperbolic functions
       ✔️   How to do integration

WMP! #41 says to:

Happy solving!

Check back on Friday, March 12th 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! 📓✏️

⬇️

And we're back to get this problem done. So, let's see... The problem is asking us to find the arc length of a function on a closed interval. There is a formula for such a situation.

In regards, to our problem, we need to identify the function, take its first derivative, and use the interval. Starting with the function, we're dealing with hyperbolic cosine who has a first derivative that is hyperbolic sine.

Now that we have the first derivative of the function and the interval, we can plug the information in the formula and perform the integration. Remember, we will come out with a numerical answer that represents the length of the function (a curve) for the specified endpoints; it makes that we are working on a definite integral.

As you can see, the arc length of f(x) = cosh(x) on [0, ln 2] is 3/4 units. Let me point out that a scientific or graphing calculator can easily evaluate hyperbolic functions for you. However, I still evaluated it by hand so you can see exactly where the final answer comes from. To assist in evaluating by hand, the exponential representation of hyperbolic cosine is used.

Here is an image of the given function and the part of the function for which we found the arc length.

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

▪️ We're you able to find the arc length?

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

Thanks for solving with me this week!

Moving right along to WMP! #42. 👩🏿‍🏫

Cheers!

The Younge Lady

Sunday, November 8, 2020

Weekly Math Problem! #27

Hyperbolic Functions. So....about hyperbolic functions...👀 They're not things that I have worked with a lot, however, they are similar enough to trigonometric functions that I don't feel too intimated to work with them. I've had the opportunity to work with them more this semester because of tutoring. So I though it would be fun to have them as part of this week's WMP!

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

        ✔️   Differentiation rules
  ✔️   Pythagorean identities
  ✔️   Factoring techniques

WMP! #27 wants us to...

Happy solving!

Check back on Friday, November 13th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

Differentiation time! The first thing I do is rewrite the function in a form that allows me to take the derivative more comfortably. What do I mean by that? Well, the second term in the function is rational, so one may be inclined to use the quotient rule to take its derivative. However, I am not so inclined. 🤫 I try to avoid the quotient rule whenever I can. 🤫

By rewriting the second term with a negative exponent, I can take the derivative of it using the power rule. Now, that that's complete, let's get to the taking the derivative of the function.

That's the derivative. It has been taken and simplified, ever so slightly, after applying the applicable differentiation rules. ...But I'm not going to leave my answer like this. Nope! There is more simplification that can be done. Here goes:

These are the kind of problems professors like to give. 👀 Let me tell you why. Are you 👂listening? Good. There was more work involved in simplifying the derivative compared to the work needed to find the derivative. In general, points are often lost in the simplification process. In my opinion, the differentiation process in this week's problem wasn't too bad. I or anyone could easily have made an error in the simplification process. How?

By not knowing you can split the second term in the derivative up as a product of two factors, where one of them can be rewritten as a single hyperbolic function--tanh(x).
By not knowing you could factor the derivative.
By not knowing/being familiar with Pythagorean identities which allow for the derivative to be further simplified.

In a case like this, and lots of other problems, the algebra is where students make mistakes...not the calculus.

▪️ What is your relationship with hyperbolic functions like??

▪️ How did you do this week's problem??

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