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! 📓✏️

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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

Weekly Math Problem! #13

Integration. Are you familiar with CLEP exams? I've heard the acronym before but didn't know what it stood for or what kind of exams they were. Now, I know. I got to find out a little more about CLEP by working with a student who will be taking a CLEP Calculus exam. CLEP stands for College-Level Examination Program. CLEP exams allow students to "demonstrate their mastery of introductory college-level material and earn college credit". There are 34 CLEP exams one can take. To get more information about CLEP exams, click here.

In helping a student prepare for the CLEP Calculus exam, the question below showed up in some practice questions.

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

    ✔️  How to solve a definite integral

    ✔️  Substitution method in integration

    

I present to you WMP! #13...

Happy solving! 

Check back on Friday, July 31st for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Here is plot of the function on the domain:

**This plot was created using Geogebra's Graphing Gaclulator.

As you can see, it's a semicircle. I can find the area of this via geometry with a straightforward calculation.

Then, there is the calculus way to do this problem. In the world of calculus, trigonometric substitution is the method to use. I decided to use the cosine substitution, because I already used the sine substitution.

As you can see, I arrived at the same answer. For me, this trig sub problem wasn't so bad to complete. However, I feel like the ones that involve tangent/cotangent, won't be so easy. 🤷🏿‍♀️ I'll present one in a future WMP.

◾️ Do you understand how to use trigonometric substitution?? 

◾️ If you've taken calculus before, what is your favorite topic from calculus?? 

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

Tired yet? No? GREAT! 😉

Up next...WMP! #14 😁

Cheers!

The Younge Lady