Weekly Math Problem #78

2nd Derivative. If you're anything like me, then you may treat finding the derivative of a function like solving a puzzle. I find a lot of puzzles fun...until I get annoyed at how long it takes me to do it, if it's taking me too long. 🙄 Typically, finding the first derivative of function is not too bad. Finding the higher order derivatives is where some of the challenge may come in. This depends on what type of function the previous derivative yields. This week, we'll just go up to the second derivative. (If you want to go higher, let me know.)

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

       ✔️     Differentiation rules
       ✔️     How to find a derivative

            

WMP #78 says ...

Happy solving!

Check back on Saturday, April 2nd for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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Here we go. If you remember how derivatives work, then you'll remember that you have to find the first derivative before finding the second derivative. So let's do that. To find the first derivative of the function, the product rule is required.

Applying the product rule to h(x) is as follows:

Now that we've found the first derivative, we can find the second derivative...which is that the problem asked us to find. The first derivative consists of two terms--each needing the product rule to find its individual derivative. I use a green plus sign "+" to show the two terms, and their respective product rules are below their respective  diagonal arrows.

After applying the product rule twice on the first derivative and combining like terms, we have the second derivative, h''(x). 

 

▪️ Did you get the same answer for your second derivative?

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

Thank you for solving with me this week. ✏️
We're on to WMP! #79. 🤓

Cheers!

The Younge Lady

Weekly Math Problem #72

Limits. I found this week's problem in my personal archives. 😁 I'm inspired by students I'll be working with that are learning limits. I can recall learning the concept of what a limit is and how to evaluate the limit of various types of functions. Honestly, I learned limits more after my Calculus 1 course as tutor than when I was enrolled in the course. Repetition really is the key 🗝 for me. Seriously, if I don't use it, I can definitely lose it. **Whispering** "This is why I started my blog." 😊

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

        ✔️     Substitution        
        ✔️     Simplifying rational expressions
        ✔️     Web Resource: Limits (Evaluating)

             

WMP! #72 want us to...

Happy solving!

Check back on Saturday, February 5th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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To begin the process of finding the limit, we'll use the substitution method. 

Unfortunately, the substitution method yields an indeterminate form expression. What does that mean? 🤷🏿‍♀️ This means that the method used doesn't tell us whether or not the limit exists. Another method for evaluating the limit needs to be used to determine whether or not the limit exists.

The numerator of the expression is a cubic polynomial that can be factored. The form is a difference of cubes. When the cubic expression is factored, you will see that one of the factors matches the expression in the denominator. **Yes!** So, we'll simplify the expression. This time, the substitution method will work.

Nice. The limit does exist, and it's equal to 3. 

Another method that can be used to evaluate a limit that yields and indeterminate form is L'Hôpital's Rule. It involves using derivatives and yields the same results. Take a look...

I didn't do it here, but you can graph the original expression and the simplified expression to see what their graphs look like. Then verify that as x approaches 1 from the left and right, the output value is 3.

▪️ Were you able to find the limit?
▪️ Did you do something else to find the limit? If so, please share. (No judgment.)

▪️ 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! #73. 

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 #51

Differentiation. Coming back to a calculus classic this week--the derivative! I feel like calculus helps my mind to feel nice. 😌

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

        ✔️     How to find the derivative of a polynomial

WMP! #51 says...

Happy solving!

Check back on Friday, May 28th 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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We've been asked to find the derivative of a polynomial written in product form. This lends itself very easily to the product rule for differentiation. (Also, this is the way to do it because, expanding the polynomial is too time consuming. 😩) This requires us to identify the factors of the polynomial so that we can tag them accordingly. The formula below has f and g in it but, other identifiers can be used. After identifying the factors, I find the derivative for each one, then plug in all components into the formula.

As soon as you complete the substitution, you have found the derivative. The remaining work that's done is for simplification and better presentation of the derivative. 

▪️ Were you able to find the derivative??

▪️ 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 #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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Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 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! #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