Taylor Series. How did I get to this topic for this week? Well, it was inspired by my learning of Fourier series over the summer. (Yeah...so, we'll talk about that in another post.) I do remember learning about Taylor and Maclaurin series in college, but I can't say that it's a topic that has come up in calculus tutoring sessions. Nonetheless, I want to remember how to do these types of problems, so here it is as a WMP!
To solve this week's problem in completion, you need to recall the following math skills:
✔️ How to find derivatives
✔️ How to evaluate functions
✔️ How to work with a series
WMP! #16 says to...
Happy solving!
Check back on Friday, August 21st for the solution, which will be posted below ⬇️.
Shameless plug: Follow me on Instagram @TheYoungeLady
✏️📓 Solution Time! 📓✏️
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It's been super long since I've solved a problem like this. Since the Taylor series is centered around x = 0, the problem is really a Maclaurin series for the function. First, I went through the process of calculating the derivatives and evaluating the function and its derivatives at x = 0.
These derivatives got out of hand quickly and are annoying to calculate by hand. If you keep going, you will see that the power of the polynomial part of the derivatives just keep increasing. This also means that evaluating these derivatives at x = 0, yield zero. I wasn't really getting anywhere, so I had to do a little research for help. Then I found a technique someone mentioned, which made sense but I didn't realize I could employ it here--substitution.
Substitution allows you to work with a simplified version of the original function. Actually, you're working with the outer function whose series can more easily be found by hand.
Once the series has been found, just undo the substitution that was done in the first place. Then you'll have your answer.
WolframAlpha has a Taylor Series Calculator widget. I discovered this in my research. Check it out and see what you think. Did I use it? Of course I did! I needed reassurance. Don't judge me. 😜
Here is the graph of the original function with one approximation:
**This plot was created using Geogebra's Graphing Gaclulator.
I did more work than I thought I would, but it was worth it. My 🧠 is getting 💪🏿.
◾️ Did you know how to solve this problem right away, or did you have to do some research like I did??
◾️ Comment below with your responses and let me know what you thought about this week's problem.
Moving right along to WMP! #17
Cheers!
The Younge Lady
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