Wednesday, August 9, 2017
LAC 2017: Power Series, DiffEqus & Euler's Identity!
Power Series, DiffEqus & Euler's Identity!
Welcome to Life After Calculus 2017 (LAC2017)! Yes, there is life after Calculus! If you don't believe it, take a look at this blog's sidebar for evidence. This year must be the 9th LAC I've recorded on YouTube and BlogSpot.
DAY01: 2017 AB2
The first 3 days were about solving the latest AP Calculus Part IIA Free Response Questions (FRQ) which are calculator active.
DAY02: 2017 AB2BC1
These FRQs are meant to be completed with a Graphing Calculator (GC). Now-a-days said GC would most likely be a TI84C or a TI nSpire CX CAS.
DAY03: 2017 BC2
However, our LAC final project was about using coding to solve these sorts of questions instead. So we used Python and the latest version of SAGE. SAGE is a Computer Algebra System (CAS) based on python with lots of other FLOSS math utilities built in to solve problems in Calculus and even higher order Mathematics. BTW, I use FLOSS every day, don't you? FLOSS stands for Free Linux Open Source Software.
So, on the fourth day I went back to basics and talked about the 7 arthimetic operators that Python is based on: addition, subtraction, multiplication, decimal division, integer division, integer remainders and exponentiation. Python also has conditional and looping syntax that's very easy to learn and use. In fact, if you are working with sequences and series, Python's List data structure is very useful!
DAY05A: Complex Powers
It soon became apparent to me that some of my students didn't really know how to exponentiate. My students were fine calculating powers as long as the exponents were real. For example, 2^2 aka real^real, (2i)^2 aka imaginary^real and (2+2i)^2 aka complex^real were all familiar operations we could do by hand sans technology! Then I showed my students powers with imaginary exponents and complex exponents on SAGE and asked how we could accomplish these calculations by hand?
DAY05B: DeMoivre's Theorem
On the fifth day we derived Euler's Identity using McLaurin Power Series and used said identity to compute each complex power mentioned above by hand! I was running out of school days so I didn't get to DeMoivre's Theorem (Day05B) which is a nice extension of Euler's Identity making powers and roots of complex bases very easy. That would have been our sixth day but I had to skip it. See the video above for completeness sake!
DAY06: DiffEqus & Power Series
On our sixth and final day in LAC 2017 we talked about solving Variable Separable Differential Equations (DiffEqus) without separating the variables and without Anti-Differentiation! In fact, all we did was differentiate known McLaurin Power Series! Then we found we could solve non-variable separable DiffEqus the same way. We even solved 2nd order DiffEqus!
This year we had a lot less time after the AP Exam since ETS moved our exam to the second week. Also, we had our annual Math Movie Marathon every Monday and Friday. I had planned for 12 Days Of Calculus after the exam using SAGE. So here's the rest of what I planned for your viewing pleasure. Maybe we'll try this version of our final project again next year.
Since we covered Calc I and Calc II this year, most of my students will take Calc III (Vector Calculus) next year. That's why I was planning to cover Vectors on Day07 and Matrices on Day08 using SAGE (should be 2 or more days each).
DAY09: Surveyor's Formula
Finally, the ninth day was going to be devoted to a cool application of Vector Cross Products in 3D, namely the Shoelace Algorithm aka the Surveyor's Formula aka Gauss's Method of Polygonal Areas. Please see the video above for my proof of this algorithm. Like I said, my proof is based on Vector Cross Products. I don't think I've seen this proof anywhere else!
That reminds me, if you want to see the Free Fall Model with Air Resistance DiffEqu Solution, by hand without tech mind you, we did that in the Physics LAC listed in the side bar. I've never seen this solution anywhere else either! Don't forget my preCalc video on graphing Rotated Conic Sections in Polar Mode on a GC which isn't in any textbook I've used either. Aren't you lucky, getting all these original mathematics concepts here for free?
DAY10A: Encryption By Hand!
The tenth day was going to be devoted to a cool application of Matrices, namely Encryption! We were going to make our own version of the Enigma Device cracked by Alan Turing during World War II! This lesson was to have a non-tech half (Day10A) as well as a tech aka SAGE half (Day10B). I just recorded these two screencasts for the first time so you can see what I mean!
DAY10B: Encryption By SAGE!
Since I broke this up into 12 videos, I think of this as the 12 Days After Calculus! I hope to find 12 days to do this project justice next year as we only really completed 6 days. We did have fun with our Math Movie Marathon too including: Hidden Figures (Katherine Johnson, Dorothy Vaughan & Mary Jackson), The Imitation Game (Alan Turing), The Man Who Knew Infinity (Srinivasa Ramanujan), Proof (Fiction) as well as Stand And Deliver (Jaime Escalante). So we did justice to our Math Movie Marathon. Somehow we didn't get to A Beautiful Mind (John Nash), but there's always next year! Maybe we'll through in The Martian (Fiction) for fun!
Hope you enjoyed our final project!
Have a great rest of the Summer!!