In Aviation an aircraft’s stability characteristics can be represented by a graph with this shape. The Cessna 172 has this desirable characteristic:) where X is the airplane’s pitch attitude and Y being time. If trimmed for level flight and you pull back abruptly on the yoke and let go, the plane will pitch back down negative and back positive for a short time, it will oscillate but quickly stabilize back to level flight. This is called a Positive Dynamic Stability. The opposite of this graph is called a Negative Dynamic Stability and a plane with this characteristic we’re in level flight and you let go of the pitch control, even a small amount of turbulence will cause the plane to oscillate, pitching up and down more and more until eventually the wing would stall. I believe fighter jets are dynamically unstable due to the design trade off making it wayyy more maneuverable. But this would require constant control adjustments to maintain a stable pitch attitude… or roll and yaw for that matter. This is way the flight computer adjusts the control surfaces to maintain stability. The pilot’s controls actually tells the computer what they want the plane to do and the computer makes it happen.
Lovely stuff and brings me back. I had a passionate HSC 4 Unit maths teacher back in the 80s in regional Australia. Sometimes we used to distract him by mentioning flying and he'd regale us with tall and true tales! In yr 12, he really passed on some golden wisdom, which was very helpful to me with many years of pure and applied maths at uni the following years. You are giving wings to your students Eddie! Cheers - David
@@nickyang1143there are other constants as you make your model more sophisticated and describe a system better and it’s technically phase shifted because the convention is cosine but lightly damped harmonic motion is an example. If you take a pendulum and model air resistance, the displacement over time oscillates back and forth(the cosine or sine aspect) but the magnitude of displacement decreases over time and approaches 0(e^-x aspect)
not the age demographic at all for this (22years old) but love his videos. amazing communication and teaching. it's fun watching easy concepts get explained by a pro. cheers!
Back in the day, I remember plotting these types of functions (products and quotients, composition of various elementary functions) in Extension II and my teacher had an approach that did not typically involve calculus because the graph could technically be resolved "by inspection" and a choice table of function values ("anything times 1 is itself", "anything times -1 is negative itself", "anything times 0 is 0", dividing by a very large number tends to 0, and dividing by a very small number tends to infinity")
Did extension maths more than a decade ago and barely any of this makes sense anymore 😢 Eddie is a legend tho and every student deserves a teacher with his passion and enthusiasm for education
I think one nuance not captured here is that the curve is tangent to e^(-x) at pi/2 and to - e^(-x) at 3 pi/2. I'm not sure if the examiners wanted that.
I like a trick - multiplying by \sqrt{2}/\sqrt{2} you can turn the trigonometric part into \sqrt{2}cos(x+\pi/4) - which is equal to zero when x+\pi/4 = \pi/2 or 3\pi/2 or x = \pi/4 or 5\pi/4
I remember seeing this question whilst I was running out of time. I looked at and was instantly like "bro this graph is gonna take too long and be way too convulated to understand at a first glance. Just leave it and maybe come back"... I did not have enough time to come back... Edit: Nevermind, maybe I did do this question. I vaguely remember it but I also remember one that looked similar where we had to graph it towards the end of the test so I might be getting them confused. Maybe I just didnt do the graphing portion and just did the S.P version as I felt more confident doing that part and viewed it as easier.
When we did this in my HSC in 1968, there were no calculators and no formula sheets allowed....not in maths nor in physics. You had to show the full workings on a split page and all the formulas...about 40 of them... you had to remember and understand how they were derived.
@@ti84satact12 I'd highly disagree, memorizing and critically thinking to solve the problems without any support increases your ability to do said problem solving and promotes actually learning the material to where it never leaves you and stays with you for decades. I'm a high school math teacher and I promote the use of pencil and paper calculations and problem solving.
@@j.pesquera I’m a math teacher also and I believe there are things that you can do/see more easily with technology than you can with pencil and paper! Btw technology doesn’t have to interfere with critical thinking!
Wait till you get to University Applied Mathematics and start learning: Laplace Transforms, Z Transforms, Fourier Transforms, Mellin Transforms, Henkel Transforms in the context of solving partial differential equations and then the fun begins.
What was the question again please sir? My brain melted about 45 seconds in. I just don't get equations and never will - my brain doesn't work that way. Good on those of you who can understand this! (I'm jealous in the nicest possible way).
i Definitely drew that graph wrong, i drew a sin wave with the graph touching zero at 0 and 2pi (i also use my calculator to guess for what values of x (sinx=cosx) instead of simplifying to tanx=1, i still got the same values just took me anouther 5 minutes to figure out
The old right angled triangle of unit length around the right angle and thus where the hypotenuse is in length the square root of 2 and the angle 45 degrees hence both sine and cosine give 1/root 2@@theneekofficial8829
I have a question about the area of a triangle "If I gave 3 coordinates on the certesian plane and told you to find the area of the triangle made by this and I also told you that you can't use linear algebra or trigonometry to find the area." How would you solve that
I actually have another solution with the unit circle. When rearranging the equation, you can stop at sin(x) = cos(x), because with the unit circle it becomes apparent, that this is only the case at these two angles (top right and bottom left). No need to draw the tangient graph, just the unit circle, which i think is much more intuitive :) The rest then follows exactly as you did
I mean maths is maths. as long as the numbers add up, you're good. oh also we learnt in grade 8, as just a memorised value, like times tables, trig values for 0, pi/6, pi/4,pi/3 and pi/2 plus everything that comes with those ref angles, so imagine drawing a tangent graph at all if you're gonna call the unit circle method a baby. all you need are your memorised, clearly superior exact values!@@Sir_Isaac_Newton_
From the way he keeps saying "students will be familiar with..." I think this isn't a class with students, but a teaching workshop with other teachers demonstrating how to break this down on a more understandable level.
40^0.75 isn't really a nice value. You can rewrite as (2^3*5)^(3/4) = 2^(9/4)*5(3/4), if you try to approximate it you can get 2^2*5*2^(1/4)/5^(1/4). Writing 2^(1/4) as approx 1.19 which is close to 1.2 (sqrt 2 approx 1.414 and 1.2^2 = 1.44 which is close) and 5^(1/4) as approx 1.5 (1.5^2 = 2.25, 2.25^2 = 5.0625 which is close to 5), so you get 2^2*5*1.2/1.5 = 2^2*5*0.8=2^2*4 = 16. The actual value is 15.905 so that's pretty close. 81^1.25 or 81^(5/4): 81^(1/4) = 3 so 81^(5/4) = 3^5 = 243
@@armstrongtixid6873 That's a good method for the approximation, although if I'm doing it in my head, I would think of 2^(9/4) * 5^(3/4) as 4*2^(1/4) * 5/5^(1/4) = 20 * 2^(1/4) / 5^(1/4) as it's easier for me to hold the 20 in my head while I work out 2^(1/4) / 5^(1/4). That can be done either as you did (i.e. 1.2/1.5 = 0.8) or by taking the fourth root of 0.4, which I'd do by considering the square root of 40 = 6.4 roughly, and then the square root of 64 = 8, so I get 0.4^(1/4) = 0.8 approximately. So 20 * 0.8 = 16. I guess everybody has their own tricks and preferred methods. I memorised a lot of square roots, so that's the easiest route for me, mentally.
He sticks to the Australian NSW content and very rarely (and I mean very rarely) does he do something from outside of the curriculum he teaches. So there is a high chance he will never do any JEE problems.
@@kyal I hope not, otherwise he's really patronising them. What sort of maths student needs to have explained that ab=0 only happens when a=0 or b=0? Or that e^(-x) is never 0?
I don't know how I got here, and just seeing it gave me immeasurable anxiety. Your presentation somehow managed to get me through it though, and that's major lmao.
Any1 know how to get all the answers for math, can't sleep if I don't know 😥 (standard 2 btw, seems like we got away with a way easier test than u advanced ppl) 👀
Eddie is a great teacher, but honestly, this question was not well explained. 1stly, the relative concepts need to be well introduced; 2ndly, the testing points should be identidied clearly, so that students would know where is the direction to fight for; 3rdly, when to the calculation part, it shouldn't be calulated direclty, there are many useful formaulors for people to use. Last but not least, every problem explaination should be concluded with the recap of skills gain, not just let it go.
I worry about a maths lecturer who thinks that a damped oscillation is a "weird shape". Maybe he needs to get out into the real world where damped oscillations are commonplace?
He's a high school mathematics teacher. Not a uni lecturer or college professor. The NSW HSC course only looks at functions in seperate not products of two functions, this question is to test the high achievers that have a deeper understanding of functions.
@@izaakvw So he's teaching a course that is examined by questions that are not on the syllabus? NSW needs to get a grip on what's being taught in those schools.
@@RexxSchneider You are taught how to differentiate and find the turning points using the product rule but you have to know how to sketch this function. But subbing values in your calculator helps with that.
@@izaakvw In any applied maths course at this level, students should be able to recognise expressions representing damped harmonic motion. And there's absolutely no reason why a maths teacher should miss an opportunity to reinforce that. It's not "weird".
A rather tedious plotting exercise for an exam, waste of time really once you've solved the first part. But this is typical schooling, to turn everything into pointless grunt work. The point is to prepare you for your life in the commercial sector, as a mindless tool with no thoughts or desires of your own. And the guy is so enthusiastic about it, that's what gets me, as if he's solving world hunger. What a joke.
