A faster way to derive the formula of the curvature of y=f(x) (no vector calculus)
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- Опубликовано: 10 июн 2024
- Big thanks to @ianfowler9340 for this wonderful way to come up with the formula for the curvature of y=f(x), which doesn't require vector calculus.
Check out the vector calculus way for the curvature of r(t)=x(t)i+:y(t)j • Proving the curvature ...
What is curvature? • What is curvature? (in...
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Wow, thanks for the shout out. Did not see that coming. Very nice of you to do that.
I need to thank you for always leaving thoughtful comments! 😃
probably the curvature is equal to 0 when the function is linear or constant
You can summarise the two with linear and affine linear.
1:25 Straight lines have 0 curvature as θ does not change. Or one can utilize the formula, and say curvature is 0 at points of inflection where second derivative = 0
From the formula, we can see that the curvature will be 0 only when y' ' = 0. But it is also true that a necessary (but not sufficient) condition for an inflection point that y' ' = 0. So when y = f(x) has an inflection point at x = a, f ' '(a) = 0 and the curvature is always 0 at that inflection point (a,f(a)). All inflection ponts have 0 curvature. The straight line (see comments below) qualifies because y ' ' = 0 for ALL points on the line.
Necessary but not sufficient means that we can have y ' ' = 0 but not have an inflection point. f(x) = x^4 is an popular example. f ' '(0) = 0 but (0,0) is NOT an inflection point for y = x^4. But the curvature at (0,0) is still 0 ! So cool. When we solve f ' '(x) = 0 to find the possible x-values of any inflection points we need to use the 2nd derivative test to check that the concavity changes sign to the right and left of the candidate. One of the very few times in high school math where the converse is not necessarily true.
Love your new channels. Many topics covered at many different levels. Well done.
BTW, have you ever heard of the THIRD derivative test for inflection points? I have never been able to find anyone on RUclips that has covered this test.
Thank you! Yes I have heard and used the 3rd derivative test (same examples as the one you provided) but never made a video on that.
In France, in the equivalent of the SAT (le bac) we have to present an oral about one of our two principal subject, and you just convinced me to present my oral about "How mathematics can help doctors to find out if you have a problem with your back"
vector calculus way for the curvature of r(t)=x(t)i+:y(t)j ruclips.net/video/0TbYtbcxmm0/видео.html
I like the little detail that the "k" for curvature in "Thanks" is in black, otherwise it's a very good video
where can i get that euler's number poster? i want it
Eid Mubarak!
Awesome ❤
Mathematics is really the study
of equation.
You're not allowed to memorize how the other people reach to their conclusion ( end product formula).
They need to know how to reach to the same conclusion and use their own way of math.
That involves substituting a familiar term ( or expression) into an equation, so that they reach the same conclusion (or solution).
I kind of like this kind of philosophy. When you don't remember things.
That missing bracket is bugging me immensely!
Where?