How to prove the formula of a curvature of a function (without using vector calculus)

Рет қаралды 8,069

bprp calculus basics

Күн бұрын

Пікірлер: 23

@ianfowler9340 7 ай бұрын

Wow, thanks for the shout out. Did not see that coming. Very nice of you to do that.

@bprpcalculusbasics 6 ай бұрын

I need to thank you for always leaving thoughtful comments! 😃

@ianfowler9340 7 ай бұрын

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 KZbin that has covered this test.

@bprpcalculusbasics 6 ай бұрын

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.

@bitoty9357 7 ай бұрын

probably the curvature is equal to 0 when the function is linear or constant

@MH-sf6jz 6 ай бұрын

You can summarise the two with linear and affine linear.

@cyrusyeung8096 7 ай бұрын

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

@bprpcalculusbasics 7 ай бұрын

vector calculus way for the curvature of r(t)=x(t)i+:y(t)j kzbin.info/www/bejne/ZoXFiqeYmN2go5I

@pyrelord6882 6 ай бұрын

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"

@keescanalfp5143 2 ай бұрын

9:05 : “and then divide …” - we are afraid to miss your mentioning that x" is just zero , after having already written down this before , of course .

2 ай бұрын

Very nice explanation, very good topic, very good attribution.

@nanamacapagal8342 6 ай бұрын

I like the little detail that the "k" for curvature in "Thanks" is in black, otherwise it's a very good video

@s从 7 ай бұрын

where can i get that euler's number poster? i want it

@connor-t3o6m 2 ай бұрын

let me tell you something. I cannot follow any of this. You all say it's easy. No doubt to all of you that it is indeed easy. However., no amount of learning or study on calculus, derivatives etc would be of benefit to me. This type of mathematics cannot be learnt. It is a gift. You all have this special gift. You were born with this gift. I only wish I was as well.

@mohammadshinwari9643 6 ай бұрын

Eid Mubarak!

@herbie_the_hillbillie_goat 3 ай бұрын

Some people are of a mind that for the surface of the Earth, K=0. 😁

@Iomhar 7 ай бұрын

That missing bracket is bugging me immensely!

@ensiehsafary7633 6 ай бұрын

Where?

@keescanalfp5143 2 ай бұрын

@@ensiehsafary7633, perhaps between sec²(theta) and d(theta)/dx , later d(theta)/ds . ? from about 2'51 for many minutes .

@keescanalfp5143 2 ай бұрын

@@ensiehsafary7633, or might be at 6'oo under the √ sign at the very end . ? we missed them too , but not all to bad .

@abacaabaca8131 7 ай бұрын

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.