How to find the curvature of y=x^2 at (1,1)

This Calculus tutorial discusses the difference between the slope and the curvature of y=x^2 at (1,1).
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#calculus #bprpcalculus #apcalculus #tutorial #math

[@skylercourville6941]

I just want to throw this out there, thank you for bringing in some Calc 3 content! Please keep it coming.

[@robertpearce8394]

Thanks. I always think that it is a good idea to start with a straightforward example.

[@ErikOosterwal]

Because the numerator of the Kappa function for y = x^2 is a constant, the maximum curvature is found where the denominator is at its minimum. As x increases, the denominator also increases and squaring x in the denominator converts all negative values of x to positive values, so the minimum value of the denominator and maximum value of K(x) is when x = 0. 🤔

[@adrified9352]

Biggest K(x) for x^2 is 0. The graph of K(x) will look like a normal distribution peaking at K(0) = 2. This is found using 1st derivative test.

[@Flash_Skaterr]

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[@joeythreeclubs]

Maximum curvature of y=x² is 2 at x=0. Determined by the 1st derivative test.

[@Ninja20704]

It can be done without calculus as well. For all real x, 4x^2 >= 0 1 + 4x^2 >= 1 (1+ 4x^2)^(3/2) >= 1 So k(x) = 2/[(1+4x^2)^(3/2)] x=0

[@joeythreeclubs]

@Ninja20704 Wow! That's a lot faster than 1st derivative test. Can't really say we're not using calculus though, since we already used it to find the equation for curvature.

[@quidest5]

Btw the radius of curvature r = 1/k, that is the radius of tangent circle for that point

[@z000ey]

Yes! What I'd love to know is what happens with the curvature sign when you've a curve that changes direction, i.e. y=x^3, what will the sign be at x=0? Is it both - and +, due to it being curved to different sides but by the same amount?

[@coshy2748]

For y=x^3, K(x) = 6x/(1 + (3x^2)^2)^(³/²). At x=0, K(0) = 0. The curvature is like a straight line. This is due to a point moving along the curve is coming out of a concave down curvature and into a concave up curvature of equal magnitude.

[@z000ey]

@@coshy2748 yeah I see it, it's fully flat in (0,0), but would it be a "plus/minus" zero, or just a "plain" zero?

[@joeythreeclubs]

​@z000ey it would be a zero with no sign

[@bjornfeuerbacher5514]

@@z000ey What does "plus/minus zero" mean? Zero has no sign.

[@z000ey]

@@bjornfeuerbacher5514 ​ very true! stupid question... but in some cases there exists a plus or minus zero, when the answer is not equal zero but very close to zero, and then you don't put "=" equality but the sign for "approximately equal to", "≈", as it can be approximately equal but larger or approximately equal but smaller than zero.

[@maburwanemokoena7117]

In probability theory what would curvature of a density function tell us? Am looking at this formula and it seems to resemble the coefficient of skewnes

[@ianfowler9340]

So here is a good problem. Find the differential equation that must be satisfied in order for y = f(x) to achieve maximum curvature. Hint: Take the y ' ' definition and differentiate wrt x. It's pretty easy to do. You can drop the absolute value so you can also get a minimum curvature. But a curvature of -2 is really the same amount of bend as +2, just concave down. It also makes it easier to differentiate. y ' ' ' [ 1+ (y')^2] - 3 y ' ( y' ' )^2 = 0 must be satisfied for max/min curvature. i.e You know k(x) = y' ' / [1+(y')^2] ^(3/2) . Find k ' (x) and set the numerator = 0. This will generate this diifferential equation. Now here comes the really cool problem. Graph the cubic f(x) = x^3 - 3x^2, take a look, and then make a good guess as to what you think the x - value for maximum curvature might be. Now apply the differential equation to the curve f(x) = x^3 - 3x^2 and find the EXACT x-value for maximum curvature. Then aprroximate and check out your approximation. The answer may surprise you. It sure surprised me. But then you stop and think about it for a minute and it hits you - the perfect ahha moment. It makes perfect sense. Those kind of moments are what makes all this so much fun. Cheers.

[@Lussimio]

Why was the numerator 2 for K(0) and not 0?

Because the curvature formula uses the second derivative of the function in the numerator. And the second derivative of x^2 is 2 (constant function) so in case of the parabola, the curvature formula gives 2 as the numerator everywhere, only the denominator changes depending on x (because there the fist derivative if used and it's not a constant in case of the parabola).

[@Yahia.Alwaeli]

What is the unit of curvature?

[@bprpcalculusbasics]

rad/unit of the length

[@Yahia.Alwaeli]

@@bprpcalculusbasics thank you!

[@clementfradin5391]

Can you show the proof of the formula of the curvature pls 🙏

[@griffinf8469]

He actually did do a proof. You can scroll through his videos and you’ll find it. He did it not too long ago.

[@clementfradin5391]

@@griffinf8469 oh thanks you !

[@yassiryassir-rp4to]

again

[@VictorZheng-sc5sc]

first liked😇

[@billnguyen4697]

is this taught in calc 3 only.

[@sebastiangudino9377]

This concept is taught in the context of multivariable calculus, so you might even see this in a completely separate class from regular calculus depending on your curriculum