Understanding The Third Derivative Geometrically

We explore the geometric meaning of the third derivative by seeing what happens when it is positive/negative. Our geometric interpretation can be stated rigorously, and we provide a simplified proof of this.

Further reading:
Byerley, C., & Gordon, R. A. (2007). Measures of Aberrancy. Real Analysis Exchange, 32(1), 233-266.
Gordon, R. A. (2004). The aberrancy of plane curves. The Mathematical Gazette, 89(516), 424-436.
Schot, S. H. (1978). Aberrancy: Geometry of the third derivative. Mathematics Magazine, 51(5), 259-275.

00:00 Intro
00:36 Intuitive sketches
02:24 Geometric interpretation
04:24 Simplifying assumptions
06:18 An equivalent geometric setup
08:01 Statement of result
08:52 Proof
11:20 Third derivative = 0 case

[shlokdave]

Pretty much how we define and look at skewness. That one is the third moment around the central tendency, ie the mean. The fourth derivative mirrors the properties of kurtosis ie the spread around the tails. Its nice to see how everything ties up so nicely. Thank you for a nice video. Perhaps a segment on skewness would add to the understanding plus application of the concept.

[FildasFilipi]

"the rate of increase of inflation was decreasing"

- Richard Nixon 1972, the first use of the 3rd derivative in economics 😃

[txikitofandango]

The way I always thought of the 3rd derivative:
When I'm cruising at constant speed in my car, my back's position is at zero relative to the seat back. When I accelerate, the seat back squishes proportionally. So the seat squish and car acceleration differ by two derivatives. Therefore, the third derivative of car motion is related to the speed of seat squish.

So 1 m/s³ of car jerk equals so many cm/s of seat squish. Roughly.

[Simpson]

I love it :D

Positive slope: position starts negative, then reaches zero

Positive curvature: slope starts negative, then reaches zero

Positive "skew:" curvature starts negative, then reaches zero

[eliyahzayin]

This made a surprising amount of sense to me. A while ago I tried to find the "asymmetry" of a curve and found that the most sensible definition is the derivative of curvature with regards to arc-length. Since curvature is analogous to the second derivative, then this asymmetry would be analogous to the third derivative

[jens]

Your choice of topics and case studies are so tasteful and refreshing! Thank you❤

[emanuellandeholm]

In mechanics, there's this entire hierarchy for time derivatives of displacement: position, velocity, acceleration, jerk, snap, crackle, pop. I'm not even kidding. :D

[shadow-htgk]

Great channel man, definitely underrated

[bernardlemaitre]

Thanks ! I Always ask me what should be useful in the third derivative, now i have the answer !

[Khashayarissi-obyj]

beautiful video.
With luck and more power to you.
hoping for more videos.

[ianfowler]

I'd like your take on the following:

Third Derivative Test for Inflection Points.
For y = f(x) at x = c:
(c, f(c)) is an inflection point iff f ' '(c) = 0 and f ' ' '(c) <>0
If f ' '(c) = 0 and f ' ' '(c) = 0 then no conclusion can be drawn. In this case you must go back to the 2nd derivative test and check the concavity around x = c.

This is just anther way of saying that the slopes of the tangent reach a max or a min at the point of inflection.

[Essentialsend]

have been looking for this a while. Thx :-)

[kindreon]

Wow that's a sick visual representation. I'm working with something related to this in cognitive science surprisingly. It's kind of like how fast you're pushing your foot down on the gas pedal.

[Generalth]

Very cool! Although I'm curious how often we'd have a situation where we're dealing with the third derivative but we don't want to reach for a full algebraic approach yet. Anyway, I wonder if this can be extended any further.

[tuongnguyen]

Can the third derivative be used for optimization

[alexismiller]

8:42 no assumption about Taylor series are needed; little-o notation gets the job done too
btw is the eps'eye'lon pronunciation of epsilon common somewhere? It's the first time I hear it

[TheMrbigfresh]

Perhaps you could do a video on a geometrical understanding og fractional derivatives aswell? :D That topic boggles my mind.

[behzat]

So, the 3rd derivative of a circle arc at any point is 0. Am I missing something?

[kepler]

What would happen if the 100th derivative is positive

[conanedojawa]

‏‪11:07‬‏ but maybe the remaining terms be greater than that term ? Who know!