The Third Derivative

Question

What is a practical function of the third derivative? For example, the function $y=x^3+x^2+x+1$ has a third derivative of $d^3y/dx^3=6$. What is the practical application of this? I know that the first derivative of a function $f(x)$ is how the function is changing. Given that $f'(x) = g(x)$, I know the change in $g(x)$ is equal to $f''(x)$. That is the change of the change.

Does $f'''(x)$ have any practical application that is not miniscule?

@alexjo I meant "What is the practical application of this". Thank you. — 1089, Mar 10 '16 at 06:20
Your assumption that functions with constant i derivatives are useless, is incorrect. — Narasimham, Mar 10 '16 at 08:39
@Narasimham, and where does he state that functions with constant derivatives are useless? — Eric S., Mar 10 '16 at 08:51
variation is practical (function,application) = non-miniscule — Narasimham, Mar 10 '16 at 09:26

score 3 · Accepted Answer · answered Mar 10 '16 at 06:03

3

The third derivative is sometimes called jerk because it is the rate of change of acceleration if you interpret the function you are considering as position. If you were riding in a vehicle driving on a straight road such that the position of the vehicle from its starting position was equal to $y$, the fact that the third derivative is positive indicates that the pressure between your back and the seat would be increasing with time.

answered Mar 10 '16 at 06:03

math wannabe

696

Then thats a physics related answer how to apply it to maths as OP hasnt written anything about mechanics – Archis Welankar Mar 10 '16 at 06:21
Sorry @idonutunderstand, that is slightly confusing. – 1089 Mar 10 '16 at 06:21
1

I disagree with Archis. The question did not limit application to just math. So, this answer seems fine to me. As for OP understanding, that's a different problem. If I could vouch for the answer, then I would up vote. – Em. Mar 10 '16 at 06:28
@1089 If you can explain what is confusing I will try to clarify. – math wannabe Mar 10 '16 at 07:03
I mean it technically still has the effect of jerk to the function, it tells you how much the curvature of the function is changing as x changes – Triatticus Mar 10 '16 at 08:50
@probablyme You're an applied math major but you can't vouch for an elementary property of the derivative? – math wannabe Mar 10 '16 at 20:21
@idonutunderstand I sort of get it now. – 1089 Mar 10 '16 at 21:46
Thank you everyone for answering. – 1089 Mar 10 '16 at 21:59
Also he asked about the practical application of third derivative how do you explain a "practical" application without any involvement of physics – bulbasaur Mar 11 '16 at 13:40
@probablyme Point taken. I guess I was just surprised that you didn't learn about this particular application when you learned about derivatives in Calculus, because in my Calc. classes we did a fair amount of applications to different things. – math wannabe Mar 12 '16 at 23:15

score 0 · Answer 2 · answered Mar 11 '16 at 13:28

In "strength of materials" (subject studied in mechanical engineering or civil engineering), studying deflection of beams, the shear force calculation requires the third derivative of the displacement functions of a beam.

See the example 7.5 in this link

http://www.me.mtu.edu/~mavable/Book/Chap7.pdf

or

http://www.nptel.ac.in/courses/Webcourse-contents/IIT-ROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect30%20and%2031/lecture30%20and%2031.htm

*

Besides there is the fourth derivative used in the calculation of beams on elastic foundations.

http://nptel.ac.in/courses/105106049/lecnotes/mainch11.html

The Third Derivative

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