Expert: Scotto - 7/20/2009

Question
QUESTION: Question:

Function f(x) = x^4-3x^2
1st Der     f ‘ (x) = 4x^3-6x
2nd Der    f ” (x) =12x^2-6
3rd Der    f ’”(x) = 24x
4th Der    f ””(x) = 24

Questions:
1). When the 3rd derivative intersects (POI) the 2nd Derivative at
x = -0.22474, what is happening?
2).  Where is maxiumum “jerk” (3rd derivative) occurring on the position curve f(x)=x^4-3x^2?


ANSWER: 1) At the point where the acceleration (f"(x)) is the same as the derivative of acceleration, that is where the speed at which the object is increasing in speed is the same point at which the increase in the increase in speed is the same as the increase in speed.

2) The value of the derivative is ever increasing.
This means that x could be as far as to the right as it would go.
If there is no limit, x could go off to infinity.


---------- FOLLOW-UP ----------

QUESTION: Dear Expert, Scotto...Thank you for your answer

Answer 1 does not make sense, sorry.  The function f(x) is
taken from a roller-coaster design by Crompton Engineering (they build about 80% of the coasters in this counrty).  The function is actually one segment of a coaster design, I found this out through some investigation after the problem was given to our class by our prof.

Where f"'(x) = f"(x) at that POI (plot these graphs if you wish), this is a point of inflection leading to a concavity (down and then upward) in f"(x) acceleration producing a parabolic acceleration effect.  At this intersection of jerk and acceleation (POI) do we have maximum phsical jerk, or at the point where f'"(x) jerk curve intersects both the position curve f(x) and velocity curve f'(x) which is at cordinate (0,0) origin????

Your second answer is great thank you, in fact jerk f"'(x) would continue on in a linear function until reaching an upper acceleration limit which is NOT the speed of light anymore, since there is no upper boundry on the speed of light (source Bell Labs, 2001)

Answer
OK, I thought it didn't matter that much to me, but now I've go to know - what is POI?  Point Of Inflection?
If it were, a point of inflection is where f" is 0.

The point of cancavity in any curve is where f"(x) is 0.
If f"(x) > 0, the function is concave up and
if f"(x) < 0, the function is concave down.

Note that I'm not sure if POI is the point of inflection, since it was also stated the f"'(x) = f"(x) at this point.

It would be great if you woul rephrase what is being asked into smaller questions so I can comprehend what is being looked for in them.