Can the third derivative be used in describing the curves of power functions?

Question

I graphed a lot of power functions on my calculator, from X^.01 to X^2.3 to X^5, etc., and it seems between 0 and 1, where no matter what the two points are always the same, there is a bend that determines the the shape of the curve. This bend seems very defined, and it seems it must be important mathematically. I know that maximum and minimum are determined by first deriviative, and inflection points by second, is this bend point determined by third derivative. IF not, is there a way to find the exact point where the bend is the most pronounced?

Answer 1

Cool, you are onto something:

1 is a transition point in the sense that

for x < 1 , x^k decreases
for x > 1, x^k increases.

and of course, 1^k = 1 for any k.

As k get larger
x^k hugs the x axis up to just below 1 and then has a tremendously large slope as it zips through 1 and continues.

For 0< k < 1
a type of mirror image behavior occurs:
x^k hugs the y axis until it gets close to 1 and then
flattens out as it zips nearly horizontally through 1.

Whereas for k =1 , you have the plane old straight line through the origin.

If you have a graphing calculator, it's fun to graph separately

x^k for k= 0.1, 0.5, 2, 10 and fool around with the curves.

The 'bend' that you see if roughly the rapid change in the slope and so is the change in the first derivative, ie, the second. There is information about shape available from the third.

Btw, in engineering the derivative of acceleration is known as 'jerk'. Think about it. It's what it feels like when you undergo a sudden change in acceleration.

Answer 2

i think the bend is more pronounced for x^n as n gets larger and larger

you're right, every graph like that passes through the points (0,0) and (1,1). the higher the degree, the flatter the curve will get and thus the closer the bend will come to being a 90 degree angle.

i doubt the third derivative will mean that much in this case.

Answer 3

The zeroth derivative (or the function itself) signifies the exact value.
The first derivative signifies the increase/decrease of value.
The second derivative signifies the increase/decrease of the increase/decrease of the value. (or simply the concavity of the value)
The third derivative signifies the increase/decrease of the increase/decrease of the increase/decrease of the value (or the increase/decrease of the concavity of the value)

Therefore, your "bend points" are the points where the increase/decrease of concavity changes/switches otherwise.

^_^

Answer 4

polynomials are the finest to tell apart: the rule of thumb for differentiating polynomials is that if f=x^n; f' = nx^(n-a million) f'=20x^4 + 36x^3 +12x^2 f''=80x^3+108x^2 +24x f'''=240x^2 +216x+24