inflection point of a curve

[PG1995]

Hi

Before I go on to make some queries about inflection point I need to understand some of the terminology used in this context. I was reading this Wikipedia article and it says this:

In algebraic geometry an inflection point is defined slightly more generally, as a point where the tangent meets the curve to order at least 3...

What does the part in bold mean? Please let me know. I have googled for it but didn't find any clue. Probably, I wasn't using a right search phrase. Thank you.

Regards
PG

 

[JimB]

Explanation by example:

A Quadratic Equation Y = Ax^2 + Bx + C is Second Order

A Cubic Equation Y = Ax^3 + Bx^2 + Cx + D is Third Order

A Quartic (?) Equation Y = Ax^4 + Bx^3 +Cx^2 + Dx + E is Fourth Order

So "Order at least 3" means that it has as a minimum an x^3 term.

JimB

 

[PG1995]

Thank you, Jim.

It was simple! I was looking at it from different angle.

Best wishes
PG

 

[MrAl]

Hi,

The inflection points (if any) are found by setting the second derivative to zero and finding the roots. Thus it would have to be an equation of 3rd degree or higher in order to have something to set to equal to zero and still make sense.

 

[PG1995]

Hi

I don't really expect a reply to this post. But if you think you have something which I might useful then you are always welcome. **broken link removed**, I have tried to summarize important points about inflection point, first and second derivative tests, etc. which someone like me could find helpful. You can also see the attachment in a PDF format here. Note: **broken link removed** is the missing paragraph from the main document. Also give it a read when going through the main text. Thank you.

References:
1: https://en.wikipedia.org/wiki/Second_derivative_test
2: https://en.wikipedia.org/wiki/First_derivative_test
3: https://www.electro-tech-online.com/custompdfs/2012/10/lecture8.pdf
4: https://en.wikipedia.org/wiki/Convex_function
5: https://en.wikipedia.org/wiki/Concave_function
6: https://en.wikipedia.org/wiki/Inflection_point

Regards
PG

 

[MrAl]

Hi again,


The simplest explanation of an inflection point is that it is a point where the graph changes from concave upward to concave downward or vice versa.

The simplest explanation of the derivatives comes from simple motion of an object (like a projectile) where we have:

1. distance s
2. velocity ds/dt (first derivative of distance with time)
3. acceleration d^2s/dt^2 (second derivative of distance with time)
4. jerk or jolt d^3s/dt^3 (third derivative of distance with time.
5. I read jounce is the fourth derivative.
and it is easiest to think of the higher derivative as just being the change in the next lower derivative, so the jerk is just the change in acceleration with time. The acceleration of course makes the velocity change, just like the velocity makes the distance change. If you look at a derivative and then the next one just below the very next one such as acceleration and distance, it becomes a little more difficult but then you can think of it as the acceleration is making the distance change faster and faster as time progresses (assuming a positive value).
And the functional outlook is similar too, for example the third derivative will tell us about the inflection points in the velocity, similar to how the second derivative tells us about the inflection points in the distance.

You can then start to link up to something like F=ma, where if we have a constant acceleration and constant mass this means we are pushing on an object with a constant force and it is moving through space, but since acceleration is the first derivative of velocity, it's the slope of the velocity so if it is positive then the graph of the velocity is increasing, which means the object is going faster and faster. If we add a jerk, the object will go faster than faster-and-faster because a will increase (even with constant jerk).

When you look at these derivatives in terms of distance in the motion, it starts to make a lot of sense.

Oh yeah, if you want to fully explain "the point where the tangent meets the curve", we can start with a curve that is first concave downward and then later some time changes to concave upward.
As we start at the left side of the curve, we can track the tangent to that curve and we see slope getting less and less positive, until it reaches the peak and then it is zero, then continues to get less and less positive, but only up to the inflection point where "the tangent meets the curve" and then the slope starts to get more and more negative. The tangent runs into the curve (at a very local point) because the tangent is straight and even a small distance away it intersects the curve itself. It never intersects the curve (locally) before or after again that unless it hits another inflection point.
If a car were traveling along the path of the curve, it would be constantly turning right but then once it hit the inflection point it would have to start turning left or it would run off the road. So from an intrinsic point of view, it's when the intrinsic angle changes polarity.

 

[PG1995]

Thank you, MrAl.

After reading your post, I realized that I had missed one interpretation of the inflection point so I have updated the information there. Thanks.

Best wishes
PG