What can we learn from higher derivatives in calculus?

[EchoRush]

TL;DR Summary

A quick questions about the meaning of derivatives?

As I have said before, I am in calculus class for the first time. I am doing really well in the class, however because of how my mind works, I’m always asking questions to know more, even when it’s too advanced for me. I just like to ponder and think about “what if” I know it’s probably not good to do this, but i can not help but wonder about things. Can you tell me if these two statements are correct and if I understand then and also can you help me with my additional though for the third point?

1. first derivatives tell us about a function increasing or decreasing on an interval (it changes form increase/decrease at critical points)

2. second derivatives tell us about a function being either concave up or concave down on a interval (changes from concave up/down at inflection points)

3. After I learned those two statements in class, I got thinking, what does a THIRD derivative tell us about a function? My guess is it only applies to graphs of functions who are 3-dimensional? What does a third derivative tell us about? What about fourth, fifth and so on derivatives?

 

[fresh_42]

One and two are correct. The third derivative, if it equals zero, tells us where saddle points are, locations which are locally flat for a moment. They are not restricted to higher dimensions. Dimension and differentiation are not related. Higher derivatives tell something about the form of the curve, e.g. if it is skew somehow.
3rd is called jerk, 4th jounce or snap, the 5th crackle, and the 6th pop.

I would recommend some Wikipedia pages to learn more about it, and:
http://wordpress.mrreid.org/2013/12/11/jerk-jounce-snap-crackle-and-pop/https://en.wikipedia.org/wiki/Jerk_(physics)https://en.wikipedia.org/wiki/Jouncehttps://en.wikipedia.org/wiki/Pop_(physics)

 

[WWGD]

This helps me: seeing derivative as a (local) rate of change. Then the second derivative tells you how the rate of change itself is changing. Is speed ( change of position with time constant?). For the third: is the change of velocity (i.e., the acceleration) constant? Of course, it is difficult to continue this chain indefinitely, but it seems helpful as far as it goes.

 

[PeroK]

3. After I learned those two statements in class, I got thinking, what does a THIRD derivative tell us about a function? What does a third derivative tell us about? What about fourth, fifth and so on derivatives?

The nth derivative tells you the nth term in the Taylor series expansion. If you wanted to approximate a function on a small interval (about $x = 0$, for example) by a polynomial, then the nth term in the polynomial would be $\frac{f^{(n)}(0)}{n!}$, where $f^{(n)}(0)$ is the nth derivative evaluated at $x = 0$.

For example, for small $x$:

$f(x) \approx f(0) + f'(0)x$

Or, a better approximation is:

$f(x) \approx f(0) + f'(0)x + \frac{f''(0)}{2}x^2$

Or, even better again:

$f(x) \approx f(0) + f'(0)x + \frac{f''(0)}{2}x^2 + \frac{f'''(0)}{6}x^3$

And so on. The Taylor series mentioned above is in fact an infinite series of this form.

 

[jedishrfu]

If you're truly curious about Calculus then I'd suggest watching the 3Blue1Brown series on YouTube:



This is the first video in a series. Check it out. It may answer questions you haven't asked yet.

By the way, this is a good and bad habit to let your mind wander. I would suggest focusing on the material capturing as much as you can before wandering off. A simple example from distracted driving illustrates the point:

Imagine you're driving on some road to school and it reminds you of a country road in Ireland. In that brief moment that you are reveling in its Irish beauty, your distracted mind misses the critical change in traffic up ahead and now you find yourself in deep danger.

The same effect happens in school causing you to lose some key point that the prof has hinted at. These will build up and come back to haunt you as you become an upperclassman and courses are built on what you've learned already and are much tougher to get through.

In one of my Calculus classes on double integrals, I remember the Prof saying that sometimes you need to integrate over y before you integrate over x. A trivial remark at the time but his Friday quiz had such an example and having heard what he said I luckily figured it out where others didn't.