Formalizing Understanding of Relationships between f(x), f'(x) and f''(x)

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In summary: Thanks for the suggestion!In summary, Deus Abscondus is seeking to become proficient at looking at and working out what a function would look like, in basic outline, from its first and second derivatives, and is asking for help from others in finding formalized principles or generalizations of the same kind.
  • #1
DeusAbscondus
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Hi folks,
I am seeking to hone my skills & deepen my knowledge and understanding of the relationship between f(x) and f'(x) and f''(x)

To this end, I have made a workhorse cubic with four sliders (using geogebra) attached to co-efficients a, b, c and d respectively.

My study plan is to hide f(x), manipulate the sliders (values of co-efficients) thus transforming f(x) in various ways, then trying to deduce what f(x) must now look like and trying to sketch it myself.

Two questions occur to me:

1. How can I best systemize any learnings thereby gained, in terms of rules of transformation which I can formulate, as I manipulate the co-efficients? and,

2. What principles do I need to arrive at to be a good sketcher of f(x) from f'(x)Thanks for any help, suggestion or directions you can offer.

PS I have included inline a screenshot of the geogebra worksheet I have made and would also welcome any comments as to any errors to correct, areas of obvious improvement etc

PSS I have asked identical question in geogebraforum and am awaiting a response.
 
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  • #2
In the same vein, is my comment attached to sketch below generally valid for all cubic functions and their $f'(x)$
 
  • #3
DeusAbscondus said:
Hi folks,
I am seeking to hone my skills & deepen my knowledge and understanding of the relationship between f(x) and f'(x) and f''(x)

To this end, I have made a workhorse cubic with four sliders (using geogebra) attached to co-efficients a, b, c and d respectively.

My study plan is to hide f(x), manipulate the sliders (values of co-efficients) thus transforming f(x) in various ways, then trying to deduce what f(x) must now look like and trying to sketch it myself.

Two questions occur to me:

1. How can I best systemize any learnings thereby gained, in terms of rules of transformation which I can formulate, as I manipulate the co-efficients? and,

2. What principles do I need to arrive at to be a good sketcher of f(x) from f'(x)Thanks for any help, suggestion or directions you can offer.

PS I have included inline a screenshot of the geogebra worksheet I have made and would also welcome any comments as to any errors to correct, areas of obvious improvement etc

PSS I have asked identical question in geogebraforum and am awaiting a response.

Hi DeusAbscondus, :)

So are you trying to understand how to graph a cubic function taking into account its first and second derivatives?

Kind Regards,
Sudharaka.
 
  • #4
Sudharaka said:
Hi DeusAbscondus, :)

So are you trying to understand how to graph a cubic function taking into account its first and second derivatives?

Kind Regards,
Sudharaka.

Hi Suharaka,

It's more like I want to become proficient at looking at gradient function and working out what function would look like, in basic outline; and vice-versa.

To become proficient, I presume I need to understand some generalized principles of the kind which I set forth below in my graphs (or ones like them)

So, I am asking others here if they can suggest formalized principles or generalizations of the same kind; for instance, another I've learned since making my last post is this:
if a graph is increasing across an interval, then the curvature of the second derivative will be positive; and negative, if the curve is decreasing across an interval

I want to garnish as many of these ideas as possible, generalize them into rules, make memory cards of them using geogebra and build a file of them as a study aid for my upcoming exam/test in 3 weeks.

Regs,
Deus Abs
 
  • #5
DeusAbscondus said:
Hi Suharaka,

It's more like I want to become proficient at looking at gradient function and working out what function would look like, in basic outline; and vice-versa.

To become proficient, I presume I need to understand some generalized principles of the kind which I set forth below in my graphs (or ones like them)

So, I am asking others here if they can suggest formalized principles or generalizations of the same kind; for instance, another I've learned since making my last post is this:
if a graph is increasing across an interval, then the curvature of the second derivative will be positive; and negative, if the curve is decreasing across an interval

I want to garnish as many of these ideas as possible, generalize them into rules, make memory cards of them using geogebra and build a file of them as a study aid for my upcoming exam/test in 3 weeks.

Regs,
Deus Abs

Have you come across the First derivative test and the Second derivative test? These are useful in sketching the graphs of most of the functions that you will encounter.
 
  • #6
Sudharaka said:
Have you come across the First derivative test and the Second derivative test? These are useful in sketching the graphs of most of the functions that you will encounter.

Hi Sudharaka,
yes, these are the current focus of our course at school;
I was just hoping to derive a few more "tricks" to employ when looking at a function and trying to discern its gradient.

thanks for the response and the urls which are helpful,

Deus Abscondus
 
  • #7
DeusAbscondus said:
Hi Suharaka,

It's more like I want to become proficient at looking at gradient function and working out what function would look like, in basic outline; and vice-versa.

To become proficient, I presume I need to understand some generalized principles of the kind which I set forth below in my graphs (or ones like them)

So, I am asking others here if they can suggest formalized principles or generalizations of the same kind; for instance, another I've learned since making my last post is this:
if a graph is increasing across an interval, then the curvature of the second derivative will be positive; and negative, if the curve is decreasing across an interval

I want to garnish as many of these ideas as possible, generalize them into rules, make memory cards of them using geogebra and build a file of them as a study aid for my upcoming exam/test in 3 weeks.

Regs,
Deus Abs

You might also check out posts 8 through 11 of the http://www.mathhelpboards.com/f10/differential-calculus-tutorial-1393/. I go into a fair bit of detail in graphing functions, as that is an important application of derivatives.
 
  • #8
Ackbach said:
You might also check out posts 8 through 11 of the http://www.mathhelpboards.com/f10/differential-calculus-tutorial-1393/. I go into a fair bit of detail in graphing functions, as that is an important application of derivatives.

I will, thank you kindly.
Just the thing.

edit: 30 mins later
Just now reading through #8: love it, really! so clear.
Question Achbach: could you direct me to convenient source of help for the logical signifiers you are using with which I am unfamiliar
Example:
[FONT=MathJax_Main]lim[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]L[/FONT][FONT=MathJax_Main]iff[/FONT][FONT=MathJax_Main]∀[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]∃[/FONT][FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main]such that if[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]then[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math]L[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main].[/FONT]​
Inverted A = ? (is it like the universal: "For every ..." in this case: "For every epsilon"?
Backfacing E = ? (is it like existential quantifier: "There is a ..." in this case "There is a number M" ??

thx for the help

Regs,
Deus Abs
 
Last edited:
  • #9
DeusAbscondus said:
I will, thank you kindly.
Just the thing.

edit: 30 mins later
Just now reading through #8: love it, really! so clear.
Question Achbach: could you direct me to convenient source of help for the logical signifiers you are using with which I am unfamiliar
Example:
[FONT=MathJax_Main]lim[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]L[/FONT][FONT=MathJax_Main]iff[/FONT][FONT=MathJax_Main]∀[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]∃[/FONT][FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main]such that if[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]then[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math]L[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main].[/FONT]​
Inverted A = ? (is it like the universal: "For every ..." in this case: "For every epsilon"?
Backfacing E = ? (is it like existential quantifier: "There is a ..." in this case "There is a number M" ??

thx for the help

Regs,
Deus Abs
Yes, that is more or less what they mean.
 
  • #10
I typically read $\forall$ as "for all", and $\exists$ as "there exists". They are the universal quantifier and existential quantifier, respectively. As for a good logic website, if I were you, I would pm Evgeny.Makarov for more info - he is the undisputed logic guru around here. I'm sure he could point you in a good direction.
 
  • #11
Ackbach said:
I typically read $\forall$ as "for all", and $\exists$ as "there exists". They are the universal quantifier and existential quantifier, respectively. As for a good logic website, if I were you, I would pm Evgeny.Makarov for more info - he is the undisputed logic guru around here. I'm sure he could point you in a good direction.

'Preciate it. My formal logic from 30 years ago (an introductory course, neglected since then) is welling back up through the mists of memory, now that I turn the fading beam-pole of my attention towards the alcohol-sodden past...(Giggle)

thanks, I'll plague Evgeny (if I need to pursue this further) who has already proved his worth by helping me in a number of ways.

Have a good one Ackbach,
Deus Abs
 

FAQ: Formalizing Understanding of Relationships between f(x), f'(x) and f''(x)

What is the relationship between f(x), f'(x), and f''(x)?

The relationship between f(x), f'(x), and f''(x) can be described using the derivative, which represents the slope of a function at a specific point. The first derivative, f'(x), represents the rate of change of a function, while the second derivative, f''(x), represents the rate of change of the first derivative. In other words, f''(x) is a measure of the curvature of the original function f(x).

How do you formally define f'(x) and f''(x)?

f'(x) can be formally defined as the limit of the difference quotient as h approaches 0. This can be written as f'(x) = lim(h->0) [f(x+h) - f(x)]/h. Similarly, f''(x) can be formally defined as the limit of the difference quotient of f'(x) as h approaches 0. This can be written as f''(x) = lim(h->0) [f'(x+h) - f'(x)]/h.

How are f(x), f'(x), and f''(x) related to each other?

f(x), f'(x), and f''(x) are related through the process of differentiation. The first derivative, f'(x), represents the slope of the original function f(x), while the second derivative, f''(x), represents the rate of change of the first derivative. In other words, f''(x) is a measure of the curvature of the original function f(x). Additionally, the third derivative, f'''(x), represents the rate of change of the second derivative, and so on.

How can understanding the relationships between f(x), f'(x), and f''(x) be useful?

Understanding the relationships between f(x), f'(x), and f''(x) can be useful in many ways. For example, it can help in identifying the critical points of a function (points where the slope is 0 or undefined), determining whether a function is increasing or decreasing, and finding the concavity of a function. This knowledge can also be applied to real-world problems in fields such as physics and economics.

Can the relationship between f(x), f'(x), and f''(x) be represented graphically?

Yes, the relationship between f(x), f'(x), and f''(x) can be represented graphically. The original function f(x) can be graphed as a curve, f'(x) can be graphed as a line representing the slope of the curve at each point, and f''(x) can be graphed as a curve representing the rate of change of f'(x). This graphical representation can help in visualizing the relationships between the three functions and understanding their behavior.

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