How to Change the Independent Variable from Time to Position in a Function?

[awholenumber]

Functions are pretty simple things , they just express a relationship between two different quantities

How do i express this function in terms of y(x) = something ?

 

[Daniel Gallimore]

I'm assuming you know that it doesn't matter whether you call a function $j(t)$ or $y(x)$ as long as you define what $j$ and $t$ or $y$ and $x$ are supposed to represent.

If what you want is a simple formula for $j(t)$ like $j(t)=t^2$, then you would first need to specify a function $s(t)$, $v(t)$, or $a(t)$ and do some differentiation.

 

[FactChecker]

Functions are pretty simple things , they just express a relationship between two different quantities

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View attachment 199363

How do i express this function in terms of y(x) = something ?

Which function do you mean? There are 4 functions mentioned, s(t), v(t), a(t), j(t). The only one that is not defined is s(t). s(t) must first be defined before numerical values can be obtained for the others. But even without a definition of s(t), the others are correct definitions of velocity, acceleration, and jerk, assuming that all the derivatives required exist.

As @Daniel Gallimore says, once you have a formula for s(t), one can hopefully differentiate to get the formulas for the other functions.

EDIT: Sorry. I lost track of the title, which says that the goal is to put everything in terms of position rather than time.

 

[SlowThinker]

How do i express this function in terms of y(x) = something ?

Some time ago I derived, not quite rigorously, that
$$\bar v(s)=\frac{dt}{ds}=1/v(t)$$
$$\bar a(s)=\frac{d^2 t}{ds^2}=-a(t) \bar v(s)^3$$
$$\bar j(s)=\frac{d^3 t}{ds^3}=-j(t) \bar v(s)^4-3\bar a(s)^2/\bar v(s)$$
(I didn't check the last one because I didn't need it)

 

[awholenumber]

Which function do you mean? There are 4 functions mentioned, s(t), v(t), a(t), j(t). The only one that is not defined is s(t). s(t) must first be defined before numerical values can be obtained for the others. But even without a definition of s(t), the others are correct definitions of velocity, acceleration, and jerk, assuming that all the derivatives required exist.

As @Daniel Gallimore says, once you have a formula for s(t), one can hopefully differentiate to get the formulas for the other functions.

EDIT: Sorry. I lost track of the title, which says that the goal is to put everything in terms of position rather than time.

Well my original question was called , Some doubts about functions ? the rest was added by the mods ...

There are 4 functions mentioned, s(t), v(t), a(t), j(t)

How do i turn these functions into f(x) form . so that its easier for me to understand ...

I want an example from these type of functions , so that i can learn to differentiate it , integrate it and i would even like a differential equation sort of equation from it

 

[SlowThinker]

Well my original question was called , Some doubts about functions ? the rest was added by the mods ...

There are 4 functions mentioned, s(t), v(t), a(t), j(t)

How do i turn these functions into f(x) form . so that its easier for me to understand ...

I want an example from these type of functions , so that i can learn to differentiate it , integrate it and i would even like a differential equation sort of equation from it

I'm not sure what you want, or if you know what you want. These function already are in f(x) form, except x is renamed to t.
If you want it more concrete, you need to specify one of the functions, e.g. v(t)=3t+sin(2t), and compute the rest using differentiation or integration:
$$s(t)=\int_0^t 3\tau +\sin(2\tau) d\tau=\frac{1}{2} (3t^2-\cos(2t))+s_0$$
$$a(t)=\frac{dv}{dt}=3+2\cos(2t)$$
$$j(t)=\frac{da}{dt}=-4\sin(2t)$$
Obviously you need to learn integration and derivation first before you can use it.

 

[awholenumber]

Look mostly i am trying to understand this in terms of ,

Functions are pretty simple things , they just express a relationship between two different quantities

In a strict mathematics sense, y is just a variable. When someone writes "y=f(x)", it means that the value of y depends on the value of x, which is another variable. That is, for different values of x, there is a function, called f(x), which determines the value of y.

The x variable is therefore called the "independent" variable, while the y variable is called the "dependent" variable because it's value "depends" on the value of x

My main doubt was how do i write everything here in x,y coordinate ?

y(x) = x
y(x)= dy/dx
y(x) = d²y/dx²
y(x) = d³y/dx³

is this correct ?

 

[SlowThinker]

My main doubt was how do i write everything here in x,y coordinate ?

y(x) = x
y(x)= dy/dx
y(x) = d²y/dx²
y(x) = d³y/dx³

is this correct ?

I believe your confusion is pretty deep
1. You don't have to name all variables $x$ and $y$. s=s(t) is perfectly fine.
2. Coordinates are something completely different. The excerpts you are showing have time (which I believe is not treated as a coordinate in this case), and only one coordinate, s.
3. What you wrote doesn't make any sense. $y$ can't be 4 different things at once.

 

[awholenumber]

I was just trying to understand this

 

[FactChecker]

An equation like v(t) = ds/dt can be misleading because it implies that velocity is a function of time, which may be wrong.
It would be a better definition of velocity to say v = ds/dt, (without implying that the velocity is a function of t).

Suppose that we are dealing with the velocity of something going through a dense liquid that does not depend on time, but does depend on the density, ρ, of the liquid.
Then we would still define velocity v = ds/dt. It's just that after we take the derivative, it will be a function of ρ instead of t.
For a particular density ρ=ρ₀ we could then say that v(ρ₀) = ds/dt|_ρ=ρ0. In general we can say that v(ρ)=(ds/dt)(ρ).

So the conclusion is that velocity is always the derivative of position with respect to time. That is true whether the resulting derivative is a function of time or not. After seeing what variables remain in the derivative, you can determine what variables it is a function of and plug in values.

 

[UsableThought]

@rosekidcute, it might help if you provide the sources for these two cropped images that you keep re-posting. That the fonts are different suggests they are excerpts from two completely different textbooks. What are these textbooks?

Related to this - when you ask the following about the second excerpt -

My main doubt was how do i write everything here in x,y coordinate ?

- this suggests that you don't understand what these functions are trying to describe, but are trying to shoehorn them, unexamined, into a previously learned definition from a different source and different context. But surely understanding of these functions for what they are ought to come first? E.g. one could go to Kahn Academy and study their presentation of acceleration and velocity: https://www.khanacademy.org/science...l-motion/acceleration-tutorial/v/acceleration

A more general comment: From previous posts of yours that I've read, you seem to be jury-rigging your self-taught math curriculum by piecing together different sources. That's what I'm doing too (you can search for my various math-related posts or read my https://www.physicsforums.com/members/usablethought.611113/#info), but I think that in attempting this, we have to be careful not to get lost or fall into gaps between different sources. Here you seem to have created such a gap and leaped into it.

I remember a recent thread where you were advised to find a master means of coordinating your curriculum; have you done so yet? Without that, you will keep wandering off course like this and slow your progress way down.

 

[awholenumber]

@rosekidcute, it might help if you provide the sources for these two cropped images that you keep re-posting. That the fonts are different suggests they are excerpts from two completely different textbooks. What are these textbooks?

Related to this - when you ask the following about the second excerpt -- this suggests that you don't understand what these functions are trying to describe, but are trying to shoehorn them, unexamined, into a previously learned definition from a different source and different context. But surely understanding of these functions for what they are ought to come first? E.g. one could go to Kahn Academy and study their presentation of acceleration and velocity: https://www.khanacademy.org/science...l-motion/acceleration-tutorial/v/acceleration

A more general comment: From previous posts of yours that I've read, you seem to be jury-rigging your self-taught math curriculum by piecing together different sources. That's what I'm doing too (you can search for my various math-related posts or read my https://www.physicsforums.com/members/usablethought.611113/#info), but I think that in attempting this, we have to be careful not to get lost or fall into gaps between different sources. Here you seem to have created such a gap and leaped into it.

I remember a recent thread where you were advised to find a master means of coordinating your curriculum; have you done so yet? Without that, you will keep wandering off course like this and slow your progress way down.

UsableThought , One is an online material and other is from a different math forum

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/functions/function1.html

Yes , I have been self learning math exactly that way , that is what forums are useful for in my opinion . Text books are dense and you cannot simply rely on one text to learn all these things
I have been narrowing it down like that for sometime , My list keeps improving .
My list of things i should be following looks like these .Arithmetic
Algebra
Trigonometry
Differentiation
Integration
Differential equation

Arithmetic
https://www.wyzant.com/resources/lessons/math
Lots of factoring examples
http://www.mathhands.com/046/
Trigonometry
Differentiation
Integration
If you do an image search for this , "Eeweb.com maths" . There is a nice list of things

As for differential equations ,
Mod note: Deleted much of the rest of this post as being irrelevant to the question in this thread.Thanks a lot for the reply . i have to re read this a couple of times .
I was trying to figure out equations from it so that i can use those examples of motions to learn differentiation , how it affects the graph . Integration , How an integration question regarding "jerk" might look like , What the end result might look like etc .
Also how a differential equation might look like in these contexts of the equations of motions .
Thus , i thought i might have a little bit of physics too to work with , Thereby this whole thing might be a little bit more interesting than the usual plain mathematics .

 

[SlowThinker]

Yes, I have been self learning math exactly that way , that is what forums are useful for in my opinion . Text books are dense and you cannot simply rely on one text to learn all these things.

I disagree on this. In a forum, you spend days on every little thing, because it takes very long to figure out all the prerequisites that you have skipped. And not only your days but ours as well.
A book or an online course has things ordered in the right way.
I find it best to view recordings of actual university courses, because the teacher has a lot of experience in explaining things, and if something is not clear, usually someone will ask about it.

You may watch one of the earliest Veritasium videos. If the book is hard to understand, it might be because you are actually learning something.

 

[FactChecker]

I was trying to figure out equations from it so that i can use those examples of motions to learn differentiation , how it affects the graph . Integration , How an integration question regarding "jerk" might look like ,

Integration has such a smoothing effect that jerk is very hard to see on a graph of position versus time.
jerk => integrates to acceleration => integrates to velocity => integrates to position
In fact, suppose you had an infinite jerk where acceleration instantly changed from 0 to 1. I bet no one could spot it in a graph of position:

j(t) = 0 if t≠0, +∞ if t=0 (such that its integral is 1);
a(t) = 0 if t<0, 1 if t ≥0;
v(t) = 0 if t<0, t if t≥0;
s(t) = 0 if t<0, t²/2 if t≥0

 

[awholenumber]

Thanks

Maybe i should start from scratch once again .

https://www.khanacademy.org/math/calculus-home/differential-calculus
https://www.khanacademy.org/math/calculus-home/integral-calculus
https://www.khanacademy.org/math/differential-equations

 

[Mark44]

Yes , I have been self learning math exactly that way , that is what forums are useful for in my opinion . Text books are dense and you cannot simply rely on one text to learn all these things

I have been narrowing it down like that for sometime , My list keeps improving .
My list of things i should be following looks like these .

Arithmetic
Algebra
Trigonometry
Differentiation
Integration
Differential equation

"Text books are dense" - so what?
"you cannot rely on one text to learn all these things"
This scattershot approach to learning mathematics will not do you any good. The disciplines you list above are not independent of one another. If you don't understand arithmetic, you won't be able to make any progress in algebra. Without a solid foundation in algebra and trig, studying calculus (differentiation and integration) will be a waste of time.

In this thread you are asking very basic questions about the relationship of variables in a function. Until you understand this, you don't have a hope of understanding calculus, let alone differential equations.

Thanks a lot for the reply . i have to re read this a couple of times .
I was trying to figure out equations from it so that i can use those examples of motions to learn differentiation , how it affects the graph . Integration , How an integration question regarding "jerk" might look like , What the end result might look like etc .

Until you understand basic concepts about functions and graphs, which is what you're asking about in this thread, you won't understand differentiation concepts such as velocity, acceleration, and others.

Also how a differential equation might look like in these contexts of the equations of motions .
Thus , i thought i might have a little bit of physics too to work with , Thereby this whole thing might be a little bit more interesting than the usual plain mathematics .

If you build a house and the foundation isn't solid, the house will fall down. You are trying to put the roof on your house before you have laid the foundation, and built the framing and walls.

 

[FactChecker]

@rosekidcute , I have always liked the Schaum's Outline series of books because they have a lot of worked examples and exercises. There are some for precalculus, calculus, and differential equations. They tend to be fairly inexpensive compared to textbooks.

 

[UsableThought]

Text books are dense and you cannot simply rely on one text to learn all these things

"Text books are dense" - so what?

If the book is hard to understand, it might be because you are actually learning something.

I agree w/ SlowThinker and Mark 44 here. To help me in my own math studies, I have dipped into some books on "how to learn math"; and they all emphasize (as do good math teachers) that learning math takes time and hard work; most of us are not going to race through material we are learning for the first time, or even material we once knew but have largely forgotten. Myself, I find it's more enjoyable to take time learning rather than rush it.

Also I really agree w/ this other comment by SlowThinker - I have bolded the part that is especially so. I thought of mentioning this and am glad someone else did. People enjoy helping others learn; but the enjoyment diminishes if the person they are trying to help habitually creates unnecessary difficulties.

In a forum, you spend days on every little thing, because it takes very long to figure out all the prerequisites that you have skipped. And not only your days but ours as well.

 

[haushofer]

If you build a house, you start with a solid base of concrete. If concrete is too dense for you, you should stay away from building houses.

 

[awholenumber]

I had confusions about dependent and independent variables .Anyway i have improved my knowledge of these this much , which is a big progress for me ...

 

[UsableThought]

Please feel free to delete it if i have messed up the thread

I'm not arguing for deletion, but posting an "image and links salad" made up entirely of other people's words does not suggest that you yourself have learned anything other than how to cut and paste. It's impossible for others to guess what you might know if you deliberately avoid expressing ideas in your own words & deliberately avoid demonstrating any actual mathematics on your own.

I'm not one of the math mentors here, just a fellow student; however, speaking for myself, I'm not going to comment further on threads you start; not unless your approach changes. I feel as if I would be helping you deceive yourself.

 

[awholenumber]

Some doubts about functions... (changing the independent variable from time to position)

Now i know what this is , which is a big thing for me

The mods are probably going to delete all those things i arranged in order to understand some fundamental concepts , They are again going to call that irrelevant

But all those things were very important for me

 

[FactChecker]

is it possible to rewrite the above equations in terms of f(x) = something ? y(x) = something ? or y=f(x) ?

You can use whatever variable and function names you want, but you should be clear about what they represent. Is there some reason that you prefer 'f' to a variable like 'v'? Or 'x' to a variable like 't'?
Other than that, I can't figure out what you are asking.
Keep in mind that an equation like v = ds/dt is not a formula that you can plug numbers into. If that is what you want, you need the formula for s(t) so you can take a derivative. Once you do take the derivative with respect to t, you may end up with a formula no longer involves t. That is why I don't like the link saying v(t) = ds/dt.

 

[Mark44]

I'm not arguing for deletion, but posting an "image and links salad" made up entirely of other people's words does not suggest that you yourself have learned anything other than how to cut and paste.

Yes, exactly. A recent post that was just such an "image and links salad" has been deleted (for the second time).

The mods are probably going to delete all those things i arranged in order to understand some fundamental concepts , They are again going to call that irrelevant

They are irrelevant. If you don't have a good understanding of algebra, which is evident from recent posts in this and other threads, you are wasting your time trying to learn about more esoteric things, such as calculus and differential equations.

Posting a bunch of images you found online is NOT how you learn mathematics.

 

[awholenumber]

OK , I was just looking for an easy way . Strictly going through the texts is going to take a lot more time

Now that i have understood independent and dependent variables , Let me see what i can read and understand from this book now , W. Boyce, R. Diprima - Elementary Differential Equations

Thanks for the replies .

 

[Mark44]

Now that i have understood independent and dependent variables , Let me see what i can read and understand from this book now , W. Boyce, R. Diprima - Elementary Differential Equations

Boyce and DiPrima is going to require a huge amount more than just understanding the difference between independent and dependent variables.

I would suggest some more along the lines of a precalculus book, where they talk about functions of various kinds, and function notation.

OK , I was just looking for an easy way . Strictly going through the texts is going to take a lot more time

Well, yes. As the Greek geometer Euclid is said to have told Ptolemy I, "there is no royal road to geometry." The same applies to an easy way of learning mathematics in general.

 

[awholenumber]

Thanks for the suggestion Mark44 ,

Has anyone seen this website ?

http://17calculus.com/precalculus/
http://17calculus.com/
http://17calculus.com/differential-equations/
Looks really nice for beginners