Calculus & Diff. Eqn: Beginner Qs on Function, Derivative & Gradient

[kupid]

I have some beginner doubts about Calculus and Differential equations .

Is a function always a curve ?

Doesn't a function already has a slope ?

d/dx of a function gives the gradient of the curve between two points ?

The derivative ,d/dx ,The gradient , is the rate of change of a function between two points ?

Do i have to imagine a function as a point like object moving that have a gradient or a rate of change between only two points ?

What is the use of simply finding the rate of change of a curve at two points ?

Can we somehow predict the rate of change for the rest of the points ?
Now for differential equations ...

It is an equation involving an unknown function and its derivatives

The derivative ,d/dx ,The gradient , is the rate of change of a function between two points ?

And we are supposed to find that unknown function that produced that rate of change of a function between two points ?

Again , Can we somehow predict the rate of change for the rest of the points ?

 

[Jameson]

Hi kupid, (Wave)

I'll try my best to cover the first part of your questions. These are good things to ask by the way!

Is a function always a curve ?

There is a precise definition of a function, but it does not always have to be a curve. Lines like $f(x)=x+1$ for example is a function and isn't a "curve". So in general, a function can be a curve or not, and curves aren't always functions.

Doesn't a function already has a slope ?

d/dx of a function gives the gradient of the curve between two points ?

The derivative ,d/dx ,The gradient , is the rate of change of a function between two points ?

Do i have to imagine a function as a point like object moving that have a gradient or a rate of change between only two points ?

Ok, so here we are getting at the core of the derivative. Between two points on a curve there is a slope, or rate of change, that we can calculate as $\frac{\Delta y}{\Delta x}$. This could be useful for a lot of things like "How fast was the car going on average between points A and B" or "How much profit per year did the company make between 1900-2000?" but with calculus and derivatives we are trying to find the instantaneous slope at one point. This is done by taking two points and shrinking them closer and closer and closer together, until they approach the same point. The goal is to find a rate that captures the slope of a function at a particular point.

 

[kupid]

Thanks a lot for explaining very important basic concepts , so most of the time it is simply about one point . Usually one point does not give us any information , so we take another point as part of the process of finding the derivative .

I was thinking really hard to find some sort of practical application , suddenly i was thinking about a ball rolling through these slopes between these two points like for example How fast does that ball role through that one point ..

OK , That makes sense

 

[Dethrone]

Thanks a lot for explaining very important basic concepts , so most of the time it is simply about one point . Usually one point does not give us any information , so we take another point as part of the process of finding the derivative .

I was thinking really hard to find some sort of practical application , suddenly i was thinking about a ball rolling through these slopes between these two points like for example How fast does that ball role through that one point ..

OK , That makes sense

The irony is that it's hard to find anything for which the concept of a derivative cannot be applied, but generally understanding why the derivative is useful is not apparent until you progress further into your studies. Most of the time, we are concerned with finding the rate of change at one point. If you're driving, you'd be concerned about how fast you're going at a particular "instant" -- that is, the velocity of your car. The technology you used to type your question, whether it's a computer or phone, consists of integrated circuits. Derivatives or differential equations arise if you were to design the circuit -- the concept of a transfer function may be important, which relates the $n$th order relationship between the input and output. One important application of the derivative is optimization, and in a generation where artificial intelligence is becoming more and more prevalent, the application of machine learning in Siri, or movie recommendations, facial recognition, google translate, etc. relies heavily on derivatives. Machine learning is in fact an optimization problem. So yes, derivatives, or more generally calculus, is used everywhere: in business, finance, physics, all branches of engineering, etc.

 

[kupid]

Thanks for the reply Rido12,

but with calculus and derivatives we are trying to find the instantaneous slope at one point

The irony is that it's hard to find anything for which the concept of a derivative cannot be applied, but generally understanding why the derivative is useful is not apparent until you progress further into your studies. Most of the time, we are concerned with finding the rate of change at one point. If you're driving, you'd be concerned about how fast you're going at a particular "instant" -- that is, the velocity of your car. The technology you used to type your question, whether it's a computer or phone, consists of integrated circuits. Derivatives or differential equations arise if you were to design the circuit -- the concept of a transfer function may be important, which relates the $n$th order relationship between the input and output. One important application of the derivative is optimization, and in a generation where artificial intelligence is becoming more and more prevalent, the application of machine learning in Siri, or movie recommendations, facial recognition, google translate, etc. relies heavily on derivatives. Machine learning is in fact an optimization problem. So yes, derivatives, or more generally calculus, is used everywhere: in business, finance, physics, all branches of engineering, etc.

I am still not sure which point that is , i see the function going through a lot of points . I am still confused .

 

[MarkFL]

There is an error in the working by first principles, because if given:

\(\displaystyle f(x)=x^3\)

then:

\(\displaystyle f'(x)=3x^2\)

What this means is that for any value of $x=a$, the instantaneous slope of $f$ at $(a,a^3)$ is $3a^2$.

[DESMOS=-10,10,-1005,995]y=x^3;y=3a^2\left(x-a\right)+a^3;a=0[/DESMOS]

Move the slider to see the tangent line respond to different values of $x$. :)

 

[kupid]

Thanks Mark

This expression 3x² is the derivative for the function, and we can find the slope of the tangent at any point on the curve by plugging in the x value of the coordinate. ?

 

[MarkFL]

Thanks Mark

This expression 3x² is the derivative for the function, and we can find the slope of the tangent at any point on the curve by plugging in the x value of the coordinate. ?

Yes, in my previous post, I used in the graph the fact that at the point $(a,a^3)$ on the curve, the slope of the tangent line will be $3a^2$. So, armed with the point and the slope, I used the point-slope formula to obtain the tangent line:

\(\displaystyle y=3a^2(x-a)+a^3\)

Then, by use of the slider (used as a parameter of the tangent line), you can view the tangent line at varying points along the curve. :D

 

[kupid]

Thanks Mark ,

OK , let me try one more example .

Find the equation to the tangent line to the graph of f(x) = x² + 3x at (1,4)

f(x) = x² + 3x

We find the derivative using the power rule for differentiation

f'(x) = 2x +3
Plug in our x coordinate into the derivative to get our slope

f'(1) = 2(1) + 3

f'(1) = 5

Now we can use point slope form to find the equation of the tangent line. (1,4) is our point and 5 is our slope

y-y₁ = m (x - x₁ )

y-4 = 5(x-1)

y= 5x -1

Can i get a Desmos graph like in post #6 , one more time . When i tried to make the graph with these values it is not working .

 

[MarkFL]

Okay, to construct the dynamic tangent line graph for this function, we begin with the point:

\(\displaystyle (a,a^2+3a)\)

and the slope:

\(\displaystyle 2a+3\)

And we construct the tangent line using the point-slope formula:

\(\displaystyle y=(2a+3)(x-a)+a^2+3a\)

[DESMOS=-10,10,-3,50]y=x^2+3x;y=\left(2a+3\right)\left(x-a\right)+a^2+3a;a=1[/DESMOS]

 

[kupid]

Thanks a lot Mark ,There is lot to learn from this thread :)

 

[HallsofIvy]

There have been plenty of answers to these, but I just have to put in my oar.

I have some beginner doubts about Calculus and Differential equations .

Is a function always a curve ?

I would say that a function is never a curve! A "function" is a set of order pairs. The graph of a function is a curve. The graph of a "linear function" is a straight line which some people would say "doesn't curve" but I am willing to leave the verb alone and let the noun be used more generally!

Doesn't a function already has a slope ?

I'm not sure what "already" is intended to mean here! Strictly speaking, "slope" is only defined for straight lines, i.e. linear functions.

d/dx of a function gives the gradient of the curve between two points ?

No, it doesn't. The derivative of a function, at one point is the slope of the tangent line to the curve at that point.

The derivative ,d/dx ,The gradient , is the rate of change of a function between two points ?

No, it isn't. It is the rate of change of a function at a single point. That doesn't fit the basic idea of "rate of change" as requiring a change which is why we have to use limits to generalize the idea of "rate of change".

Do i have to imagine a function as a point like object moving that have a gradient or a rate of change between only two points ?

You don't have to imagine it like that but it helps some people to think of it that way.

What is the use of simply finding the rate of change of a curve at two points ?

That would depend upon your purpose in working with that curve. For example, if the "curve" is the graph of "distance as a function of time" then the rate of change of a curve between two points is the average speed between those two points. And if it is a graph of "speed as a function of time" then it is the average acceleration.

Can we somehow predict the rate of change for the rest of the points ?

Not unless we have some other information about the function itself. The derivative at a given point tells us only the rate of change at that particular point.

Now for differential equations ...It is an equation involving an unknown function and its derivatives

Generally, yes. Although we can write a differential equation to illustrate some fact about a known function as well.

The derivative ,d/dx ,The gradient , is the rate of change of a function between two points ?

This is the same question as before. No, it is not. The derivative is found at a single point.

And we are supposed to find that unknown function that produced that rate of change of a function between two points ?

No, a differential equation does not deal with two points! And, while, typically, in a beginning differential equations course, we are asked to "solve" a differential equation, that is not the only possible reason for writing a differential equation. For example, in more advanced classes we may use a differential equation that we cannot solve to get information about possible solutions.

Again , Can we somehow predict the rate of change for the rest of the points ?

If you are given a differential equation that holds for all points, then it gives immediately (we do not need to "predict") the rate of change for all points.

 

[kupid]

HallsofIvy ,

Thanks a lot for all those important points

Maybe i should look at examples of calculus and differential equations from this website

https://www.khanacademy.org/math/calculus-home

https://www.khanacademy.org/math/differential-equations

:)

 

[MarkFL]

HallsofIvy ,

Thanks a lot for all those important points

Maybe i should look at examples of calculus and differential equations from this website

https://www.khanacademy.org/math/calculus-home

https://www.khanacademy.org/math/differential-equations

:)

We have a large repository of worked problems of all kinds here:

http://mathhelpboards.com/questions-other-sites-52/

Please feel free to post in any thread there with questions you might have regarding the solutions posted. :D

 

[kupid]

Thanks a lot Mark , I will look into it when i can find some free time :)