Some questions while I self-learn Applied Calculus

In free time I start to solve differentials and integrals, I am doing fine, I just follow rules and solve the tasks.
I start solve some applied calculus tasks, but I dont really understand why for exmple second derivative represent acceleration, why first is speed, why I need to derivate Area=axb to find a and b with max area of recntangle if I know diagonal..etc

I know that differential is ratio of delta y / delta x. If I change x how much is change y, or simple tangent angle..
Can someone explain this?

What texts (physics and calculus) are you using?

user079622 said:

I dont really understand why for exmple second derivative represent acceleration, why first is speed,

Derivatives of displacement with respect to time, you mean. It's important to say what you are differentiating and with respect to what.

How fast you are going is a measure of the rate of change of your position. Rate of change of a function (position) as you vary something (time) is what derivatives measure. Acceleration is the rate of change of velocity with respect to time - so ditto.

user079622 said:

why I need to derivate Area=axb to find a and b with max area of recntangle if I know diagonal..etc

Imagine doing it the hard way - draw all the possible rectangles, measure the area as you vary ##a##. You could draw a graph of area as a function of one side length. Note that the maximum area is the top of the curve you draw on the graph. So the slope is changing from positive to negative at that point - it's zero. So to find the maximum area you take the derivative of that graph and set it to zero.

Vanadium 50 said:

What texts (physics and calculus) are you using?

I learn and solve tasks from youtube videos, I also bought book mathematics for engineering university, but I didnt learn from this book yet.

Do you think is better to learn from book or what is best source to learn?

Ibix said:

Derivatives of displacement with respect to time, you mean. It's important to say what you are differentiating and with respect to what.

How fast you are going is a measure of the rate of change of your position. Rate of change of a function (position) as you vary something (time) is what derivatives measure. Acceleration is the rate of change of velocity with respect to time - so ditto.

Imagine doing it the hard way - draw all the possible rectangles, measure the area as you vary ##a##. You could draw a graph of area as a function of one side length. Note that the maximum area is the top of the curve you draw on the graph. So the slope is changing from positive to negative at that point - it's zero. So to find the maximum area you take the derivative of that graph and set it to zero.

I rember from my university time(long time ago) integral is area under curve, so wondered why I derivate function A=axb for this...
I am struggle to connect what has rate of something change(differential) with area, speed..

user079622 said:

I start solve some applied calculus tasks, but I dont really understand why for exmple second derivative represent acceleration, why first is speed, why I need to derivate Area=axb to find a and b with max area of recntangle if I know diagonal..etc

Derivatives are rates of change of some quantity with respect to another. Velocity is the rate of change of position with respect to time (also described as the time rate of change of position). If s(t) represents the position at time t, the velocity v(t) is the time rate of change of position. (Note that physics books often use s to represent position -- I don't know why.) So ##v(t) = \frac d {dt}(s(t))## also written as s'(t).
Acceleration is the time rate of change of velocity, so ##a(t) = \frac{d^2}{dt^2}(s(t)) = \frac d{dt} v(t)##, also written as s''(t) or v'(t).

user079622 said:

I know that differential is ratio of delta y / delta x.

No, what you're talking about is the derivative, which is different from the differential of something. Also, the derivative is a limit.
If y = f(x), the derivative is ##\frac{dy}{dx} = \lim_{h \to 0}\frac{f(x +h) - f(x)}h##.
The differential of y is defined as ##dy = f'(x) dx##.

user079622 said:

I learn and solve tasks from youtube videos,

Not the best source.

user079622 said:

Do you think is better to learn from book or what is best source to learn?

From a reputable textbook.

user079622 said:

I rember from my university time(long time ago) integral is area under curve, so wondered why I derivate function A=axb for this...

You don't "derivate" a function -- you differentiate it. Doesn't make sense, but it is what it is.

I don't understand what you mean when you say you "derivate function A = axb for this." Another member has already mentioned what happens when you integrate a function. The most basic example of integration is to find the area under a curve.

Mark44 said:

I don't understand what you mean when you say you "derivate function A = axb for this." Another member has already mentioned what happens when you integrate a function. The most basic example of integration is to find the area under a curve.

I think OP is referring to maximizing Area - as for a rectangle when given some constraint such as perimeter.

I can say with a high degree of confidence that you will make more progress with a good textbook than a bunch of YouTube videos.

user079622 said:

... I rember from my university time(long time ago) integral is area under curve, so wondered why I derivate function A=axb for this...
I am struggle to connect what has rate of something change(differential) with area, speed..

The relationship between derivatives and integrals (that the integral is the anti derivative) is "the fundamental theorem of calculus." Look it up.

Mark44 said:

I don't understand what you mean when you say you "derivate function A = axb for this." Another member has already mentioned what happens when you integrate a function. The most basic example of integration is to find the area under a curve.

I dont know what is difference of differential vs derivation , I mean on sign ´
English is not my first language

SammyS said:

I think OP is referring to maximizing Area - as for a rectangle when given some constraint such as perimeter.

Diagonal is 4cm, I must find sides a and b for rectangular with max area.
A=a x b --> MAX
d^2=a^2 x b^2

Vanadium 50 said:

I can say with a high degree of confidence that you will make more progress with a good textbook than a bunch of YouTube videos.

But problem in book there is no professor who explain, you must grasp everything alone.
Khan academy?
https://www.khanacademy.org

user079622 said:

But problem in book there is no professor who explain, you must grasp everything alone.

And when you ask questions of a recorded video, does it answer you back? And for how long has this been happening?

Vanadium 50 said:

And when you ask questions of a recorded video, does it answer you back? And for how long has this been happening?

No. How long I study calculus? Few days

user079622 said:

I dont know what is difference of differential vs derivation , I mean on sign ´

I explained the difference between a derivative and a differential in post #5.

user079622 said:

Diagonal is 4cm, I must find sides a and b for rectangular with max area.
A=a x b --> MAX
d^2=a^2 x b^2

One of the most common applications of differentiation is to find the maximum value of some function. To lessen the chance of confusion I'm going to call the sides l and w (for length and width). For this problem, since the diagonal is 4 cm, then ##w^2 + l^2 = 16##, so ##l^2 = 16 - w^2 \Rightarrow l = \pm \sqrt{16 - w^2}##
We can assume that neither side is negative, so ##l = \sqrt{16 - w^2}##

Then we can write the area of the rectangle as a function of side w as ##A(w) = w \cdot \sqrt{16 - w^2}##.
A maximum value occurs at a point (w, A(w)) at which

• A'(w) = 0 or
• w is an endpoint of some restricted domain or
• a point at which A(w) is undefined.

user079622 said:

But problem in book there is no professor who explain, you must grasp everything alone.
Khan academy?

Khan Academy is better than most, but IMO, not as good as a textbook. As pointed out already, after you have watched a video, if you have a question, you can't ask the video for an answer to your question.

Mark44 said:

I explained the difference between a derivative and a differential in post #5.


One of the most common applications of differentiation is to find the maximum value of some function. To lessen the chance of confusion I'm going to call the sides l and w (for length and width). For this problem, since the diagonal is 4 cm, then ##w^2 + l^2 = 16##, so ##l^2 = 16 - w^2 \Rightarrow l = \pm \sqrt{16 - w^2}##
We can assume that neither side is negative, so ##l = \sqrt{16 - w^2}##

Then we can write the area of the rectangle as a function of side w as ##A(w) = w \cdot \sqrt{16 - w^2}##.
A maximum value occurs at a point (w, A(w)) at which

• A'(w) = 0 or
• w is an endpoint of some restricted domain or
• a point at which A(w) is undefined.

Why you use A(w) and A(ω) ? Is this tipfeler?

Mark44 said:

Khan Academy is better than most, but IMO, not as good as a textbook. As pointed out already, after you have watched a video, if you have a question, you can't ask the video for an answer to your question.

Yes but you cant ask question to a book as well, plus book has lots of proofs that is not maybe necessary.

user079622 said:

Why you use A(w) and A(ω) ? Is this tipfeler?

I used A(w), where w represents the width of the rectangle, because I didn't want to have both A and a in the same equation. I didn't use A(ω) (ω is lower-case Greek ##\Omega##). Maybe you thought that the w in the LaTeX expression looked like ##\omega##.
No, it wasn't tipfeler (or in English, typo).

user079622 said:

Yes but you cant ask question to a book as well, plus book has lots of proofs that is not maybe necessary.

Most of the calculus textbooks I've seen (lots) have very few proofs.

Mark44 said:

Then we can write the area of the rectangle as a function of side w as
##A(w) = w \cdot \sqrt{16 - w^2}##
A maximum value occurs at a point (w, A(w)) at which

• A'(w) = 0 or
• w is an endpoint of some restricted domain or
• a point at which A(w) is undefined.

So, @user079622 , can you differentiate the expression ##A(w) = w \cdot \sqrt{16 - w^2}##

to get ##\frac {d} {dw} (w \cdot \sqrt{16 - w^2}) ##

You will need to use the product rule (https://en.wikipedia.org/wiki/Product_rule)

Mark44 said:

Most of the calculus textbooks I've seen (lots) have very few proofs.

Vanadium 50 said:

And when you ask questions of a recorded video, does it answer you back? And for how long has this been happening?

Do you think that book is even better option from watching class video lecture, what site/channel do you suggest?
For example like this

gmax137 said:

So, @user079622 , can you differentiate the expression ##A(w) = w \cdot \sqrt{16 - w^2}##

to get ##\frac {d} {dw} (w \cdot \sqrt{16 - w^2}) ##

You will need to use the product rule (https://en.wikipedia.org/wiki/Product_rule)

Yes I can do it.

Official lecture series from a reputable university are a good option. Especially if they follow a textbook that you have.

There are also some good unofficial sources, like this guy:

https://thebrightsideofmathematics.com/

The main problem is that these series rarely give the viewer access to problem sheets. I quite like pausing the video and trying to figure out the next bit myself. If you can do that, then you definitely understand the material.

PeroK said:

Official lecture series from a reputable university are a good option. Especially if they follow a textbook that you have.

There are also some good unofficial sources, like this guy:

https://thebrightsideofmathematics.com/

The main problem is that these series rarely give the viewer access to problem sheets. I quite like pausing the video and trying to figure out the next bit myself. If you can do that, then you definitely understand the material.

Something like this?

user079622 said:

I learn and solve tasks from youtube videos, I also bought book mathematics for engineering university, but I didnt learn from this book yet.

Do you think is better to learn from book or what is best source to learn?


I rember from my university time(long time ago) integral is area under curve, so wondered why I derivate function A=axb for this...
I am struggle to connect what has rate of something change(differential) with area, speed..

You'll have to figure out and tweak over time the learning method(s) that work best for you.

I am lost in this letters notation, such a shame that didnt use only one universal and the most simplest, like prime..
https://en.wikipedia.org/wiki/Notation_for_differentiation

user079622 said:

I am lost in this letters notation, such a shame that didnt use only one universal and the most simplest, like prime..
https://en.wikipedia.org/wiki/Notation_for_differentiation

It's a language. Don't synonyms make a language richer?

Hill said:

It's a language. Don't synonyms make a language richer?

Why call it df/dx if I can write simple and less confusing f´(x) ?

user079622 said:

Why call it df/dx if I can write simple and less confusing f´(x) ?

Many reasons. For example, sometimes you need to make a manipulation such as to go from ##\frac {df} {dx} = g(x)## to ## df = g(x) dx##.

Hill said:

Many reasons. For example, sometimes you need to make a manipulation such as to go from ##\frac {df} {dx} = g(x)## to ## df = g(x) dx##.

What this letters "d" mean?

user079622 said:

Why call it df/dx if I can write simple and less confusing f´(x) ?
What this letters "d" mean?

The notation ##\frac{dy}{dx}## is so common that there is nothing to be gained by complaining about it.

user079622 said:

What this letters "d" mean?

Infinitesimal change.

PeroK said:

The notation ##\frac{dy}{dx}## is so common that there is nothing to be gained by complaining about it.

Here is some different letters

Ibix said:

Derivatives of displacement with respect to time, you mean. It's important to say what you are differentiating and with respect to what.

How fast you are going is a measure of the rate of change of your position. Rate of change of a function (position) as you vary something (time) is what derivatives measure. Acceleration is the rate of change of velocity with respect to time - so ditto.

Imagine doing it the hard way - draw all the possible rectangles, measure the area as you vary ##a##. You could draw a graph of area as a function of one side length. Note that the maximum area is the top of the curve you draw on the graph. So the slope is changing from positive to negative at that point - it's zero. So to find the maximum area you take the derivative of that graph and set it to zero.

I will stop watching youtube and try learn from book.

When I learn from university book, does it make sense to write notes in my notebook something that is already written in book or this is just waste of time?

What learning strategy do you suggest?

​

(for sure I will solve tasks in notebook..)

user079622 said:

No. How long I study calculus? Few days

This is rather enlightening to those offering advice to you.

What is your mathematical background?

What math have you studied previously?

What mathematical skills do you have?

user079622 said:

I will stop watching youtube and try learn from book.

When I learn from university book, does it make sense to write notes in my notebook something that is already written in book or this is just waste of time?

What learning strategy do you suggest?


(for sure I will solve tasks in notebook..)

Sorry, but this , learning style and methods, is something you need to develop by/for yourself and tweak it as you go.

user079622 said:

Here is some different letters

Some of the notation in the image you posted I've never seen before.
For example, I've never seen ##f_{'}## to represent ##\frac{\partial f}{\partial y}##, ##f_{'}^{'}## to represent the mixed second partial derivative, or ##f_{''}## to represent ##\frac{\partial^2 f}{\partial y^2}##.
I don't believe any of these notations are used in any modern calculus textbooks, with "modern" defined as being published in the last 125 years.

SammyS said:

This is rather enlightening to those offering advice to you.

What is your mathematical background?

What math have you studied previously?

What mathematical skills do you have?

I had calculus at university 30years ago, all these years I wasnt in touch with this..

Mark44 said:

Some of the notation in the image you posted I've never seen before.
For example, I've never seen ##f_{'}## to represent ##\frac{\partial f}{\partial y}##, ##f_{'}^{'}## to represent the mixed second partial derivative, or ##f_{''}## to represent ##\frac{\partial^2 f}{\partial y^2}##.
I don't believe any of these notations are used in any modern calculus textbooks, with "modern" defined as being published in the last 125 years.

Learning from book is very hard, it is hard to figure out what you need to do with this "hieroglyphics".

I find it easier to learn when person solve the task and explain step by step what is he doing and why.

user079622 said:

Learning from book is very hard, it is hard to figure out what you need to do with this "hieroglyphics".

I find it easier to learn when person solve the task and explain step by step what is he doing and why.

Presumably this is why people part with considerable sums of money to be taught at university.

PeroK said:

Presumably this is why people part with considerable sums of money to be taught at university.

Yes there is good reason why professors exists, otherwise everyone would study alone.

just one example:
In book I cant find how they get du, because book skips steps.

Instead just watch one video and find how he get du.. everything is clear now