Awesome way of introducing calculus! Would love a video like this working into the fundamental theorem of calculus, you always have a nice way of putting things for new leaners
Amazing video, and probably the best 5 minute introduction to calculus I’ve seen. To anyone watching this and wanting the same type of content giving you an intuitive insight into what calculus is, but far more in depth and covering more calculus topics, 3blue1brown has a series called “the essence of calculus” that is truly outstanding
@@dxdux I unfortunately don’t as of now, it’s something I’ve been looking for, however. I’m a new math major and have been looking for resources that aren’t videos (I like videos but they’re not usually great to learn from for me) and aren’t textbooks. Books that I can just sit down and read, and that discuss math concepts at a conceptual level, without actually doing lots of math. I do plenty of math already, but gaining insight beyond crunching the numbers is what I’m after, and so far it’s mostly come from my professors or me stumbling into understanding of my own while doing the numbers. There’s a good handful on proof writing that are like that, but I’m still on the lookout for books like this on other topics, like real analysis, abstract algebra, etc. If I stumble across one for calculus, I’ll try and update you here though, and hopefully others come along with recommendations as well!
Awesome video Andy! I love problem solving in Math and the way I was taught is it's always about how to solve the problem. But not why we solve the problem. Or what is even the problem at a zoomed out level. I want to understand the basics and principles of concepts like Calculus and this did a great job of it. Thanks!
Wow! That makes SO MUCH SENSE! Wish I could have seen this... 50 years ago! Thank you. Nice to revist something that is really useful in this world... and that I barely understood.
as i have gotten older and gotten farther away from my high school days, i have gone to realize and fee that a lot of the more complicated maths and such is just long-form logic puzzles, and I come to understand them a lot more. great stuff here, Andy Math. Thank you.
I wish math was more like this when I was in school. Instead all I got was confusing math problems at 8am or a computer program that just refused to accept an answer that turned out to be correct
I was an engineering student in 1994 as well. MatLab sucked so bad. I barely made it through Calc 3 but everything else was fine. Ended up with a math minor somehow.
I just passed my Differential Calculus class. I'm familiar with the practical uses of Calculus, like minimizing costs or maximizing the volume of your DIY box, but this is my first time actually understanding how it can be used to calculate rates of change. I just assumed that it's a magical process which can automatically generate us numbers we want if we tweak some formulas. This is really cool.
@@TransientNeuroticIt's called an optimization problem. You can use a function to model the cost for each choice of some independent variable, and then you can use calculus to find a minimum point of that function, which tells you which choice will minimize the cost.
Where were you in 1992 when my calculus teacher couldn't explain this all year? I would have been able to avoid summer school if I had learned it this way. Brilliant!
Thought calculus was like some hard near impossible stuff to even get the basics of until now theres probably more complex problems but I'm starting to understand it now..
So is the slope of the tangent line at (1,1) 2x then, and what does f’(x)=2x mean it applies to the whole graph of you only use one point to plug it into, or is that only the slope of the graph at (1,1) ?
i really like using a graphical representation for this. if you graph x^2 and 2x, you'll notice that on the graph of the derivative (2x) at any given x value you have the slope of the tangent line. on the line 2x, you have a point at (1,2) and (2,4). the equation for the tangent line of x^2 at x=1 is y=2x-1, at x=2 it is y=4x-4. if i didn't make it clear, the graph of the derivative gives the slope of the tangent line at any given x value in a function f(x). also by the way, once you get past the first section in calculus you no longer manually compute derivatives. there's a bunch of rules that you can use. in this case, instead of doing all those calculates i immediately knew that the derivative of x^2 is 2x because for a given expoential function x^n, the derivative is n*x^(n-1). for something like 3x^3, i can immediately calculate the derivative to be 9x^2. constant multiples (cx, where c is a constant number) aren't evaluated in the derivative by the way, as long as they are multiplying something. the derivative of c + x would be 0 + the derivative of x, but the derivative of cx would be c* the derivative of x. quick little intro. it's pretty cool.
Mr Andy could you do a calculus sum for a period of x to ...say.... x +10 ? That would mean doing 10 calculations, iterating the calculation at x + 1, x + 2, and so on....and what would be the real world application.
The function f is defined by the equation f(x) = x^2. Whatever you plug in, you just replace every x with that value. For example: f(2) = 2^2 f(7) = 7^2 f(372) = 372^2 f(π) = π^2 So, if we plug in x + h: f(x + h) = (x + h)^2
2:18 Absolutely not. The point isn't to see what happens when h is equal to 0; the point is to see what happens as h gets closer and closer to 0. That's how a limit works. If you just plug in h = 0 to (f(x + h) - f(x)) / h, you get (f(x + 0) - f(x)) / 0 = (f(x) - f(x)) / 0 = 0 / 0, which is undefined. And no, I'm not nitpicking technicalities solely for the sake of nitpicking technicalities. It is absolutely essential that anyone learning calculus truly understands what a limit is. This is why it's important to be careful with our choice of words. Yes, this is corrected at 2:53, but the video should've just not had the incorrect information in the first place.
I think you meant one of the following: 1) d/dx x^2 = 2x (the derivative with respect to x of x squared is two x). 2) For f(x) = x^2, f'(x) = 2x (for f of x equals x squared, f prime of x equals two x). What you wrote implies that there exists some function f, and when you apply its derivative f' to the expression x^2, you get 2x. This is probably not what you meant. However, it does give rise to an interesting question: What is the function f? The equation you gave is called a functional equation, and we can solve for the function. From here, I'll assume we're working in the real numbers. So, let's start by figuring out what the derivative of f is. How do you take the value a^2 and transform it into the value 2a, for any constant a? Note: I'm talking about *values*, not functions. It's an important distinction. We're assuming we plugged in some constant value for the variable a beforehand, and we want a function that'll take a^2 to 2a no matter the value of a. We want it to take 7^2 to 2(7), we want it to take 83^2 to 2(83), and so on. Assuming nonnegative values of a, we just have to take the square root, then double the result. This suggests the following relation for f': f'(x) = 2√x. Unfortunately, this doesn't work for negative values of a. You'd have to use f'(x) = -2√x in that case, but our function f' can't give two outputs for a single input. So, let's just add the restriction that a ≥ 0. Actually, we were using x in the original equation, so it's x ≥ 0. Okay, moving on. Now we can find the function f by antidifferentiation: f(x) = ∫f'(x) dx = ∫2√x dx = 2∫x^(1/2) dx = 2(2/3)x^(3/2) + C = 4/3 x^(3/2) + C So, the function f is defined by f(x) = 4/3 x^(3/2) + C, for some arbitrary constant C. Thus, we have solved for what f must be in the equation f'(x^2) = 2x, assuming x ≥ 0. Using essentially the same logic, if x ≤ 0, we can instead define f by f(x) = -4/3 x^(3/2) + C.
Being able to measure how something is changing is pretty important in almost every field of science. The derivative, and calculus in general, allows you to do this. Pretty cool!
Why does every "Intro to Calculus" vid I ever see just jump right into the math without defining exactly what Calculus is? I remember my first day of AP Calc, my teacher must have said it 10 times to drill it into our hesds: Calculus is the mathematics of motion and change.
I want every math student to see this 👌
Awesome way of introducing calculus! Would love a video like this working into the fundamental theorem of calculus, you always have a nice way of putting things for new leaners
Saving it in my playlist "math lore" 😂
Lol no, this is math gameplay
Amazing video, and probably the best 5 minute introduction to calculus I’ve seen.
To anyone watching this and wanting the same type of content giving you an intuitive insight into what calculus is, but far more in depth and covering more calculus topics, 3blue1brown has a series called “the essence of calculus” that is truly outstanding
Thanks!. Do you know of any book with the same premise?..I am more interested in math in general but from zero and in a logical and fun way.
@@dxdux I unfortunately don’t as of now, it’s something I’ve been looking for, however. I’m a new math major and have been looking for resources that aren’t videos (I like videos but they’re not usually great to learn from for me) and aren’t textbooks. Books that I can just sit down and read, and that discuss math concepts at a conceptual level, without actually doing lots of math. I do plenty of math already, but gaining insight beyond crunching the numbers is what I’m after, and so far it’s mostly come from my professors or me stumbling into understanding of my own while doing the numbers.
There’s a good handful on proof writing that are like that, but I’m still on the lookout for books like this on other topics, like real analysis, abstract algebra, etc.
If I stumble across one for calculus, I’ll try and update you here though, and hopefully others come along with recommendations as well!
This is such a great video, short, straight to the point and intuitive.
Man that was brilliant the way you make it so intuitive using visuals is remarkable cant wait to see the part on integration
Awesome video Andy! I love problem solving in Math and the way I was taught is it's always about how to solve the problem. But not why we solve the problem. Or what is even the problem at a zoomed out level. I want to understand the basics and principles of concepts like Calculus and this did a great job of it. Thanks!
I love that you don't gloss over a single step.
Wow! That makes SO MUCH SENSE!
Wish I could have seen this...
50 years ago!
Thank you.
Nice to revist something that is really useful in this world... and that I barely understood.
I was searching for a video about this and you posted at the perfect time
How fascinating, although I already did my class, it’s cool to get a reminder of all the theory I went through.
as i have gotten older and gotten farther away from my high school days, i have gone to realize and fee that a lot of the more complicated maths and such is just long-form logic puzzles, and I come to understand them a lot more. great stuff here, Andy Math. Thank you.
Thank you so much ! I always had a hard time to understand the link between limits and derivative. This is so well explained !
Really amazing, nice and helpful explanation of calculus .. much apreciation!
2:44 - 2:50 god the passion with which he delivers this information is infectious
That was mind-blowing.
I wish math was more like this when I was in school. Instead all I got was confusing math problems at 8am or a computer program that just refused to accept an answer that turned out to be correct
I was an engineering student in 1994 as well. MatLab sucked so bad. I barely made it through Calc 3 but everything else was fine. Ended up with a math minor somehow.
This is known as the First Principle from which all of differential Calculus flows. Excellent explanation!
Thanks Andy, I'm now prepared for my highschool Calculus course
I used to hate Calculus. But you along with The Math Sorcerer have made Calculus something I love!!
Eddie Woo is really good as well
I just passed my Differential Calculus class. I'm familiar with the practical uses of Calculus, like minimizing costs or maximizing the volume of your DIY box, but this is my first time actually understanding how it can be used to calculate rates of change. I just assumed that it's a magical process which can automatically generate us numbers we want if we tweak some formulas. This is really cool.
Minimising costs?
Can you elaborate?
U passed differential calculus but have no idea about rates of change? Your teachers have failed you.
@@TransientNeuroticIt's called an optimization problem. You can use a function to model the cost for each choice of some independent variable, and then you can use calculus to find a minimum point of that function, which tells you which choice will minimize the cost.
@@isavenewspapers8890 Wish I could understand how to do something like that lol
@@TransientNeurotic There are good resources online if you want to learn.
I’ve been wondering why we’ve been learning about (x+h) - (x) over h. Thanks for the explanation!
Actually, it's (f(x + h) - f(x)) / h.
Thanks for a great lesson. Much appreciated
how exciting. looking forward to more of this
I love all your videos, they are super helpful
Thanks Andy! This helped me understand derivatives better.
Amazing explanation. I really appreciated
this is very helpful for someone like me who is new to calculus
spectacular!!!!
This was amazing
Love this!
Andy litterally woke up and gave you three months of math class for free
Epic Andy moment
How exciting!
How exciting.
Where were you in 1992 when my calculus teacher couldn't explain this all year? I would have been able to avoid summer school if I had learned it this way. Brilliant!
very very clean
Really how excited ❤
Congratulations for your cristal claire explenation.
Very great..
Love the sayan hairstyle!
Dude u legit showed the importance of why derivation of formulas exist ... Legit taught maths the way it should be taught
Amazing❤
5 minutes compared to 1 year in school. Great job! :)
you are a master, i realy mean it
Amazing, it would be nice to have something like this but for the Fundamental Theorem of Calculus.
im using calculus to calculate the slope of my life going downhill
Mind = blown !!
Yoo this is soo nicee
I was really confused but that was a good explanation
He explains the fundamentals of calculus in 5 minutes. This video is not recommended for someone who learns this subject for the first time
More calculus, please.
Thought calculus was like some hard near impossible stuff to even get the basics of until now theres probably more complex problems but I'm starting to understand it now..
absolutely gets a million times harder, this is the very first thing you learn in calculus.
@@zydn Wow, how encouraging...
@@zydnHyperbole much?
God bless
"How exciting!"
So is the slope of the tangent line at (1,1) 2x then, and what does f’(x)=2x mean it applies to the whole graph of you only use one point to plug it into, or is that only the slope of the graph at (1,1) ?
i really like using a graphical representation for this. if you graph x^2 and 2x, you'll notice that on the graph of the derivative (2x) at any given x value you have the slope of the tangent line. on the line 2x, you have a point at (1,2) and (2,4). the equation for the tangent line of x^2 at x=1 is y=2x-1, at x=2 it is y=4x-4.
if i didn't make it clear, the graph of the derivative gives the slope of the tangent line at any given x value in a function f(x).
also by the way, once you get past the first section in calculus you no longer manually compute derivatives. there's a bunch of rules that you can use. in this case, instead of doing all those calculates i immediately knew that the derivative of x^2 is 2x because for a given expoential function x^n, the derivative is n*x^(n-1). for something like 3x^3, i can immediately calculate the derivative to be 9x^2. constant multiples (cx, where c is a constant number) aren't evaluated in the derivative by the way, as long as they are multiplying something. the derivative of c + x would be 0 + the derivative of x, but the derivative of cx would be c* the derivative of x.
quick little intro. it's pretty cool.
*Andy Math:* 💁🏼♀️ 5m!
*Other Teachers:* 🤦🏼♀️♾️…
Great
i wish andy was my math teacher
nice explanations
👏👏 this shit gonan be crazy goddamn
Mr Andy could you do a calculus sum for a period of x to ...say.... x +10 ? That would mean doing 10 calculations, iterating the calculation at x + 1, x + 2, and so on....and what would be the real world application.
Is he replacing f(x+h) /plugging in for x …(x+h)^2 bc x is basically the same thing as x+h based on the earlier change ?
The function f is defined by the equation f(x) = x^2. Whatever you plug in, you just replace every x with that value. For example:
f(2) = 2^2
f(7) = 7^2
f(372) = 372^2
f(π) = π^2
So, if we plug in x + h:
f(x + h) = (x + h)^2
@@isavenewspapers8890 you know what’s funny ? Since I posted that I learned algebra lol
Like I took a class and I thought about this video and realized how dumb my question was fr
@@ToTheWolves Ah, I see. I'm glad you advanced in your math education. But I will say, I don't want you ever to feel dumb for trying to learn.
What’s that lil swiggly line that I see in calculus problems
Why is this not how it's taught everywhere?!
Limx-0 f(x+h)-f(x)/h is called the first principal method
انت راجل جدع ❤
Thank u for listening to the video I'm understand
Quick correction at 4:42, you said derivative of X is 2x, I think you meant to say the derivative of the function with respect to x is 2x
You need a follow up with integrals or more harder derivatives
Calc I: 😊
Calc II: 💀
Damn i wanna be like u in my country. I think I'll copy u some day
BRO I NEVER GOT CALCULUS UNTIL NOW. NOW I UNDERSTAND IT WTF KSKFDJASLDJFPQEWROJDA
I saw the thumbnail and thought it was a meme oh I'm too far in
Lost me at the beginning 😢
You know what you're talking about.
I love maths
Who made one line so complicated
Maybe I gotta go back and study more algebra
@andy math can you please explain how instantaneous rate of change is used in real life
Best example i can think of is engineering and physics. If the values arent constant in formulas u need calculus.
Well, for example, it can be used to find the velocity of a moving object, like a car. Velocity is just the rate of change of position.
isnt it woderful, i pay a uni tax just so that this guy can explain me calculus
You’re so cute without a hat. You’re also cute with a hat, but still .
y=mx+c ✅
Part 2 - Integration!
2:18 Absolutely not. The point isn't to see what happens when h is equal to 0; the point is to see what happens as h gets closer and closer to 0. That's how a limit works. If you just plug in h = 0 to (f(x + h) - f(x)) / h, you get
(f(x + 0) - f(x)) / 0
= (f(x) - f(x)) / 0
= 0 / 0,
which is undefined.
And no, I'm not nitpicking technicalities solely for the sake of nitpicking technicalities. It is absolutely essential that anyone learning calculus truly understands what a limit is. This is why it's important to be careful with our choice of words. Yes, this is corrected at 2:53, but the video should've just not had the incorrect information in the first place.
I'm surprised that he didn't tease f'(x^2)=2x. Love the Videos though
I think you meant one of the following:
1) d/dx x^2 = 2x (the derivative with respect to x of x squared is two x).
2) For f(x) = x^2, f'(x) = 2x (for f of x equals x squared, f prime of x equals two x).
What you wrote implies that there exists some function f, and when you apply its derivative f' to the expression x^2, you get 2x. This is probably not what you meant.
However, it does give rise to an interesting question: What is the function f? The equation you gave is called a functional equation, and we can solve for the function. From here, I'll assume we're working in the real numbers.
So, let's start by figuring out what the derivative of f is. How do you take the value a^2 and transform it into the value 2a, for any constant a? Note: I'm talking about *values*, not functions. It's an important distinction. We're assuming we plugged in some constant value for the variable a beforehand, and we want a function that'll take a^2 to 2a no matter the value of a. We want it to take 7^2 to 2(7), we want it to take 83^2 to 2(83), and so on.
Assuming nonnegative values of a, we just have to take the square root, then double the result. This suggests the following relation for f':
f'(x) = 2√x.
Unfortunately, this doesn't work for negative values of a. You'd have to use f'(x) = -2√x in that case, but our function f' can't give two outputs for a single input. So, let's just add the restriction that a ≥ 0. Actually, we were using x in the original equation, so it's x ≥ 0. Okay, moving on.
Now we can find the function f by antidifferentiation:
f(x)
= ∫f'(x) dx
= ∫2√x dx
= 2∫x^(1/2) dx
= 2(2/3)x^(3/2) + C
= 4/3 x^(3/2) + C
So, the function f is defined by f(x) = 4/3 x^(3/2) + C, for some arbitrary constant C. Thus, we have solved for what f must be in the equation f'(x^2) = 2x, assuming x ≥ 0. Using essentially the same logic, if x ≤ 0, we can instead define f by f(x) = -4/3 x^(3/2) + C.
i swear ur an ai andy
My man don't know whole square of a+b. LOL😅😅😅😅
Or maybe it was intended to show how to actually obtain the value logically instead of just blindly applying a rule.
Are you a teacher?
Ok but why tho
Being able to measure how something is changing is pretty important in almost every field of science. The derivative, and calculus in general, allows you to do this. Pretty cool!
Instantaneous rate of change is stupid . Just use derivative.
hooooooooooly crap
Why does every "Intro to Calculus" vid I ever see just jump right into the math without defining exactly what Calculus is? I remember my first day of AP Calc, my teacher must have said it 10 times to drill it into our hesds: Calculus is the mathematics of motion and change.
Am I the only one who is obsessed with the echo? It's so annoying.
Would love to watch an entire series on calculus🤌🤞
How exciting.