I’m 76 years old and I always wondered what calculus was. This is the closest I’ve gotten. I am going to keep trying until I completely understand. Please publish more o these examples. You are a good teacher.
This is hands down the best explanation of The Fundamental Theorem of Calculus that I've seen. The reason is because you explain the WHY behind it all and give a real world example of how it is applicable and WHY its needed. Thank you for the video!
Of all the Calculus videos I've seen on UA-cam, yours are definitely my favorite. Concise, clear, conceptual - they're really good for understanding the concepts. I'm going to school for engineering and plan on viewing your Physics videos soon! Right now, I'm hoping to survive Calc. 2 online over the summer... Thanks!
I am 60yrs old. As a kid, I was a maths wizz and spent my working life as a betting shop manager. I have always been comfortable with probability theory; but calculus always bemused me. This is excellent!
very good explanation . now i have got the sense of using calculus. though i was able to solve problems in my schooldays i was not able to understand it in reality . we blindly used formulas, and how to solve typical problems just to score for exams not enough time to think over it ,due to law imagination power , and due to pressure of completing the courses . basically the purpose of calculus were not taught . and this still may be a problems for some students.. THANK YOU SIR.
J K J yes thats a problem with math in general. Some people are able to instantly click with the concept but some like me spend countless hours to understand but end up memorizing how to do it instead of understanding it. Videos like these help alot.
Veey true and this is one of the reason why students hate science classes.. the application part of it is missing (so that makes science classes look solid and horrible)
The explanation excellent for those that already have enrolled or take a course on Integral Calculus, not for those who doesn't. JUst a comment: Constant aceleration doesn't mean that the veocity doesn't change, it will change since there is acceleraion. Thanks for this excellent video.
I've been teaching for 25 years, and the past 12 years have been Introductory Calculus and APCalculusAB, and I want to tell you that this is an outstanding video of FTC Pt.1. Fantastic job! Looking forward to checking out your others, which is why I subscribed. :-)
Loving these videos. I had to leave school at 14 - 15 and have been using these as prep for the Uni entrance exam next year for a Bachelor of Engineering , keep up the great work.
It often helps to think of it from top-down instead of bottom-up. Let's say you have a function that gives the area under a graph up to any point on the x axis. Take for example the area (A) of a triangle formed under the line y = x. Its area will be 1/2bh, i.e. 1/2 x^2. Now consider how A changes with regard to x, i.e. dA/dx. It's x, the same equation (y = x) as the upper boundary line. If you don't know the original area function, you get back to it by integrating this line equation.
awesome...the most lucid, direct, clear explanation EVER !!...SO many thanks for this excellent demonstration of what was once a mind boggling concept... !! very much appreciated !!
I always thought Khanacademy was good while although slow, but this is so much better, more professional, and both neat and concise. I know I'm subscribing.
Only If I had a physics mentor like you I would have been doing a course to be a physicist instead of engineering but I am happy that I found someone who can even teach physics to toddlers
Weight is typically defined as the force of gravity on an object, and the calculation is W = mg, in which m is the mass, and g is the acceleration due to gravity. In the metric system, that's kg times m/s^2, which works out to force in Newtons. In the English system the units can be a little confusing.
@derekowens, surely you are the bestest tutor that I have seen so far. The way you explain makes maths soo easy. If you were my primary school teacher and taught me this at the age of 7, I am sure I would of passed Calculus course even then, But I have to say I owe you for your time and doing this for students. Thanks a lot, ur truely a LIFESAVER!
You are correct, there certainly should be a constant! However, when we are calculating a _definite_ integral, the constant disappears. It disappears because it would show up once in g(b) and again in g(a), and we subtract. I'm going to redo these videos soon, and I'll address the constant of integration when I do.
I just bumped into your video by accident. I must say it was excellent. I have been studying calculas on-line and I think your video is the best I have seen. I have subscribed to your site. Thank you.
@@nahrafe Yessir I am! Though I definitely did not take the math route, hahaha. I am going to Rhema Bible College. And 6 years ago lol, it's been a little while
Excellent. He has a good voice and is very concise. Took me a while to get that dx means derivative of x. I didn't notice what dx is, only saw what its anti-derivqtive g is.
Acceleration = change in velocity / time. Therefore, change in velocity = acceleration * time. As the area under the graph also equals acceleration * time, it represents the change in velocity.
You are correct. That is the KEY issue, and in fact the physics of motion was one of the key motivators for the development of calculus. That is essentially one of the problems that Newton himself was thinking of when he produced this. I do cover the physics of motion in more detail in other parts of the course, though, just not all in this video.
Dude that lecture blew my mind I haven't taken calc 1 yet but I've looked up diif quotient and out of curiosity anti derivitves. I wasn't sure how you got the anti derivitives to plug into the equation but I knew you did and everything else was easy to follow.
honestly I know this is just pure calculus 2 but now I see how calculus based physics makes more sense than just using algebra formulas and plugging in numbers, calculus rules. I need calculus for my major computer engineering tech and this is a good course for that major
@Kaiyazu Yes, the capital F notation is fairly common, and I see that used some on AP exams also. The concept, though, is what is critical, and the goal is for it to make sense, in either notation. Glad you liked the video! DO
0:58 I see, because even if you trying to find the area of figures without curves say a rectangle, this do work as well. Say the area of rectangle with length 3 and width 4 which =12 Here f(x)= 3 a=0 b=4 and the integral (or anti derivative )of 3 = 3 x 3 x 4 - 3 x 0 = 4 here we finding the area under f(x) or y=3 (a vertical line) from x=0 to x=4 so yes it works. And if we are doing a triangle (area of triangle base x height all divided by 2) with base 3 and height 3 where f(x)= x and a =0 and b= 3 then the integral of x = x^2/2 3^2/2 - 0^2/2 = 9/2-0 = 4.5. here it work not triangle as well as rectangle.
I don’t know if it’s because I live in Europe, but here we put a « +c » by every primitive function we calculate. Just because the derivatives of x^3/3 and x^3/3 + (a constant number like 1,2,3,4,...) are the same: x^2. To me this seems quite important. Great video still
I agree with some comments below which state in school the mechanics of operations are taught. But where and how is the actual equation generated? Where did a=1.2t squared come from as an example? How is the original curve found? Without figuring out how to generate the function the mechanics could just as easily be done by a computer and plugging in values. It seems to me the development of the actual function is the first step to solving the problem: Which I will add totally illudes me.
I'll try to weigh in on this. In a given situation, the actual function comes from an analysis of the particular situation. In this example we simply started with a given function. Regarding where the function actually comes from in the real world: In some simpler situations, the function is easily intuited from certain known facts. In simple cases involving a constant rate of change, for example, it may be easily seen that the function is clearly linear with respect to time. in more complicated situations, we have multiple variables and varying rates of change. In these cases an analysis of the situation leads us to a differential equation which then needs to be solved. Finding and solving the DE is a more advanced topic, which is typically introduced a couple of chapters after the Fundamental Theorem, and covered in more detail in later courses. After third semester calculus, students often take a full course in differential equations. Personally, that was the hardest math course I took. One fact not often emphasized is that many situations are actually extremely complicated, with too many variables or too many unknowns, and we simply can't model it effectively without certain simplifying assumptions. In some other cases we can produce a DE describing the situation, but can't easily solve the DE to find the function.
for a case, A rocket is lift off. The location of rocket from the starting point is measured using distance meter at every micro seconds. Now, the data gives rocket distance verses time. Using arithmetic operators, the velocity of rocket may calculated. however, it may not be accurate. somewhat, a function is created. then, think on how to develop acceleration vs time curve? Curves, can be developed using athematic operators. It depends on how much accuracy the market need. In a shop, seller can measure ideal 1kg mass with error of 20%. The population in the area is okey with that. market balance occurred. However, In some field, more accuracy is required to achieve or demonstrate some products or services. In this need, human explore any ideas that fulfills his satisfaction at his understanding about nature.
Nice Videos , but you missed a small thing which is , when you calculated the anti-derivative of x^2 , you have forgotten to add the constant of the integration C . of course this constant would be neglected when we take the definite integral as C-C=0 , but it may be important point to be mentioned for the beginners who face fundamental theorem of calculus for the first time . This is of course Great Video so keep up the good work! Regards.
Last video I watched last night before I went to bed. Enjoyed it immensely. Going to watch the others now. Would say something about the word 'anti-derivative' but that is more like that old 'tomay-toe'/'tomaa-toe' arseholery that leads to folks calling 'the whole thing' off. We used the word 'primitive' where I read maths, but the definition of that is what you say. Excellent.
I should have pronounced antiderivative differently. Thought about re-recording it for that reason but I never had time. Glad you liked the video, though!
In the rocket example, why does the area under the curve represents the speed of the rocket at the end of the 7th second? Wouldn't the value of the y axis be the speed?
Wolfgang, In this example we have a graph of acceleration vs. time, so in this case the value on the vertical axis is the value of the acceleration at any moment. The area under the graph is the change in velocity. If we have a graph of velocity, then the slope of the velocity graph would be the acceleration. Hope that helps. D.O.
Your method is great but I think you should share the workbooks for free. You would do a big favor to people and specially to those with limited resources. Take care.
Newton's attempt at quantifying energy/force fluxions......or energy as the sum of the forces exerted......integral=sum total of forces exerted from time a to time b.
Thanks very much, and if I remember, I do address the Constant of Integration in a later video in this series. And yes, it's an important for beginners, and an easy item to miss.
(I hope the following questions won't offend your minds) While I do understand the idea of calculus in general, at some point I start to be baffled. That point is when I try to apply units. The solution for the first example is 7/3 - but of what? I assume the answer: of any unit that the axis x and y have, but squared. But it doesn't click with me really. The rocket example is difficult for me in a different way: why is the answer to "how fast" buried in the area of this 1.2t^2 fragment? Why does area depict velocity here?
pls see what are in x and y axis. In simple, Area means, multiplication of Length and width . X axis shows time, and y axis shows acceleration. Now, time X acceleration is velocity ?
I just stumbled across your channel, and I have to say your way of explaining this stuff is absolutely fantastic. Thank you for taking the time to make these videos. They're a great resource for people like me who got off to a rocky start in mathematics. Cheers!
Could you please tell me what program did you use for this video? It's really helpful to understand. I like the function of changing colors and instant redo functions.
Mr. Derek Owensthank you for this detailed explanationhowever, I think, when there is a function given as a derivative, the area under the curve of that function is the distance
+Hassan Alanazi You're welcome! If the function is a graph of velocity vs. time, then the area under the curve would be the distance or the displacement. That would be one specific application of the concept.
I’m 76 years old and I always wondered what calculus was. This is the closest I’ve gotten. I am going to keep trying until I completely understand. Please publish more o these examples. You are a good teacher.
Thanks for such a thoughtful and encouraging comment!
This is hands down the best explanation of The Fundamental Theorem of Calculus that I've seen. The reason is because you explain the WHY behind it all and give a real world example of how it is applicable and WHY its needed. Thank you for the video!
Of all the Calculus videos I've seen on UA-cam, yours are definitely my favorite. Concise, clear, conceptual - they're really good for understanding the concepts. I'm going to school for engineering and plan on viewing your Physics videos soon! Right now, I'm hoping to survive Calc. 2 online over the summer... Thanks!
Very good. Thank you.
I wish college professors would take the time to teach like you do.
reviewing this after 35 years for my son - wish I had a teacher like this
and explanations like this
Most of our teachers memorised the formulas
I FINALLY get this, I wish online classes were just watching your videos, because it's SO much more helpful than just a wall of text. THANK YOU!
I am 60yrs old. As a kid, I was a maths wizz and spent my working life as a betting shop manager. I have always been comfortable with probability theory; but calculus always bemused me. This is excellent!
very good explanation . now i have got the sense of using calculus. though i was able to solve problems in my schooldays i was not able to understand it in reality . we blindly used formulas, and how to solve typical problems just to score for exams not enough time to think over it ,due to law imagination power , and due to pressure of completing the courses . basically the purpose of calculus were not taught . and this still may be a problems for some students.. THANK YOU SIR.
J K J yes thats a problem with math in general. Some people are able to instantly click with the concept but some like me spend countless hours to understand but end up memorizing how to do it instead of understanding it. Videos like these help alot.
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Veey true and this is one of the reason why students hate science classes.. the application part of it is missing (so that makes science classes look solid and horrible)
Ok I know this comment is 6 years old, but what are those spaces between the words?
What a champ you are professor!! Explicit and clear explanation without any confusion.
I'm an English teacher who avoided higher level math, but In 5 Minutes of your video I was hooked.
One of the best teacher I have seen. Mind blowing. Better than Khan academy. I would like to touch his feet in reverence. Nameste Sir.
I actually searched for your channel
I read physics from your channel some 8 years ago
Still the best channel
Beautifully clear and concise. Bravo 👏 and thanks 🙏
Excellent presentation. I feel I understand the Fundamental Theorem in a much deeper sense. Thank you.
are u so stupid
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The explanation excellent for those that already have enrolled or take a course on Integral Calculus, not for those who doesn't. JUst a comment: Constant aceleration doesn't mean that the veocity doesn't change, it will change since there is acceleraion. Thanks for this excellent video.
I've been teaching for 25 years, and the past 12 years have been Introductory Calculus and APCalculusAB, and I want to tell you that this is an outstanding video of FTC Pt.1. Fantastic job! Looking forward to checking out your others, which is why I subscribed. :-)
Loving these videos.
I had to leave school at 14 - 15 and have been using these as prep for the Uni entrance exam next year for a Bachelor of Engineering , keep up the great work.
It often helps to think of it from top-down instead of bottom-up.
Let's say you have a function that gives the area under a graph up to any point on the x axis. Take for example the area (A) of a triangle formed under the line y = x. Its area will be 1/2bh, i.e. 1/2 x^2.
Now consider how A changes with regard to x, i.e. dA/dx. It's x, the same equation (y = x) as the upper boundary line.
If you don't know the original area function, you get back to it by integrating this line equation.
awesome...the most lucid, direct, clear explanation EVER !!...SO many thanks for this excellent demonstration of what was once a mind boggling concept... !! very much appreciated !!
I always thought Khanacademy was good while although slow, but this is so much better, more professional, and both neat and concise. I know I'm subscribing.
Thanks for the Great job with the video, Derek. After years of working up to Calc III, this is the first time the fundamental theorem made any sense.
Only If I had a physics mentor like you I would have been doing a course to be a physicist instead of engineering but I am happy that I found someone who can even teach physics to toddlers
can we take a moment to appreciate that perfect ellipse at 1:14
Weight is typically defined as the force of gravity on an object, and the calculation is W = mg, in which m is the mass, and g is the acceleration due to gravity. In the metric system, that's kg times m/s^2, which works out to force in Newtons. In the English system the units can be a little confusing.
Thanks very much for the encouraging comment! I'm very glad you enjoyed the video!
@derekowens, surely you are the bestest tutor that I have seen so far. The way you explain makes maths soo easy. If you were my primary school teacher and taught me this at the age of 7, I am sure I would of passed Calculus course even then, But I have to say I owe you for your time and doing this for students. Thanks a lot, ur truely a LIFESAVER!
A very perfect video. It explains in a very simple way
Such a clear video, even clearer than the Kahn Academy video, and that's quite a statement, because Kahn academy videos are usually outstanding.
Brilliant explanation, this put so much of Calculus 1 and Physics in perspective for me... awesome work!!!!!
You are correct, there certainly should be a constant! However, when we are calculating a _definite_ integral, the constant disappears. It disappears because it would show up once in g(b) and again in g(a), and we subtract.
I'm going to redo these videos soon, and I'll address the constant of integration when I do.
The constant c you are talking about? 🤔
Superb
Simply Superb explanation Sir.....👍
Thank you, thank you! I'm very glad you liked it.
I teach classes to homeschool students. I have live classes in the Atlanta area during the school year, and online courses available year round.
Hello Mr. Owens, I understand everything except how did you get x^3 or x3. Did you add x2 dx together to get x3 or did you multiply?
@@megatton7207 there is a general way for getting there that's simple
∫ x^n dx= x^(n+1)/(n+1) (when n≠1)
I just bumped into your video by accident. I must say it was excellent. I have been studying calculas on-line and I think your video is the best I have seen. I have subscribed to your site. Thank you.
I cant wait and subcribed..
These so good teachings even monkey could understand. 1000 thanks for this guy
OMG, I haven't even taken Calculus, yet I understand it clearly. Well done sir
Im in 8th grade taking geometry right now and this just blew my mind how many variables to the whatever
hi
Lel I am in 7th grade and I am learning Calculus
Saaaaame but I’m in 7th taking geometry
Hi, now you must be on college.
@@nahrafe Yessir I am! Though I definitely did not take the math route, hahaha. I am going to Rhema Bible College. And 6 years ago lol, it's been a little while
Thanks SIR you did your best l like your way of teaching thanks
Excellent. He has a good voice and is very concise. Took me a while to get that dx means derivative of x. I didn't notice what dx is, only saw what its anti-derivqtive g is.
Thanks for the excellent video. Very concise and to the point with a good example!!
Nice explanation 🙏🙏🙏🙏🙏🙏 sir...
Yes, you nailed it. That's a more difficult problem, but it could be solved later in the course.
"Calculus is special." It stands out from all the other branches in math. Calculus is king."Very fascinating."
Acceleration = change in velocity / time. Therefore, change in velocity = acceleration * time. As the area under the graph also equals acceleration * time, it represents the change in velocity.
You are correct. That is the KEY issue, and in fact the physics of motion was one of the key motivators for the development of calculus. That is essentially one of the problems that Newton himself was thinking of when he produced this. I do cover the physics of motion in more detail in other parts of the course, though, just not all in this video.
Awesome! Thank you very much, I have to say, you're on par with KhanAcademy when it comes to clarity and organization with your problems.
wow, you did a better job than kahn academy. very clear and quick
Dude that lecture blew my mind I haven't taken calc 1 yet but I've looked up diif quotient and out of curiosity anti derivitves. I wasn't sure how you got the anti derivitives to plug into the equation but I knew you did and everything else was easy to follow.
I love you! Everyone made this so complex but you kept it really simple!! Thank you!!
honestly I know this is just pure calculus 2 but now I see how calculus based physics makes more sense than just using algebra formulas and plugging in numbers, calculus rules. I need calculus for my major computer engineering tech and this is a good course for that major
Good Job preofessor
Great video. I understand this concept much better now, thank you.
Well explain very clear to understand
@Kaiyazu Yes, the capital F notation is fairly common, and I see that used some on AP exams also. The concept, though, is what is critical, and the goal is for it to make sense, in either notation. Glad you liked the video!
DO
Good math lesson.thanks for vdo
Because when you integrate variables to a power you add one and divide by the
new variable.So x^2 becomes x^3/3.
Thank you Father.
tnk u so much . i didn't understand till now after watching this video i understood perfectly.
0:58 I see, because even if you trying to find the area of figures without curves say a rectangle,
this do work as well. Say the area of rectangle with length 3 and width 4 which =12
Here f(x)= 3 a=0 b=4 and the integral (or anti derivative )of 3 = 3 x
3 x 4 - 3 x 0 = 4 here we finding the area under f(x) or y=3 (a vertical line) from x=0 to x=4
so yes it works.
And if we are doing a triangle (area of triangle base x height all divided by 2) with base 3 and height 3
where f(x)= x and a =0 and b= 3 then the integral of x = x^2/2 3^2/2 - 0^2/2 = 9/2-0 = 4.5. here it work not triangle as well as rectangle.
Extremely clear, thanks a lot! Great refresher.
OMG- now it all makes sense.
thanks very much for keeping it simple.
I don’t know if it’s because I live in Europe, but here we put a « +c » by every primitive function we calculate. Just because the derivatives of x^3/3 and x^3/3 + (a constant number like 1,2,3,4,...) are the same: x^2. To me this seems quite important.
Great video still
Thank you soooo much!!! It's a amazing thing you're doing making all these videos for everyone!!:D You're great at explaining!!
Yes.. indeed.
Gee, you make it so easy to understand.
You make Calculus sound great. Thanks.
I found this video very helpful and clear. Thank you very much!!
sir you are so amazing teacher
Great video and explanation. A+
Derek, these videos are GREAT. Very clear and articulate. Nice work!!
Thanks
Excellent Vid - thank you!!!!!!
thank you very much ...im 60 and heard first time abaut non constant acceleration..
Big thanks from Ireland, the fundamental principle was well outlined with nice examples
Regards Tom
Really brilliant love it more more ..please.👍
Very nice and clear presentation. Thank you.
Watched the series and it is very good ! Thank you !
I agree with some comments below which state in school the mechanics of operations are taught. But where and how is the actual equation generated? Where did a=1.2t squared come from as an example? How is the original curve found? Without figuring out how to generate the function the mechanics could just as easily be done by a computer and plugging in values. It seems to me the development of the actual function is the first step to solving the problem: Which I will add totally illudes me.
I'll try to weigh in on this. In a given situation, the actual function comes from an analysis of the particular situation. In this example we simply started with a given function.
Regarding where the function actually comes from in the real world: In some simpler situations, the function is easily intuited from certain known facts. In simple cases involving a constant rate of change, for example, it may be easily seen that the function is clearly linear with respect to time.
in more complicated situations, we have multiple variables and varying rates of change. In these cases an analysis of the situation leads us to a differential equation which then needs to be solved. Finding and solving the DE is a more advanced topic, which is typically introduced a couple of chapters after the Fundamental Theorem, and covered in more detail in later courses. After third semester calculus, students often take a full course in differential equations. Personally, that was the hardest math course I took.
One fact not often emphasized is that many situations are actually extremely complicated, with too many variables or too many unknowns, and we simply can't model it effectively without certain simplifying assumptions. In some other cases we can produce a DE describing the situation, but can't easily solve the DE to find the function.
for a case, A rocket is lift off. The location of rocket from the starting point is measured using distance meter at every micro seconds. Now, the data gives rocket distance verses time. Using arithmetic operators, the velocity of rocket may calculated. however, it may not be accurate. somewhat, a function is created.
then, think on how to develop acceleration vs time curve?
Curves, can be developed using athematic operators.
It depends on how much accuracy the market need.
In a shop, seller can measure ideal 1kg mass with error of 20%. The population in the area is okey with that. market balance occurred.
However, In some field, more accuracy is required to achieve or demonstrate some products or services. In this need, human explore any ideas that fulfills his satisfaction at his understanding about nature.
Perfect sir
Excellent videos - thanks so much.
Nice Videos , but you missed a small thing which is , when you calculated the anti-derivative of x^2 , you have forgotten to add the constant of the integration C . of course this constant would be neglected when we take the definite integral as C-C=0 , but it may be important point to be mentioned for the beginners who face fundamental theorem of calculus for the first time . This is of course Great Video so keep up the good work! Regards.
Last video I watched last night before I went to bed.
Enjoyed it immensely. Going to watch the others now.
Would say something about the word 'anti-derivative' but that is more like that old 'tomay-toe'/'tomaa-toe' arseholery that leads to folks calling 'the whole thing' off. We used the word 'primitive' where I read maths, but the definition of that is what you say.
Excellent.
I should have pronounced antiderivative differently. Thought about re-recording it for that reason but I never had time. Glad you liked the video, though!
Derek Owens
Nah - sounds fine to me. And yes - they _are_ really awesome videos.
In the rocket example, why does the area under the curve represents the speed of the rocket at the end of the 7th second? Wouldn't the value of the y axis be the speed?
Wolfgang, In this example we have a graph of acceleration vs. time, so in this case the value on the vertical axis is the value of the acceleration at any moment. The area under the graph is the change in velocity. If we have a graph of velocity, then the slope of the velocity graph would be the acceleration. Hope that helps. D.O.
But then you used the word, maths and I had to call the whole thing off. :-P
You are a legend!
Very clear.
Your method is great but I think you should share the workbooks for free. You would do a big favor to people and specially to those with limited resources. Take care.
greenyblu he is all ready contributing to our education
🙏👌 clearly & very good to declare the topics ❤️
Having little bit upper concept on calculus best fit your teaching.
How do you know g of x is equal to x cubed over 3?
x cubed over 3 is the ANTI-DERIVATIVE of x squared
Just add 1 to the power of the function
Integration of x²=x²+¹/2+1
using integration formula guys!
Thanks man,, Great Teaching
Thank you so much , you are my calculus teacher ^^
Took this in college and I got a "mercy " pass. Whew!
Newton's attempt at quantifying energy/force fluxions......or energy as the sum of the forces exerted......integral=sum total of forces exerted from time a to time b.
Thanks
Thank you u are a really good teacher :D
Thanks very much, and if I remember, I do address the Constant of Integration in a later video in this series. And yes, it's an important for beginners, and an easy item to miss.
Awesome video. What is the software used by the way or is it any software??
(I hope the following questions won't offend your minds)
While I do understand the idea of calculus in general, at some point I start to be baffled. That point is when I try to apply units. The solution for the first example is 7/3 - but of what? I assume the answer: of any unit that the axis x and y have, but squared. But it doesn't click with me really.
The rocket example is difficult for me in a different way: why is the answer to "how fast" buried in the area of this 1.2t^2 fragment? Why does area depict velocity here?
pls see what are in x and y axis. In simple, Area means, multiplication of Length and width .
X axis shows time, and y axis shows acceleration. Now, time X acceleration is velocity ?
I just stumbled across your channel, and I have to say your way of explaining this stuff is absolutely fantastic. Thank you for taking the time to make these videos. They're a great resource for people like me who got off to a rocky start in mathematics. Cheers!
Could you please tell me what program did you use for this video?
It's really helpful to understand. I like the function of changing colors and instant redo functions.
Mr. Derek Owensthank you for this detailed explanationhowever, I think, when there is a function given as a derivative, the area under the curve of that function is the distance
+Hassan Alanazi You're welcome! If the function is a graph of velocity vs. time, then the area under the curve would be the distance or the displacement. That would be one specific application of the concept.