I am an English language teacher working in an Italian high school. You are not alone. I understand your anxieties. I can assure you that gradually, your anxieties about grammar and language learning will fade away. Grammar difficulties are overcome with time, patience, and practice. The same goes for writing compositions. The important thing is to believe in yourself and be patient. Take care and all the best.@@GG-ud1ib
I appreciate the explanation of the real life purpose of calculus!! I struggled to understand why I would ever need to use this when I learned it in school. I got decent grades but only because I blindly followed the formulas. Having this understanding of practical use would have made a huge difference to my interest. Thanks x 100😊
Thank you. Studied calculus decades ago, never understood it or its usefulness, and miraculously passed the course. I've always said the higher you go in math the more it's like a foreign language and less like math. I like math because it's concrete. No BS. Thanks for your channel.
I learned it in 1997 through a professor with two PhDs who taught us “his way”. I quickly forgot setting the moment after in part his final exam and never heard it again. I’d love to learn it…again.
Great job! I like the keep-it-simple aspect of your approach. You're getting some good questions and comments. They are really making an effort to follow and understand. Kudos!
Thank you so much! You're teaching technique is impeccable! You put it in day-to-day language and simplify it so much! Back in high school my math teachers were unable to adequately explain it to my thickened skull! One was just expected to accept the formulas and that's it! I just couldn't comprehend it! I was always interested but couldn't wrap my mind around it! Perhaps when I retire I will do some math courses! Thank you again for your service!
Thank you for your explanation video. I always thought I was stupid at Math in school because I thought everybody but me understood the mechanics behind every rule. If I had known that I just needed to apply the rules I wouldn't have thought I was so stupid. Years later had to take Math at College. Learned the rules and got an A. Not a Math guy, but it got me through.
Interesting that in the UK when I was young, calculus started at year 10 for those taking advanced maths at GCE 'O' level (taken at age 15/16) and started in year 11 for those us studying physics etc to GCE 'A' level (taken at age 17/18). Freshman year at university seems a late start to me for students who enjoy maths.
I hardly remember studying calculus. I got some help after school but I was unsuccessful. That was 36 years ago. Maybe if I had taken it later I would have been allright.
That is the operation called integration. I can't explain why it works, but Newton and Liebnitz figured it out. When you integrate a function you can use that integral to get the area under the curve.
I put it on 2x playback speed and I absorbed all the material like a sponge. It's very thoroughly explained and presented, thanks.. Honestly, on regular speed and with the long intro I was about to fall asleep, but I skipped that and sped it up and it sank right in. So it's actually good stuff.
I'm assuming that the addition of 1power, to X to the 2nd power, is because the area being found covers 1 unit on the X axis? So if the area to to be found was between the 2 and 4 on the X axis, than it would have been increased by 2 powers, making it X to the 4th power? Or am I crazy?
You are looking for a deeper explanation of the Power Rule. For the given function, (x^2) the new formula (called the anti-derivative) will always be (x^3/3). It doesn't change. (Won't go into why that works, here. Maybe try Khan academy ) Your region of interest, in this case from x=2 to x=4, are called the limits of integration, and define where you evaluate (plug into) your formula for the anti-derivative: To calculate the area between your new limits of integration, say between 2 and 4, you will plug in the value of the upper limit 4^3/3 minus the value of the lower limit (2^3/3). The formula for the anti-derivative only depends on the form of the original function. It does not depend on the region you are integrating over. You adjust the region you are integrating over by plugging the desired limits into the formula for the anti-derivative. You get the formula for the anti-derivative by applying a simple rule, in this case the Power Rule, to your original function, that is all he is saying here.
Great video. Gave you a like, as in all your videos. I use many of them in my free Summer STEM classes. However at 4:37 one tiny note: When you reviewed the formula for area of a Triangle, you didn't show what was the base and the height. I know it's a nit, but since I teach STEM to six students at a time from varoius grade levels at once, I can't miss anything (even in review), else it confuses some. Your videos are so great, just some feedback to hone them to perfection. [smile]
The most important part about this should have been the concept of limits. Yes, differentiation and integration were developed at rather different points in history, but the linking of the two as inverses is what's powerful about calculus. Calculus is about rates and time.
I'm stuck on one part. You drew an equilateral triangle but gave the formula for a right triangle, and it got me thinking that the equilateral triangle, being 2 right triangles back-to-back should have an area of 2(1/2bh). But then we are left with b*h, aka l*w which is the area of a rectangle. Where am I going wrong, here?
I dropped out of high school after 9th grade and got my GED. then later I went to university and did remedial classes for math. Unfortunately I had a terribly ineffective immigrant geometry/trigonometry professor who did not teach from the textbook. After that I took precalculus and was unable to understand anything. Passed with a C somehow. Lost my scholarship because of it. Later I tried to go back and I have to take calculus for my major but I have no foundation for getting started with it. I've dropped the class twice now. It's very frustrating. I want to teach myself calculus before I try to go back. Fortunately I have internet access. Eventually it could happen. Thanks for the video.
Yes, I am not against foreigners at all, but a hard subject like this is impossible when you can hardly understand the words spoken , because of the accents/ weird sounding pronunciation! !!!
@@tesmith47 that's entirely the problem. If he had been teaching from the textbook I might have been able to piece it together. But between the heavy African accent and not teaching from the textbook, it was pretty much impossible
This was interesting. I have actually done this on grid paper for some project I was working on to get an estimate for materials. I had no idea this was calculus. I no longer fear it. 😂 Thanks. And you don't have to be smart. You just need common sense.
Why did you put two and three where you did? Could they have been different numbers? Do those numbers matter? Seems like intervals along the x axis could be been anything you wanted. And 6.3 units? What are the units? I dont understand this at all. I understand the equations you did but how is 6.3 useful? An even more basic explanation would help, I reckon.
This wasn't a good introduction or summery. Calculus is the math of change, and algebra is the math of lines and constants...also, you ignored trig, which is the math or circles, waves, and triangles.....Trig is very important in calc.
In line with other 'explanations' of calculus this video presents the rules but makes no attempt to explain the rules. Actually I could explain the rules with respect to y = x² (by calculating average gradients) but it does not work so well for y = x³ and higher indices and not with more complex expressions.
For calculating the properties of random strange shapes vector calculus might be a good option. You can construct parameterisations that make pretty weird shapes
Then you approximate the shape as the sum of known, well behaved shapes. That way you can apply the simple rules to the well behaved functions! Sometimes, you have to add an infinite number of well behaved shapes to get it to match just right! We call this an infinite series expansion. (You'll learn these tricks later on (great question, by the way) such as Taylor series or Fourier series expansion. (This may sound intimidating, but Fourier series, for example, is really just a bunch of sine waves) The idea is just the same: add simple functions together to get where you're going! 🙂
To understand that, you actually have to take the course - it is called the Power Rule, and is one of the most widely used rules in intro calculus. For this video, he just wanted to show you that if you follow a simple rule, you can transform a formula (x^2) into another one (X^3/3) that will quickly give you the area under the curve. And therefore, that calculus doesn't have to be intimidating. 🙂 Can you guess the form of the rule? You actually said it - Increase the exponent by 1, (and divide by n+1). This works for all simple polynomials. Most intro calc students memorize it like this, and it is very easy to apply. However, understanding why the rule works... is a bit more involved... ... but it's a really cool proof!
@@choister522 Thanks. I always loved math but my last course was Trig in college, just short of pre calculus, but I always needed to know why something was done, didn't accept rules without explanation. Thanks again.
The Rectangle use is irrelevant . Just tell us that is how Newton figured it out to start to develop calculus but just gives us the rules that we must use to determine the answer. Rectangles like limit theorem is confusing the issue.
WAIT A MINUTE! WHO CAME UP WITH THE FOUNDATIONS OR FORMULAS IN THE FIRST PLACE? IE. CALCULUS X2+ 1= X3 ETCETERA. THEN THEY CAME UP WITH NEW ADVANCED FORMULAS. Why and how would they figure these things out, a recipe? Are they merely working to make a new language? If so, why couldn't they figure out a completely different language to make things work? For instance, you started out with math, algebra 1 & 2, Calculus etcetera. If you speak French and you're explaining all this stuff. I don't know French, I wouldn't understand it. How would these formulas start over from the beginning and be explained in a completely different way? Then more and higher advanced formulas would be invented. Then, how would the in depth awareness of something like technical game analysis be applied as awareness or knowledge of awareness and then figure how to learn or see the origin of further advancement? [I'll keep my day job].
Why didn't; you explain dx? You said it was a language but you never told us how to interpret dx. I appreciate you not confusing the issue with talk about limits as it is totally unnecessary and confusing to understanding Calculus.
Not arbitrary. What it does is to find the area under a curve or graph for which you have a formula - the curve can be a straight line, a parabola, a circle, a sinusoid etc. Once you have the formula for the curve, you can look up the table of integrals to get the "rule", for example if the curve is y = x^n, then the integral is x^(n+1)/(n+1) . After this, to find the area under the graph above the x-axis from say x=3 to x=5, just substitute x=5 and x=3 in the integral fromula and find their difference.
dx is an infinitesimally smal division of the x axis such that it is almost 0. Area in general is the y dimension multiplied by the x dimension. Since the y is changing as per the equation for the curve, to find the area from say, x=3 to x=5 manually, you would first find the first y point by substituting x=3 in the equation and multiply that with a dx of your choice, say dx=0.001 (1 milli-unit of x, whether meter, seconds or whatever) to find the area of the first infinitesimal (very small) rectangle. Then you will increment x to 3+dx = 3+0.001 and find the area of the next small rectangle and so on till after adding up 2000 such rectangles, you get an approximate area under the curve. The integral formula and the integration within bounds (x=3 to 5 here) makes the manual process very accurate by mathematically making use of dx that is very very small (refrerred to as dx tends to 0, or dx is very close to 0 so that it is not the 2000 divisions that we used in the manual integration we did above, but infinite umber of small rectangles are used to find the area).
Thank you. I had math anxiety when I was young. I wish math videos like this would have been around back then.
I still have anxiety connected to grammer, more accurately writing, composition... heck language in general.
I am an English language teacher working in an Italian high school.
You are not alone. I understand your anxieties. I can assure you that gradually, your anxieties about grammar and language learning will fade away.
Grammar difficulties are overcome with time, patience, and practice.
The same goes for writing compositions. The important thing is to believe in yourself and be patient.
Take care and all the best.@@GG-ud1ib
I appreciate the explanation of the real life purpose of calculus!! I struggled to understand why I would ever need to use this when I learned it in school. I got decent grades but only because I blindly followed the formulas. Having this understanding of practical use would have made a huge difference to my interest. Thanks x 100😊
Thank you. Studied calculus decades ago, never understood it or its usefulness, and miraculously passed the course. I've always said the higher you go in math the more it's like a foreign language and less like math.
I like math because it's concrete. No BS.
Thanks for your channel.
I learned it in 1997 through a professor with two PhDs who taught us “his way”. I quickly forgot setting the moment after in part his final exam and never heard it again. I’d love to learn it…again.
So true.
As a former teacher of math & science, MS 10 years, I really like this channel. I learn, myself, with every watch.
You have a very good way of explaining it. A lot of videos jump from A to Z too quickly.
Fascinating. Great explanation. Thank you!
Great job! I like the keep-it-simple aspect of your approach. You're getting some good questions and comments. They are really making an effort to follow and understand. Kudos!
Thank you so much! You're teaching technique is impeccable! You put it in day-to-day language and simplify it so much! Back in high school my math teachers were unable to adequately explain it to my thickened skull! One was just expected to accept the formulas and that's it! I just couldn't comprehend it! I was always interested but couldn't wrap my mind around it! Perhaps when I retire I will do some math courses! Thank you again for your service!
I appreciate you not confusing the discussion with Limits.
Thank you for your explanation video. I always thought I was stupid at Math in school because I thought everybody but me understood the mechanics behind every rule. If I had known that I just needed to apply the rules I wouldn't have thought I was so stupid. Years later had to take Math at College. Learned the rules and got an A. Not a Math guy, but it got me through.
Thank you professor! I have always feared Calculus, but you made it very clear.
God bless your heart!
Great explanation, thanks.
Awesome. Thank you. Well explained. I’d like to go into more of these and into area of cubic weird shapes.
I am 59 years old and never got to learn calculus. I was able to understand everything in this video. Thanx very much. Well done!
Interesting that in the UK when I was young, calculus started at year 10 for those taking advanced maths at GCE 'O' level (taken at age 15/16) and started in year 11 for those us studying physics etc to GCE 'A' level (taken at age 17/18). Freshman year at university seems a late start to me for students who enjoy maths.
I hardly remember studying calculus. I got some help after school but I was unsuccessful. That was 36 years ago. Maybe if I had taken it later I would have been allright.
At 5:30 I would call that shape a 'Non-Linear Rectoid' a rectangle with one or more non-linear sides.
Nicely explained.
Thanks.
Why did you add 1 and make it X cubed.
That is the operation called integration. I can't explain why it works, but Newton and Liebnitz figured it out. When you integrate a function you can use that integral to get the area under the curve.
@@danolson1467 But there must be a reason why the number is one and not some other number.
I thought the E in STEM represented engineering, not education.
Yeah, I’m not sure if he did that on purpose because he’s a teacher……or if it was a Freudian slip.
If you do a search for the beaning of STEM, you get Engineering not education.
It is. I prefer STEAM.
Engineering.
I put it on 2x playback speed and I absorbed all the material like a sponge. It's very thoroughly explained and presented, thanks.. Honestly, on regular speed and with the long intro I was about to fall asleep, but I skipped that and sped it up and it sank right in. So it's actually good stuff.
I started to like math because of you 👍
Many thanks.
7:00😇 I am happy. I understand how to do it now, thanks a lot. Make more examples. 👍👍👍😇😇😇🙏🙏🙏
Could you please tell which app you are using for writing notes on your tablet?
I'm assuming that the addition of 1power, to X to the 2nd power, is because the area being found covers 1 unit on the X axis? So if the area to to be found was between the 2 and 4 on the X axis, than it would have been increased by 2 powers, making it X to the 4th power? Or am I crazy?
You are looking for a deeper explanation of the Power Rule. For the given function, (x^2) the new formula (called the anti-derivative) will always be (x^3/3). It doesn't change.
(Won't go into why that works, here. Maybe try Khan academy )
Your region of interest, in this case from x=2 to x=4, are called the limits of integration, and define where you evaluate (plug into) your formula for the anti-derivative:
To calculate the area between your new limits of integration, say between 2 and 4, you will plug in the value of the upper limit 4^3/3 minus the value of the lower limit (2^3/3).
The formula for the anti-derivative only depends on the form of the original function. It does not depend on the region you are integrating over. You adjust the region you are integrating over by plugging the desired limits into the formula for the anti-derivative.
You get the formula for the anti-derivative by applying a simple rule, in this case the Power Rule, to your original function, that is all he is saying here.
@@choister522 thank you. I just got notification of you post. Sorry for the delay of my responce.
Great video. Gave you a like, as in all your videos. I use many of them in my free Summer STEM classes. However at 4:37 one tiny note: When you reviewed the formula for area of a Triangle, you didn't show what was the base and the height. I know it's a nit, but since I teach STEM to six students at a time from varoius grade levels at once, I can't miss anything (even in review), else it confuses some. Your videos are so great, just some feedback to hone them to perfection. [smile]
The most important part about this should have been the concept of limits. Yes, differentiation and integration were developed at rather different points in history, but the linking of the two as inverses is what's powerful about calculus. Calculus is about rates and time.
I totally agree. Understanding the concept of limits is essential.
I'm stuck on one part. You drew an equilateral triangle but gave the formula for a right triangle, and it got me thinking that the equilateral triangle, being 2 right triangles back-to-back should have an area of 2(1/2bh). But then we are left with b*h, aka l*w which is the area of a rectangle. Where am I going wrong, here?
Very good explanation. One note: STEM = Science Technology Engineering Math
I aced calculus without knowing what I was talking about!
This helps sososossosos much
This is the newest math video I've ever seen
I dropped out of high school after 9th grade and got my GED. then later I went to university and did remedial classes for math. Unfortunately I had a terribly ineffective immigrant geometry/trigonometry professor who did not teach from the textbook. After that I took precalculus and was unable to understand anything. Passed with a C somehow. Lost my scholarship because of it. Later I tried to go back and I have to take calculus for my major but I have no foundation for getting started with it. I've dropped the class twice now. It's very frustrating. I want to teach myself calculus before I try to go back. Fortunately I have internet access. Eventually it could happen. Thanks for the video.
Was hoping for a success story man.. you can do it!
Yes, I am not against foreigners at all, but a hard subject like this is impossible when you can hardly understand the words spoken , because of the accents/ weird sounding pronunciation! !!!
@@tesmith47 that's entirely the problem. If he had been teaching from the textbook I might have been able to piece it together. But between the heavy African accent and not teaching from the textbook, it was pretty much impossible
This was interesting. I have actually done this on grid paper for some project I was working on to get an estimate for materials. I had no idea this was calculus. I no longer fear it. 😂 Thanks. And you don't have to be smart. You just need common sense.
More please
You had me for about 10 minutes before my ADHD kicked in.
!0 minutes is actually quite a long time for me. 🙂
AT 11:15 you say that Y=X^2. How do you know that? Why not Y=X , e.g.? Or Y=X^3? Anybody know?
It is the simplest version of a parabola. He didn't just make it up.
I did very well in Calculus in high school, then aced it in college. Now it's like Greek to me.
Why did you put two and three where you did? Could they have been different numbers? Do those numbers matter? Seems like intervals along the x axis could be been anything you wanted. And 6.3 units? What are the units? I dont understand this at all. I understand the equations you did but how is 6.3 useful? An even more basic explanation would help, I reckon.
This wasn't a good introduction or summery. Calculus is the math of change, and algebra is the math of lines and constants...also, you ignored trig, which is the math or circles, waves, and triangles.....Trig is very important in calc.
I couldn't believe he didn't cruch all of calculus in a 20:00 minute video either
In line with other 'explanations' of calculus this video presents the rules but makes no attempt to explain the rules. Actually I could explain the rules with respect to y = x² (by calculating average gradients) but it does not work so well for y = x³ and higher indices and not with more complex expressions.
Calculus was not taught in my high school. Students had enough trouble with Algebra.
Dentists refer to "calculus" as that stuff they scrape off teeth. 😂😂😂
But what if it was just a random shape without any know function?
For calculating the properties of random strange shapes vector calculus might be a good option. You can construct parameterisations that make pretty weird shapes
Then you approximate the shape as the sum of known, well behaved shapes. That way you can apply the simple rules to the well behaved functions!
Sometimes, you have to add an infinite number of well behaved shapes to get it to match just right! We call this an infinite series expansion. (You'll learn these tricks later on (great question, by the way) such as Taylor series or Fourier series expansion. (This may sound intimidating, but Fourier series, for example, is really just a bunch of sine waves) The idea is just the same: add simple functions together to get where you're going! 🙂
Why did he add 1 to the exponent 2 to make it 3?
To understand that, you actually have to take the course - it is called the Power Rule, and is one of the most widely used rules in intro calculus. For this video, he just wanted to show you that if you follow a simple rule, you can transform a formula (x^2) into another one (X^3/3) that will quickly give you the area under the curve. And therefore, that calculus doesn't have to be intimidating. 🙂
Can you guess the form of the rule? You actually said it -
Increase the exponent by 1, (and divide by n+1).
This works for all simple polynomials.
Most intro calc students memorize it like this, and it is very easy to apply.
However, understanding why the rule works... is a bit more involved...
... but it's a really cool proof!
@@choister522 Thanks. I always loved math but my last course was Trig in college, just short of pre calculus, but I always needed to know why something was done, didn't accept rules without explanation. Thanks again.
The Rectangle use is irrelevant . Just tell us that is how Newton figured it out to start to develop calculus but just gives us the rules that we must use to determine the answer. Rectangles like limit theorem is confusing the issue.
Trigonometry
Ok job
WAIT A MINUTE! WHO CAME UP WITH THE FOUNDATIONS OR FORMULAS IN THE FIRST PLACE? IE. CALCULUS X2+ 1= X3 ETCETERA. THEN THEY CAME UP WITH NEW ADVANCED FORMULAS. Why and how would they figure these things out, a recipe? Are they merely working to make a new language? If so, why couldn't they figure out a completely different language to make things work? For instance, you started out with math, algebra 1 & 2, Calculus etcetera. If you speak French and you're explaining all this stuff. I don't know French, I wouldn't understand it. How would these formulas start over from the beginning and be explained in a completely different way? Then more and higher advanced formulas would be invented. Then, how would the in depth awareness of something like technical game analysis be applied as awareness or knowledge of awareness and then figure how to learn or see the origin of further advancement? [I'll keep my day job].
i'm in 6th grade
You hate mixed fractions
😇
Why didn't; you explain dx? You said it was a language but you never told us how to interpret dx. I appreciate you not confusing the issue with talk about limits as it is totally unnecessary and confusing to understanding Calculus.
We old ladies remember the days when girls were
thought to be lousy at math and were treated accordingly. ☹️
Rules seem arbitrary. Hard to understand.
Not arbitrary. What it does is to find the area under a curve or graph for which you have a formula - the curve can be a straight line, a parabola, a circle, a sinusoid etc. Once you have the formula for the curve, you can look up the table of integrals to get the "rule", for example if the curve is y = x^n, then the integral is x^(n+1)/(n+1) . After this, to find the area under the graph above the x-axis from say x=3 to x=5, just substitute x=5 and x=3 in the integral fromula and find their difference.
Disappointed. Why not explain the significance of the "dx"
dx is an infinitesimally smal division of the x axis such that it is almost 0. Area in general is the y dimension multiplied by the x dimension. Since the y is changing as per the equation for the curve, to find the area from say, x=3 to x=5 manually, you would first find the first y point by substituting x=3 in the equation and multiply that with a dx of your choice, say dx=0.001 (1 milli-unit of x, whether meter, seconds or whatever) to find the area of the first infinitesimal (very small) rectangle. Then you will increment x to 3+dx = 3+0.001 and find the area of the next small rectangle and so on till after adding up 2000 such rectangles, you get an approximate area under the curve.
The integral formula and the integration within bounds (x=3 to 5 here) makes the manual process very accurate by mathematically making use of dx that is very very small (refrerred to as dx tends to 0, or dx is very close to 0 so that it is not the 2000 divisions that we used in the manual integration we did above, but infinite umber of small rectangles are used to find the area).
Difficult and only smart people know how to do.
Know nothing about it