I used to excel in math, but because of some tough times I had to pull myself out of school for 2 years to work. I forgot everything I knew about math and having to take calc 2 without any refresher, I thought I was completely hopeless. The professor I have just dived right into calc 2 as if all the students took calc 1 with him. Well I didn't. Not being able to understand anything that was written on the board, I wanted to cry. But here you are, an angel disguised as a math professor. All hopes are not lost. :) Thank you.
Requirements for a function to have an inverse 0:00 > Trig functions generally aren't 1:1. But restricting the domain allows us to treat them as inverse functions y = sin (x) 4:00 > What does the inverse do? 5:30 y = cos (x) 10:20 y= tan (x) 14:15 y = csc (x) 17:00 y = sec (x) 22:50 y = cot (x) 27:30 w/ Examples 30:00 Compound trig functions 38:00 Properties 46:00 Inappropriate Domain issue 50:00 Derivatives of Inverse trig functions 52:30 Proofs 56:00 cosine inverse [implicit just means chain rule] > using an identity 58:30 [cosine y] > using triangles 1:01:00 [secant y] w/ Examples 1:07:30 Harder problem 1:10:30 Even harder problem 1:16:00 Harder still 1:22:15 Sidebar: using triangles solve cos-1 (x^2) 1:26:00 > 'some fancy maths' 1:30:05 Integrals of inverse trig functions 1:32:20 w/ Examples 1:35:30 Third example rewriting the function into the right form 1:43:00 > In this example the function was rewrite in the 1/1+u^2 form so that it could be integrated using the tables. Important to consider algebraic manipulation, substitution etc... look for patterns. Last example 1:47:00 [using substitution twice]
I can't believe I just watched 2 hours worth of Calculus lecture in my spare time. It's truly a testament to how inspiring you are as a professor. Thanks for all the hard work you've put in to making these videos! :D
@Jessica Pennington A lot of math teachers have this annoying tendency to be super-formal and super-serious, and obsessed with the most general forms possible of everything so that the actual intuition is lost.
I understand you a lot better than my professor. I come to his class and take some notes, but ultimately, right after class, It's with you that I learned! THANKS A LOT! Not just an inspiration and a role model mentally, but physically as well! Who in earth could have a nice physique while teaching math? hahhaha
Professor Leonard teaches math "as Raphael painted pictures, Michelangelo carved marble, Beethoven composed music, or as Shakespeare wrote poetry." In short, thank you for being one of the best educator!
I wanted to state what a fantastic professor you are. My current professor seems to gloss over topics without ever going over edge cases or techniques to use for solving more difficult problems. You have saved me from a very difficult situation. These lessons are excellent! Very clear, and we'll explained
Covered the exact same content that I had in my first class except this time my head didn't explode. I'll be using your videos all semester and I hope you know what a great service you are doing for everyone!
WHY NOT PLUS OR MINUS FOR THE PROOF? for those who might be wondering about the proofs: for example, why cosy = sqrt(1-sin^2y) and not cosy = + - sqrt(1-sin^2y), after some thinking, here is my intuition: because we are trying to find the derivative of cos^-1x as the problem, we can see from a sketch of the inverse of cos that the slope throughout the domain must be negative, and since it was -1/cosy, we only needed to plug in a positive value for cosy to have a negative slope, so basically it all just depends on the range and domain. --> to be more exact, in case if that wasn't clear, because we got -1/cosy and want to simplify, we know that x can only take positive values because of the restrictions of domain of the trig functions, thus we can only pick the positive value to plug in. EDIT: if this helped you plz upvote, if you want a better explanation comment below and i might catch you :)
I totally love your simplicity as a teacher. The inverse trig intergration method is way better than the a and u method where you have to memorize two separate formulas for each. Thanks
I have to say thank you so much you really have helped me so much with eng maths.. honestly can't say it enough thank you. Your teaching style is amazing.
This video is 5 years old today, and I would just like to say that I am thankful for your work. Your videos are crucial to my math education in college. Thank you Professor!
@@rul1175 Oh man. I owe this guy for my college math grades, but now that I’m more mature I can learn for myself, and I would take this knowledge in very differently. If you’re watching this, good luck! Understand math, don’t learn it.
You are the greatest teacher i ever had till now. Your classes are even better than those of Khan Academy!! You are the true saviour of The CalculusVerse.
I have a calc 2 exam tomorrow. I barely did any hw because the wow xpac release. I realized this morning that I am a fool as I sat down to "study." Now, after finding this video I feel there still may be hope. And yes, I think I have learned my lesson.
This lesson isn't hard at all as long as you stick to what has said and memorized the graphs and the domains of the functions that he just drew :) Thanks again Professor Love
Professor Leonard, thank you for an awesome explanation and analysis of the Inverse Trigonometric Functions in Calculus Two. The graphs and derivation are important in understanding Inverse Trigonometric Functions. Instead of memorizing these important Transcendental Functions, students should derive them from scratch with the aid of Trig Identities and the Right Triangle. There are no errors in this TH-cam video.
Sir , a video providing *detailed explanation* on applying *properties of inverse trigonometric functions* to solve equations would be really helpful. Thank you.
This is an appreciation post..if you found yourself here... u wont regret it!! stick to him... after 2 Fs in calculus i got a C+ after watching his videos... thankyou!! ❤❤❤❤❤❤❤❤❤
I was trying to understand around the time 34:46 the numbers you were saying on the unit circle. Tell me where u have a video on an explanation on that
Unit circles are used to derived the trig functions, and so are discussed in a trigonometry class or in the trigonometry section of a "Precalculus" class. Unfortunately, Prof. Leonard has not posted videos of him teaching the Precalculus class.
my prof is left handed and stands in front of his work while hes writing and explaining it so i can't see anything and thus dont understand anything so i feeeeeeeel this
An easy way of memorizing the inverses of CSC and SEC is comparing them to the derivatives of cos and sin and appyling the changes which are : we change the signs of the outside and the inside which is the sign under the root and don't forget to multiply x in the denominator
At 38:00 when you talk about inverse COS, would you not have you calculator in degrees for COS(.6)? You said radians and im pretty sure you would use degree.
At 1:09:53...isn't d/dx [5x]^2= d/dx [25x^2]=50x, by the chain rule ? And at 1:14:42 isn't the square of the square root [2T+1] just equal to 2T+1, and therefore it's derivative is equal to 2 ? I am confused !
LMAO 6 years ago, but ya I caught that too, and if anyone has any doubts I have a comment on it, but heres the explanation: (its kinda hard to explain it with words but I tried my best) WHY NOT PLUS OR MINUS FOR THE PROOF? for those who might be wondering about the proofs: for example, why cosy = sqrt(1-sin^2y) and not cosy = + - sqrt(1-sin^2y), after some thinking, here is my intuition: because we are trying to find the derivative of cos^-1x as the problem, we can see from a sketch of the inverse of cos that the slope throughout the domain must be negative, and since it was -1/cosy, we only needed to plug in a positive value for cosy to have a negative slope, so basically it all just depends on the range and domain. --> to be more exact, in case if that wasn't clear, because we got -1/cosy and want to simplify, we know that x can only take positive values because of the restrictions of domain of the trig functions, thus we can only pick the positive value to plug in.
He is very good at teaching. If you want to understand better take notes and try to solve examples before he does. And I think in 1:15 he missed a little square power.
for the exercice at 1:22:30 => is it possible to do it otherwise ? I did it by setting COT(....) = (Y) into COT^-1 (Y) = (....) and taking the derivative with respect to x of both sides. We will then have an implicit derivative on the left side and a normal derivative on the right side. We will use the formulas on the board for both sides and we solve for dy/dx and then replace Y by COT(....). This way I got (dy/dx)=[2x(COT^2(COS^-1(x^2))+1)]/[sqrt(1-x^4)]. It may seem different but it's the same derivative function as the one found in the video but just wrote in another way....
Does anyone know where the actual lessons are located? If it's close to me I'mma head my ass over. I've been wanting to finish my math forever but actually want to have the credits. I'm done highschool and wanna do academic upgrading but most places around me have horrible teachers.
Hey Professor Leonard, thank you so much for having these videos up! I'm starting to understand Calc 2 a lot more than I used to thanks to you! I have one quick question on derivatives of inverse trig functions. Does the chain rule still apply if there are no x's [Ex: sin^-1 (1/2)]?
I know this is pretty late, but yes you still apply the chain rule in this instance. the derivative of your example ends up being 0 though, as it is a constant
About the derivative of the inverse trigonometric function of sinx (sin^-1 x) there is another proof (I guess): So the derivative of a an inverse function is equal to: (f^-1)' = 1 / (( f' o f^-1)) = 1 / ((sin)' o (sin^-1))(x) = 1/(cos(sin^-1(x)) = 1/(sqrt(1 - sin^2(sin^-1x)) = 1/(sqrt(1-x^2)) Please correct me if i'm wrong
anyone understand 46:00 ? i dont know how they find y using the unit circle and im getting tripped up. seems pretty easy but i might've missed something
I have a question on your last problem. I put it in my tnspire cas cx and get arctan sqrt(9x^2-1). Is there another way of doing these and if so, do you have a video on how to do it? Btw I want to thank you for unknowing helping me get through this first year of Calculus. I was going to give up half way through until I found your videos and turned my grade completely around. Next week is the final. Thanks
I used to excel in math, but because of some tough times I had to pull myself out of school for 2 years to work. I forgot everything I knew about math and having to take calc 2 without any refresher, I thought I was completely hopeless. The professor I have just dived right into calc 2 as if all the students took calc 1 with him. Well I didn't. Not being able to understand anything that was written on the board, I wanted to cry. But here you are, an angel disguised as a math professor. All hopes are not lost. :) Thank you.
+Hui Yi Chen You're asian, youll be ok
Lol, the stereotype is real
+Dan Luckman I wish youtube dislikes did something
Gosh guys... I'm pretty sure he was joking... Its not the end of the world... Its not even a harmful joke...
how did it go? did you succeed?
Requirements for a function to have an inverse 0:00
> Trig functions generally aren't 1:1. But restricting the domain allows us to treat them as inverse functions
y = sin (x) 4:00
> What does the inverse do? 5:30
y = cos (x) 10:20
y= tan (x) 14:15
y = csc (x) 17:00
y = sec (x) 22:50
y = cot (x) 27:30
w/ Examples 30:00
Compound trig functions 38:00
Properties 46:00
Inappropriate Domain issue 50:00
Derivatives of Inverse trig functions 52:30
Proofs 56:00 cosine inverse [implicit just means chain rule]
> using an identity 58:30 [cosine y]
> using triangles 1:01:00 [secant y]
w/ Examples 1:07:30
Harder problem 1:10:30
Even harder problem 1:16:00
Harder still 1:22:15
Sidebar: using triangles solve cos-1 (x^2) 1:26:00
> 'some fancy maths' 1:30:05
Integrals of inverse trig functions 1:32:20
w/ Examples 1:35:30
Third example rewriting the function into the right form 1:43:00
> In this example the function was rewrite in the 1/1+u^2 form so that it could be integrated using the tables. Important to consider algebraic manipulation, substitution etc... look for patterns.
Last example 1:47:00 [using substitution twice]
hope you have an amazing day and god bless
danke
I can't believe I just watched 2 hours worth of Calculus lecture in my spare time. It's truly a testament to how inspiring you are as a professor. Thanks for all the hard work you've put in to making these videos! :D
+Arlo haha literally his lecture is addictive.
@Jessica Pennington
A lot of math teachers have this annoying tendency to be super-formal and super-serious, and obsessed with the most general forms possible of everything so that the actual intuition is lost.
hello fellow mathematician
If you could do videos on physics, I think my life would be complete.
Grayfox772 ilectureonline This guy is the physics version of Prof. Leonard
NOT IN THE CASE OF BODY STRUCTURE OF COURSE! LOL
Grayfox772 Hell yeah
@@madalincalamanciuc6656 "welcome to ilectureonline" Van Biezen is a God. (almost as Godly as Leonard)
Trust me you are going to love Walter Lewin for physics then
His videos should be more popular, this guy really knows how to teach his stuff. I wish my college professor was as good as him :(
skip to 51:39 for the calculus part and you already know the basic trig
Derivatives start at 1:07:28
Integrals start at 1:32:17
I understand you a lot better than my professor. I come to his class and take some notes, but ultimately, right after class, It's with you that I learned! THANKS A LOT! Not just an inspiration and a role model mentally, but physically as well! Who in earth could have a nice physique while teaching math? hahhaha
Professor Leonard teaches math "as Raphael painted pictures, Michelangelo carved marble, Beethoven composed music, or as Shakespeare wrote poetry." In short, thank you for being one of the best educator!
💀
I wanted to state what a fantastic professor you are. My current professor seems to gloss over topics without ever going over edge cases or techniques to use for solving more difficult problems. You have saved me from a very difficult situation. These lessons are excellent! Very clear, and we'll explained
Your a gift from god. My Calc 2 teacher is a fool without these videos I would 100% be failing.
Covered the exact same content that I had in my first class except this time my head didn't explode. I'll be using your videos all semester and I hope you know what a great service you are doing for everyone!
You know he is a great teacher when I'm here studying watching his videos, 8 years after he posted this.
WHY NOT PLUS OR MINUS FOR THE PROOF? for those who might be wondering about the proofs: for example, why cosy = sqrt(1-sin^2y) and not cosy = + - sqrt(1-sin^2y), after some thinking, here is my intuition: because we are trying to find the derivative of cos^-1x as the problem, we can see from a sketch of the inverse of cos that the slope throughout the domain must be negative, and since it was -1/cosy, we only needed to plug in a positive value for cosy to have a negative slope, so basically it all just depends on the range and domain. --> to be more exact, in case if that wasn't clear, because we got -1/cosy and want to simplify, we know that x can only take positive values because of the restrictions of domain of the trig functions, thus we can only pick the positive value to plug in.
EDIT: if this helped you plz upvote, if you want a better explanation comment below and i might catch you :)
nice username lol
also very good explanation, i forgot that derivative was a slope lmao
@@erahamzah6983 lol thanks
@@erahamzah6983 thanks lol it was lowkey hard to explain without writing on paper or smthing T_T
I'm going to show this to my grandchildren one day.
Literally a life saving video
I totally love your simplicity as a teacher. The inverse trig intergration method is way better than the a and u method where you have to memorize two separate formulas for each. Thanks
I have to say thank you so much you really have helped me so much with eng maths.. honestly can't say it enough thank you. Your teaching style is amazing.
Thank you so much Professor Leonard! You are awesome.
Thank you so much for posting all of these. Excellent teacher.
This video is 5 years old today, and I would just like to say that I am thankful for your work. Your videos are crucial to my math education in college. Thank you Professor!
Its 10 years now.
@@rul1175 Oh man. I owe this guy for my college math grades, but now that I’m more mature I can learn for myself, and I would take this knowledge in very differently. If you’re watching this, good luck! Understand math, don’t learn it.
You are the greatest teacher i ever had till now. Your classes are even better than those of Khan Academy!! You are the true saviour of The CalculusVerse.
hello prof. i am from Philippines ive always watched your tutorial video..'' and it make sense to me''...i learned a lot from you honestly.
keep on uploading videos ur a superman disguised as a math teacher
Am really glad because the presentation was marvelous and I wish you could also do it for other subjects like Chemistry.
He`s a legend!!!!Finally i understand inverse function!!wow!!you`re really have a lot of positive vibes!!thanks a lot!!
Thank you! My instructor put up the triangle, and I had zero clue where he got it from! Thank you thank you thank you!!!
I have a calc 2 exam tomorrow. I barely did any hw because the wow xpac release. I realized this morning that I am a fool as I sat down to "study." Now, after finding this video I feel there still may be hope. And yes, I think I have learned my lesson.
Summer B. Did you pass?
Summer B. Rip
i like how a flick of a button can bring captain america in a Calculus class, u are awesum thank u
This teacher teaches stuff better than my profrssor. I did not even know that i have spend 4 and half hours doing these. Until i finishespd two videos
math final in 3 days, will be binge watching! great explanations!
Really impresive
This man is a national hero. Thank you professor.
This lesson isn't hard at all as long as you stick to what has said and memorized the graphs and the domains of the functions that he just drew :) Thanks again Professor Love
you have absolutely no idea of how much the trig lecture helped me! thank you so so much
What a Great Teacher!
Extremely good professor.
great video
Professor Leonard, thank you for an awesome explanation and analysis of the Inverse Trigonometric Functions in Calculus Two. The graphs and derivation are important in understanding Inverse Trigonometric Functions. Instead of memorizing these important Transcendental Functions, students should derive them from scratch with the aid of Trig Identities and the Right Triangle. There are no errors in this TH-cam video.
Such a life saver...
You're the mathematical legend of the world 🙏🏿👍🏿
Job well done professor!
Thank you sir thanks🙏🙏🙏🙏🙏🙏🌹🌹🌹🌹🌹🌹❤🌹
BIG THUMB UP !!!!!!
You are a very fine teacher!
If I pass calculus it is ENTIRELY because of you
Amazing stuff yet again!
great one
Awesome class.
Sir , a video providing *detailed explanation* on applying *properties of inverse trigonometric functions* to solve equations would be really helpful.
Thank you.
This is an appreciation post..if you found yourself here... u wont regret it!! stick to him... after 2 Fs in calculus i got a C+ after watching his videos... thankyou!! ❤❤❤❤❤❤❤❤❤
Dude thanks for saving my academic year!! I salute you
I was trying to understand around the time 34:46 the numbers you were saying on the unit circle. Tell me where u have a video on an explanation on that
Unit circles are used to derived the trig functions, and so are discussed in a trigonometry class or in the trigonometry section of a "Precalculus" class. Unfortunately, Prof. Leonard has not posted videos of him teaching the Precalculus class.
my saviour !!! thank u sm
Hello sir, I am watching from Bangladesh 🇧🇩
why is there no absolute on the (1-x^4) in the last derivative example?
You're awesome and very talented, wish my prof taught like this!!
I’ve sat through your classes practicing problems on my whiteboard. Thank you for these
dude YOU ARE THE BEST
When your math teacher doesn't speak English and you have to get TH-cam to teach you smh.
my prof is left handed and stands in front of his work while hes writing and explaining it so i can't see anything and thus dont understand anything so i feeeeeeeel this
LMAO SAME
SAME
literally same omg
felt that when i first been taught math in english in college lol
Thank you very much ❤️❤️❤️you’re the best
Thank you so muchhh for teaching me calculus...it means a lot.
ringing phones make him crazy but u r the best Mr. Leonard
An easy way of memorizing the inverses of CSC and SEC is comparing them to the derivatives of cos and sin and appyling the changes which are : we change the signs of the outside
and the inside which is the sign under the root and don't forget to multiply x in the denominator
For those who know your pre calc and trig well, the video starts at 51.35
WOW. You are amazing man. Thank you so much for these videos.
At 38:00 when you talk about inverse COS, would you not have you calculator in degrees for COS(.6)? You said radians and im pretty sure you would use degree.
THANK YOU!!!! Especially for showing the proofs. I feel like I finally understand!! :)
At 1:09:53...isn't d/dx [5x]^2= d/dx [25x^2]=50x, by the chain rule ? And at 1:14:42 isn't the square of the square root [2T+1] just equal to 2T+1, and therefore it's derivative is equal to 2 ? I am confused !
it’s 5 because we calc just the differential of 5x
Prof: "We'll start off easy and gradually build up to where its harder"
Student: "You mean more fun?"
🤣🤣🤣
I wish you were my professor i love you thank you pray for me on finals I am currently failing calc 2
i'm not sure why I'm learning this in calculus 1 but regardless,i'm happy this video is here!!!!!!
1:42:45 third part starts here
I just stumbled upon ur channel and holy HELL who do I pay to have you teach me math lol
Show hands if you actually show hands when he asks you to show hands.
you are legend !!!!!
In 59:51 you didn't put the plus and minus in front of the root. I don't understood why :(
LMAO 6 years ago, but ya I caught that too, and if anyone has any doubts I have a comment on it, but heres the explanation: (its kinda hard to explain it with words but I tried my best) WHY NOT PLUS OR MINUS FOR THE PROOF? for those who might be wondering about the proofs: for example, why cosy = sqrt(1-sin^2y) and not cosy = + - sqrt(1-sin^2y), after some thinking, here is my intuition: because we are trying to find the derivative of cos^-1x as the problem, we can see from a sketch of the inverse of cos that the slope throughout the domain must be negative, and since it was -1/cosy, we only needed to plug in a positive value for cosy to have a negative slope, so basically it all just depends on the range and domain. --> to be more exact, in case if that wasn't clear, because we got -1/cosy and want to simplify, we know that x can only take positive values because of the restrictions of domain of the trig functions, thus we can only pick the positive value to plug in.
1:33:20 integration part
He is very good at teaching. If you want to understand better take notes and try to solve examples before he does. And I think in 1:15 he missed a little square power.
Still 2 years later, Professor Leonard saves me again.
Do you have a video on work using integrals
Awesome video as usual! Fart at 1:38
?prof wher we can get extra exsercise about calculusbook
Thank you generous man
for the exercice at 1:22:30 => is it possible to do it otherwise ? I did it by setting COT(....) = (Y) into COT^-1 (Y) = (....) and taking the derivative with respect to x of both sides. We will then have an implicit derivative on the left side and a normal derivative on the right side. We will use the formulas on the board for both sides and we solve for dy/dx and then replace Y by COT(....). This way I got (dy/dx)=[2x(COT^2(COS^-1(x^2))+1)]/[sqrt(1-x^4)]. It may seem different but it's the same derivative function as the one found in the video but just wrote in another way....
Before the y=cot(csc) prof. Leonard goes what do you see someone in the room goes “2chainz” i literally laughed out loud😂
I CANT BELIEVE THAT I FAILED CALCULUS 1 ........ IF IVE WATCHED THIS PROFESSOR FROM THE BEGINING I WAS GOING TO SCORE BETTER GRADES
are the inverse trig differentials only for radians?
Does anyone know where the actual lessons are located? If it's close to me I'mma head my ass over. I've been wanting to finish my math forever but actually want to have the credits. I'm done highschool and wanna do academic upgrading but most places around me have horrible teachers.
went over this in calc 1 for maybe 40 minutes. wow....
Audible groans at 1:31:20 when he says he wants it simplified to the MAXXX lol
So, is this the only video he covers inverse trigonometric functions with differentiation & integration?
~ 13:00, that graph of y=cos^-1 x cannot be correct. something seems wrong. opinions anyone?
Hey Professor Leonard, thank you so much for having these videos up! I'm starting to understand Calc 2 a lot more than I used to thanks to you!
I have one quick question on derivatives of inverse trig functions. Does the chain rule still apply if there are no x's [Ex: sin^-1 (1/2)]?
I know this is pretty late, but yes you still apply the chain rule in this instance. the derivative of your example ends up being 0 though, as it is a constant
About the derivative of the inverse trigonometric function of sinx (sin^-1 x) there is another proof (I guess):
So the derivative of a an inverse function is equal to:
(f^-1)' = 1 / (( f' o f^-1)) = 1 / ((sin)' o (sin^-1))(x) = 1/(cos(sin^-1(x)) = 1/(sqrt(1 - sin^2(sin^-1x)) = 1/(sqrt(1-x^2))
Please correct me if i'm wrong
I love u papa leonard
what do you mean by one to one?
hero
anyone understand 46:00 ? i dont know how they find y using the unit circle and im getting tripped up. seems pretty easy but i might've missed something
I have a question on your last problem. I put it in my tnspire cas cx and get arctan sqrt(9x^2-1). Is there another way of doing these and if so, do you have a video on how to do it? Btw I want to thank you for unknowing helping me get through this first year of Calculus. I was going to give up half way through until I found your videos and turned my grade completely around. Next week is the final. Thanks