Calculus 1 Lecture 1.5: Slope of a Curve, Velocity, and Rates of Change

/ professorleonard
Calculus 1 Lecture 1.5: Slope of a Curve, Velocity, and Rates of Change

[@osk213]

I'm so engaged in this that I even raised my hand when he asked if everyone felt comfortable with what was just explained. This dude rocks.

[@saragerdbo]

Lol same, and in the school library

[@sandguardian5868]

Same

[@mhammamabid2275]

I almost did it at public, good thing i had a good self control.

[@kentbray7535]

Wow me too!

[@mattnelson3247]

I did it at home, then remembered I wasn't actually in his class :/ lol

[@strangesignal9757]

Never thought I'd be learning proper calculus from Clark Kent, but here I am. Stay fighting for education!

[@citizenseducated8390]

2:13 Conceptual overview of section 4:00-14:59 Graph & discussion of the tangent line problem 15:00 Transitioning from secant line to tangent line using limits 16:42 Writing out slope of tangent line 18:13-29:15 Ex: Finding the equation of the tangent line to y=x^2 at (1, 1) 30:22 Isn’t that neat though? 30:30 Ex: Finding equation of tangent line to y=3/x at (3, 1) 44:04 Ex: Finding slope of tangent line to y=sqrt(x) at ANY point 59:14 Average velocity 1:02:22 Ex: Finding average velocity of S(t) = 1 + 3t - 2t^2 on [1, 3] 1:10:31 Instantaneous velocity and how much time elapses in an instant 1:14:48 Ex: Finding instantaneous velocity S(t) = 500 - 16t^2 at t=5 seconds 1:39:23 Discussion of rates of change 1:43:00 Formal definition of average rate of change 1:44:42 Formal definition of instantaneous rate of change 1:46:37 Ex: Finding average rate of change of f(x) = 3x^2 - 4 on [2, 5] and instantaneous rate of change at x=-2

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you sir are a legend

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you are the real MVP

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ROFL at "Isn't that neat, tho" because honestly that's how I react at every 'aha' moment

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did ya give it?

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@@zakaria23045 how much did you donate

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[@Mark-nm9sm]

53:00 i think he says, "20,000 views, won't even get there.." or something similar. Currently it has 400k views and hes almost 1 Million subscribers! And its non other than his amazing teaching and math skills that got him here. Stay strong and keep beleiving 💪

[@ranonymous27]

pretty sure he said "nothing, just want to get there, personal goal" after someone asks "what happens at 20 thousand?" Nevertheless, he's awesome at teaching

[@tinoballesteros]

Brilliant teacher that makes everything so easy to understand. What a hero.

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if i was in class with this guy i would be thrilled... his methodology of teaching in a way to build you up to recieve the next lesson is phenominal.... he doesnt seem to leave anyone behind .... he is hilarious with his nerdy math .... and i feel he honestly is excited about the math as a tool ... (and i agree!!!!), there is only one thing that tricks my mind, and it is the fact that when i look at him i think hes not a math teacher... but then i hear the way he talks and he speaks nerd!!!!! ( confusing but amusing).... to RECAP!! ::click::: ...... LMFAO ......he rocks and suprises.... \m/, go professor superman i mean professor leonard... !!!

[@piyush6631]

52:27 "Do you need a tissue" "Yeah" "Yeah, I don't have one" lmao you're so awesome

[@philippjohannsen6217]

I finally understand calculus (at least this part). I feel like I get much more out of his videos than going to my lectures! God save Professor Leonard!

[@hannahwilliams1029]

Hey, Professor Leonard. I know this video is old, but I do hope you read this. I'm an upcoming Computer Science major with a Math minor this fall. I took Calculus I my junior year of high school but had to take Statistics my senior year because my school did not offer Calculus II. I passed the AP test and received college credit and all that good stuff. Since I had a year off from "traditional" math with statistics, my calculus skills got a little rusty in the down time. I'm watching this whole playlist in preparation for Calculus II in the fall since I refuse to retake Calculus I. I had an amazing Calculus I teacher whom I hold in high respect. I also had an amazing Statistics teacher whom I also respect greatly. I always loved math, but unfortunately Statistics killed my love for the great subject. It's just not my thing. You, sir, have restored my love for math. I had forgotten the joy I received from studying Calculus. Thank you for being the wonderful, well-paced, thorough teacher you are. I hold you at the same level of respect that I hold my former math teachers. You, sir, are a blessing to students, and I wish you all the best in your future endeavors. Godspeed, my friend.

[@mattlink9222]

How'd calculus II go? I'm in the same boat, refusing to take calculus I again after years of taking it.

[@nomuscleasianman]

As I was taking calculus, many years ago, I found that my understanding of the lectures in class was somewhat slow. Naturally, I am probably a slow learner. I don't mean to say that the material was presented in my class was immensely difficult, but the concepts were just hard to grasp at first. I would take around 2-3 extra hours of studying the material on my own before class and afterwards to fully catch up. At this point, I would have said I was around 70% of full understanding. Watching these videos now makes everything so clear, and I wish all math professors would teach in a similar manner as Professor Leonard.

[@bxav2813]

I just blindly followed instructions and formulas as to how to compute for a slope of a tangent line. Seeing how you explained the "why" just gave me mathgasm!!!

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Guys. It seriously makes me SO ANGRY AND FRUSTRATED that my professor for this class is HORRIBLE! All he does is read off of slides and NOBODY understands anything! THANK YOU SO MUCH PROFESSOR LEONARD. I seriously learn more form you in 10 minutes than an entires week worth of lecture of my "Professor"

[@DrumsTheWord]

I'm back here after a little break continuing with these lessons and I'm kicking myself for not coming back sooner. Prof Leonard makes this stuff so easy to understand and is such an engaging teacher.

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We was talking in our university about you !! The students who have issue with calculus start watching your videos and all they said the instructor is incredible !! He is fantastic instructor better than almost all instructor in our university !! Best wish is to you from Middle East - Bahrain !! :) #IT_Students

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Prof. Leonard is the Best! He even got me a little excited learning how to find the slope of a curve. I couldn't believe it so I put it in my handy-dandy TI-83 Plus and graphed the slope equation at 54:57. Boom it was real! I found the slope of a curve. Kind of cool:)

[@chrisoflaherty84]

Despite the fact that I'm very critical, and point out minor details, I do think Professor Leonard is an excellent teacher. Anyone could learn calculus if this guy is teaching. I wish it was a little faster at times, but everyone gets hung up on different parts, and I never get hung up on anything with these lectures because of the methodical and detailed progression through the subject matter.

[@greense65]

I agree. With a proper precalc background (which, unfortunately, many people lack), there are few people who could not learn calculus from Prof. Leonard. Therefore, I appreciate that he is slow, methodical, and tries not to leave people behind.

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If you want him to speed up, just speed up the video by 1.25 or 1.5 or 2. Some of us need him to go this pace.

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I cannot express how thankful I am. You are saving me big time. I feel like in class we just write notes and then are left to figure concepts out on our own. ITS IMPOSSIBLE..you make it possible, Thanks!

[@personalnothin]

am i the only one who raises their hand when prof ask if we are understanding this or not? nah tbh prof is the best teacher ive seen on yt , i like the way he teach ad his lectures are helping me alot to understand calculus. wishing the best for you man

[@upsidedownChad]

i also always raise my hand lmao

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I love crossing stuff out in equations. It feels almost like cleaning the house - get rid of a bunch of crap, and make things look neater.

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10 years and still relevant you are a hero

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[@MrGlockshna]

You are the reason I will pass my calculus class. My professor does a very poor job tying all of these concepts together. I've started watching your lectures before we cover the content in class and now what he's trying to explain makes so much more sense.

[@Shapelander]

Just watched the first 30 minutes of this. First time Calculus student and I've never felt so excited doing math. Things really clicked for me. Thank you, Professor!

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Your lessons are great. Looking forward to completing Calculus 1, going to Calculus 2 and your planned release of videos on Calculus 3 in 2016.

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A brilliant, kind and generous teacher!

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Clark Kent of math.

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Woah I’m in a time machine.. how’s life

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Prof. Leonard I can't even begin to explain how much I appreciate your channel. From a student's perspective I feel like I found a goldmine. Thank you for uploading and it is a pleasure to watch someone who is passionate about the topic. Truly makes learning it even more enjoyable.

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Wow! Thank you so much Prof. Leonard. That was the BEST!!! explanation that I've ever heard to teach the whole finding the slope of a tangent line problem. I can't believe it Whew! I had a lightbulb moment. You're a Saint. Thanks again.

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Thank you so much, Professor Leonard! I'm preparing for Calculus 1 in Fall. Wish me luck. :)

[@NAVYCORPSMAN906]

Me too…I am just gonna do HW and take tests for this class…Just try to learn these lectures in two weeks…and I am ready…hehehehe..Good luck…😊

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13 years ago I flat lined on the OR table. When I woke up I could not read or write. I have a PhD in P-Chem. I am going back to grad School. I am watching Proff Leonard and Leanard Susskind. Susskind's Grad students are very talkative. In calculus most students are only taking the class because they have to. I have taught chemistry so I know. Leonard is getting his students to open up. If I can get this level of excellence in teaching. I am going to go for my PhD in Math this time.

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How is your Ph.D. in mathematics? I am planning on doing one as well but I am a bit reluctant. Would like to provide me with any advice, I would appreciate it.

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still waiting on that advice ngl. Anyways did you do it?@@a_a1561

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@@a_a1561what school?

[@seabasschukwu6988]

Nerd

[@vladislavmatiusenco1089]

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[@prudenciomendezjr2518]

Professor Leonard is an example of an effective calculus professor in a small city of Merced. One doesn't have to go to a well known university to learn the trade as well as one would in Merced.

[@MakoMori]

I wish the amount of enthusiasm you have for talking about calculus was a requirement for professors to demonstrate in order to teach. My current Calculus teacher is awful. Doesn't explain it in ways I can understand. You brought me back to highschool Algebra II. My teacher there loved algebra II. I initially didn't care about it, but his enthusiasm was infectious. He was able to explain things in a way I could eventually understand. It made me love algebra. My current teacher's poor teaching methods made me want to give up. I couldn't understand anything even after a week of tryhing to study it myself. I wish I didn't waste $3k on one class for a teacher who isn't proficient in teaching. Thank you for existing. Otherwise I don't think I would be able to be a doctor.

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I like the way you are explaining , I wish you were my Calculus Teacher but anyway I guess By watching your videos that's really how I am going to pass my semester.

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[@aliabrahimi3711]

At 1:06:33 professor says that point one (1 , 3) is y2-y1 over x2-x1, you should not get confused how one set of point can be both x2-x1 and y2-y1. It is actually x2-x1 and will become y2-y1 once we plug it into the equation S(t)=1+3t-2t^2.

[@kentbray7535]

Thank you for clarifying that.

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Body Sounds 3:17, 49:43, 51:04, 52:05, 52:15, 53:03, 58:43, 1:03:18

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lol

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[@cornelxbox25]

6:38 You meant to say that there will be a very very small difference as we let Q approach P infinitely close and I see you corrected that later in the video. It will be so small that we would disregard it but we know point Q will never be equal to P (both in x and y values) as that will mean we will no longer have 2 points which are needed to find the slope. No matter how close point Q gets to P, it will never become P and there will be a difference between the 2 points. I really enjoyed this lecture. It's a blessing to have this resource available.

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as a physics major, you've truly saved my life. the hero we all need...

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Did you graduate yet?

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@@fratakkaya667I did

[@eddysato8430]

Students: How is Calculus going to be helpful in real life? Regular Teacher: It's gonna help you get into Calculus 2 and so on. Professor Leonard: Solving Instantaneous velocity, we can know the exact speed of impact on a windshield of a car and that way, we can measure up if the windshield can hold up or not against the falling item. You know a professor is good when they actually incorporate math into real life examples, unlike other teachers who just say that math is required in certain degrees.

[@rogu3dog]

Everything just clicked for me finally!

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Ganto ka gwapong instructor, perfect attendance ako neto lagi!

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my calculus instructor says its better to learn the material on your own and then come to class to do the homework in groups, so I'm over here trying to teach myself calculus but you make it a whole lot easier :) thanks

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You have solidified my understanding of the derivative sir. From deriving it to proving why it works. Thank you so much.

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Thank you so much! You give not only examples to your problems but provide a deeper understanding to how you solve the problem. You are actually making math fun

[@melkenhoning158]

for Stewart's calculus 8th edition these topics are covered in 2.1. It's such a poorly written chapter section it's essentially impossible to learn from without a professor right next to you. Thank you prof Leonard for doing all of this.

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Best video ever.We just muuged up formulas during our college days.Watching your series has created curiosity in my mind and interest for calculus.a big thank you Leonard Sir.Your videos are great help to me for Gate engineering exam,India.

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This man a decade after posting this lecture is about to hero ball me into passing Calculus

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I know many have said this, but this guy is amazing and teaches way better than my Calc 1 professor

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I must keep commenting as i go through this series! it gets really interesting when you do derivatives of a function instead of calculating using the difference quotient.

[@tutingrubena1584]

and 55:02 is actually what you get if you differentiate √x 😮😮 Amazing! Now I just have to thank you sir for letting me know the concept behind the formulas our past prof. just teach us how to use but not the actual thing what we're doing. Thanks a lot 😊

[@mattnelson3247]

I love how he said that his personal goal for the video was 20k views and it now has 384k... I think he's a better educator than he even realizes ;)

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Finally i was waiting for this point to start where i would understand the need of calculus , u put it all together in such a good way , thank u so much for this and i couldnt get much better , u r a perfect example of a teacher

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You remembered me of my professor of Fluid Mechanics In Alexandria university in Egypt, he has the same style of your way of explaining and disassembling the difficulties Check him Dr. Ihab Adam of Fluid Mechanics ❤❤ You both are great ❤

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Can someone give this guy a world best teacher award please!

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thank you sir for the calculus lecture,the fact that you make it so easy to understand am so grateful.may GOD bless you and answer all your secret prayers.

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[@oneofyk]

Thank you for explaining the reasoning behind the definition of the derivative. It really helped.

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This guy rivals Sal from Khan Academy(IMO)

[@PimpMatt0]

This guy makes the abstract aspect of Calculus easy to understand. Some of us can't think abstractly instantaneously on the spot because somewhere in our childhood, we forgot to develop those skills.

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Leanord is better than Sal

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sal is nothing in front of him

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r yall 3 y/o kids? just stfu and be grateful that they're kind enough to offer lessons for free

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Thanks for all your help Professor Leonard

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Wow this saved me from failing my mid-term. THANK YOU

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Thank you so much, I'm taking Cal 1 at my University and your videos are helping me a lot, because if I don't really understand the material in class I can just watch your video to get a better understanding on it. Thank again.

[@Muhammadilyas-gv8tm]

thank you soo much sir... i failed two time in calculus but inshALLAH this time i will clear this subject

[@seabasschukwu6988]

Did you pass?

[@AnkurSingh-mk9rc]

You are the legend man , 53:00 don't worry about subscribers professor

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if there were many like you a lot of people would love math, and the world would have alot of mathematicians!

[@Opium41462]

finally i understand Instantaneous velocity problems

[@screamingseaman6507]

-Do you need a tissue? -Yeah -I don't have one This is the kind of content I came here for

[@chrisoflaherty84]

On square roots there is actually a solution set. Saying that the square root of 25 is 5 is incomplete. The square root of 25 is 5 and -5. However, the derivative was still correct at the end because both the function and the derivative included a square root therefore the derivative would have accounted for both the positive and negative slope.

[@intemperate7]

chris oflaherty square roots are always positive. you must mean the negative root of a function. for example, sqrt(x^2)= |x| because x might be negative. 25= (-5)^2 but sqrt((-5^2))=|-5| = 5. another way to see this are the semi-circles. sqrt(16-x^2) covers the upper half (with respect to the x-axis) of a circle with radius 4 (graph it), the other half is below the x-axis and comes from the negative root; that is, from -sqrt(16-x^2). Hope this helps.

[@seabasschukwu6988]

Woah I’m in a time machine.. how’s life

[@StaticBlaster]

54:26 this is the most exciting part of the entire video.

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I apprechiate you to share those wideos. Tomorrow I have a quiz and with the help of your advanced level of teaching skills, I'll be successful.

[@shjaaz4139]

A Real HERO for himself.

[@DevelopmentPill2002]

Dear Professor Leonard (and any other boffin who may see this ;) ) P.S. Prof: I'm sorry about spamming these on your videos. I'm just in desperate need of help. I understand if you can't provide it personally - you seemingly have a lot on your plate as is! However, I am hopeful that at least somebody knows how to get around this. I am studying a Calculus I course at my university, but the work is wayyy more rigorous than how it is laid out in the textbook, or even how it is on the internet. Professor Leonard has helped me a lot in getting me to understand the basics and my marks have gone up by 5-10% ever since. But I still can't understand some concepts in the calculus context. (ie. Triangle inequality, bijection, invertible, and many others). For a better idea of what I am complaining about, here is a OneDrive link with my previous homework assignments: @t Thank you so much for anyone who may help me! Also, thank you to Professor Leonard for giving the motivation and confidence to see that I can get around this huge obstacle. I may not be around it yet, but you have at least given me the confidence and have picked me up when I was down

[@janedoe-vd3hn]

I went in string postulate land. Its the same as the big bang postulate it is not a Theory until the maths works! Einstein's 2nd law of Thermo is a paradox. The Law quite simply prevents you from doing what the law state. Quite nicely done. Proff Leonard you are very very good. You have me LOL at time To get through limits was the best. It is so boring to relearn but there is something I will miss. So as a middle aged chemist trying to relearn the most important class. I think your style #1.

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Thank you Professor Leonard, All of your lectures are very helpful !

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Sir u know ur maths um enjoying all of them professor

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This guy is an amazing teacher

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This guy really just proves that most teacher's aren't bad, they just don't put in a lot of effort to teach, then struggle to comprehend why their student's don't put in a lot of effort to study.

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I used to be addicted to porn, now I'm addicted to math and going to university !

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Best teacher without a doubt

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[@getezra1]

the slope created by two different size shoes [f(s1), f(s2)].

[@Riley4Prez]

I never understood what “h” was until now. “Aha” moment over here =)

[@shadowstar6841]

Went years just "plugging in the numbers" and wondering why I still sucked at math. Apparently, knowing why you're plugging in numbers (and getting better at algebra) is the difference of a letter grade or three

[@nabeeledrees5776]

really really Im so grateful to you professor the are amazing lectures

[@amyreynolds1079]

please confirm that the average rate on the last problem was 21, and the instantaneous rate was 12.

[@ksrajavel]

Was the answer to the final question on instantaneous rate of change equal -12?

[@carolinegrosch4735]

Thank you for these videos!! You're a life saver!

[@firelordzaki1600]

@3:17 try sneezing in class in 2020 smh if these guys only knew what the future would become lol

[@icono__7136]

I've been thinking the same thing the entirety of these lectures lmao

[@Peter_1986]

Next time I see people freak out just because I coughed or sneezed, I am literally gonna start _warn_ people around me _before_ I cough or sneeze after that. Might sound ridiculous, but seriously, it actually seems kind of necessary.

[@snipinglive5798]

At 47:31 thank u for explaining why that’s wrong... kept doing f(x)+h instead of f(x+h)

[@someone-en1gg]

1:48:17 negative time?

[@GonzoBeCrazyYo]

so you can use the difference quotient to find both instantaneous and average rate of change. My teacher is making us do it different. for average rate she is giving us the y-y1=m(x-x1).