Calculus 1 Lecture 2.4: Applications of the Derivative

[@ethanaurellano6847]

Timestamps: (1:35) Example: DVD Sales (17:10) Overview of Velocity, Acceleration, and Jerk (22:40) Example: Find Acceleration & Jerk (27:50) Example: Find Max Height (32:20) Example: Managerial Economics

[@mohamedelmoukhtarsfifa9863]

Life saver

[@anonymoususer4356]

Some teachers can't articulate these things clearly. YOU can! Thank you!

[@famidaislam5542]

Its sad how the views decrease as the lectures progress...

[@Peter_1986]

Seems to me like they increase actually, at least for the next several videos after this one.

[@restitutororbis964]

Yup, i guess they decrease because some people don`t dive too deep into calc, its usually just to gain some knowledge on the main topics that being differentiation/integration or to review. Since i`m not in a uni i treasure every single one of these.(Calculus 2 is hell tho,atleast integration,even if leonard is teaching it)

[@rockfordlines3547]

The first videos are the most important because they talk more about fundamentals. Product rule and quotient rule aren't difficult with a solid grasp on the basics.

[@bebianbecolli1642]

Yo what if someone made a function modeling views per video in the series as of a function of time that would be awesome

[@RubiMercuri]

Brooo

[@motorlover2126]

you're the best teacher I've ever seen, keep it up

[@MDE123]

For physics geeks like me it's very interesting that he discusses the third derivative of position as a function of time: jerk or the change in acceleration. Most people only experience that when you begin accelerating in a car or stop suddenly or on an amusement ride. Constant jerk like in the example he gives would be experienced on a spinning ride where you feel yourself being pressed deeper and deeper into your seat as the speed at which it spins increases assuming the speed increases uniformly. (in a spinning ride you are accelerating even when the rotational speed is constant so an increase in rotational speed is a change in acceleration and therefore jerk (you don't have to feel "jerked around". )If you listen to the cockpit transmissions of the Apollo 17 launch, commander Gene Cernan describes the build up in g forces as the saturn V's acceleration increases. This is also an example of jerk.

[@ambreentariq1423]

I love pesudo forces, such an insanely cool subject matter. Your way of describing them is great.

[@honeybadger6878]

Professor Leonard, at minute 32:37, it is Managerial Economics. I am in Calc I and Managerial Econ at the same time and Managerial Econ is full of calculus.

[@hasselflores]

I hate application problems they're hard for me to understand but you have done an awesome way at explaining them clearly. I get it now. thank you

[@Peter_1986]

It's nice to see a math professor admit having difficulties with spelling. Really demonstrates that spelling is not some sort of indicator of overall intelligence, like some language cops want to believe, and it's definitely not a laughing matter either, especially not for actual dyslexic people.

[@paulroscoe39]

Um. What.

[@kevivarma2759]

The example of jerk literally blew my mind. It makes so much sense but it's so unintuitive

[@luckytinashe2264]

honestly i was happy to get these tutorials. May God continue to use you Prof

[@ameeranoon3913]

After watching multiple videos on applications of derivatives, I only found this one good. Thankyou for making it. And I must say that you are a handsome professor. ☺️

[@Chelsea2009FC]

Saw the whole 40 minutes of this, thanks!

[@collar50]

These are excellent lessons. The calculus, of course, is completely correct. I do want to say, I think his total cost function at the end of the video is a little bit inconsistent with what's normally assumed. Total Cost (and each of the fixed and variable portions of it) should never be allowed to be negative. Likewise, marginal cost should never be allowed to be negative either (the additional cost associated with producing a little bit more of a product should never be negative). You could maybe view subsidies as causing an exception to these rules. Usually, marginal cost is assumed to be initially decreasing due to gains from specialization (as we produce more output, we use more variable inputs to production -> i.e. hire more workers. Those inputs (workers) can then be used in the ways that they are best at). Then, we would normally see marginal cost increasing due to diminishing returns after a certain point -> i.e. as we keep on trying to produce more and more output, we have to use more and more variable inputs to production, and these overload our fixed inputs (as we keep on hiring workers, there are eventually too many of them crammed into our one factory). So marginal cost could probably be something quadratic, which means that total cost would have a leading term to the 3rd power. Sometimes, if we don't care that much about cost, we might simplify, and assume that marginal cost is constant. In that case, total cost should probably have a leading term to the 1st power. I'm sorry for this; I'm a teacher too, but not as good as Professor Leonard. He's great at modelling concepts for the students, and preemptively identifying common pitfalls. His lessons are instructive in more ways than one.

[@natalieelskamp03]

Great video! A couple of notes. First, at 15:58: This was explained somewhat incorrectly. The moment the DVDs are released, the rate of sales is not the highest. That happens at T=1, as we found earlier. It’s the rate of *change* in sales that’s the highest. Also, at 37:08, you’ve calculated the cost of producing the 101th item, the *next* item. Well, roughly. It shouldn’t make much a difference. The derivative method of calculating the cost of the next item isn’t perfect since it neglects the fact that the output is still changing over the item-to-item interval.

[@jz1224]

Are you sure about your first point? You seem to be implying that the rate of dvd sales is maximal at T=1. But I don’t think that’s correct. The number of sales is maximal here, not its rate of change. To find the maximal rate of change you would need to use the second derivative.

[@rwharrington87]

s(t) - position s'(t) - velocity s''(t) - acceleration s'''(t) - jerk s''''(t) - snap or jounce s'''''(t) - crackle s''''''(t) - pop Mind blown.

[@aram5642]

It's even hard to wrap your head/imagination about the concepts after acceleration. [Well, jerk is easy to grasp]

[@095_gloriagalores2]

wowowowowowowoowwww!!!!

[@sdparsons]

Thanks Prof Leonard! I am finding your videos enjoyable and helpful, much appreciated! They are an invaluable resource

[@AlAminKhan-on4ts]

i wish i had met with these videos 2 years ago, btw THANK YOU VERY MUCH SIR FOR YOUR EFFORT

[@IronEducation]

"Aren't you having fun today?" Me:Yes!

[@MultiSergio900]

never seen a swole professor haha

[@nighthawkmoneybaggs1660]

Freakin' YOLKED!

[@kostyay.4071]

kzbin.info/door/H6Oc4MAJzzmK0SM805ZnjQ

[@QwertyClipz]

10 years later, wonder if he’s getting those DVDs

[@tommicheal118]

Thank Professor Leonard, this was really helpful.

[@UnforsakenXII]

Every single time he chooses feet instead of meters, I get slightly bothered as chem major. LOL. Great teaching!

[@cilva7able]

I would say the same thing as you as a physics student. Why would a student of chemistry care? If this were about chemical reactions and he was using Fahrenheit then you should be bothered.

[@UnforsakenXII]

It was just a preference. Once you get to upper division physical chemistry and you deal with modelling crystal lattices or if you have to get a statistical average distance for some many electron element, then you would probably deal with some form of meter. (Although, I guess usually people use angstroms) but the feet thing is still eeeeeeeeeeee.

[@dankatedoru1026]

Andres Franco Valiente both physics and chemistry majors should understand and enjoy that he uses ft instead of meters. For both it gives them a chance to practice their conversion skills 😎 chem majors can convert ft ---> meters ---> nm ---> (mew)m and then practice their significant figures!

[@robertflores5520]

how high can you really be if you're watching calc 1 videos?

[@davidsullivan1254]

You should just be annoyed as a human, that a single group of humans are too arrogant to use the metric system like all the other humans in the world :P

[@Hdbcanfb12838]

Best calc teacher ever

[@miqomargaryan15]

I think that in 37:31 plugging 100 we didn't get the cost. Because if we want to find the cost while selling 100 item we should plug 100 in original equation

[@uEthanol]

it's the cost of producing an extra item, not the total cost of items

[@TINTINMIKMIK]

@25:29 the unit for jerk is m/s³ or ft/s³.

[@isruabebaw6743]

I wish i had a teacher this good

[@denizyildirim4047]

i wish professor leonard also taught intro to econ :)

[@jeancreamintime5633]

he is looking for the mean jerk time, silicon valley did it as well

[@jonahgo7743]

Simply the best

[@heather0572]

Could you please turn on captions for this video?

[@lorenzozanella6561]

Professor, at 9:09 could you dist. the denominator?

[@Sss-vs7pu]

Lorenzo Zanella of course. No need to bother, though

[@fargo7018]

Applications are so fun!

[@tomjenkins6728]

At 14:44 of this film, item 1. Right side of board, "how fast will sales increase when" the word following when what is it?

[@thegallery7657]

I have a exam on jan 12.. My teachers methods are too complicated. 2 mins into ur video and its really different.. Thank you for this and ill update after my test.. :)

[@nel385]

How did your test go?

[@BenMaina-l2k]

@@nel385 wondering the same. bro forgot to update

[@maps.lourine]

@@BenMaina-l2klmao same😢

[@usansew782]

Thank you prof leonard ,from sri lanka

[@SuperEliasc]

Dear Professor, It is a little confusing the the fact that you chose to plug in "0" in order to get how fast would the sales increase when the DVD is released. I say this because "0" represents the idea that the DVD has been out on the market, if this time is zero then there can't be any sales, right? Thank you

[@FernandaMilagres]

Dear Elias Cohen, zero in this case don´t represent sales (S(T)). Zero is the time when we started to count, when it begins, when the movie is released. (T=0,1,2,3,4,5....). Best regards

[@beezball38]

Elias Cohen amazingly enough, something can have an "instantanious rate of change" even when nothing has happened yet. like a windup toy being help above the ground

[@nickspringer7494]

It is approaching zero, so it isnt actually zero.

[@carultch]

@@beezball38 In mathematical models, yes. In reality, this isn't necessarily the case. Nothing really starts at an abrupt increase from rest to an instantaneous velocity. There will be some smoothing (i.e. high order differentiability) behind the scenes, which is numerous orders of differentiation beyond velocity, that we just don't bothering with, in these mathematical models.

[@StrangeS]

At 11:17, it's written on the board that the tangent has slope of 0. But does not that literally means the second derivative of the function is 0? Am I interpreting it wrong here?

[@XavierLogan-cr9uq]

I'm watching this video in 2024

[@SakibLH44]

what do you drive??

[@kelvinxg6754]

honda city

[@vikatochophy4552]

Pendrive

[@edastrophe]

OMG I loved the way you're teaching (and of yours muscles lol)! Thankssss♥

[@stevedl3150]

`As good a maths teacher as any.

[@LinhNguyen-ei9cz]

This tutorial is so useful for my tomorrow midterm test. And teacher is so hot =)))

[@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

[@The_Angry_BeEconomist]

boost is addictive!

[@ctamai]

Hey Prof. Leonard - You didn't find the max height of the firework (33:10). I found it by plugging in t=8sec into the OG position function and got 1024. is that correct?

[@Aj3.14]

It is 3072

[@sanketshah8153]

Thanks Sir The video was pretty helpful :)

[@infinitezer3950]

i wish i watched this series instead of wasting my time trying to learn in collage

[@grrrrsty]

1. Is there an explanation as to where 7T/T^2+1 came from in the first place?

[@FernandaMilagres]

Jeff Gerstemeier It is an example. To calculate this in real life you have to know about math modeling and time series.

[@Aurora-bv1ys]

He made it up

[@AhmD_ameen]

Thank you professor

[@ronelico]

Helps a great deal. Thorough!

[@siddhantchettri2655]

Thank you

[@pkmkid84]

28:48 I actually calculated the max height using t = -b/2a = -256/(2*-16) = 8.

[@llh6910]

9-11the grade students who didn't learn Calculus yet can solve this quadratic function and find out the max. height, but they have to memorize that t = -b/2a

[@mayaporov8086]

This is not max height it is the time at which you will attain max height so we have to use formula -D/4a

[@mta7221]

the hottest teacher ever and it doesn't affect his awesomeness

[@aymanzaman1666]

I am bit confused in marginal cost??? Is this some stuff other than the company already making with a slight improvement in it??? Then why we take derivative of the original equation of the stuff making the company??? Just to vanish the constant???

[@theuserfromwakanda5497]

The marginal cost is the additional cost supported by the company with each additional item produced. In other words it is "what amount of money it is gonna cost if I produce 1 more item". C(x) was the cost equation (the total cost to produce x items). The derivative C'(x) gives us the slope of C(x), or how the cost changes as you produce one more item => the marginal cost.

[@daylyght3654]

I understand how to work out the equation { s(t) = 7t/t^2+1) } and this is very useful when I thought about it in real world. Example sell your dvds within one year of the release date. but my question is how do we generate the equation itself in the first place in real world scenarios? I'm guessing we have to generate the equation before we solve it right?

[@h-67shreeshaa54]

wow interesting question....i think in real world situation,function is picewise..

[@rwharrington87]

Good question

[@horizonbrave1533]

At 28:00 why is -16t^2 of all numbers used for the affect of gravity? and where did the 256 come from? Why these numbers?

[@carultch]

The free fall acceleration due to gravity on this planet, is 9.8 m/s^2, which is experimentally determined. Translate the numerator from meters to feet, and you'll get 32 feet/s^2. Since this is the acceleration, we integrate it once relative to time to get velocity, and we integrate it again to get displacement. Integrating twice generates a 1/2 coefficient on this term. Thus, the 32 ft/s^2 turns into 16 ft/s^2 as the leading coefficient on t^2 in the position vs time equation. Had this problem taken place on the moon instead of Earth, we'd use 2.6 in place of 16, for these problems gravity problems.

[@carultch]

As for 256, he chose that as the initial speed in ft/s, so that the problem would be easy to solve. 256 is an integer multiple of 16. This way, the time to reach maximum height, would be a number you could calculate without tedious arithmetic.

[@horizonbrave1533]

@@carultch thanks! All these years later it was still bugging me lol

[@jaysmith788]

Thanks for the info

[@leenasaif4897]

What is the textbook that you are referring to?

[@georgebaker1615]

Still trying to understand the DVD?

[@TINTINMIKMIK]

@19:36 should be Acceleration = V'(T) = S''(T)

[@hanhtooaein6368]

Which secotions of Stewart 's Calculus Early Transcendentals 9th edition do I have to work out related to this lecture?

[@yunfeichen9255]

There is one thing about marginal cost that doesn't make sense to me????? Once you produce more than 800 copies and you plug it into the formula -0.4(x)+200, you get a negative marginal cost. What does this mean, does it mean you actually make 0.4 dollars more for each item you produce after the 800th one????????

[@snipinglive5798]

Yunfei Chen The function probably has a restriction passed 500. ( the maximum) which is found when the slope is zero ,or in this case x=-b/2a

[@feoleb]

you're the best

[@AbdulaiNyei-c6u]

Excuse me sir, how did you get the DVD sales which is= S(0)=7?

[@ifeelepic11]

Hi! Could you please turn on auto subtitle? Because English is not my native language , the sub. help me so much. Thanks for videos!

[@briansullivan2664]

How are these functions created in real life, is there a branch of mathematics that assigns formulas to real world problems so these techniques can be applied? Some mathematical modeling field or something. I have seen formulas derived theoretically in physics, but how are these math functions assigned in real life? Otherwise, calculus is just sitting in as mathematical sophistication with any proper useage right?

[@follklor9976]

professor leonard i would be appreciate if you can record video about supremum and infimum on calculus for is really hard to get it thank you so much for your great generous to put all of these videos for free god bless you

[@hanoroam2259]

dear leo. give us home work too

[@Kamaneleilehua]

TECH NERD can I give you my homework

[@pinkyc7328]

Max height will be when you plug in 8 for t, in s(t)

[@kamenfaiz]

real life Clark Kent helping us out weather it is math or cost of doing bussiness

[@sarbajitpanda7869]

Like how do we know that this is the equation of the scenario how we came up with it sir @leonerd

[@carultch]

If you are talking about the example for the DVD sales, it's complicated, and the details are beyond the scope of this lecture. If you study economics, you might see where that equation comes from, or it might just be a curve-fit to a historical trend with no analytic basis for it. For the purpose of this class, it is just a hypothetical example, and the numbers in the equation are made up. For the equation about motion and gravity, it comes from integrating the constant of 32 ft/s^2 twice, and solving for your constants of integration from the initial conditions (speed = 256 ft/s, initial position = 0). Why 32 ft/s^2? That's the acceleration of gravity on this planet.

[@noorii3535]

Sir when calculating I made this result : -7T^2 + 7 / (T^+1)^2 INSTEAD of : 7-7T^2? I mean obviously there is a difference between the two.. How do you know that you have to take the second before the first (in this case you take 7 before 7T^2. I have no idea if this made any sense. :/

[@johnwinchester9422]

Noo Rii -7T^2 + 7 is the same as 7 - 7T^2. You didn't make any mistakes. He just wanted to be fancy about it, and put the seven before the T. But it doesn't mater. Think of it like this: -a + b is the same as b - a. In this case, a would be -7T^2 and b would be 7.

[@noorii3535]

Hello John, Thank you so much for explaining, the answer is highly appreciated.

[@johnwinchester9422]

Noo Rii No problem. Glad I could be of help.

[@joshl3013]

So I can understand when the slope change sign from positive to negative, we should have a peak. But I think it doesn't guarantee at that point the slope will be 0, it that peak is a sharp point, does it count as a peak? Graph like this: /\

[@thesematter]

How was the first equation related to marketing DVDs derived? (Unless it's answered after 16 minutes). The original equation 7t/(t^2+1)

[@StuBonham]

Its just a mathematical model, maybe obtained empirically; the derivation is not important for the purposes of the lecture. Think if it as "just an example" of real-word application of derivatives.

[@chelecovers6893]

for the last problem, can someone tell me why we're not plugging 100 back into the original function?

[@snipinglive5798]

chelecovers that would give u the cost of making 100 things... not the cost of making the 100th thing. Hoped this helped

[@vlogsbyrow]

What would a negative marginal cost (i.e. C'(x)

[@craigasaurus2025]

😮 *Fascinating Lecture* ... 🧮 🤔💭 😬📝 I love taking ... *Duh* - rivatives. 🤭

[@nickgerr1991]

gawddd, dvd sales. Does anyone still buy those?

[@tutingrubena1584]

sir🖐️ how can we create a function or a model describing a situation? let's say, that marginal cost function, how did they get that???

[@matts9216]

In the DVD problem (using the original formula not the derivative) would S(1) represent the number of DVDs sold after 1 year? (multiplied by a constant that was simplified of course like 1 = 1,000,000 units)

[@TINTINMIKMIK]

@20:21 should be Jerk = V''(T) = S'''(T)

[@Ireemo1]

can someone tell me wheres the linearization and differentials video

[@timberwolvesfan18]

+Rim Oo He skipped that section. It should be after Related Rates, but at the end of that video he informed his class that he would skip that section. I know that PatrickJMT and KhanAcademy covers them pretty well, although I prefer Professor Leonard.

[@Fsp01]

dymm this math dude is ripped

[@kyaiaiakyaiaia4230]

Merci beaucoup sensi~

[@masterofmath449]

+Kyaiaia Kyaiaia Vous êtes le bienvenue

[@arjnath]

Thanks very much for the video. So in the question of DVD sales, the peak happens at t = 1 year and if I substitute t = 1 in S(t) that would amount to peak sales of 3.5mn (Say) in year 1. But at t = 0, we find that the rate of sales would be 7m/T by setting S'(t) = 0. My understanding is that the highest rate of sales would probably be at t=0, and this rate decelerates as T approaches 1 where it becomes zero. When we say it peaks at T=1 year, we mean that the cumulative sales from t = 0 to t= 1 is 3.5m and bulk of the sales happened in the early part of the year and slowly decelerated to the end of year. I can understand that the 7m/T rate at t = 0 but unable to visualize the meaning. Does it mean that at the point of launch t=0, and if the same rate of 7m/T continued till the end of year 1, we would have sold 7m instead of 3.5m. The 3.5m figure is because the rate of sales decelerated and that kind of makes sense. So the next question is, if some one asks me what was the sale at launch - how can I get that number ? I have the rate of change at t=0, but how to arrive at the number of copies sold at t=0. Thanks very much in advance !!

[@carultch]

Assuming you don't have returns counting as negative sales, this probably refers to the sales rate. As in how many millions of copies they are selling per year. The area under this curve, from t=start date to t=end date, would indicate how many copies they sell during a given time interval. Let's say that S has units millions per year, and t has units of years. This function peaks at t=1 and S = 3.5. The area from t=0 to t=1, would correspond to how many copies they sell in the first year. Then the area from t=1 to t=2 would correspond to how many copies they sell in the second year. And so forth. . Take the peak at S=3.5 million per year. It might be more practical to understand this in terms of sales per day. 1 million sales per year corresponds to 2740 sales per day. You would then divide number of days by 365 to get time t in years. Translating the units, we get 9589 copies per day, on the day of greatest demand, which is 1 year after release. From t=0 to this day, the number of sales per day, ramps up to this value. 6 months after release at t=0.5, the function value of 2.8 million per year, corresponds to 7670 sales per day. 18 months after release at t=1.5, the function value is 3.23 million/yr, which corresponds to 8850 sales per day. The sales per day decays thereafter.

[@chrisweber759]

The cost curve should be positive instead of negative, I think that's why he was confused. that was a revenue curve. Also, its cost/ revenue per item produced not total cost/ revenue. Total lines would be positively sloped the whole way through because the more pillows my factory produces the higher the total cost. In rare cases will it slope back down

[@adrianloh6919]

Find max height, s(8)=-16(8)² +256(8)=1024....i think they forgot to find the max height....

[@guessilldie6867]

15:47

[@yareink6478]

ofc im here because i need to study , not to see his arms

[@georgebaker1615]

I get it I understanding the Different?

[@AlexWong-ss5jd]

Doesn't derivative collect basis house or mall.

[@pinkyc7328]

At 32 mins, question is find max height but the answer is t equal to 8 sec, should question changed to when the maximum height will be reached

[@kshitizshrestha9398]

8 sec is the time taken to reach the maximum height. Put 8 in s(t) and you'll get maximum height.

[@sardin3829]

superman teacher

[@vinzchannel01]

The derivative of my gains right now are negative. :,( Pls Professwole Leonard teach me how to make my d^2gains/dx^2 shoot up positive

[@23wajiha]

32:59

[@Digiphex]

Twin turbo, hmmm. BMW 335 coupe I would say?

[@shanaya5855]

😍😍😍

[@martinsimbona6145]

Hey you those nine dis-likes, what exactly are you disliking? The calculus course, Proff? Or are you just disliking KZbin for shitty interface? What exactly, are you disliking the fact that your PCs are too slow to load the vid in time? Or maybe just disliking the fact that we as social animals have no free will at all?

[@FormulaDZM]

Clearly Professor Leonard drives a Mk. IV Supra . . .

[@int16_t]

I'm here because of quarantine.