Calculus 1 Lecture 2.8: Related Rates

[@wolfpatrol318]

If KZbin had some kind of KZbin education award, I'd nominate Prof Leonard for Calculus. Best teacher ever; using KZbin to educate students!

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Yea better than PatrickJMT, always skipping steps and shit cause he's "lazy".

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literally i was just about to comment that

[@markbouwman7847]

not even the comments section on a calculus one course is immune from people calling each other "assholes" and "stupid".

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They should make an education section yo! Finding these help a lot with concepts !

[@goodvibesatalltimes6033]

I received a 98% on my calculus exam and the professor at my school is the "tough" one. Thank you for these videos Professor Leonard.

[@ProfessorLeonard]

Great Job!!!!

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@@mosamokebe3198 Boo-hoo, cry about the missing 2%

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genuis

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@@joudborno1622 Are ya alright, bud?

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@@SparkHorror-r5c no. I wish I can complain about getting a 96

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This Man is out here saving lives....proves that not all heroes wear capes.

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Red Fish u speak the truth bro

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Dude he is clark Kent! Just freaking look at him!

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he looks like superman icel

[@Miokien]

I don't want to speak to soon, but your lectures are making calculus feel like a breeze. I managed to get an A on my last exam covering much of derivatives. Your lectures gave me the knowledge base needed to learn things you didn't cover, like logarithmic differentiation, derivatives of logs, etc. and now I am extremely comfortable with derivatives, thus having no trouble learning ahead of the class... It will feel soo great to go in to lecture tomorrow actually understanding what the professor is teaching. In my opinion, the key thing you do that makes it so easy to learn is taking the time to explain things, be it explaining things "in English," writing down your explanations, or doing many examples and explaining the steps throughout each one. For this lecture I liked starting off with the difficult example, because understanding it allowed the simple examples to reinforce what I gained from the difficult example.

[@ProfessorLeonard]

Great! Very happy to have helped. Now, about Log Diff, etc. You are in what's called and "Early Transcendentals" Calculus 1 course. We cover that particular material in Calculus 2. So.... Just watch Chapter 6 from my Calculus 2 Playlist and you'll be all set. Best of luck!!

[@D3f3ndY0urFr3d0m]

Professor Leonard Thanks for your lectures. My professor isn't good at explaining things, especially concepts. He spends considerable amount of time each lecture covering the previous lecture's homework problems. It's frustrating that he teaches students to solve and not the underlying ideas. Also he doesn't do it step by step like you do here, often times students gets confused and then he spends more time going back explaining it. You're the epitome of what a good teach style is.. and which every math professor should emulate or at least try to emulate. Also, would you mind sharing what book your classes use? I'd like to self-study and it'd help greatly if I used textbook as a complimentary guide. Thanks again for being our superman!

[@samanrai2647]

Why did the answer came in 0.72 radian i mean it could have been in angles as well.........i'm confused at it.

[@mandelyu4153]

9/125 = 0.072

[@grahbuckfan6]

Thank you for this! I was wondering if you covered this.

[@AmyLikesQuestions]

EVERYTIME he erases the board my heart is so happy. He erases EVERYTHING. Finally! A teacher who doesn't leave spots.

[@Peter_1986]

My math teacher was extremely bad at erasing everything - he left lots of ugly dots and lines everywhere. He even forgot to erase the board for other classes a couple times and just left everything on the board, lol.

[@deltafoxone]

I think it's not the most important thing in this video

[@Peter_1986]

@@deltafoxone It does mean something though, because it can be quite distracting when there are ugly dots left on the board. So it is very relieving that Professor Leonard always erases everything on the board.

[@biawa2979]

What do you think all the training is for?

[@idkusername2981]

I thought it was going something like I understood everything he erased but oh well...

[@reid211]

Just so you know... YOU are my Calculus teacher! I go to class and listen to what is said and I come home and you actually teach me what I need to know to pass the class. I made an 86 on my first exam.. which was given a week after I found your videos... I made a 96 on my last exam after watching your videos for a whole test cycle. You're doing GREAT GREAT THINGS Sir!

[@iylaedwards4069]

Related rates oh my god I didn't know how to solve any problem in it ANY problem so I was sure to get no less than a zero lol. I tried meeting up with a friend to help me but that didn't happen bc a sand storm came and we couldn't get out of the house anyways I thought of just not being there on the test day and be absent but knowing that makeup tests are usually harder I tried KZbin. And you were like a god gift sent from heaven lol. I watched your video and only your video. Today the results of the test came .... I GOT A FULL MARK WHAT IS THAT ?! DOES A "FULL MARK" EVEN EXISTS IN CALCULUS!!! I told all my friends about you and everyone was like "right okay professor Leonard from now on" bottom line is THANKYOU fr thank you. People as good at teaching as you, would charge the students but yours are free so Thankyou ( I'll say it over and over lmao) greetings from Kuwait, Middle East.

[@ali2a429]

شوفلي حل مع ذا السكشن 🤣💔

[@jtparton]

"Show of hands if this is what you got." *raises hand*

[@shannin2165]

same!!!! I do this all the time lol

[@UROOZFATIMA190]

Same here 😅

[@gulnaroguz2510]

I do it too with imagination that I am on hid real class.

[@Peter_1986]

I love the rocket problem at 52:00. It's such a wonderful combination of Arithmetic, Algebra, Geometry and Trigonometry, which are basically the prerequisites for Calculus.

[@COOKIEMONSTER90]

Just plug in 4000ft in the formula and check the result :)

[@anonymoususer4356]

This is 8 years ago? These lectures are better and more valuable than NDSU provides today. Thank you P.L.!

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95 % acceptance rate...

[@nilsguitarsandmandolins5662]

That walk-though of the "oil spill" problem was incredible. Saying that you do a "great job" of coming up with ways of explaining things would be an understatement. I really appreciate all the time an effort you've put into these lessons to make them so dense, comprehensive and easy to follow. When you're explaining these things, you annotate the steps so well and in such a way that it's like the concepts teach themselves. This makes it so much easier to retain all this information. Thanks!

[@ProfessorLeonard]

You're very welcome!! I'm certainly glad that you find the videos so helpful and thanks for taking the time to comment :)

[@MarioLoco03]

I wish this guy was my math teacher...I really like math but my current calc teacher is not good at explaining things to us. I have viewed A LOT of professor Leonard's videos because the explanation is much clearer.

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I got 100% on my second Calculus I exam. Never in my life have I ever got 100% on a math exam. Mahalo Prof. Leonard!

[@ProfessorLeonard]

Great job!!! That's really excellent :)

[@FOSology]

I like your lectures because you remind us of the little things, like the origin of ft^2 in an area problem. Explaining those small things helps pave a clear road to the main goal of learning the calculus. Usually my mind can get clouded by worrying about things not associated with the problem at hand, but you do a pretty good job of making sure that doesn't happen

[@iñigote]

Introduction to the Concept of Related Rates ● [0:00]. Introduction to the concept of related rates. ○ Definition of related rates: how a formula relates to its change over time. ○ Examples: ■ Cost as a function of time. ■ Profit as a function of time. ■ Volume in expansion or contraction as a function of time. ○ Explanation of the objective of related rates: relating a formula to time. Introductory Example: Cone with Water Leak ● [1:02]. Introductory Example: Cone with a Water Leak ○ Description of the problem: ■ A cone with a water leak at the bottom. ■ The water level in the cone changes over time. ○ Variables to consider: ■ Height of the water (h). ■ Radius of the water surface (r). ● [2:45]. Objective: Find the rate of change of the volume of water with respect to time (dV/dt). ● [3:34]. Considerations: ○ The height (h) and radius (r) of the water change as the water drains. ○ The volume of water (V) depends on h and r, and thus also changes over time. ● [4:13]. Steps to Solve the Problem: ○ Step 1: Find a formula that relates the involved variables. ■ In this case, the formula for the volume of a cone: V = (π/3)r²h. ○ Step 2: Differentiate the formula implicitly with respect to time (t). ■ Justification: Volume (V), radius (r), and height (h) are functions of time. ■ Use the chain rule to differentiate each variable with respect to t, resulting in terms like dV/dt, dr/dt, and dh/dt. ■ [5:35] Take advantage of Implicit Differentiation △ Why Use Implicit Differentiation in Related Rates Many related rates problems present relationships where variables cannot be easily isolated. Implicit differentiation allows us to handle these complexities by: - Treating dependent variables (like y) as functions of independent variables (like x or t). - Applying the chain rule to account for the interdependence of variables. Example Insight: In the cone with a water leak example, the volume V depends on both the radius r and the height h, which are themselves changing over time. Implicit differentiation facilitates finding how V changes with time without needing to isolate r or h. ● [8:27]. Interpretation of the Derivative Terms: ○ dV/dt: Rate of change of volume with respect to time. ○ dr/dt: Rate of change of radius with respect to time. ○ dh/dt: Rate of change of height with respect to time. ● [10:00]. Use of the Product Rule when Differentiating the Volume Formula. ○ d/dt [V] = d/dt [(π/3)r²h] ● [13:33]. Information Needed to Solve the Problem: ○ Value of the radius (r) at a given moment. ○ Value of the height (h) at a given moment. ○ Rate of change of the radius (dr/dt). ○ Rate of change of the height (dh/dt). ● [14:54]. Transition to Simpler Examples to Illustrate the Concept of Related Rates. ○ [15:26]. Example: Find dy/dt when y = x³ ■ Find dx/dt at t = 1 ■ Assume that y and x are functions of time (t). ■ Differentiate the equation implicitly with respect to t, resulting in dy/dt = 3x² dx/dt. △ [17:34]. Additional Information Needed!: - Value of x at a given moment (e.g., t = 1). - Value of dx/dt at the same moment (e.g., t = 1). △ Find this out: - If x = 2 and dx/dt = 4 at T = 1. Example: Oil Spill ● [19:46]. Example: Oil Spill ○ Description of the problem: ■ An oil spill is expanding in a circular shape. ■ The radius of the spill is increasing at a constant rate of 3 feet (0.91 m) per second. ○ Objective: Determine how quickly the area of the spill is increasing when the radius is 30 feet (9.14 m). ● [23:19]. Steps to Solve the Problem: ○ Step 1: Assign letters to the variables: ■ t for time. ■ A for area. ■ r for radius. ○ Step 2: Identify the formula that relates the variables: ■ In this case, the area of a circle: A = πr². ○ [27:12]. Step 3: Differentiate the formula implicitly with respect to time (t): ■ dA/dt = 2πr dr/dt ○ Step 4: Identify the given rates in the problem: ■ dr/dt = 3 feet per second (rate of change of the radius). ○ Step 5: Substitute the known values into the differentiated equation and solve for the unknown rate of change (dA/dt): ■ dA/dt = 2π (30 feet) (3 feet/second) = 180π square feet per second. Example: Camera Following a Rocket ● [30:18]. Example: Camera Following a Rocket ○ Description of the problem: ■ A rocket is launched from the ground vertically. ■ A camera on the ground must follow the rocket by adjusting its elevation angle. It is a distance of 3000 feet (0.91 km) from the rocket. ■ The rocket ascends at a speed of 600 feet (0.18 km) per second when the rocket is at 4000 feet (1.22 km). ○ Objective: Determine how quickly the elevation angle of the camera must change to keep the rocket centered in the frame when the rocket is 4,000 feet high. ● [31:43]. Considerations: ○ The height of the rocket (h) changes over time. ○ The elevation angle of the camera (θ) must change to follow the rocket. ● [32:29]. Steps to Solve the Problem: ○ Step 1: Assign letters to the variables: ■ t for time. ■ h for the height of the rocket. ■ θ for the elevation angle of the camera. ○ Step 2: Find a formula that relates the variables: ■ Use the tangent function: tan(θ) = h / 3,000 (where 3,000 is the constant horizontal distance between the camera and the launch point). ○ [36:30]. Step 3: Identify the given rates in the problem: ■ dh/dt = 600 feet (0.18 km) per second (rate of change of the rocket's height) when h = 4000 feet (1.22 km). ○ [44:40]. Step 4: Differentiate the formula implicitly with respect to time (t): ■ d/dt [tan(θ)] = d/dt [(1/3000)⋅h] △ sec²(θ) dθ/dt = (1/3,000) dh/dt ○ Step 5: Determine the value of sec²(θ) when the rocket's height is 4,000 feet (1.22 km): ■ Use the Pythagorean theorem to calculate the hypotenuse of the right triangle formed by the rocket, the camera, and the ground. ■ Use the definition of secant (sec(θ) = 1/cos(θ)) to calculate sec(θ). ■ Square the value of sec(θ) to obtain sec²(θ). ○ Step 6: Substitute the known values into the differentiated equation and solve for the unknown rate of change (dθ/dt): ■ dθ/dt = [(1/3,000) (600 feet/second)] / (5/3)² = 9/125 radians/second ■ Convert the rate of change to degrees per second: (9/125 radians/second) ⋅ (180°/π) ≈ 4.13°/second

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thanks Professor Leonard; you made me actually like math which is something no teacher has been able to do for me :)

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istg he is the best teacher ever. i never thought i'd enjoy studying math but i literally get so happy while watching these videos!! thank you so much mr. leonard :')

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Thanks a lot, Professor Swole!!

[@3rdDegree_]

I have been in a bit of slump because of related rates, this video finally gave me that "AH HA!" moment. Thank you so much

[@renewd]

Your attention to detail by writing out every single step is helpful for deep understanding of these concepts. Thank you.

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Professor Leonard, you have been great help this semester my teacher always says that I'm beyond my grade level at my senior year maths class, and it's all thanks to your videos!!! thanks a lot for everything.

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God bless you Leonord, carrying me upon your glorious biceps to mathimatical Valhalla. Thank you so much for existing.

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Prof is saving my future! Future civil engineer here! Thank you sir Leonard!

[@nicolasmabey1959]

I've never understood related rates in my life, but after watching just 6 minutes of this video I could literally teach a whole class about it. Amazing professor! unlike the ones here at UCLA.

[@joshuasen1331]

Thank you very much professor. Because of you, I got a 4 on the BC exam with a 5 AB subscore. My calculus teacher was very grouchy and impatient, but you explain this very lucidly, and I highly appreciate how you record your lectures to help random schmucks like me.

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NO WAY ME TOO BRO I USED TO STUDY MECHANICAL ENGINEERING IN CUKUROVA aswell

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Sa kral

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This explanation made this type of problem so much easier to understand. This confused the heck out of me in class and it took forever to really grasp it. I wish I had seen this when I was learning it in class, but now that I have I'm using this to review for my final tomorrow :) Thank you for all you do for others. Your videos are very informative :)

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the best professor of all times

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because of you I got a really good grade for my first midterm. Your videos are beyond helpful professor Leonard. I highly appreciate the information

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Let's put calculus aside here for a sec. The way he taught us to derive the volume of a cone was godly. That's what I love about this dude; he doesn't just spew out info, he explains why it is what it is.

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One thing I like about your lecture is you take the time to walk through the little things on the side. It is helpful in understanding

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This is incredibly helpful, both in how detailed the explainations are, leaving no parts to be guessed about rather than understood, and in the way the information is given overall, by which I mean a little joke here, a little gesturing there, working with the audience, involving them in the process. This is the type of teaching I adore most, with no presentation skills ignored or overlooked and no assumptions made about neither the students`fluency in the area, nor the complexity of a particular concept. Shortly, most useful and inspiring. Thank you very much!

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I wish I would have discovered this channel before my last calc exam- 35%. But after watching one of his videos I just got a 100% on my homework. Thank God

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You really are saving lives as in literally, not figuratively.

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I just wanted to say thank you for your videos! You are helping me so much in my math class in college! My teacher just zooms through everything because he assumes we all had calc. in high school (I took statistics instead); and is rather condescending and unhelpful when you go to see him in office hours!

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Your are an amazing teacher. Thanks for doing this for who doesn't understand in college so they can watch your video and get this at home. I personally really appreciate that.

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Hi Sir, Im from India..I literary loved the way you explaned such a complex topic.

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Oh I thought Clark Kent is working at a newspaper company.

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No money in Journalism any more so he changed his name and moved to Merced to teach Calculus.

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Why can't my professor be like this. Your explanations are just perfect and simple as hell. After watching this video I nailed all my hw. Thank you

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You've saved my life, I've got an exam tomorrow and I'm binging all day!!

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8 years later...

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i'm a 1st year computer engineering student and i've always struggled with math even in highschool. my professor at college explains things in a way that i cant understand and i dont know what id do if i hadnt found your channel! my calc1 midterm is tomorrow and i know that any good grade i might get on it will all be thanks to you!

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How’d you do ?

[@agtl1711]

@@UNKNOWN-xp6iq better than i had been expecting (got 43 out of 100) but in all honesty, i doubt i wouldve even gotten a 20 if i hadn't come across professor Leonard's lectures

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43 is bad @@agtl1711

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Superman saved my life, by teaching me calculus!!!

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I failed my fist test even though I watch his videos, but it probably cause it my fist time taking the class. I still love his videos. It’s okay not to get things right the first time. Just keep at it and eventually it will make sense. Don’t give up!!

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You are very effective at getting the points across to students. I hope that you continue making videos like this. They help tremendously.

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Thank you so much this was better than my lectures or textbook or any other KZbin videos you're the goat

[@famidaislam5542]

Funny I am procrastinating on my maths study by learning this which is in my next year... But hey at least I am not playing games Thanks a lot for this videos prof

[@travisroot9741]

This is a legit explanation of related rates. Thank you.

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Amazing feeling when you're finally able to actually grasp the things instead of just being able to apply some techniques.

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Thank you for putting this on KZbin! I appreciate it!

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Props to this professor! Keep doing what you're doing sir! Helping countless college students everywhere, myself included! thank you! For some reason I decided online calculus would be fun this semester ---___---

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His tip on putting large portions in brackets is a lifesaver!

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It’s like Clark Kent quit his job and started teaching calculus

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FACTSSSSSSS

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Got an F on my test. Couldn't focus on what you were teaching cause those juicy biceps grabbed all my attention

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Thank you Professor Leonard! You make thing easy and make math very simple. Very nice done. Happy I found your videos.

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Thanks again for explaining all these themes in a perfect manner. God bless!

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I love your teaching style. You are amazing professor Leonard.

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literally saving my life i love his lecturesss :))) !!!

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thanks a lot Professor Leonard!!!!!!!!!! after watching your video.i got 100% score

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I cant express how much this helped me with this chapter. very happy, thank you for posting

[@DrumsTheWord]

Does anyone know why in the rocket problem, the answer came out initially as Radians?

[@whalesAREforever]

it's because trigonometric functions like cosine, secant etc. are defined in terms of radians. Hope that makes sense.

[@fungiuse]

Excellent lesson.... Congrats, Super Man !!!

[@dmjoy85]

Best description of how the notation ties everything together I found that so confusing before

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simply awsome you are

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you are the best math teacher. I loved your lectures

[@magicmuk]

Please make a Linear Algebra series PLEASE. My dad GOAT was Michael Jordan, my goat is Proffessor Leonard.

[@1light4love]

"What's 1h/dx? I sunk your battleship" *BOOM.* Rate: *Related*

[@johnnolen8338]

@45:56 "... the reason we're figuring out that distance [length of the hypotenuse] is because we have to now find what secant theta is." Why does one have to know anything about the hypotenuse in order to determine sec 𝛉? There is the Pythagorean identity 1 + tan^2 𝛉 = sec^2 𝛉 , and you already have tan 𝛉.

[@goodlack9093]

Love these lectures!!! Can't express enough gratitude for this free content!!!

[@niallspotat0e823]

thank you for explaining what the dr/dt and all of that mean. thats what i needed

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Wow! Thank you so much for this video. I sat through the entire thing without getting lost.

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congratulations on 600k subscribers

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thanks a lot for ya lecture. mr. macho teacher

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Wow! lecture was so good it felt like i was actually there lol. Great job prof thanks alot.

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I am preparing for Praxis...thanks for the teaching..very helpful!!! :)

[@peterpalumbo1963]

I was in Alaska around the time of the EXON Valdez spill. What I remember was the stillness, just about all life had vacated the area. It took time for it to return.

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thanks a lot prof leonard! you are a real superhero!!!!!! I wonder if any students fails in your class!!! having such a cool math Prof!

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I just missed my 2.8 calculus class and you saved me : ) Thanks professor

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Wow! 7 years ago? This is superman!

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this man understand every single word he says and write that what our teacher should learn ' our teacher just read the law explain the steps without saying WHY

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Keep the videos coming. I am learning so much.

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In egypt we study this lesson at school not at university So It's hard for us to understand this lesson easily. It's hard for us to understand this lesson easily. Thx for this High-end explanation

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You need to be the most famous youtuber of all time! Unfortunately, in the world we live in, education is not given the value it deserves.

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I have a Calc final this Friday and I've learned absolutely EVERYTHING that I didn't understand all semester just by watching your videos and you have no idea how happy and grateful I am YOU DA BOMB.COM

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he explains everything very well and simple. awsome!!

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Dude your lectures are so good!!! Thank you so much!

[@m-akinga-mazingt-hingsh-ap7411]

Hey Professor, You're lectures are AWESOME! It's definitely a great thing having a source like this!... Side note: At 0:03 When you say "What we're talking about" sounds like a sound clip for a sports movie!

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You are so amazing professor

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The magician!! Thank you so much Prof!

[@shep7484]

Thanks again for a great lecture. You really have a knack for getting the point across. Unless I misunderstood something around 43:44 I believe you mixed apples with oranges when you took the derivative of the tan Θ to get sec^2 Θ * dΘ/dt. To me it seems that should be sec^2 Θ * d tan Θ/dt. To get the rate of change of the angle should involve arc sec to get the angle rather than the sec rate of change. Jerry E Shepherd

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Very professional teacher!

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Best professor in the history 💛💙💚💜

[@keithcoup6442]

I wonder if those students know how lucky they were! Thanks prof Leonard