Related Rates #1 Problem Using Implicit Differentiation
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- Опубликовано: 22 окт 2024
- Related Rates in Calculus: The Two Ship Problem
In this calculus video we have two ships with one initially being 100 km due west of the other. One travels north at rate of 25 kph while the other travels south at a rate of 35 kph. We want to know the rate at which the distance between them is changing some time later.
What You’ll Learn:
Understanding Related Rates: Discover the principles behind related rates problems in calculus.
Step-by-Step Approach: I’ll guide you through the process, which includes:
Creating a Diagram: Visualizing the scenario helps clarify relationships between variables.
Labeling Rates: Identify and label all rates involved in the problem.
Finding the Equation: Establish the equation that relates your variables.
Taking the Derivative: Differentiate with respect to time (d/dt) to find the relationship between the rates.
Substituting Values: Plug in specific information to solve the problem.
Why Watch This Video?
Ideal for Students: Perfect for high school and college students studying calculus and related rates.
Clear Explanations: Easy-to-follow instructions that simplify complex calculus concepts.
Enhance Your Problem-Solving Skills: Build confidence in tackling related rates problems in calculus.
📈 Don’t Forget to:
LIKE this video if you find it helpful!
SHARE with classmates or friends who want to master related rates!
SUBSCRIBE for more calculus tutorials, problem-solving techniques, and educational content!
Timestamps: 0:00 Introduction to Related Rates 1:30 Overview of the Baseball Diamond Problem 2:15 Creating the Diagram 3:00 Labeling Rates 4:30 Finding the Equation 5:45 Taking the Derivative 6:30 Substituting Values 7:15 Conclusion and Key Takeaways
#RelatedRates #Calculus #TwoShipProblem #MathTutorial #EducationalContent #LearningCalculus #ProblemSolving #HighSchoolMath #CollegeCalculus #Differential Calculus #VisualLearning #MathematicalConcepts
Hi all! Wanna help a RUclips education OG? Please post comments, questions and anything else on your mind in the comment section! so, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly as it helps me :)
I cross my fingers every single time I type in a confusing topic into the RUclips search bar with your name after it hoping that there are a multitude of videos to help me understand. Thank you for digitally aiding me in all of my math classes :)
Twas the eve of the AP Calculus exam...
+Rocky Randhawa twas the morning of the calculus final.
i got a 4, i could've gotten a 5 if i knew about this channel earlier..
Rocky Randhawa me too in 2018!
me too but make it 2020
You Smart. You Loyal.
Big Ups My Dude.
Did the drake vocals come in yeeeet?
When I watch these videos I wonder why I waste time going to class.
DREDS is the perfect acronym. Definitely feeling the "DRED" lol
I watched like all of your videos and got a 96% on my final xD You da best!
I wished you were my teacher
I DRED my exam tomorrow... but a little less after this! Thanks
did you pass?
Passed the class with an A! Phew!
Nat Cat great job! My final is in two days, hopefully I get an A too
hour and a half till i fail my midterm, lets gooooooo
how did you do?
I'm hoping for a C but could be worse or better.
TheAlmightyNivs keep me updated. I'm curious.
alright haha i shall let you know
sqielyr 52% lol
One thing I really love about maths is that the concepts don't change Even after a long time. I'm watching this video 12 years later and it's refreshing my mind.thank you for making everything looks so easy to do
This video's more than 10 years old and it still provides better information than my teacher.
Ok im in the library at OSU and not only did this help immensely but it was word for word the problem that I was working on in my book. Great explanation. Thanks a lot!
I literally did this EXACT problem 3 days ago and was so confused...looked at the solution and it didn't help a great deal either. You did an awesome job explaining the process, thank you!
all hail the math king
thank you so much. I have a test tomorrow on implicit differentiation and related rates. this helped me out so much.
I'm so glad you're left-handed; so much easier to see what you're writing.
When you do the trick with flipping the triangle to make one large right triangle, you are looking at the system as if you are on boat A. The side of the triangle x+y represents the relative distance between the ships as a function of time.
Never learned D.R.E.D.S., but I'm now glad that I know this before going into my Cal I test in 30 minutes. Thanks!
this is the same textbook that i used in grade 12 calculus!!
@kwarkn no. you should not think about positive and negative as on an x-y plane. if the lengths are increasing as time progresses, the derivative with respect to time should be positive (to indicate it is increasing)
I despise related rates. You make them less despicable.
Sarah W despicable me
Less DREDful?
Thanks Patrick for all your help over the past couple years, your hard work has made a serious difference to my capability. Glad to hear you are still kicking out the tutorials, there is still so much of your material to explore!
I have been asking everyone what is a good way to remember the steps on related rates problems! DREDS is perfect. Now I can do the whole problem without feeling lost. I'll tell everyone that acronym. Thanks so much!
holy crap Patrick, no matter what year of math I have taken, you always made videos on every section in my textbook. I am in first year uni now, and we're doing related rates, so this video was very helpful. Also when I was grade 10, I remember you doing a "completing the square" video which helped me out a ton. you are great
You just saved my grade and got me ungrounded. I would pay you if I wasn't broke.
@patrickJMT I love your responses almost as much as I love your videos, they're always so polite they just make me smile!
glad you like them... but everyone makes mistakes, me for sure!
however, i can either delete the video or at least make annotations if it is a 'small' mistake! : )
I just might pass my calculus test....thank you!!!
we should have our campus send the tuition to you rather than to our teacher :)
We've been studying related rates in my calc. class and this problem seemed very familiar - I found this exact problem in my book! This video was made in 2008 and my calc book was made in 2018. 10 years later they're still using the same material from old books but selling it with a new price tag.
welcome to over priced textbooks
Thank you for providing these related rates videos. They are the clearest I've seen.
You're an excellent teacher. If I pass my calculus final in the morning I will owe it all to this channel.
This is the EXACT same problem im doing in my book thanks. It makes sense now
I have been watching my teacher do these for 2 classes now...and you actually showed problem solving steps..I hate my professor. She just implies that we know EVERYTHING up to this point. She never recalls anything for us, and she just skips over steps. Thanks for the vids man
@durrthock that excuse does not work on me. i took 18 credit hours in college while working 40 hours a week and still found time to study the stuff i needed to study. better to say: it was not that important to me.
It's amazing how you can explain something that would usually take about an hour to explain in about 10 minutes.
@zero6140 i am not allowed to encourage people to click the ads in my videos ; )
Yes, I understand that they were opposite directions, but I was merely commenting on how it might be confusing to a viewer that usually denotes x and y to perpendicular axes, and might instead use a and b rather than x and y to denote that sort of thing.
Thanks a lot!
my pleasure!
patrickJMT 6 years later and youre still helping people pass Calculus classes. Bravo.
Thank you sit! Dreds is a lifesaver on these
Another method which may be worth mentioning (that doesn't use implicit differentiation) is to note that the height of your right triangle, x+y, increases at 60km/h, or can be expressed as just 60t.
Using Pythagoras the same way, you get
z = [ (100)^2 + (60t)^2 ] ^0.5
Differentiate with respect to t, substitute t=4, and you're done!
This method should be easier, since you had to use some form of z(t) in the end anyway. I suspect you were forced to diff. implicitly. :)
Good job on the video!
THAT'S THE EXACT PROBLEM!! MIND BLOWN..and extremely grateful
@seifdeiab no. it is zero. what is the derivative of 10,000?
OMG this is crazy! I take on online calc course and the practice problems and the tutorial aren't working but this is the EXACT problem I needed help on.
Instead of it making an "hourglass" shape, he flips the right angle that A makes over the line Z to create a better shape (right triangle) to work with for him to be able to use the Pythagorean formula. This changes the shape, but leaves the ending area of it the same. Hope this helps!
Ok nerds, I was stuck on the part where he uses (x+y)^2 . WHY? Well we're setting up an equation that we'll then differentiate. You should NOT think of x and y as merely distances traveled. They are the RATES at which those distances are traveled. So we can't just lump them together and get something like z^2=x^2+100 because they are different terms, different variables because they're different rates. One is moving faster than the other. Hope this helps someone!
You're amazing! My brain was hurting after doing this exact problem. You summarized it for me in less than 10 mins. I thank you for that sir.
One tip that has helped me with the implicit differentiation aspect is: Always use the chain rule with individual variables. It can get pretty confusing trying to keep track of which is the dependent variable, which are independent, etc. But if you always use the chain rule (meaning the derivative of x is dx/dt etc.) you can't go wrong, because if you do so on the independent, the derivative is just 1 anyways. Just a tip that may help some people.
You saved my life. Ur teaching skill is way better than my calculus teacher. I finally know what's going on. Thanks s lot
why does this make more sense than any lecture I ave ever been in before
You have a real knack for teaching bro keep it up
I honestly do not know what I would do if I had not seen all your videos!
@BeBopDeluxe85 glad you like my stuff :)
show that v=root3/3 pi r^3, and find dv/dr
then it says you may assume that the volume of a cone of hight h and radius r is 1/3 pi r^2 h
then work out using answers from part i and ii the value of dr/dr wen the radus of water filled in the cone is 2
This video was extremely helpful thank you! However I find it easier to understand if you write it like this instead:
z(t)^2 = 100^2 + (x(t) + y(t)^2
Dt( z(t)^2 ) = Dt( 100^2 + (x(t) + y(t))^2)
(Using Chain rule, look up if you don't know it) =>
2*z(t) * z'(t) = 2(x(t) + y(t)) * (x'(t) + y'(t))
Then you just juggle it around with some algebra to solve z'(t). It looks neat I think and it's easy to keep track of what the diffrent variables represent, x'(t) is the change eg km/h and x(t) etc
no problem! happy to help
you've seriously helped me through this semester. thank you
A noon ship A is 100 Km west of ship B. Ship A is sailing south at 35Km/hr and Ship B is sailing north at 25Km/hr. How fast is the distance between the ships changing at 4pm?
tnx to your videos i really improve a lot in my related rates problem by watching your video you added another in my tool kit tnx
no problem. good luck with the rest of the semester
Your videos are probably going to be the difference between me getting a 3 on the AP test tomorrow and a high four or maybe even a 5. Muchas gracias.
This is seriously the most helpful video on the internet!!! Thanks Patrick!!
Thank you so much. I have been struggling to understand related rates and this video really helped me understand! Thank you so much!
Calc 1 final tomorrow... wish me luck and thanks for all your help with my homework :)
Could you not use the distance formula? Set the initial position of ship B at (0,0), the position of A at (100,0). Then find the position of each ship 4 hours later (to find the final coordinates). Once we do this, can we then take the derivative of the distance formula? Thanks!!
THANK YOU SOOO MUCH! This exact question was on my Midterm today!
I don't think you truly understand how many lives you've saved Mr.PatrickJMT... Cheers.
no, the signs would only be negative if the lengths are decreasing instead of increasing in this case
glad i am helping! make sure and still do problems though!
Thanks for posting! I thought that I had understood this concept in class, but I guess I forgot it between then and the time I started my homework, so seeing this done again helped A LOT! Thanks again! :D
Approximately 55.385 km/h
That's what I got! Yay!!!
exactly!
The math teacher we never knew we needed
no. both distances get longer with respect to time, so they should both be positive
You're amaziiiiiiiing!!!!! my confidence is boosted up for tommorows quiz, thanks to you!
you did an amazing job breaking this down
Thanks to you, I just might have a chance on passing my calculus exam.
You inspire me to get a secondary in teaching.
170/13 or approx 55.4
720/13*
saving my A since 2010... great job Patrick, I greatly appreciate it
@durrthock so?
You did a great job explaining; I understand it. Thank you.
Awesome videos. It's a nice review for my test coming up.
I'm just learning related rates, and I was wondering why do you have two one hundred squares on the bottom? Don't you use pathagrean theorem and replace that with z so you get sq.root 140^2 + 100^2, so where does the other 100^2 come from?
Ah this makes review so easy. I took calculus about 4 years ago XD.
@petermilko it was probably the yard guy, cause my wife and i both hate vacuuming!
Thank you so much for all the videos you make. They are so very helpful.
at t =4, x = 35*4 and y = 25*4. shouldn't z^2 = x^2 + y^2, so z^2 = 140 + 100 and z = sqrt(260) = 16.12? So z*4 = 64.48 km/h? Using implicit differentiation you got 55.38... I feel like everything I wrote out makes sense, but it's not matching.
WOW, YOU SAVED MY LIFE.
I've been stumped on a similar question for the past few hours now...
and now that I've watched your video, I finally understand it!!!! You're way better than my calc teacher at school x___x .. thanks for sharing your knowledge!! =D
thank you so much , you just saved my ass for tommorrows final. I get whats goin on a little more. thanks again
the one and only james stewart 5th edition!
a classic! larson's textbook is also great!
Thank you.
@Machammerballs I know, its hard to deal with trolls sometimes yet Patrick still manages to.
Awesome! You're using an older edition of my textbook, James Stewart Calculus 7th Edition. The pics on the page looked familiar, and this is p. 181 in my book. Just when I was wondering if the examples you taught would be close enough to the ones in my book. :D
you are awesome. i might actually pass my test on friday now!
You sir, are a gentleman and a scholar.
4:37 you said why don't we move X over to the side of y. This part confused me. I realize Z can't have X on the same side, but what is the exact procedure and placement of x with the Pythagorean Theorem? Is the movement of X a subtraction from Z? How exactly does y = (y+x)? I think that is a step that you should have explained instead of skipping.
@patrickjmt can you post a video of the problem you have circled (#19) in that book? About water leaking out of an inverted cone while water is being pumped into it.
thank you for all the videos, but i think you made a mistake in calculating the height (y) of the triangle. y=100, but since x is going down, and taking the base down with it, shouldnt you add x+y to get the height of the tiangle? so the height would be 240 rather than 100