Optimization Problem: Maximizing the Area of a Rectangle Under a Parabola
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- Опубликовано: 28 сен 2024
- 📏 Explore Optimization with a Rectangle Under a Parabola! 📏
In this video, we tackle an optimization problem: maximizing the area of a rectangle that is bounded below by the x-axis and above by a downward-opening parabola. This classic calculus problem illustrates how to apply optimization techniques to find the largest possible area within given constraints.
What You’ll Learn:
Understanding the Problem: Discover how to set up the rectangle in relation to the x-axis and the parabola.
Setting Up the Area Function: Learn how to express the area of the rectangle as a function of its dimensions.
Finding the Derivative: We’ll guide you through the process of taking the derivative to find critical points where the area is maximized.
Analyzing the Results: Understand how to interpret the results to determine the dimensions of the rectangle that maximize the area.
Why Watch This Video?
Ideal for Students: Perfect for high school and college students studying calculus and optimization.
Clear Explanations: Follow along with detailed, step-by-step solutions that simplify complex concepts.
Real-World Applications: Learn how optimization plays a vital role in various fields, including engineering and economics.
📈 Don’t Forget to:
LIKE this video if you find it helpful!
SHARE with classmates or friends who want to master optimization problems!
SUBSCRIBE for more tutorials, problem-solving techniques, and educational content!
#Optimization #Calculus #MaximizingArea #RectangleUnderParabola #MathTutorial #EducationalContent #LearningCalculus #ProblemSolving #HighSchoolMath #CollegeCalculus #DifferentialCalculus #RealWorldMath #AreaFunction #CriticalPoints
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Quick tip: take the second derivative (the concavity) and plug in the critical number(rad 8/3). If the number is positive, then its a min, and if its negative, then it's a max.
Calculus wouldn't be possible without you, Pat. :)
fax
My favourite lecturer from RUclips University! :-)
You don't apologize for a shameless plug, it's not shameless if you do!
seems easy. But when i do it....fucking hard.
watwat293 The story of how I found out these Videos doesn't help, you actually have do many examples to figure out how to do it on your own. I usually just use these Videos for reference incase I missed something in class.
Story of my life...
The hardest part is setting up the fucking equation. Hate this part of calc
i feel you
my life in math
@patrickJMT the best part of this video (besides the fact that the material is much easier to follow than in class) is that i can rewind it if i don't comprehend the concept the first time as opposed to raising my hand and disrupting the flow of the class every few minutes. Thanks alot for taking the time to post these, math guru!
by far, the clearest anyone has ever explained math to me. you are my hero.
I am not able to do one optimization in the James Steward Calculus book for Optimization given the ridiculous difficulty in the problems... So I watch this and continue to Newton's Method...
I hate that book, stupid proofs and useless hard examples.
Bruh i'm here for the same reason. xD The optimization ones are absolutely ridiculous, i can only do ~ 1/8 of them, and i had no difficulty doing the ones before optimization.
@@lerubikscubetherubikscube2813 Same here Lol, honestly Im really just trying to understand bits and pieces of it so I get part marks..
dude same 100%
George Hanna SAME!! We also use James Stewart
God bless this man
happy to help :) come back any time!
I'm majoring in middle school education. I've wanted to teach science, but I'm beginning to be more accepting of the possibility of teaching math someday because of all the people you're able to help. There's so much gratitude and relief on your channel.
Dude, thanks for making it so clear and straightforward! I watched one of your videos and I finally understood the concept! :)
i guess if you are a math teacher, the math noble prize is constantly yours absolutely no doubt. your usual present in youtube always make me pass my calculus. the first cheque am gonna get in my math career is will be yours. thank you so much Patrick
it's amazing how a free short video explains everything than a 80 minute paid college lecture
you not only have a gift in math but you also have a gift of teaching!!!!!!!! you have no clue how much your videos have helped me thank you so much
OMG! I saw this video last night, (a night before my test), and I swear to you my professor put the exact same problem on the exam except with different numbers! thanks to you I was able to do it without a problem!!! I know you must hear this all of the time but you help more than you know!!!!!
spent 2 weeks in math class not understanding optimization, learned it from just 2 videos of this
The fact that you are left-handed makes me really happy.
--a left-handed person
LOL, I watched about 30 of his videos and never noticed until I read your comment ... :-)
if the coefficient on the x^2 term is negative, for example:
-x^2, -4x^2 + 7x - 49, -10x^2 + 8x, etc.,
the parabola opens downwards!
@charminator123456 no, it is there
thank you sir, I came here to understand it a little bit more because I couldnt understand it from my professor. You made it a little bit clearer for me.
@metalmine1 oh, people would still complain
no problem
I cannot express how thankful I am for your videos!
a= xy is only the area of one side of the rectangle from the y axis. since its a parabola the area on both sides will be the same, so i guess u can multiply it by 2 or double it
everyone 'relearns' algebra as they go.
my suggestion would be to brush up on it over the summer know.
know basic graphs, how to solve a variety of equations involving a variety of functions
Please write the question the comment box! That would really help. AND THANK YOU FOR ALL YOUR HELP.
I promised myself I would thank you (yet again) if I got an awesome FINAL grade in my calc class!! .... And guess what, I did!! I THANK YOU from the bottom of my heart because it was your videos which truly helped me more than my professor ever did!!! :D THANK YOU PATRICKJMT!! I look forward to using your statistics videos on your website!
Thank so much,,,please I have a question,,,,Is there anyway to determine if its a maximization or Minimization based on the value of the second derivative ?....because my teacher said you have to find the second derivative then check if it's negative or positive so u can determine if it's maximum and minimum.....THANK U.
Without you I would be failing. Went from like a 65 to an 80 with your help after viewing your videos for just a month, which is a lot for AP Calc. Hopefully I can make it into the high 80's, maybe even the 90's this marking period :)
Any given topic,i watch your videos. They help. God bless You
so where would u plug in the square root of 8/3 o check if its a maximum or a minimum? by the way u explain so well.
Ughhh FML my Calc final is at 6pm! Your videos are helpful though!! :)
YOU JUST SAVED ME FROM CRYING THANK YOU
How would you do this problem if they were asking you to minimize the rectangles area?
Chris028 G you would either put your point (x,y) as close as the vertical axis or as close as the horizontal axis, but that would not make a lot of sense as a math exercise
happy that i could help!
The product of xy is the area of only one rectangle.
So 2xy would be the area of both.
"heres my shameless plug for my website" haha love it
could you have taken the second derivative to determine concavity, and therefore whether or not the point is at a max or min?
haha awesome, i was doing this exact problem when i felt like checking your vids.. BAM PatrickJMT saves my day.
You are helping me so much. Keep up the great work man.
yep! thanks for helping choo!
i got perfect on my first calc test thanks to you!!! keep up the great work! these videos really do help :)
Yes. I'm definitely interested in viewing your video concerning the application of the second derivative test to find max/min values. I would appreciate that greatly. On a more personal note, what would you say to an individual like myself who aspires to obtain a B.S. in math who also has a questionable strength in his foundation( algebra,etc). My foundaton in math can be compared to a removable and jump discontinuous function, too many gaps and holes,lol.
RUclips should make their own University!
Sorry for a noob question, but im curious why you didn't use the whole base as one x, you instead divided it into 2?
Thanks for the video. You should make more, thanks!
@patrickJMT That patrick, was badass
I was literally failing till I found these. Thank you so much!!
I tried scrolling through the comments for old questions-watching because I got calculus-but how did you get the equation for y? y = 8 - (8/3)?
I want to know how you got A=2xy, i just dont see it. I want to be able to do it by myself and I get everything else except for that part.
@MrONEMILLIONCOWS that would be a good advertising technique. no one wants 'math videos' but many would accept watching 'porn for the brain'
Probably not going to pass this part of the test but with what I learned I'm confident I'll get partial credit at least.
nice work, thank you for making it simple.
can you take the second derivative to determine that its a max?
Do you know how to do this problem if there is a different formula for the line when x is negative and a different formula when x is positive?
"What the heck are they saying"....my thoughts every single time.
After going through first and 2nd problem, I have a question, In first problem we required a minimum value and in second problem A maximum value was needed but for solving both cases we used derivative. What is the math behind?
How would you find the domain of the area function?
Oh I coulda sworn u said to find area...oops haha. I did get the dimensions, but I went on to find area. Would it be (64sqrt(6))/9 ?
"Here's my shameless plug for my website" XD
Why would that prove it's the max? I'm not doubting that is does but I just want to know why so I can comprehend better.
Hey Patrick are you a student? If so what school do you go to?
Thanks so much! I'm studying for my exam and this problem showed up... with different numbers. :D
i have been to the lecture n the professor solved similar question but i didnt get it. now am confident about myself now through this video.anyway what is your trick
Thank you.
Ugh, when you put an exclamation point after "Derivative" its like a light bulb going off in my head
Awesome! I hope you are teaching in some Academia.
this is #24 in the "single Variables calculus" 6th Edition. chapter 4.7
great thank you
Thank you! Easily understandable!
@patrickJMT No they won't. Hey, what the heck Patrick...your hand is taking up all the camera space! Geeeeez.
What is the domain in this case?
how do i know whether the value of x is maxima or minima
+Stuthi.R Raja second derivative. subsitute x, if the value is less than zero its a maximum, if its more than zero its a minimum
@suckmythach That x he wrote there is just a multiplication sign. I know its confusing, I thought the same thing at first! :)
that is why i am here : )
thx a lot!
This is so disappointingly easy. I was hoping for something harder!
You can do this with algebra. But it is complicated
Nrxt time post the problem on the board.
thanx man but i still suck at this i think im not looking at it the way everyone else is ohh well heh.
Your x result is incorrect. It has to be root 6 over 4
ditto
I love you.
ur a boss!!!
you explain this so well i feel like i am cheating :D
couldnt u take the second derivative and get A"=-12x and then plug in the critical number of sqrt 8/3
Yes, however there is no need since A' will give slopes. Knowing one critical number and the fact that slope is positive then negative, he knows he has a global maximum and therefore no need for the second derivative.
yeah but doing the second derivative is easier than testing interval in my opinion
nice! : )
ahhhh the comments... like a wave of happiness at the end of a long day. : )
glad i was able to help
My Calculus book (James Steward) has optimization problems that are close to impossible difficulty for Calc 1 students. I wish we could be given problems like this to make sure we get the concept instead of trying to remember a million unimportant trig identities.
same with me. you are so on the point for this.
If my teacher had only used the word constraint, that word really is the reason I get the topic now.
Yup. Baby Clara.
TheBetito123 You do know how to make people feel uneasy.
@@clarahav Clara?
well, understanding the problem first is the key! then i would draw a diagram. the hardest part (to me) is trying to find the equation that you need to maximize or minimize; this is often the part where you have to be resourceful!
Can you replace my current calculus teacher? Id appreciate it
teach what you love :) people need good science teachers too!
i love all of your videos, but can you please display the question in the beginning of the problem
You're a beauty! Why can't you be my prof? HAHA
Why did you add the "2x" in the "2x*root*8/3" at the end? did you not already apply the 2x at the start of the question? that's the only part that confused me haha thank you for the video! it's sooooooo helpful! =D
Dylan Holton that is not an x it is supposed to represent "by" as in 2 root 8/3 by 16/3. It was definetely a bit confusing the way he wrote it
Why do you get A=2xy and why do you know it is maximize at the first
+Dao Huyen he is using only the positive value of x, which is only half of the base, so he has to double it to get the full area.
i was laying on my side when i was thinking about it...
yes, you can use the second derivative test to determine if a critical number is a local max/min value (in most cases)
sorry your teacher is not the best!
i am happy that ! could save you!!