Optimization Problems - Calculus
Вставка
- Опубліковано 11 вер 2024
- This calculus video explains how to solve optimization problems. It explains how to solve the fence along the river problem, how to calculate the minimum distance between a point and a line, and how to maximize area while minimizing perimeter as in the case of fencing problems.
Get The Rest of the Video on Patreon:
/ mathsciencetutor
Direct Link To Part 2: Questions 13 thru 31:
bit.ly/2Y3Aem5
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Optimization Problems - Part 2 (3 Hour Video):
• Optimization Problems ...
![How to Solve ANY Optimization Problem [Calc 1]](http://i.ytimg.com/vi/cUMvwG7wvzM/mqdefault.jpg)
Access The Full Video on Patreon: www.patreon.com/MathScienceTutor
Direct Link - Questions 13 thru 31: bit.ly/2Y3Aem5
Those monthly Patreon subscriptions all are a bit too high for me, no thanks ☹.
🥲🥲🥲🥲🥲
hi
Too high😭
For a Filipino
Thanks! I am a retired chemistry and physics teacher helping a Calc AB student. Your videos refreshed my skills with very clear examples and warnings of common mistakes. BTW, I tutored Organic for several decades.
Thank you for existing. My teacher doesn’t teach and just assigns problems and trying to teach myself is really rough. Your videos make everything so much simpler.
Our professors simply link us to youtubers & other professors' videos and then assigns homework generated & graded by third-party websites. I'm beginning to wonder if these professors are worth the salaries they're making.
@@voxvolhynia5400 ironically, teachers make very little lol. Especially public school for k-12 teachers
@@beatriceines3456 Yeah, that's the flip side of it, there are plenty of teachers that do so much and truly change a student's life by being such a beneficial influence, and their income doesn't come close to reflecting that :( Some of my high school teachers really went above and beyond to make the world a better place in any way they could, but they probably get paid around the same as my current Professor Redirect. He was a friendly guy the one time I've talked with him, but he's never taught our class a single thing.
If I didn't need a degree to prove my education, I'd be able to do the exact same thing I'm currently doing, but for free, and that's what really gets to me. With the k-12 teachers you mentioned, there's no way I'd have learned anything without (most of) them. Those are the real MVPs.
we must have the same professor
@@beatriceines3456 Well they are not getting paid more for teaching critical race theory. So its a win win for the socialist heads of the school.
1:29
1.) Find two numbers whose sum is 60 and whose product is a maximum.
7:38
2.) Find two numbers whose difference is 40 and whose product is a minimum. What is the value of the minimum product?
13:09
3.) Find two positive numbers whose product is 400 and whose sum is a minimum.
16:46
4.) Find a positive number where the sun of that number and it's reciprocal is a minimum.
20:42
5.) Find the dimensions of a rectangle with a perimeter of 200 ft with an area as large as possible.
23:09
6.) A farmer has 600ft of fencing and wants to create a rectangular field along a river. He needs no fence along the river itself. What are the dimensions of the field that has the longest area?
27:01
7.) A farmer wishes to create a rectangular field along a river with an area of 10,000 ft^2. What are the dimensions of the field that will require the least amount of fencing?
30:52
8.) A farmer uses 1600 ft of fencing to enclose a rectangular area which will be divided into 3 pens. What is the maximum total area of the 3 pens that he can enclose with the limited amount of fencing that he has available?
34:48
9.) Find the point on the line y= 3x+5 that is closest to the origin.
39:53
10.) Find the point on the line y=4-x that is closest to the point (7,6).
48:27
11.) Find the point on the curve y=6x^2-x^3+10 that has the highest slope. Calculate the maximum value of the slope.
53:33
12.) A rectangle is inscribed in a semicircle with a radius of 10 cm. What are the dimensions of the rectangle that will maximize its area? Calculate the maximum area.
You're a saint You're a saint You're a goddamn saint
Thanks, gl on finals brothers
@@kyzuno5066 damn, thought i was the only one going to this guy for help on my calc final
God bless you mate!
Thank you
My biggest problem in my calculus 1 course is the algebra, and finding ways to simplify or rationalize problems so that the formulas and methods given apply. My professor always shortcuts her algebra by two or three steps, making her work hard to follow. Thank you for showing every step along the way to the solution. Your videos mean the world to me this semester, and I'm sure I'll be back next semester for calculus 2!
I just skip class for these basic courses tbh. Reading the book is enough to earn an A
@@shubhamkumar-vx4ld Please do refrain from using « you people » in the future, and a lot of people take calculus 1 and even calc 2 in high school. I’m also in college and almost every very persons I know that are going in a scientific major took calculus 1 during their senior year in high school.
Me too I really struggle with the algebra and simplifying ! I love this guy's videos he has helped me so much
@@MeroVPN i have deleted it thanks anyway we don't have to learn it was just me 😭 not able to get it as I was not preparing for school but for some competition to prestigious gov college
@@shubhamkumar-vx4ld Ahah no worries mate! Did you end up getting accepter into the schools you applied to?
I have my calculus final tomorrow and I didn’t understand my teacher’s notes and the first 25 minutes of this video was all I needed. Thank you!
How’d it go
I have my AP exam today as well
@@aliabughazaleh5541 I got a 85 on the multiple choice and a 91 on the FRQs. So it went pretty well!
Optimization problems nearly killed me back in Calc 1. I’m glad you’re doing more on this subject.
I’m an eight grader and beve it our mot I haft Toto this stuff.
@@goodplacetostart4606 oh boy
@@SDILUYNTsiu39fnd lmao
it is killing me rn cuz i dont understand what the professor is teaching, i mean, MY professor
@@goodplacetostart4606what.
Holy cow. I’ve been crying over stress about these problems since my professor teaches so poorly, and I cannot tell you how grateful I am for this video of yours! I’m appalled at how quick I got the hang of the subject. Thank you!!
@Adam Khan L
@Adam Khan L
@Adam Khan mad fucking cringe bro
@Adam Khan L
@Adam Khan you went on a date with her?!
You are not a king; you are the entire kingdom
Thank you.
JG is the BEST! HIs explanations are clear, easier to understand, and concised. Whatever I search, JG covers. Great job JG. I shall buy Calculus hoodie to support you! Thank you for saving my neck over and over again.
you literally explained optimization way better and easier than my Calculus SI leader!! Definitely gonna book mark this video so I can watch all of it whenever the semester is over.
timestamp
Sum & Product of 2 numbers
1:30 - 20:41
Fencing
20:42 - 34:49
Distance btw point & line
34:50 - 53:33
Rectangle inscribed in semicircle
53:34 - end
Kurdî?
@@studychannel7424 بەڵێ برا
@@peerankurdi are you studying here in kurdistan or another country? And i think it is not for high school right?
@@studychannel7424 im in kurdistan, its in high school in grade 12 & i also studied them back in grade 11
@@peerankurdi oh i thought you are in college because everbody here in the comments are talking about calc subject so i think they are studying for their college exams
Pray for me that I do good on my Calculus exam
Yes do the same for me as well ❤
How was the calculus exam? I hope you did well
38:53 This is one hell of a solution, but still I learned another approach. But if you want a less complicated solution, remember that when dealing with distance on a cartesian plane problems, the square of the distance formula is also maximized and minimized. Basically, if the distance formula is D=√(Xsub2-Xsub1)²+(Ysub2-Ysub1)², then the square of it is D²=(Xsub2-Xsub1)²+(Ysub2-Ysub1)². Always remember than you can always find the derivative of this function even if it's in its squared form.
Now, we need at least 1 set of point and express this function into one variable in order to differentiate. Remember that the other set of point is given at the origin-(0,0). So, D²=(Xsub2-0)²+(Ysub2-0)² or just basically D²=(X)²+(Y)².
The value of Y is already given as Y=3x+5. Substituting this expression into the equation; D² = x² + (3x+5)².
Always remember the square of a binomial in the form (a+b)²=a²+2ab+b², that just means you need to take the square of the first term, multiply the first and second term and multiply the result to 2, and square the second term. We will get:
D² = x² + 9x² + 30x + 25
Simplying this equation;
D² = 10x² + 30x + 25
Now, this looks like a quadratic equation right? But don't let the function deceive you, take note that you'll be DIFFERENTIATING the equation.
So, dD²/dx= 20x+30 and don't forget to equate to 0 to find the maximum/minimum value.
Solving for x, x = -30/20 or simplified; -3/2. Then the rest will be easy.
I saw it in the textbook, You are legend for writing this in the comment section.
I have my final this morning and this video is helping a lot
This video popped up at the right time.. we re having calculus this semester
Just what I needed with my AP Calc exam next week!
Yeah I had to go back here to learn optimization because I skipped the lesson because it was too hard long ago.
@Beckett Phoenix nice
@William Archacki Best of luck, chap.
imma take it tmr.. good luck bro
@@thecorndogslayer7638 How'd it go
I have a nelson calculus and vectors textbook which I think is standard in Ontario grade 12 calc courses, and at least 3 of these problems were taken from the textbook. This has helped me tremendously because instead of looking at the answers in the textbook, I have mr organic chemistry tutor show me how it’s done.
People say your phones Rot your brain but I’m learning more about these calculus problems from this guy than my actual teacher, thank you for existing my guy.
Here's a trick at 37:00. You don't have to find the derivative of the whole radical because the minimum of what's inside the radical is the same thing as the whole function. So you can just find the derivative of x^2+(3x+5)^2 which is 20x+30, and you get -3/2 for x as well
Very good trick!!!
May you explain for me what he called point y to be equal to the function of the line? y=3x+5 which is the function of the line
This video really covers everything regarding the optimization problems. Thanks a lot.
Good. Ahmad Bazzi has amazing optimization problem lectures.
Huh? Bro JG is the best :|
Thank you for everything you do my guy, we are so lucky to have people like you in the world ong.
I dont who you are or where you live, i just pray from my heart that you get the best of your life. Thank you for existing and making our graduation lives easier🖤
I follow another playlist for math, although I love that guy, but I do not understand a word he says.
You, on the other hand, got such a talent for teaching. I always come out of your videos understanding everything fully.
That explanation at the beginning of constraint equation and objective equation was SO clarifying!
In question 11, I think the critical points found as zero of the first derivative should be evaluated with the second derivative to classify the points where the curve will have local minimum and a local maximum and not the zeros of the second derivative function rather(those are used for checking intervals of concavity). Thank you sir
I think you have taught more students than any actual teacher on the planet. God bless
this man has covered every unit of my 4 years of math, Chemistry, Biology, and physics. how
Bro organic chemistry tutor you have saved me not only in chemistry but in calc I love u
Calc final tomorrow, you're a hero
I have been watching this guy my whole first year of college and he really is the goat. Thank you so much. with out your videos I won't have understood anything in my classes so thank you are always having a video for whatever I look up. keep up the good work.
An unsung hero that blasts us out to the stratosphere whenever we need a hand to release the pressure of last minute "cramp".
You have made it possible for me to obtain Computer Science degree, I always recommend your content, and the best thing is that the University introduced you to me.
CONGRATS ON 3 MIL🎉🎉🎉. hope we one day we can atleast celebrate these milestones but honestly im already thankful enough for the videos you post
6 million now
throughout my academic and college life I never heard of constraint equation or objective equation I just recently learned it in this video never knew there was such a thing. Massive thanks to you
why is optimizing a fence so difficult :(
I know right
You made it simpler than what my teacher discussed about this lesson.
thank you so much for doing these videos! They have helped me a lot throughout the years.
Related rates and optimization problems are definitely the hardest questions in year 1 calculus. If you understand this stuff, it's smooth sailing from here :)
my school really likes third derivatives of transcendental functions in their tests. I shudder to think how optimization tests we will have will look like
Related rates were so much easier 😔
‼‼
facts
Great. When I take Exam 3 tomorrow, I will do well.
AP calc exam in 3 days. Posted right on time. Thanks.
How did you do?
@Burhan Abdulali bad like i did (ب_ب)
What did you get Hannah
@@PHILLYMEDIC69 I got 4
@@hannahjones8115 you got a 4%?
Guy just did not only explain optimalisation he proved mathematically 30+30 =60
Math will be easy if you watching every videos of this channel, School taught us but the way we don't understand, That's Reality!!!
MERRY CHRISTMAS EVERYONE. Let's eat while we learn😅.
Bro my teacher is such garbage. Like rather than going through multiple examples and teaching us how to work these out, he just does like 1 example and wastes all class time doing the math. Like bruh were in calc 3 we all know how to do math just teach us the method. So frustrating. This guy is 100x better teacher
Same here. I feel you. My teacher isn't the best at teaching.
3M subscriber Congratulations
our best teacher
You don't understand how helpful these are. Life saver fr you're my goat
Thank you for effective presentation that provides a concise Introduction to the theory and brief descriptiom of the technique. Presentation is simply breathtaking because it assumes no previously knowledge of the subject and will be for benefit.
Thanks I have an exam in an hr this helped a lot
Professor Organic Chemistry Tutor, thank you for a lengthy and outstanding video/lecture on Optimization Problems in Calculus One. Although word problems are problematic, I found these problems/examples real helpful in learning this topic. This is an error free video/lecture on UA-cam TV with the Organic Chemistry Tutor.
Thanks for the videos, but a small suggestion, please put the questions a little lower so it doesn't get covered by the title of the video when paused in fullscreen.
Yay now you have 3 million subscribers.Congo mannnnn
you sir, are saving lives and grades
I love you so much i have a test tomorrow and ur the only one whos explanation i can actually understand
Man, I must say, I am really glad I came across your channel.
No one knows how much i love this guy
wollah! And he drops the bombshell just when I have an assignment on this!!!!
If your taking the ap calculus exam don’t put p’= some expression= 0
You effectively said p’=0 and you will get points taken off for that.
Best math channel on YT.. so far
Thank you so so much! I don't think I'd be doing this well without your help!
This is so much easier with calculus 2 omg
I have learned more about calc in the past hour then all year
This video is so helpful….it’s simple but informative! I was having trouble with my calc class but you’re saving my day
Thank you so much!
Your videos just might get me to pass first year calc
Very good stuff but you have very few videos in numerical analysis.
Life is so hard in this course without your assistance.
2:35 Constraint equation = fixed value
Objective function 2:38 = what we are trying to maximize (find first derivative, set it equal to zero, solve x or y
24:23
30:58
YEAAAAH! ORGANIC CHEMISTRY TUTOR IS THE BEST DRUNK STUDY BUDDY YEAAAAH
3:10 in the video, and "boom!" I instantly understood!
Thanks! I'm starting to get it! And seeing the power of the derivative.
If no one else got my back, I know FOR SURE that The Organic Chemistry Tutor got my back.
you are saving my calc class
it seems youtubers nowadays are teachers and vice versa. ..thanks so much
You and Batman from Slader are carrying my calc grade rn.
saaaaaaaaame literally same
You sir, are amazing and I am so thankful! Going to buy your merch!
I’m not really crazy about math. At least I wasn’t during pre-calc when we had to solve optimization problems in really clunky, unintuitive ways. Couldn’t do it without using Desmos to find a min or max really. I really did not like it.
Lemme tell you though, when I learned you could optimize algebraically using *this*? My mind was blown, man.
Nice 3m subs
He changed my life!
48:23
We can achieve the same results with less work like so:
#1 realize a line-k that is perpendicular to line-y and passes through (7, 6)
P(x1, y1) will be the closest point, intersection of line-k and line-y
#2 calculate the general form of the line-k
k = ax + b
a = - 1/(y-line a)
b = 6 - a * 7
I hope the numbers are not confusing
#3 find P(x1, y1)
k = y
x1 - 1 = 4 - x1
y1 = 4 - x1
at time 50:45 i dont understand how you determined there was a max point at x=2 from the seccond derivative, I thought the seccond derivative is only used for determining concavity no?
For problem 9 and 10, you can take the objective function for d^2 as it makes the calculations less complex. (Note: if d=0 then d^0 is also 0)
If d=0 then d^0 is actually undefined
This has been very helpful for me to solve optimization problems thank you.
One thing I don't understand is why is it that whether or not we are finding the max or the min we just set the derivative to zero and find the variable?
For example in the first problem we find the maximum product but what if the question had asked for the minimum product? Then using this method would not have given us the correct answer.
holy shit i have a cumulative final tomorrow and Optimization Problems are the only thing I don't know
Me too dude.
You are my favorite human being.
right on time, you absolute beast
For those one, who used to think or not about this question in 1:29 like this: "if we can find max of 2 number whose sum is 60, so can we find min of 2 number whose sum is 60 ?"
After take a few min to think, there's no way to figure out, cause the graph of this function just only has max value.
It's may useful or not but the lesson is i want to share my perspective on a problem to you in any kind of work.
I love you man
Hopefully one day i can donate
I live in a 3rd world country and we haven't been in school for 10 months and we're being rushed and I could barely remember anything
This video was too good
You're a real life hero
I have a cal 1 final tomorrow. I am ready.
another way to think about question 1 is setting the sum equal to 2 sides of a perimeter of a 4 sided shape. The maximum area will always be a square, therefore the 2 side lengths must be equal
I'm glad this happened to my life
I like the way of your teaching.
Thank you , my greatest tutor, you're much appreciated sir
please do gradient descent optimization video!!!
For the first 20 mins: I don't understand how you're finding both the minimum AND maximum in these problems by setting the first derivative = 0. How do you get different answers with the exact same method? I wish you had done: "Find two numbers whose sum is 60 and whose product is a minimum" after the first problem... changing the problem itself to "the difference" made it hard to follow from "max" to "min".... can we find both the max and min of the same exact problem and how? As I interpreted it, you used the exact same method from problems one to two but somehow solved for the maximum in the first problem and then the same method got you a minimum in the second... how?
When you find the first derivative and set it equal to zero, sometimes you can get 1 critical point, other times you may get 2 or more critical points. You can use the first derivative test to determine if the critical point is a maximum or a minimum. If f' changes from - to +, it's a minimum. If f' changes from + to -, it's a maximum. You definitely need to do this if you have 2 or more critical points. If there's only one critical point, the problem usually tells you what kind of critical point you're dealing with. For this, if the question says find the dimensions of a rectangle that maximizes the area, the critical point is most likely a maximum.
In the full version video, I have example problems where you have at least 2 critical points and you need to identify which one is the maximum and which one is the minimum in order to get the correct answer:
Optimization Problems - Part 2:
ua-cam.com/video/7WUeqLyunVA/v-deo.html
you saved me again. Thank you so much.
I got an exam in exactly one hour...
Edit: I got the same exam in 2h again. Why am I like this...
This is a banger, and I know bangers
Thanks! About to use this for my calc final soon 😎
Wow this has been much helpful to me,
thank you we have a quiz tomorrow and my teacher has a habit of going sideways with her given problems which makes me go crazy