I currently have a 57% in my calculus class because my professor is garbage and an unfair grader, and I need a 78% on my final to pass. I honestly feel confident in getting this score thanks to your videos. Much love :)
I can not thank you enough for these videos (and the rest of your calc 1 series for that matter). Between my best in person math teacher I've ever had and you on youtube, nothing in calculus I've experienced fazes me (yet).
I feel like my total cranial capacity is doubling for every ten minutes of watching PatrickJMT work these problems. Optimization and related rates problems seem to be about the toughest quagmires that derivative calculus has to offer.
In addition to solving the equation in terms of theta, you can also use the pythagorean thereom to find h or w in terms of the other variable, take the derivative with respect to whichever variable you chose to solve for, and then use that to find the optimal height or width. From there you can apply pythagorean thereom again to find your other variable, and then you could calculate theta by taking the arctangent of your (w/h) values.
I hope your making more money then a highschool teacher. Your videos are so easy to follow, so simple, and so helpful. I've learned more in 2 hours of watching your videos then i have from 10 classes of Calculus
Hey, Patrick I love all of your videos! Im just in pre-cal and I love optimization problems and this one blows my mind! Great job! Thanks for all of your wonderful lessons!
Patrick, I owe my A in Calculus to you. Thank you for being my primary source of calculus expertise this semester. For whatever its worth, I sent you a 15 buck donation. You're the man! Keep up the excellent work.
This guy does a really good job! I just wish he would explain setting the problem up in more detail and finish the problem completely because the goal of the problem doesn't seem as realistic as it does in my math class. For me, it doesn't just end with the critical value, but more so a perimeter, area, volume, or a more extended answer. Regardless, these videos are really helpful. Keep it up!
hi...great videos! hmmm...i think you can avoid the derivation of trigonometric stuff by using the Pythagoras Theorem...just like: total area=10w + wh then build a triangle and by the definition of pythagoras theorem: 10^2= (w^2+h^2) and get the equation of "h" now you can "plug in" it: total area=10w + w([10^2+w^2]^1/2) and derivate it... once you get the solution of the equation you solve cos^-1(the solution)
I ended up using the trig identity cos^2(theta) - sin^2(theta) = cos(2*theta). You end up with cos(2*theta) = -cos(theta), which is 60 deg. The quadratic you ended up with is easier IMO, but this other approach worked out too.
The piece of metal was given at the beginning of the problem to be a total of 30cm. And also given was the fact that the metal was to be bent into 3 equal parts. 3 x 10cm
This is definitely challenging, but man are you a good explainer! Thank you so much! Attempting to become a Materials engineer and you my man are helping me through Calc 1:)
He could not have explained this any better! This was on point! I'm seriously considering giving this guy $1 per month. Patrick, is there any way you could add a graph? Function A for 0 less than or equal to theta less than or equal to pi/2?
You have to think about the context. eg if you have profit, you wouldnt wish for your business to run at a loss. you'd want to find out where you'll get maximum profit. Same for cost of materials. if you want to build something, you'll look for the dimensions that use the least materials, right ? (because more materials = more money, which i'm sure given the current economic situation, you do not want. ) so in general, i'd say you should think "would this benefit me or would this be bad"
also cos x = 1/2 at 5Pi/3 . Just wanted to let people know. So unless you are given an interval which does not include 5Pi/3 then you would normally list both these answers
Whoops, one mistake. You would actually have to do 90 - arctan(w/h), since arctan(w/h) will give you the measure of the angle from the normal, in this case 30 degrees.
When you're dealing with cosine or sign you should use exact values. Just think the numbers prettier he don't have to deal with any signs to the negative or driving sin. That's just what I did.
Because the 100 is a constant, not a variable. You don't use product rule when taking the derivative of 2x^2 because 2 isn't a variable. Same logic applies.
Hey Patrick! In this problem we want to maximize the quantity of water i.e. the volume of the water hence how can maximising the area maximise the qty of water..since volume would be area into height of the gutter..
this video got stuck and I have been trying to replay it over and over again but it still gets stuck when its at 7:57..i really wanna watch to till the end patrickJMT
btw how did you come up with 3 10cm? is it in the given? have you considered other way of dividing the 30cm? just curious though. as always thanks for the free vids! you surely help a lot specially me!
This stuff is so hard :( It wouldn't be so bad if there were a standard or general way of doing each and every single one. :( :( :( I have a test on this tomorrow!!! And related rates!!!!! :( :( :(
wait, if you already set the rectangle to a certain height, wont it be affected when you changes the angle? i mean, the flatter the angle, the less height.
Dude is so smart. Wish i could borrow his brain for tests.
I currently have a 57% in my calculus class because my professor is garbage and an unfair grader, and I need a 78% on my final to pass. I honestly feel confident in getting this score thanks to your videos. Much love :)
I can not thank you enough for these videos (and the rest of your calc 1 series for that matter). Between my best in person math teacher I've ever had and you on youtube, nothing in calculus I've experienced fazes me (yet).
I feel like my total cranial capacity is doubling for every ten minutes of watching PatrickJMT work these problems. Optimization and related rates problems seem to be about the toughest quagmires that derivative calculus has to offer.
ah hell nah bro fr said quagmire who else but quantum physics
In addition to solving the equation in terms of theta, you can also use the pythagorean thereom to find h or w in terms of the other variable, take the derivative with respect to whichever variable you chose to solve for, and then use that to find the optimal height or width. From there you can apply pythagorean thereom again to find your other variable, and then you could calculate theta by taking the arctangent of your (w/h) values.
I hope your making more money then a highschool teacher. Your videos are so easy to follow, so simple, and so helpful. I've learned more in 2 hours of watching your videos then i have from 10 classes of Calculus
Hey, Patrick I love all of your videos! Im just in pre-cal and I love optimization problems and this one blows my mind! Great job! Thanks for all of your wonderful lessons!
glad i could help :) thanks a bunch for the donation!
Patrick, I owe my A in Calculus to you. Thank you for being my primary source of calculus expertise this semester. For whatever its worth, I sent you a 15 buck donation. You're the man! Keep up the excellent work.
You really helped me through Calculus last year. Thanks!
This guy does a really good job! I just wish he would explain setting the problem up in more detail and finish the problem completely because the goal of the problem doesn't seem as realistic as it does in my math class. For me, it doesn't just end with the critical value, but more so a perimeter, area, volume, or a more extended answer. Regardless, these videos are really helpful. Keep it up!
thanx patrick i had the last 2 problems 4 my tuts u really helped me
I just got 100% on my mid-term thanks to you sir. Thanks from Canada!
i like how 2 of the three poblems you are doing are the ones done in my calculus lecture!
You just saved my Calculus' final.I just saw all the optimization plus all the related rates videos. Totally worth it.
hi...great videos!
hmmm...i think you can avoid the derivation of trigonometric stuff by using the Pythagoras Theorem...just like:
total area=10w + wh
then build a triangle and by the definition of pythagoras theorem:
10^2= (w^2+h^2) and get the equation of "h"
now you can "plug in" it:
total area=10w + w([10^2+w^2]^1/2) and derivate it... once you get the solution of the equation you solve cos^-1(the solution)
I ended up using the trig identity cos^2(theta) - sin^2(theta) = cos(2*theta). You end up with cos(2*theta) = -cos(theta), which is 60 deg. The quadratic you ended up with is easier IMO, but this other approach worked out too.
it is one of the first trig identities that one learns; it is one of the pythagorean identities
Whoa. Impressive question! Never knew optimizations could be applied to angles
yea, i noticed that too! : )
subconscious 'up yours'? : )
or just an accident....
i wonder
thank you so much for these videos, you have really helped me with my maths! way better than my maths teacher!!!!
that's the more complex kind of optimization, there are also ones with cones and cylinders which are also very tricky
The piece of metal was given at the beginning of the problem to be a total of 30cm. And also given was the fact that the metal was to be bent into 3 equal parts. 3 x 10cm
You are incredible! We in Puerto Rico love u man!
Am I the only one who thinks the drawing looks like a pair of angry eyes ?
Now that you mention it .___.
I cannot unsee this now.
I THOUGHT IT WAS RINNEGAD TURNS SHARINGAN HAHAHAHAHA
0:44 tell me i'm not the only one that sees a face. :]
That's an interesting problem. Thanks for posting!. Pretty cool a bit of trig, calculus, trig identities. Great explanation :).
man, I need a repeat of the cosine theta and sina theta's derivaives joh
1:04 oops!
@pimp2611 those were the old youtube rules. they have since changed.
thanks ill look into that. i must have missed that class
This is definitely challenging, but man are you a good explainer! Thank you so much! Attempting to become a Materials engineer and you my man are helping me through Calc 1:)
Good luck man i am attempting electrical engeneering)
@patrickJMT Thank you again man, God bless.
He could not have explained this any better! This was on point! I'm seriously considering giving this guy $1 per month. Patrick, is there any way you could add a graph? Function A for 0 less than or equal to theta less than or equal to pi/2?
You are truly a Godsend. God Bless You. You should be teaching at an university.
You have to think about the context. eg if you have profit, you wouldnt wish for your business to run at a loss. you'd want to find out where you'll get maximum profit. Same for cost of materials. if you want to build something, you'll look for the dimensions that use the least materials, right ? (because more materials = more money, which i'm sure given the current economic situation, you do not want. )
so in general, i'd say you should think "would this benefit me or would this be bad"
0:05 the devil is in the math. Amen.
@urta93 no problem
This is EXACLTY what I need, omg thank you
glad it helps everyone!!
also cos x = 1/2 at 5Pi/3 . Just wanted to let people know. So unless you are given an interval which does not include 5Pi/3 then you would normally list both these answers
Whoops, one mistake. You would actually have to do 90 - arctan(w/h), since arctan(w/h) will give you the measure of the angle from the normal, in this case 30 degrees.
When you're dealing with cosine or sign you should use exact values. Just think the numbers prettier he don't have to deal with any signs to the negative or driving sin. That's just what I did.
Patrick JMT 2012 for president
that has to be one of the best comments ever :)
i think that is a good way to use the videos ;)
10 minutes limit !!
I would give you all my time Pat , just keep going :")
The cross-section is a trapezium. It's easier to use that and the formula for the area of a trapezium.
this will help with my exams the day after tomorrow
your videos are really usefull references, thanks a lot for your dedication
@1212naked i suggest a little shrine to start.... :)
thanks gelly : )
That rain gutter is staring at me with devil eyes.
good math videos. By the way you sound almost exactly like Mr Vandreesen from Beavis and Butthead haha
@TheDuskMar I was thinking the same thing, saves a lot of time.
-flips out the middle finger as a pointer- & -slips out a 'whoops' for reassurance of the otherwise case-
finding the area of the trapezium would've also worked
For some reason, I read the title of this video as "Optimization Problem #3 - Making it Rain"
hahah the problems looks like an angry face screaming making a rain gutter !
Awesome explanation!
Thank you!
Because the 100 is a constant, not a variable. You don't use product rule when taking the derivative of 2x^2 because 2 isn't a variable. Same logic applies.
is it just me or does the picture look like a mischievous cat? XD
anyways, thanks a lot patrickJMT! ur videos helped a lot!
Hey Patrick! In this problem we want to maximize the quantity of water i.e. the volume of the water hence how can maximising the area maximise the qty of water..since volume would be area into height of the gutter..
Thank you sir for your video!
I wish you were my calculus teacher !!
@ArrudA666 that is a restatement of the pythagorean theorem.
this video got stuck and I have been trying to replay it over and over again but it still gets stuck when its at 7:57..i really wanna watch to till the end patrickJMT
If you were to use product rule, the derivative of 100 = 0, so that would still just be 100*(the derivative of sin or cos).
very helpful videos. thank you
in 9:10 we can subtute cos(theta) by X the equation will became:
200*X^2+100X-100
Thanks for the video.
btw how did you come up with 3 10cm? is it in the given? have you considered other way of dividing the 30cm? just curious though.
as always thanks for the free vids! you surely help a lot specially me!
I wish I have good memory, I keep forgetting Identities and all the important rules.
Lmao when I heard the 10 time limit. Still saved me a decade later.
Great video :D
I loved all of it!!
thanks a lot buddy, keep it real yo
This stuff is so hard :(
It wouldn't be so bad if there were a standard or general way of doing each and every single one. :( :( :(
I have a test on this tomorrow!!! And related rates!!!!! :( :( :(
related rates for me is easier now this optmization is so hard for me
How do you know when something is asking to be maximized or minimized?
Please make video of minimization for the same problem
Couldn't you use Pythag to find the height in terms of width?
our AB Calc class went through Related Rates before going through this on a crappy textbook... nonetheless I didn't do well.
how do you do optimization problems without derivatives?
what is the question? thank you!
Why do you not have to take the derivative of 100 in front of the cos or sin. Is it not a product rule?
how do you know if the hypothenus of the triangle is 10?
wait how can you ensure that its a maximum?
i mean couldnt it be a minimum?
dont you have to do a first derivative test?
how can we check if it is the maximum or the minimum?
please tell me how to maximize the area of such a shrine
Thanks a lot! You are very helpfull!
all four teta angles are equal? how?
lol You gave us the finger. Thanks for the tutorial though
Thanks
How do I solve it if there isn't a 10 value, just a variable a for each line?
That is one angry Pokemon
our class use this kind of problem but instead of that, we’ve used the area of trapezoid.
wait, if you already set the rectangle to a certain height, wont it be affected when you changes the angle? i mean, the flatter the angle, the less height.
the only part that confused me, which i guess i never learned, was how did you know that cos^2theta + sin^2theta =1 ? is that just common knowledge?
Thank you !! I had to o the exact same problem in my Calculus test today !! It looked like i was cheating though lol
Hehe, I think it would be easier to utilize the identity sin(2x) = 2sin(x)cos(x)
These videos are why I have a 97% in Calculus. :)