Oxford Calculus: Finding Critical Points for Functions of Two Variables

University of Oxford Mathematician Dr Tom Crawford explains how to calculate the critical points for a function of two variables.
Just as the critical points for a function of one variable are found by differentiation, the same techniques can be applied to a multivariable function to determine where it is stationary.
We begin with a reminder of critical points for a function of one variable, before looking at partial differentiation of a multivariable function with a worked example.
Part 2 on classifying the critical points: • Oxford Calculus: Class...
The 3D plots used in the video are all generated by the Maple Calculator App which you can download for free from Google Play and the App Store.
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Find out more on the Maplesoft KZbin channel: / @maplesoft
Produced by Dr Tom Crawford at the University of Oxford.
Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
For more maths content check out Tom's website tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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[@Aleph0]

I love your slow, patient method of explaining! It feels like the video was really made with learners in mind. I always used to get confused about whether a critical point meant that one of the partial derivatives was zero, or both of them were zero. This video provided great intuition why the latter is true. Thanks! :)

[@TomRocksMaths]

This is really great to hear - thank you for sharing :)

[@StradexEngine]

From my perspective of things, you are the coolest person alive.

[@TomRocksMaths]

You're too kind

[@rohithnarra9026]

Man I love that navier stokes tattoo Dr Tom

[@sam-ui5lc]

I have my maths exam tomorrow and this was really helpful. Thanks a ton

[@TomRocksMaths]

Good luck!

[@TomRocksMaths]

Part 2 on classifying the critical points here: kzbin.info/www/bejne/a37CpaZuhpd_e68

[@dragoncurveenthusiast]

Thanks for this, Tom! I'm relatively familiar with differentiation and liked the exercises in high school, but it's good to get a brief refresher on partial differentiation! You are such a patient explainer, it's so enjoyable to follow you through the steps of the example.

[@TomRocksMaths]

Amazing thanks! Part 2 is now live here: kzbin.info/www/bejne/a37CpaZuhpd_e68

[@dragoncurveenthusiast]

@@TomRocksMaths Thanks :-)

[@deeppathak6204]

Amazing explanation 🙌🙌👍

[@TomRocksMaths]

Thanks Deep!

[@wingslax]

I wish you had taught me multivariable calculus when I was in college. This is great stuff!

[@TomRocksMaths]

Glad it helped Ben :)

[@sanujatharinda6525]

Amazing teaching...

[@TomRocksMaths]

Thanks Sanuja!

[@I_like_science]

I really enjoy your videos even though I never really understand what you are describing or doing on the board. They encourage me to go and learn. Your channel is small and it's definitely not because your content isn't good. I respectfully suggest you to think about approaching more people like me who are truly fascinated by math but never got the chance to pursue higher maths after highschool. I'm sure you are a very busy but consider making a series of courses that starts with high school maths and then move on to higher maths. This approach was taken by Daniel Schiffman from the KZbin channel ,The coding train. This resulted in his channel growing at a very fast pace and getting people REALLY interested in coding. Thank you for taking time out of your day to make these videos.

[@TomRocksMaths]

Thank you

[@birdost8448]

senin canını yerim, çok güzel anlattın çok teşekkür ederim.

[@rogerjohan9621]

WOW Dan TDM is teaching me maths!! (awesome video btw)

[@TomRocksMaths]

My friends agree with you...

[@michaelhall5801]

So true! The accent, tattoos and hair are all so similar XD

[@majidshajarat8120]

Very good ❤

[@jorgelechon8044]

I'm not speak English, but I could understand a little. I really enjoyed the video thanks so much.

[@ikhandlelatertiaryaccessso1214]

Thank you

[@TomRocksMaths]

You're welcome :)

[@HasekuraIsuna]

Would be great if you had shown the (-4/3,4/3) on the graph after calculating it.

[@Esternosis57]

Hello, my name is Omar from Venezuela, excellent class, when do you publish the second part? If you don't mind, could you give us an example of these solutions applied in real life

[@TomRocksMaths]

Hi Omar - part 2 will be out next Wednesday (October 14th).

[@lsubandtrumpet2014]

Where do u get -4y from?

[@desklamp701]

I'm watching your videos to make A-level maths look alot easier in comparison which kinda helps me :)) Any tips to achieving an A*?

[@TomRocksMaths]

Beyond working hard and doing lots of practice questions, exam technique is also really important for the top marks. Make sure you re-read every question so that you are answering what it is asking, not what you want it to be asking... Also, leave some time at the end to double check your answers all make sense!

[@desklamp701]

@@TomRocksMaths Thank you Tom!!

[@JetteroHeller83]

Anyone know how to change the window limits for the 3D plot in Maple Calculator?

[@TomRocksMaths]

I think it's been updated so you can just drag them now

[@deang5622]

This is school level maths that we studied at age 16-17. It is not even university level.

[@itsreeah2663]

I’m really blind and just 14 and I’m watching this

[@Synthenist]

I appreciate your ambitions, but I still do not know what your hand gestures wanna try to tell me 🙈😅 But jokes besides - I really love how you explain things, very enjoyable :)

[@TomRocksMaths]

Haha - awesome thanks

[@harshsharma5768]

We can have a inflection point where slope is not zero right?

[@TomRocksMaths]

For a true inflection point the gradient has to pass through zero. You can have curves that look like they have inflection points, but if the gradient never reaches zero then they are not true stationary points.

[@Yamil.-123]

Se ve que explicas muy bien el cálculo gracias por tomarte el tiempo de ensenar. Te agradecería mucho si pudieras poner subtítulos.

[@TomRocksMaths]

Thank you - I'm afraid I don't speak Spanish, but you are most welcome to add them yourself through community contributions!

[@Yamil.-123]

Bueno de todos modos muchas gracias

[@eulersfollower7140]

Not changing? ,do u mean the derivative is zero? Thank you for this amazing explanation🎉🎉

[@TomRocksMaths]

Yes that's correct. If a function isn't changing at a certain point, then it must have zero derivative there.

[@MarkusDarkess]

You said you like tint numbers. Plankth length. Is tied to special right triangles. Initial square has 4 special. Right triangles. So when it is cubed it has a square with bottom surface 4 special right triangles top of cube 4 special right triangles. And the 4 surfaces around the cube each have 4 special right triangles. So special right triangles of that square cubed Is 24 special right triangles. So a 1ft by 1ft by 1ft cube has a surface area. Of 24, special right triangles. The square half its size. Is 2 special right triangles. . Cubed. Has 2 special right triangles on top and bottom and the for surrounding walls. Has 12, special right triangles. So 2-D. Goes from 2-4-8-16-32-64-128,etc.. 3-D from 12-24-48-96-192,ect.. Minecraft. Cannot construct a 16,by 6 cube with out adding extra surface area. And that circles at these spots. Should be doubling in area as well. And the area of a circle is just pie. Immeasurable.. And then if you make a square. The half sized square is immeasurable. And all the perfect squares it creates. Are also immeasurable. Pythagream thearum says. As long as I have a right triangles. With 2equal sides. I can double area. 1to two. Then the next doubled square has 8x the area. Or eight special right triangles. Because 0,0/0,1/-1,0 3^2+4^2=5^2. The 2-d points Can be shifted around To make a square. One way. Then a cube. And another way makes a 2-d rectangle. But the 3-d rectangles can be made 2-ways of varying heights.

[@CommanderdMtllca]

If df/dx of y^(n) is 0 then isn't df/dy of x^(n) also 0 instead of 2x?

[@TomRocksMaths]

Yes - the 2x term in the df/dy calculation comes from the y-derivative of the 2xy term.

[@noahprentice751]

in that part of the video, he's doing df/dy of 2xy. Since df/dy of cy (where c is a constant) = c, and since 2x is a constant, df/dy of 2xy is 2x.

[@CommanderdMtllca]

@@TomRocksMaths Woops! Missed that part lol

[@SaschaRobitzki]

My z=x^2+2xy-y^2+y^3 looks very different on Maple. 😆

[@CGKittenz]

Waiiit how did we get to -4y+3y^2=0???????

[@TomRocksMaths]

The df/dx = 0 equation gives that x=-y and so we substitute this into the df/dy = 0 equation so 2x - 2y + 3y^2 =0 becomes 2(-y) - 2y + 3y^2 = 0 which gives the equation you asked about.

[@nicholasspeechley9785]

I am also single.

[@charlesabernathy5842]

Oxford?! I am in the boring New York City. Just kidding.