Tangent Plane to x^2 - xy - y^2 -z = 0

Multivariable Calculus: Find the equation of the tangent plane to the surface x^2 - xy - y^2 -z = 0 at the point (2,1,1).

For more videos like this one, please visit the Multivariable Calculus playlist at my channel.

[MathDoctorBob]

Thanks so much for the support and kind words! I plan on being around for a while. Part of the next phase will be showing other teachers how to do what I do here. Good luck with the degree.

[tiffanie]

I really want to just keep your videos rolling in the background so you can get more advertisement money to persuade you to continue to make free videos. You're doing a great job. It's nice to find someone who enjoys teaching so much to create such useful videos. Thanks! From a secondary teacher earning a second teaching degree in mathematics.

[MathDoctorBob]

Two options: 1a) as (x, y, f(x, y)), since f(2, 2) = 6, we can take the positive square root to get z = f(x, y) = 2sqrt(1+...). Same procedure, but we need the chain rule for dz/dx and dz/dy.

1b) Implicit differentiation to find partials dz/dx and dz/dy.

Normal is the cross product of (1, 0, dz/dx) and (0, 1, dz/dy) = (-dz/dx, -dz/dy, 1),
evaluated at (2, 2, 6).

2) Gradient: F(x, y, z) = z^2 - 4(1+x^2+y^2). normal = grad F = (-8x, -8y, 2z).

-Bob

[MathDoctorBob]

@hockeypros Are you looking for single variable or multivariable series? Part 6 of the Calculus playlist has the single variable theory. I have one up for multivariable series - 'Taylor Polynomial of f(x, y) = ycos(x+y).' I'm happy to add more.

[MathDoctorBob]

@TitaniumBallz You're welcome! It's always good to have a second method to check your work. - Bob

[devo]

Great Explanation! Simple, understandable and to the point, Keep up the videos.

[Andrewzoh]

Calculate the rate of change of those muscles and it gives you infinity.

But in all seriousness, thank you for your videos. Your generous efforts are much appreciated.

[johnshepard]

You look like dennis cyplenkov 😂. Nice explanation btw, my man!

[MathDoctorBob]

@quitejaded I'm happy to answer any questions you might have. - Bob

[MathDoctorBob]

Just "GRRRR!" works if you do it right.

[sufianshaikh]

ca you start posting workout videos too?

[MathDoctorBob]

More like the basements of Jersey City. Baseball bats are for hitting balls. That's an eskrima, which is for hitting heads.

[theastarmaruti]

how will you solve this?
equation of the tangent plane and the normal to the surface
z^2 = 4 ( 1 + x^2 + y^2 ) at ( 2, 2, 6 )

[MathDoctorBob]

Note that I listed two methods: using the gradient and using implicit differentiation.

[MathDoctorBob]

@hockeypros You're welcome, and thanks for the comment. Please let me know if you have any requests; I'm gearing up for another run of MV Calculus videos.

[theastarmaruti]

I've another one
solve the system of differential equations
dy/dx = xz +1, dz/dx = xy
for x=0.3 using 4th order Runge - Kutta method with y(0) = 0, z(0) = 1
I don't know about R-K method so please help... if you are familiar with this method.
thank you

[CoolDude]

The baton makes you look well bad ass.

[kukuibuss]

Ty so much! I was able to finish my homework on time thanks to you! Ugh, the book was of no help at all...

[MathDoctorBob]

I'm assuming we're both guys, so no danger of multiplying.

[theastarmaruti]

and also give another method to solve it.