Calculus 3: How to linearize a multivariable function

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Learn how to linearize the multivariable function f(x,y)=1+xln(xy-5) at (2,3). This is a question from the 9th edition Multi-variable calculus textbook by James Stewart. Subscribe for more calculus tutorials @bprpcalculusbasics

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An alternative: { pf/px is a stand-in for the partial derivative}
1) When x = 2 and y = 3 then z = 1 so P(2, 3, 1) lies on the surface.
2) Express the surface as: f(x, y, z) = 0 1 + xln(yx-5) - z = 0
3) The gradient vector: N--> = < pf/px, pf/py, pf/pz > evaluated at (2, 3, 1) will be normal to the curve at (2, 3,1) and hence be a normal direction for the tangent plane.
4) I'll leave out the details of partial derivatives as you have so kindly done most of them. Evaluated at (2, 3, 1) we get N--> = <6, 4, -1>
5) The equation of the tangent plane is now : Ax + By + Cz = Ax0 + By0 + Cz0 { or in your form A(x - x0) + B(y - y0)+ +C(z - z0) = 0 }
6x + 4y - 1z = 6(2) + 4(3) -1(1) ====> 6x + 4y - z = 23 ====> z = 6x - 4y - 23

As an added note: The expression: A(x - x0) + B(y - y0)+ +C(z - z0) = 0
is just the dot product of the 2 vectors: N--> = < A, B, C > and PP_0 --> = < x - x0, y - y0, z - z0 > being FORCED to 0 in order to force the 2 vectors to be perpendicular as the
vector < x - x0, y - y0, z - z0 > lies entirely on the plane. This was my method when I taught this in high school.

I'm curious, is the gradient method ever taught? Seems pretty clean to me. At any rate, Cheers! and Well Done - Ian

ianfowler

You beat calculus with these short videos and so we do! Thank you

matematikgokseldir

This made me wonder how you would use Euler's method in a multivariable case

Asiago

Thank you. I learned something new in math today.

JP-lzvk

would be nice a demonstration of the formula

BRaleatoriedades

When you say 'Calc 1', is this in the US education system?

iankr

The curve doesn't pass through (2, 3). Therefore the question has no answer.

kevinmadden

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