## anonymous 5 years ago Find the equation tangent plane to the graph z = (x-4)^2 + (y-4)^2 at p=(5, 6, 5)...?

1. Owlfred

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2. anonymous

All rright, here we go! let f(x,y,z)=(x−4)2+(y−4)2−z we have the following partial derivatives: ∂f/∂x=2x−8 ∂f/∂y=2y−8 ∂f/∂z=−1 gradf(x,y,z)=(2x−8)i+(2y−8)j−k gradf(5,6,5)=2i+4j−k hence an equation of the tangent line at P(5,6,5) is given by 2(x−5)+4(y−6)−(z−5)=0⟹2x+4y−z=29

3. anonymous

dw= dz + dx + dy

4. anonymous

$\partial{z}/\partial{x}|_5 = 2(x - 4)|_5 = 2$ $\partial{z}/\partial{y}|_6 = 2(y - 4)|_6 = 4$ Hence the tangent plane has gradient (2, 4). Its equation is in the form $z = 2x + 4y + p$ And it passes through (5, 6, 5), so $5 = 2*5 + 4*6 + p$ $p = -29$ Result: $z = 2x + 4y - 28$ How about that...