A surface z = f(x, y) is defined implicitly by the equation x +y+ z – e² = 3 – e. - Determine the equation of the tangent plane to the surface at the point (1, 1, 1). Write down a normal vector to the surface at (1,1, 1).

Find equations of the tangent plane and normal line to the surface x=6y^2+5z^2−459 at the point (2, 6, 7).Equation of the tangent plane (make the coefficient of x equal to 1): =0.Vector equation of the normal line: l(t)=⟨2,,  ⟩+t⟨1, ,  ⟩.HINT. Represent the given surface as a level surface of a certain function f(x,y,z), that is, re-write the equation of the surface in the form �(�,�,�)=c, where �(�,�,�) is a function of 3 variables and f is a constant. Then find the gradient vector of the function f at the given point. The tangent plane to the surface at that point is orthogonal to the gradient vector, and the normal line -- parallel to it. This allows us to write the equations of both tangent plane and normal line.

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Find the equation of the tangent plane to the surface z = ly - lz' at the point (-2,0,-4). 4x+0 Note: Your answer should be an expression of x and y, e.g. "3x - 4y +6"