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Q1. Find the tangent plane to the elliptical paraboloid z=2x^2+y^2 at the point P(1,1,3).
What you'll need:
If f has continuous partial derivatives, an equation of the tangent plane to the surface of z=f(x,y) at the point P is:
0=f_x (x_p,y_p,z_p )(x−x_p )+f_y (x_p,y_p,z_p )(y−y_p )+f_z (x_p,y_p,z_p )(z−z_p )
Visual:
Q2. Find the tangent plane to the surface x^2+2xy−y^2+z^2=7 at the point P(1,−1,3). Find the normal line through P.
Note: The normal line to the surface at P is the line through P parallel to ∇f(x_p,y_p,z_p).
⟨x,y,z⟩=(x_p,y_p,z_p )+t⋅∇f(x_p,y_p,z_p )
Q3. Find the tangent plane to the surface x^2+2y^2+4z^2=10 at the point P(2,−1,1).