Show All Notes Hide All Notes

Section 14.2 : Gradient Vector, Tangent Planes and Normal Lines

1. Find the tangent plane and normal line to \(z\,y + 4x\sqrt z - {x^3}{y^2} = 221\) at \(\left( { - 2,5,9} \right)\).
2. Find the tangent plane and normal line to \(\displaystyle {{\bf{e}}^{x\,{y^{\,2}}}} + z{y^4} = 61 + \frac{{{z^2}}}{{x + 1}}\) at \(\left( {0, - 2,6} \right)\).
3. Find the tangent plane and normal line to \(9yz - \sqrt {{x^2} - 8z} = x{y^2} - 26\) at \(\left( {3,1, - 2} \right)\).
4. Find the point(s) on \(6{x^2} + {y^2} - 3{z^2} = 4\) where the tangent plane to the surface is parallel to the plane given by \(2x + 7y - z = 6\).
5. Find the point(s) on \({x^2} - 8{y^2} - 2{z^2} = - 3\) where the tangent plane to the surface is parallel to the plane given by \( - 4x - y + 8z = 1\).