Find the tangent plane to the surface with parametric equations x u 2 , y v 2 , z u 3 v at the point ( 1 , 1 , 4 ) Solution We first compute the tangent vectors ru x u i y u j z u k rv x v i y v j z v k Thus, a normal vector to the tangent plane is as follows ru rv i j k 2 u 0 1 0 2 v 3 Notice that the point ( 1 , 1 , 4 ) corresponds to the parametric values u 1 and v 1 , so the normal vector there is Therefore an equation of the tangent plane at ( 1 , 1 , 4 ) is ( x 1 ) ( y 1 ) ( z 4 ) or

Question

Find the tangent plane to the surface with parametric equations x   u 2 , y   v 2 , z   u   3 v at the point ( 1 , 1 , 4 )   Solution We first compute the tangent vectors  ru   x u i   y u j   z u k   rv   x v i   y v j   z v k   Thus, a normal vector to the tangent plane is as follows  ru rv   i j k 2 u 0 1 0 2 v 3   Notice that the point ( 1 , 1 , 4 ) corresponds to the parametric values u   1 and v   1 , so the normal vector there is   Therefore an equation of the tangent plane at ( 1 , 1 , 4 ) is ( x 1 )   ( y 1 )   ( z 4 ) or

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Find the tangent plane to the surface with parametric equations x = u 2 , y = v 2 , z = u + 3

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