Question 19 Find a normal vector of the tangent plane to 2? + y2 + x2...
Partial derivatives Example Find the equation of the tangent plane and the normal line at the point (1,1,1) to the surface 2x2 + 2y2 + 3z2 = 6. Example Find the equation of the tangent plane and the normal line at the point (1, 2, 3) to the surface 2x2 + y2 – z2 = -3.
Find an equation of the tangent plane to the surface 2 x4 = y2 +22 at the point (1,1,1).
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the surface at the point P: (-2,1,3). Determine the equation of the line formed by the intersection of this plane with the plane z = 0. 10 marks (b) Find the directional derivative of the function F(r, y, z)at the point P: (1,-1,-2) in the direction of the vector Give a brief interpretation of what your result means. 2y -3 [9 marks]...
Question 10 Find a tangent vector of the curve 7 (t) = (+2, 2 sin(t), 2 cos(t)) at (0,0,2). (1,1,1) (0,0,1) (1,0,0) 0 (0,1,0) None of the above or below O (1/2,0, 1/2)
3. (a) Consider the paraboloid z = x2 + y2 Find a unit vector normal to the surface of this paraboloid at the point P = (x, y, z) = (1, 2,5). (b) Consider a vector field ä = (xy2 + z)i + (xy + 2)9 + xk where, as usual, i = Î. Ì = û and k = 2 are the unit vectors. Show that a = Vº for some scalar field o.
([8]) Find the point on the surface z = x2 + 2y2 where the tangent plane is orthogonal to the line connecting the points (3,0,1) and (1,4,0). Useful formula: The curvature of the plane curve y = f(x) is given by k(x) = \f"|(1 + f/2)-3/2, ([9]) Use spherical coordinates to find the volume of the solid situated below x2 + y2 + 2 = 1 and above z= V x2 + y2 and lying in the first octant.
Find an equation of the tangent plane to x2+y2+z2=34 at the point (3,4,3).
Find a normal vector and an equation for the tangent plane to the surface: x3 - y2 - z2 - 2xyz + 6 =0 at the point P : (−2, 1, 3). Determine the equation of the line formed by the intersection of this plane with the plane x = 0. [10 marks] (b) Find the directional derivative of the function F(x, y, z) = 2x /zy2 , at the point P : (1, −1, −2) in the direction of...
9a. Find a normal vector to the tangent plane to the surface x = y2zs at (1,-1,-1). 35 b. Find the equation of the tangent plane to the surface x=y'7 at the point (1,-1,-1).
Find the equation for tangent plane and the normal line to the surface with equation x2 +972 +922 = 22 at the point P(2, 1, 1).