Directional Derivatives And Gradient Vector

(requires JavaScript)

  1. Find the directional derivative of fxy=5x4y at the point 41 in the direction corresponding to the angle θ=π6 .
    5163+14
    1. Find the gradient of fxy=5xy24x3y ,
    2. Evaluate the gradient at the point P12 ,
    3. Find the rate of change of f at P in the direction of the vector 5131213 .
    1. fxy=5y212x2y10xy4x3
    2. 416
    3. 17213
    1. Find the gradient of fxyz=x+yz ,
    2. Evaluate the gradient at the point P131 ,
    3. Find the rate of change of f at P in the direction of the vector 273767 .
    1. 1zy2x+yz
    2. 1134
    3. 2328
  2. Find the directional derivative of the function fxyz=xy+z at the point 411 in the direction of the vector 123 .
    9214 .
  3. Find the maximum rate of change of fxy=y2x at the point 24 and the direction in which it occurs.
    42 and 11 .
  4. Find the maximum rate of change of fxyz=tanx+2y+3z at the point 511 and the direction in which it occurs.
    14 , 123 .
  5. Find the directions in which the directional derivative of fxy=x2+sinxy at the point 10 has the value 1 .
    01 and 4535 .
  6. Find the equations of the tangent plane and the normal line to the surface x22y2+z2+yz=2 at the point 211 .
    4x5yz=4 and x24=y15=z+11
  7. Find the equations of the tangent plane and the normal line to the surface xz=4tan1yz at the point 1+π11 .
  8. Let fxy=x2+4y2 . Find the gradient vector f21 and use it to find the tangent line to the level curve fxy=8 at the point 21 . Sketch the level curve, the tangent line, and the gradient vector.
    48 and x+2y=4 .
  9. Find the points on the ellipsoid x2+2y2+3z2=1 where the tangent plane is parallel to the plane 3xy+3z=1 .
    ±32521025 .
  10. Find parametric equations for the tangent line to the curve of intersection of the paraboloid z=x2+y2 and the ellipsoid 4x2+y2+z2=9 at the point 112 .
    x=110t,y=116t,z=212t