I have notes and can kinda figure out some of these problems but can't figure them out completely. Please show you work, and if you can please explain your steps.
1.) find the directional derivative of f(x,y) = sin(x+2y) at (4,2) in the direction of g = (3π/4)
2.) find the directional derivative of f(x,y,z) = z^(3) - x^(2) y at (1,6,2) in the direction of v = 3i + 4j + 12k
3.) find the maximum rate of change of f(x,y,z) = x^(2) y^(3) z^(4) at (1,1,1) and the direction in which it occurs.
4.) Find the relative maximum and/or minimum and/or saddle points of f(x,y) = 2xy - (1/2)(x^(4) + y^(4)) + 1
5.) find the symmetric equations of tangent line to the curve of intersection of the surfaces at the indicated point, and find the cosine of the angle between the gradient vectors at this point:
z = sqrt(x^(2) + y^(2)) , 5x - 2y + 3z = 22, (3,4,5)
6.) find the relative maximum and/or minimum and/or saddle points of f(x,y) = 120x + 120y - xy - x^(2) - y^(2)
7.) find the dimensions of the rectangular box with largest volume if the total surface area is given as 62 m^(2)