Let f : R^{3} → R be de?ned by:
f(x, y, z) = xy^{2}+ x^{3}z^{4}+ y^{5}z^{6}
a) Compute ~ ∇f(x, y, z) , and evaluate ~ ∇f(2, 1, 1) .
b) Brie?y explain why f must be a di?erentiable function (you just need to "look" at the equations for the partial derivatives).
c) Find D~uf(2, 1, 1) where ~u is a unit vector in the direction of ~v = h4, 3,-1i .
d) Find an equation for the plane which is tangent to the surface de?ned by
xy^{2}+ x^{3}z^{4}+ y^{5}z^{6}= 11
at the point (2, 1, 1) .
e) Use di?erentials to approximate the value of f(2.01, 1.02, 0.97) . Use a calculator to
?nd a more accurate value, and compare.