let f r3 rarr r be dened

Question

Let f : R³ → R be de?ned by:
                                       f(x, y, z) = xy²+ x³z⁴+ y⁵z⁶

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

xy²+ x³z⁴+ y⁵z⁶= 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.