Let f (x, y, z) = sin(xy + z), and P be the point (0, −2, π3 ).(a) (6 pts) Compute ∇f (x, y, z).(b) (2 pts) At P , find the direction along which f obtains maximum directional derivative.(c) (4 pts) Calculate the directional derivative ∂f∂u (P ), where u is a unit vector making an angleπ6 with the gradient ∇f (P ).(d) (3 pts) The level surface f (x, y, z) = √32 defines z implicitly as a function of x and y near P .Compute ∂z∂x at P .

Let  ϕ(x,y,z)=3x2y−y3z2.𝜙(𝑥,𝑦,𝑧)=3𝑥2𝑦−𝑦3𝑧2. Then the gradient of ϕ𝜙 at (1,1,0) isa.6i^−3j^+0k^6𝑖^−3𝑗^+0𝑘^b.None of thesec.6i^+3j^+0k^6𝑖^+3𝑗^+0𝑘^d.6i^+6j^+0k^6𝑖^+6𝑗^+0𝑘^Clear my choice

Letf : R^2 -> R: (x,y) -> x^4 + y^2 + 2x^2y -1Answer the following:(a) Graph f(x, 0) against x in two dimensions and graph f(0, y) against y in two dimensions.(b) Draw the countour set f(x, y) = 3(c) Calculate the directional derivative of f at the point (1, 1) in the direction of thevector (−1, 1).(d) At the point (1, 1), find the direction of steepest ascent. That is, find the vector v∗ with length 1 that causes f to increase most rapidly.(e) Find the equation of the tangent plane at the point (1, 1) and comment on the relationship between the tangent plane and the slope of steepest ascent.

directional derivative

Find the gradients of the following functions:(a) f (x, y, z) = x2 + y3 + z4 .(b) f (x, y, z) = x2 y3 z4 .(c) f (x, y, z) = e x sin(y) ln(z)

Example 3.4 Find the directional derivative of g(x, y) = ex2cos y at (1, π)in the direction −3i + 4j.