Question

Consider the mountain known as Mount Wolf, whose surface can be described by the parametrization r(u, v) = u, v, 7565 − 0.02u2 − 0.03v2 with u2 + v2 ≤ 10,000, where distance is measured in meters. The air pressure P(x, y, z) in the neighborhood of Mount Wolf is given by P(x, y, z) = 26e(−7x2 + 4y2 + 2z). Then the composition Q(u, v) = (P ∘ r)(u, v) gives the pressure on the surface of the mountain in terms of the u and v Cartesian coordinates.

(a) Use the chain rule to compute the derivatives. (Round your answers to two decimal places.)

∂Q ∂u (50, 25) =

∂Q ∂v (50, 25) =

(b) What is the greatest rate of change of the function Q(u, v) at the point (50, 25)? (Round your answer to two decimal places.)

(c) In what unit direction û = a, b does Q(u, v) decrease most rapidly at the point (50, 25)? (Round a and b to two decimal places. (Your instructors prefer angle bracket notation < > for vectors.) û =

Answer 1