Question

Consider the mountain known as Mount Wolf, whose surface can be described by the parametrization (u, v) = (u, v, 7565 - 0.0202 - 0.0312) with v2 + 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) = 32el-7x2 + 4y2 + 2z). Then the composition Q(u, v) = (Por)(u, v) gives the pressure on the surface of the mountain in terms of the u and vCartesian coordinates. (a) Use the chain rule to compute the derivatives. (Round your answers to two decimal places.) (50, 25) = .15 aQ(50, 25) = .15 x (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 0 = (a, b) does Q(u, v) decrease most rapidly at the point (50, 25)? (Round a and 6 to two decimal places. (Your instructors prefer angle bracket notation <> for vectors.) Q=

Answer 1

Solution : Given Blu, v) - <u, V, 4565 - 0.024²-0.031" x zu y=v 2= 7565 -0.02 U?-0.0312 PCX,Y,Z) = 32647xº+ 4y?+22) QIU,V) - 320 Hu?) + 472+2 (1565-0.0247-0.0312) Q(u, v) = 32 & 1.0 (7.04 u’+3.9402715130) da = 32.41.04 u?+3.94 v++15130) -14.06u) -0 OQ -32 & 7:04u73,94V*415130) GSSV) 70 At (50,25) - 32 € 7.04u++3.94 - 32 £7.04 (50)*+ 3.94125)*715130] ne = 328 7.5 = 32 (0.000553) = 0.017696 (Q) 3Q (50,25) = 0.017696 [-14.08(50)] = 0.017696 (-704) = -12.457984 OQ (50,25) = 0.017696(7.88(25)] = 0.011696 (191) - 3.486112 ду

(b) VQ - OQ VU + OQ ou = -12.4579 û +3,4 861 Û Greatest rate of change is magnitude of VQ (50,25) IVR (50,25)] = VE 12.45793* +(3.48613 - (155.1992 + 12.1528 - 1167.35209 = 12.9364 (0) Directim of most rapidly decrease is û i na 17Q --[12.4579 û + 3,450107 12,9364 - 0.16 30l 6+/- 0. 2011) : 0 - (0.10 301, -0.264+1)