Question

6. (10 points) (a) (6 points) The gradient of the function o(x, y, z) at (1,2,3) is the vector (2, 1, 1) and g(1,2,3) = 1 (1) (2 points) Find the equation of the tangent plane of the level surface g(r, y, z) = 1 at (1,2,3) (ii) (2 points) Find the maximum rate of change of g(x, y, z) at (1, 2, 3). hax. rarte ot change: 23 14 (iii) (2 points) Find the rate of change of g at (1,2,3) in the direction (-1, 2,-1). Rofe of charge in (-1,2,-1)- 1. 2,3) (-2, 2,-1) (b) (4 points) The light intensity (measured in lux) in a e+e A South American electric eel is swimming in the pond along a path with position after t seconds given by the function c(t) (r(t), y(t), z(t)). Suppose c(5) (1,0,2) and c'(5) = (3, 9T, 2). What is the rate of change of the light intensity along the eel's path pond is given by f(x,y,z) after 5 seconds?

Answer 1

egvation the tangent plane 61) The gx(1,2,3) (2-1)y(,2,3) y-2) 2(1,2,3(z -3= c 9.4,2,3)= 2, (,2,3) = g,2,3) 2 (x- y-2)+ \(2-3J o 3y(1,2,3)= 32C1,2,3) 2,,> as The plane 7 2x t of change of3()at rate i) The maximum (1,2,3) z1+zl+gc

The rate of change of at C,2,3) inthe olirection ,2,1) cirection of (, 2,-3) Vector in the <2--, 2, 3) -2,0 V2) j2 <-2,01- -4 25 <- 2 2Js 2 > The rate of change in at (1,2,3) (,2,3)- V< 2,, 2 at t 5 b) gradient of fat ,02) /3 The f = 4

- /2 fy (1,0,2 -1/2 K /2 -Le 2. e ,2) 2 (3,9,2) Cs=Vq81+Y The rate of change VI3t 81A2 in f along affer S4 A 2k |13+81R /2 e -2 3K - 2 e/2 3A 13+ 81