Question

Problem 2. Suppose that gx, y, 2) is a function with the following properties: -1,2,-4 5, 9(-1,2,-4)-2, 9u(-1,2,-4) -3, and 9.(-1,2,-4) 6 Answer the following questions, and carefully justify all your answers. (a) Estimate the value of g(-0.8,2.1,-4.2). near y (b) In what direction from the point (-1,2, -4) does g decrease the fastest? What is the rate of change in this direction? (c) Which of the following objects exist(s)? Justify your answers, and find an equation for the ones that exist. The tangent plane to the graph of g(a, ,) at (-1,2,-4). The tangent plane to the level set g 5 at (-1,2, -4). at (-1,2,-4). Justify your reasoning carefully part (c) at a 40 angle and is closer to Vg than it is to-Vg? the partial derivatives of h and those of g? Why does this make sense? + ck is tangent to the level set g-t (d) Find all possible values of c, if any, such that the vector -3 (e) At the point (-1,2, -4), what is the rate of change of g along a vector that intersects the plane from (f) Find the partial derivatives of the function h, y, a) (, y,a)2. What relationship do you see between

Answer 1

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