Question

Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+4 (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z0 0, 20 2, 204 (Note: Use set notation, and draw a single contour diagram.) (v) Without finding Vg, on your diagram from (iv) mark the direction of Vg(,1), 20 40 (vi) Describe the symmetries of g.). (b) (i) Sketch the cross-sections of g with zo 0, V3 on a single diagram. (ii) Sketch the cross-sections of g with yo 0,1,+2 on a single diagram (iii) Sketch the graph of g. (You should not spend too long on your sketch. Perhaps add some words to describe what you think the surface looks like.) (c) (i) Find Vg(x,g) (ii) Calculate the directional derivative of g at the point (2,1) in the direction α (iii) What is the direction of greatest increase of g at (2,2)? (i) Find 9zr, 9ry, 9%w (ii) Write down the equation of the tangent plane to g at (2, 1) (iii) Find the second order Taylor polynomial of g about the point (2, 1) (d)

Answer 1

f is not 1:1 as (1,0)まCo, 1)