For each of the following surfaces, find the tangent plane and normal line at the point indicated, verifying that the point is in the surface:
a) x2 + y2 + z2 = 9 at (2, 2, 1)
b) ex2 + y2 − z2 = 0 at (0, 0, 1)
c) x3 − xy2 + yz2 − z3 = 0 at (1, 1, 1)
d) x2 + y2 – z2 = 0 at (0, 0, 0)
Why does the procedure break down in (d)? Show by graphing that a solution is impossible.
e) xy − z = 0 at (x1, y1, z1), where x1y1 = z1
f) xy + yz + xz = 1 at (x1, y1, z1), where x1y1 + y1z1 + x1z1 = 1
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