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Homework 7, Due by email to TA Nov 4

  1. (1 pt) What is the normal vector to the sphere {$x^2+y^2+z^2=4$} at the surface point {$(x,y,z)$}?
  2. (1 pt) What is the normal vector to the paraboloid {$z=2x^2+y^2$} at the surface point {$(x,y,2x^2+y^2)$}?
  3. (4pt, this question will take some thinking).
    Imagine that you have an infinitely large room illuminated by one infinitely long row of point lights. This figure shows a side view of the room.
    The lights are h above the floor and are 1 meter from each other. Assume that the ceiling above the lights is black and that no light reflects off of anything.
    An object at distance d from a light gets illuminated with a brightness 1/d2 .
    Each point on the floor is illuminated by all the lights, but more brightly by the closer lights.
    A point p directly below a light will be a little brighter than a point q halfway between two such points. That is the problem --- we want the floor (at least the part directly below the line of lights) to be evenly lit, at least within 10%.
    However, the higher the line of lights, the more evenly the floor will be lit.
    Your question is to tell us what is the minimum value for h so that the line of the floor below the line of lights is evenly lit within 10%.
    E.g., the brightness at p is {$$ \sum_{d=-\infty}^{\infty} \frac{1}{h^2+d^2} $$}