Quiz (Directional Derivatives and the Gradient)

Show all work and justify each step.

(1)  Compute the directional derivative of the given function at the given point in the direction of the given vector.

    (a) , and

    (b) and

(2) Find the gradient of the given functions.

    (a)

    (b)

(3) Compute the directional derivative of the given function at the given point in the direction of the given vector.

    (a) , and

    (b) and

(4) Find the direction from in which the given function increases most rapidly and compute the magnitude of the greatest rate of increase.

    (a) at

    (b) at

(5) Find a unit vector that is normal to the given graph at the point on the graph.  

    (a) the circle

    (b) the ellipse

(6) Find a unit vector that is normal to each surface given at the prescribed point, and the standard form of the equation of the tangent plane at that point.   

    (a) at

    (b) at

(7) Find the equation of the tangent plane in standard form for the given surface at the point

    (a)

    (b)

    (c)

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