Directional Derivatives and the Gradient Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Find the gradient of the function

(2) Find the gradient of the function

(3) Find the gradient of the function

(4) Find the gradient of the function

(5) Compute the directional derivative of the function at the point in the direction of the vector

(6) Compute the directional derivative of the function at the point in the direction of the vector

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

    (a) , and

    (b) and

(8) Find the gradient of the given functions.

    (a)

    (b)

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

    (a) , and

    (b) and

(10) 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

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

    (a) the circle

    (b) the ellipse

(12) 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

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

    (a)

    (b)

    (c)

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