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Quiz #4
NAME:
#1. Find a tangent plane to the surface which is defined implicitly by the equation
xy 2  xz 2  y 3 z  4 at the point (1, 2, 0) .
#2. Find the directional derivative of f ( x, y, z )  xy  xz  yz in the direction of
u  3i  4 j  12k at the point (1, 2, 2) .
#3. Let z  f ( x, y ) be a function with f (1,5)  (3, 2) . Find the maximum value for the
directional derivative of f at the point (1,5).
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#4. #7. Look at the path given by x (t )  (t cos t , t sin t ), 0  t  6 .
a. Sketch the path using arrows to indicate direction of travel. Label scales on the x- and yaxes.
b. Calculate the velocity and speed at t  2 .
c. Sketch the velocity vector from b. with initial point on the path.
d. Which will be larger:  ( ) or  (5 ) . Explain your reasoning.
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#5. Let f be a function of two variables that has continuous partial derivatives and consider the
points A(1,3), B(3,3), C (1, 7), and D(6,15). The directional derivative of f at A in the direction of
AB is 3 and the directional derivative of f at A in the direction of AC is 7. Find the directional
derivative of f at A in the direction of the vector AD .
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