MVC
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Sample Quiz
NAME:
#1 Find the directional derivative of F ( x, y)  sin( x 2  y 2 ) in the direction of the vector
 

v = i + j, at the point (  ,  ).
#2 Write a formula for
w
if w  f ( x, y ) and x  x(r , s, t ), y  y (r , s, t ) .
s
#3 Prove that if z  f ( x, y ), with x  r cos  and y  r sin  , then
2
1  z 
 z   z   z 
       2

r   
 x   y   r 
2
2
2
xe y
and P = (0,1,1). Find u such that Du [f(0,1,1)] is a maximum. Evaluate
2z 2  3
Du [f(0,1,1)] for this vector u .
#4 Let f(x,y,z) =
#5 Find a tangent plane to the surface which is defined implicitly by the equation
xy 2  yz 2  zx 2  3 at the point (1, 2,1) .
#6.
The figure below shows the level curves of a function z  k  x, y  .
In this plot, the z-values increase as you move up and to the right.
a.
b.
On this contour plot, sketch vector v with initial point (2, 2) in the same direction as
k  2, 2  .
On this contour plot, sketch a vector w with initial point (2, 2) so that Dw k  2, 2  0 .
That is, the directional derivative at (2, 2) in the direction of w is 0.
k
k
c. For this contour plot, which has the greater value at the point (3, 1),
or
?
x
y
Briefly explain your choice.
#6. Parameterize the straight line segment from (0,0) to (1,2) in terms of the arc-length parameter s.
#7. Look at the path given by x (t )  (3sin 2t , 4 cos 2t ), 0  t   .
a. Sketch the path using arrows to indicate direction of travel.
b. Calculate the velocity, speed, and acceleration at t 

.
4
c. Sketch the velocity and acceleration vectors from b. with initial point on the path.
#9. Find the curvature of the path x (t )  (t 2 , 2t 2 ,3t 2 ),0  t  5 at t  2