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Quiz #4
NAME:
#1
Find the directional derivative of F ( x, y)  xye y



v = i + j , at the point (1,0).
#2 Write a formula for
2
in the direction of the vector
f
if w  f ( x) and x  x(r , s, t ).
t
 yx zx
yx
zx
,
Suppose that w  f 
and v 
.
 is a differentiable function of u 
xz 
xy
xz
 xy
Prove that:
w
w 2 w
x2
 y2
z
0
x
y
z
#3
#4. For the contour map for z  f ( x, y ) shown below, estimate each of the following quantities.
Explain briefly how you are getting your answer.
(a) f x (1, 2) and f y (1, 2)
(b) f (1, 2)
(c) Du  f (1, 2) , where u is a unit vector in the direction of f (1, 2)
(d) Sketch the vector f (1, 2) on the contour map for f using (1, 2) as the initial point.
(e) Sketch a unit vector v with initial point (1, 2) such that Dv  f (1, 2)  0 .
6
4
4
2
0
2
4
6
5.
Find an equation for the tangent plane to the surface xy z  1  z  5 at the point (4, 1, 3).
#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 )  (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.