Download Test #2

Calculus 284
Test #2
1.
Name ________________
Find the parametric equation of the tangent line to the curve described by
r  t   cos ti  sin t j  2tk at the point  0,1,   .
2.
Find
3.
Let
8
T  0  , N  0  and B  0  for r  t   ln ti  t 2 j  tan 1 tk .
F ( x, y)  y  x ln( x  y) . Find the following if possible.
a) Sketch the domain of F ( x, y ) .(7pt)
-8
4
-4
4
8
-4
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b) Find the first partial derivatives of F ( x, y )
(8pt) _________________
c)
Find the linearization F ( x, y ) at the point
of e .
(0, e 2 ) . Leave your answer expressed in terms
(7pt) _________________
4.
Find the position vector
vector v(0)  i 
r (t ) of an object that has an acceleration vector a(t )  i with initial velocity
j and initial position vector r (0)  k .
(10pt) ________________
5.
Let
f ( x, y, z )  e yz cos x . Find the following:
a)
The gradient vector
f .
(9pt) _________________
b) The derivative of f at ( ,1, 2) in the direction of  2, 
7,5  .
(9pt) _________________
c)
Find
f
t
where
x  sin t , y  st 4 , z  s 3
(9pt) _________________
4. Set up but do not evaluate an integral that would compute the length of the curve
x  t 4  t , y  2t 3 , z  tan 1 (t 2 ) from (0,  2,  4 ) to (2, 2,  4 ) .
(8pt) _________________
5.
Sketch the following surfaces: (12pt)
a)
 x2  y2  z2  1
b)
z
f ( x, y)  1  x 2
z
y
x
y
x
6. Find all critical points and classify them as a maximum, minimum or a saddle point for the function
F ( x, y )   13 x 3  xy  12 y 2  12 y .
(15pt) ________________
Calculus 284 Fall 2003
Test #2
Name ________________
r  t   t 2 i  2t j  ln tk from the point 1, 2,0 to  e 2 , 2e,1 .
1.
Find the length of the arc
2.
If we know the following information about the position vector r 1  2i 
velocity vector
j  4k with the
v 1  i  3 j  k and acceleration vector a 1  i  2k Find the following:
a)
T 1
b)
B 1
c)
aT When t  1
t  1.
3. The equation of the normal plane to a curve is ax  by  cz  1 at the point r 1 . How are the
d) The equation of the osculating plane on the curve when
vectors
4.
T 1 , N 1 and B 1 related to this plane?
Let
f  x, y, z   x3 y  y 2 z 2 . Find the following
a)
The directional derivative of f at the point
 2,1,3 in the direction of the vector
a  i  2 j  2k .
b) Find the direction that f increases the most rapidly at
c)
5.
If
The equation of the tangent plane of the surface
 2,1,3 .
17  x3 y  y 2 z 2 at the point  2,1,3 .
f  x, y   tan 1 ( y 2 / x) , find f  x, y  . Your answer should have proper fractions that are
reduced.
6.
7.
8.
9.
10.
11.
12. Let
w  f  r  s, s  t , t  r  . Show that
w w w


0
r s t
13. Find the local maximum and minimum values of
f  x, y   x2  y 2  x2 y  4 .
14. Use Lagrange multipliers to find the extreme values of the function
f  x, y   x2  2 y 2 on the
x2  y 2  1 .
15. The length l , width w and height h of a box change with time. At a certain instant the dimensions
are l  2m and w  h  3 and w and h are increasing at a rate of 3m / sec and l is decreasing at
a rate of 1m / sec . At that instant find the rates at which the surface areas are changing.
16. The length and width of a rectangle are measured as 30cm and 24cm , respectively, with an error in
measurement of at most 0.1cm in each. Use differentials to estimate the maximum error in the
circle
calculated area of the rectangle.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27. If we know that the position vector, velocity vector and acceleration vectors are given as
r  0   i  2 j  2k , v  0   2i  k and a  0   2 j  3k . Find the following:
a).
b).
c).
T  0
(4pt) _________________
N  0
(5pt) _________________
B  0
d). The curvature of the graph at the point
1, 2, 2
(5pt) _________________
.
(4pt) _________________
e). The component of acceleration in the tangent direction when t  0 .
(4pt) _________________
28. Find the symmetric equation of the tangent line to the curve described by
r  t   cos ti  sin t j  2tk at the point  0,1,   .
(8pt) _________________
1
29. Let F ( x, y )  3
 ln( xy ) . Find the following if possible.
x y
d) Sketch the domain of F ( x, y ) .(5pt)
e)
Find
F
(2,1)
x
(8pt) _________________
30. Find an equation of the tangent plane to the surface
sin  xz   4cos  yz   4 at the point  ,  ,1 .
(8pt) _________________
31. The legs of a right triangle are measured to be 3 cm and 4 cm , with a maximum error of 0.05 cm
in each measurement. Use differentials to approximate the maximum possible error in the calculate
value of the area of the triangle.
7.
Let
(8pt) _________________
f  x, y, z   e x y 3 z .
a). Find the directional derivative of f at the point
 2, 2,1 in the direction of  2, 2,1 to
1, 2, 2 .
(8pt) _________________
b). Find a vector in the direction of greatest increase at
 2, 2,1 .
(3pt) _________________
c). Using the chain rule find
f
2
if x  ln  t  , y  ln  5t  , z  ln  t  .
t
(8pt) _________________
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Match the following graphs with the appropriate equation. (8pt)
b)
f  x, y     x 2  y 2 / 3 
c)
f ( x, y )  ln  2 x 2  y 2 
d)
f  x, y  
e)
f  x, y   8e
1
9x  y2
2

 x2  y 2

8
Let
g ( x, y)  4 xy  x 4  y 4 .(14pt)
a). Find all critical points of the surface.
b). Classify each critical point in part a) as a maximum, minimum or a saddle point.