STUDY GUIDE _
FINAL EXAM
241 C
12/15/05
Do any five problems. You may do one, two or three additional problems for extra credit.
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Problem #1: Given that w  f (x, y, z) , x  x(u,v), y  y(u,v), z  z(u, v) :
w
w
(a)-[8 pts.]- Find
and
using the appropriate chain rule.
u
v
w
w
w
(b)-[4 pts.]- Find
and
and
at (u0 , v 0 ) .
u
v
v
(See prob. # 18, p. 938)
(c)-[8 pts.]- Suppose that u  f ( x, y, z ), x  x( p, r , ), y  y ( p, r , ), z  z ( p, r , )
 u u u
,
,
Find
using the appropriate chain rule.
 p r 
 u u u
,
,
(d)-[5 pts.]-Evaluate
at p  p0 , r  r0 ,  0.
 p r 
(See section 14.5. See prob. # 25 on p. 938)
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Problem #2:
(a)-[4 pts.]-Find the directional derivative of the function f (x, y) at ( x0 , y0 ) in the
direction of the vector A  ai  bj .
(b)-[4 pts.]-In what direction does the directional derivative of f (x, y) at ( x0 , y0 ) have
its maximum value?
(c)-[4 pts.]-Calculate the maximum value of the directional derivative of f (x, y) at ( x0 , y0 ) .
(d)-[5 pts.]-Find the directional derivative of the function f (x, y.z) at ( x0 , y0 , z0 )
in the direction of the vector A  ai  bj  ck .
(e)-[4 pts.]-In what direction does the directional derivative of f (x, y.z)
at ( x0 , y0 , z0 ) have its maximum value?
(f)-[4 pts.]-Calculate the maximum value of the directional derivative of
f (x, y.z) at ( x0 , y0 , z0 ) .
(See section 14.6. See Ex. 4 p. 944, and Ex. 5 on p. 945, Ex. 6 on p. 946, and ex. 7 on p. 947)
See probs. 11-17 on p. 951. See probs. #21-26 on p. 951, and #33 on p. 951.)
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Problem #3: Let S be the surface defined implicitly by the equation F(x, y,z)  0 .
(a)-[9 pts.]-Calculate F(x, y,z) .
(b)-[9 pts.]-Use the result of (c) to find the equation of the tangent plane to S at the
point ( x0 , y0 , z0 ) where F ( x0 , y0 , z0 )  0 .
(c)-[7 pts.]- Find the parametric equations of the line that is normal to the surface at (x0 , y0, z0 ) .
(See section 14.6 pp. 947-949. See probs. #39-42 on p. 952.)
Problem #4: Let z  z ( x, y ) be defined implicitly by the equation F ( x, y, z )  0 .
(a)-[15 pts.]- Calculate z x and z y by implicit differentiation.
(b)-[10 pts.]- Use the results of (a) to find z x (1,1) and z y (1,1) .
(See “Implicit Differentiation” on pp. 936-937. See Ex. 9 on p. 936. See probs. # 31-34 on p. 938.)
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C.O. Bloom
STUDY GUIDE _
FINAL EXAM
241 C
12/15/05
Problem #5 : Given f (x, y) :
(a)-[10 pts.]-Find the points where f x (x, y)  0 and f y (x, y)  0 .
(b)-[15 pts.]-Use the second derivative test to classify each extremum located in part (a).
(See section 14.7, especially Ex. 3 on p. 955, and Ex 4 on p. 956.See probs. #5-12 on p. 961.)
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Problem #6:
(a)-[25 pts.]- Use the method of Lagrange multipliers to find the relative maxima and minima of the function
f ( x, y, z ) subject to the constraint g ( x, y, z )  0 .
(See section 14.8, pp. 965-969. See probs. #7-12 on p. 971.)
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Problem #7:
(a)-[5 pts.]-Express a given double integral
f (x, y) dx dy in polar coordinates and evaluate
R
the resulting polar integral.
(See section 15.4. See Ex. 1 on p. 1005, Ex. 2 on p. 1006. See probs. # 9-16 on p. 1008.)
(b)-[10 pts.]-Express a given triple integral
f (x, y,z ) dx dy dz in cylindrical coordinates and evaluate

R
the resulting cylindrical coordinate integral.
(c)-[10 pts.] Express a given triple integral
 f (x, y,z ) dx dy dz in spherical coordinates and evaluate
R
the resulting spherical coordinate integral.
(See sect. 15.8, especially Exs.#1-4. See probs. #33-34, and 35-36 on p. 1038.)
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Problem #8:
f (x,y) ds where C is given in vector form by r(t)  x(t)i  y(t )j (a  t  b) .
(a)-[12 pts.]-Evaluate
C
(b)-[13 pts.]-Evaluate  f (x,y,z) ds where C is given in vector form by r(t)  x(t)i  y(t )j z(t)k (a  t  b) .
C
(See sect. 16.2, pp. 1062-1069, especially Exs. #1-6. See probs. #1-16 on pp. 11071-1072.)
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Problem #9:
(a)-[10 pts.]- Find the work W done by a force F  M(x, y, z)i  N(x, y,z)j  P(x, y, z)k acting along a curved path
from ( x0 , y0 , z0 ) to ( x1 , y1 , z1 ) given by the vector equation r(t)  x(t)i  y(t )j z(t)k (a  t  b) .
Hint: Evaluate W 
C
F  T ds 
b
a F  v(t)dt .
(b)-[15 pts.]- Find the flux of the vector field F  M(x, y)i  N(x,y)j outward across the closed curve
given parametrically by the equations x  x(t) , y  y(t) (a  t  b) .
b
dy
dx 
 N  dt.
Hint: Evaluate C F  n ds  a  M
dt 
 dt
(See “Line Integrals of Vector Fields” on pp. 1069-1071.See Sect. 16.4, especially Exs. #1-2.
See probs. # 1-3, 7-9, 13-16 on p. 1069.)
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continued on next page
C.O. Bloom
STUDY GUIDE _
FINAL EXAM
241 C
12/15/05
Problem #10:
(a)-[10 pts.]-Show that a given force field F  M(x, y, z)i  N(x, y,z)j  P(x, y, z)k is conservative.
(b)-[15 pts.]-Find a potential function for the given conservative force field.
(See sect. 16.3 , especially Exs. # 1-3, and #4 and 5. See probs. # 3-9 on p. 1081. See probs # 17-18 on p. 1082.)
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Problem #11:
(a)-[10 pts.]-Apply Green's Theorem to evaluate a given line integral of the form M(x, y)dx  N(x,y)dy .
C
(b)-[15 pts.]-Apply Green's Theorem to find the area of the region R enclosed by the curve given in vector
form by r(t)  x(t)i  y(t) j (a  t  b) .
1
Hint: Use the formula Area of R 
x dx  y dx and Green's Theorem.
2 C
(See sect. 16.4, especially Exs. 1-4. See probs. # 1-3 on p. 1089. See probs. #13-16 on p. 1089.)
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Problem #12:
(a)-[25 pts.]- Use Stoke’s Theorem to evaluate the line integral
F  dr ,

C
where F = M(x, y, z)i  N(x, y, z)j  P(x, y, z)k is a given vector field, and C is boundary of the
surface given implicitly by the equation f (x, y, z)  0 .
(See sect. 16.8, especially probs. # 7-10 on p.1125.)
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Problem #13:
(a)-[25 pts.]- Use the Divergence Theorem (Theorem 7 on page 1124) to evaluate the flux
 F  n dS of a given vector field F = M(x, y, z)i  N(x, y, z)j  P(x, y, z)k where S is a closed
S
surface given implicitly by the equation f (x, y, z)  0 .
(See sect. 16.9, especially Ex. 1-2 on p. 1029. See probs. #10-14 on p. 1132.)
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C.O. Bloom