STUDY GUIDE _ FINAL EXAM 241 F 12/20/04
Do any five problems. You may do one, two or three additional problems for extra credit. ______
Problem #1: Given that , :
(a)-[8 pts.]- Find and using the appropriate chain rule.
(b)-[4 pts.]- Find and and at .
(See prob. # 18, p. 938)
(c)-[8 pts.]- Suppose that
Find using the appropriate chain rule.
(d)-[5 pts.]-Evaluate at
(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 at in the
direction of the vector .
(b)-[4 pts.]-In what direction does the directional derivative of at have
its maximum value?
(c)-[4 pts.]-Calculate the maximum value of the directional derivative of at .
(d)-[5 pts.]-Find the directional derivative of the function at
in the direction of the vector .
(e)-[4 pts.]-In what direction does the directional derivative of
at have its maximum value?
(f)-[4 pts.]-Calculate the maximum value of the directional derivative of
at .
(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 be the surface defined implicitly by the equation .
(a)-[9 pts.]-Calculate .
(b)-[9 pts.]-Use the result of (c) to find the equation of the tangent plane to S at the
point where .
(c)-[7 pts.]- Find the parametric equations of the line that is normal to the surface at .
(See section 14.6 pp. 947-949. See probs. #39-42 on p. 952.)
Problem #4: Letbe defined implicitly by the equation.
(a)-[15 pts.]- Calculateby implicit differentiation.
(b)-[10 pts.]- Use the results of (a) to find.
(See “Implicit Differentiation” on pp. 936-937. See Ex. 9 on p. 936. See probs. # 31-34 on p. 938.)
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STUDY GUIDE _ FINAL EXAM 241 F 12/20/04
Problem #5 : Given:
(a)-[10 pts.]-Find the points where and .
(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 subject to the constraint.
(See section 14.8, pp. 965-969. See probs. #7-12 on p. 971.)
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Problem #7:
(a)-[5 pts.]-Expressa given double integral in polar coordinates and evaluate
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 in cylindrical coordinates and evaluate
the resulting cylindrical coordinate integral.
(c)-[10 pts.] Express a given triple integral in spherical coordinates and evaluate
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:
(a)-[12 pts.]-Evaluate where C is given in vector form by .
(b)-[13 pts.]-Evaluate where C is given in vector form by .
(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 forceactingalong a curved pathfrom to given by the vector equation .
Hint: Evaluate .
(b)-[15 pts.]- Find the flux of the vector fieldoutward across the closed curve
given parametrically by the equations , .
Hint: Evaluate
(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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STUDY GUIDE _ FINAL EXAM 241 F 12/20/04
Problem #10:
(a)-[10 pts.]-Show that a given force field 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 .
(b)-[15 pts.]-Apply Green's Theorem to find the area of the region enclosed by the curve given in vector form by .
Hint: Use the formula and Green's Theorem.
(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 ,
where is a given vector field, and is boundary of the
surface given implicitly by the equation .
(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
of a given vector field where is a closed
surface given implicitly by the equation .
(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