Math 212, Fall 2011
Name: __________________________________
Exam 2
1. [18 points (9 pts each)]
(a) Find the equation of the tangent plane to the surface B#  CD œ " at the point a$ß #ß %b.
(b) Find parametric equations for the line tangent to the curve ra>b œ Øsin >ß $/> ß /#> Ù at the point a!ß $ß "b.
Math 212, Fall 2011
Exam 2
2
2. [10 points] In the figure to the right, a circle of radius 1 is
tangent to a perpendicular pair of line segments. Find the
coordinates of the center of the circle.
H3, 1L
H0, 0L
3. [10 points] Find the distance from the line ra>b œ Ø#  %>ß &  >ß $  >Ù to the plane B  #C  #D œ $.
Math 212, Fall 2011
Exam 2
3
4. [10 points] Let T be the plane #B  $C  D œ &, and let U be the plane C  $D œ "". Find the equation of
the plane through the point a"ß #ß $b that is perpendicular to both T and U.
Math 212, Fall 2011
Exam 2
4
5. [16 points (8 pts each)]
(a) Evaluate ((( <$ D .Z , where I is the region inside the cylinder B#  C# œ " and between the planes
I
D œ ! and D œ ".
(b) Evaluate ((( 3 .Z , where L is the solid hemisphere defined by B#  C#  D # Ÿ % and D € !.
L
Math 212, Fall 2011
Exam 2
6. [6 points] Use Green's Theorem to evaluate the integral
( sinˆB .B  BC .C,
G
#‰
5
y
H1, 2L
H3, 2L
H1, 1L
H3, 1L
#
where G is the closed rectangular curve shown in the figure to the right.
x
7. [6 points] Use the Gradient Theorem (the Fundamental Theorem for Line Integrals) to evaluate the integral
#
( #BC .B  ˆ"  B ‰ .C,
G
where G is the arc of the hyperbola B#  C# œ * staring at the point a$ß !b and ending at the point a&ß %b.
Math 212, Fall 2011
Exam 2
8. [12 points] Evaluate ( C# .=, where G is the arc of the curve C œ /B from a!ß "b to a"ß /b.
G
6
Math 212, Fall 2011
Exam 2
9. [12 points] Evaluate ((( B sin D .Z , where X is the triangular
7
H0, 2, ΠL
H2, 0, ΠL
X
prism shown in the figure to the right.
H2, 2, ΠL
H0, 2, 0L
H2, 0, 0L
H2, 2, 0L