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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