METU NCC Mat 120 Spring 2009 Final

# METU NCC Mat 120 Spring 2009 Final - function f over the...

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Dept./Sec. Time Duration Signature : : : : : : Last Name Name Student No M E T U Northern Cyprus Campus Math 120 Calculus for functions of several variables Final Exam 05.06.2009 09: 30 120 minutes 7 QUESTIONS ON 4 PAGES TOTAL 100 POINTS 1 2 3 4 5 6 7 Q1 (15=6+9 pts.) Test whether the following series converge or diverge, and explain your answer. (a) n =1 ( - 1) n n n + 1 (b) n =1 ln ( n ) n 2 + 2 n - 1

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Q2 (15 pts.) Using Analytic Geometry, ﬁnd the surface area of the tetrahedron bounded by the plane 2 x + 3 y + 4 z = 24 and the coordinate planes in the ﬁrst octant. Q3 (10 pts) Find the line integral H C F · dr , where F ( x, y ) = (sin ( y ) - y sin ( x )) i + (cos ( x ) + x cos ( y )) j is the vector ﬁeld and C is the path parametrized as r ( t ) = e t sin ( t ) i + e t cos ( t ) j , 0 t 4 π .
Q4 (15=5+8+2 pts.) Let f ( x, y ) = xy be a function deﬁned on the region x 2 + 4 y 2 16. (a) Find and classify all critical points of the function f over the interior region x 2 + 4 y 2 < 16. (b) Use the method of Lagrange multipliers to ﬁnd the maximum and minimum values of the

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Unformatted text preview: function f over the boundary x 2 + 4 y 2 = 16. (c) Find the absolute max-min values of the function f over the region x 2 + 4 y 2 ≤ 16. Q5 (15 pts.) Find the double integral Z Z R x + 2 y (2 x-y ) 2 dxdy over the parallelogram R enclosed by the lines x + 2 y = 3, x + 2 y = 5, 2 x-y =-6 and 2 x-y =-3 ( Hint: Use the linear transformation u = x + 2 y, v = 2 x-y ) Q6 (15 pts.) Express the volume enclosed by the surfaces y = x 2-1, y = 0, z = x + y + 10 and z =-x-y-4 as a triple integal and evaluate this integral. Q7 (15 pts.) Find H C (2 y-sin (sin ( x ))) dx + ( x-2 xy + cos (cos ( y ))) dy , where C is the circle x 2 + y 2-4 y = 0 oriented counterclockwise....
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## METU NCC Mat 120 Spring 2009 Final - function f over the...

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