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MultiFinalExam07 - Math 2224 Spring 2005 Common Final Exam...

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Math 2224 Spring 2005 Common Final Exam, Spring 2005 1. Consider the infinite series n = 1 ( 3 ) n / 5 n 1 . Which of the following statements is correct? (1) The series converges and has limit 5/8. (2) The series converges and has limit 15 / 8. (3) The series converges and has limit 5/2. (4) The series diverges. 2. Consider the function f ( x , y ) = x 2 + kxy + y 2 , where k is a real constant. Which of the following statements is correct? (1) For every value of k , ( 0 , 0 ) is a local minimum of the function f . (2) For every value of k , ( 0 , 0 ) is a saddle point of the function f . (3) For | k | > 2, the function f has no local minimum. (4) For | k | < 2, the function f has a saddle point at ( k / 2 , k / 2 ) . 3. The integral of f ( x , y , z ) = x 2 + y 2 + z 2 over the solid that lies above the paraboloid z = x 2 + y 2 and below the plane z = 9 equals: (1) 2 π 0 π / 4 0 3 0 ρ 4 sin φ d ρ d φ d θ (2) 2 π 0 3 0 9 r 2 ( r 2 + z 2 ) rdzdrd θ (3) 3 0 3 3 3 3 ( x 2 + y 2 + z 2 ) dxdydz (4) 3 3 3 3 9 x 2 + y 2 ( x 2 + y 2 + z 2 ) dzdxdy 4. Consider the infinite series n = 1 n + 2 n 2 + n + 1 . Which of the following statements is cor- rect?
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