M1C_Sample_Exam_2_Answers

MATH 1C: SAMPLE EXAM 2 c Jeffrey A. Anderson ANSWER KEY True/False (15 points: 3 points each) For the problems below, circle T if the answer is true and circle F is the answer is false. After you’ve chosen your answer, mark the appropriate space on your Scantron form. Notice that letter A corresponds to true while letter B corresponds to false. 1. T F If lim n →∞ a n = 0, then ∞ ∑ n =1 a n converges. 2. T F The series ∞ ∑ n =1 3 ne - n 2 diverges 3. T F If f has a local minimum at point ( a, b ) ∈ R 2 , then D u f ( a,b ) = 0 for any unit vector u ∈ R 2 . 4. T F If a n > 0 and ∞ ∑ n =1 a n converges, then ∞ ∑ n =1 ( - 1) n a n converges. 5. T F Suppose f : R 3 → R . If ∇ f = 0 at a point x ∈ R 3 , then f has a local extreme value at point x .

Multiple Choice (45 points: 3 points each) For the problems below, circle the correct response for each question. After you’ve chosen your answer, mark your answer on your Scantron form. 6. Let z = sin( x · y ) and let x = x ( t ) and y = y ( t ) be functions of t . Suppose x (1) = 0 , y (1) = 1 , x 0 (1) = 2 , y 0 (1) = 3 . Find dz dt when t = 1. A. 1 B. 2 C. 3 D. 4 E. 5 7. The series ∞ ∑ n =0 r n converges if and only if: A. - 1 < r < 1 B. - 1 ≤ r ≤ 1 C. - 1 ≤ r < 1 D. - 1 < r ≤ 1 E. r < 1 8. Find the direction of maximum increase of the function f ( x, y, z ) = x e - y + 3 z at the point (1 , 0 , 4) . A.   - 1 - 1 3   B.   1 - 1 3   C.   - 1 3 3   D.   - 1 - 3 3   E.   1 1 3   9. Which of the following series converge? 1) ∞ X n =1 1 n 2) ∞ X n =1 ( - 1) n · n ln( n ) 3) ∞ X n =1 ( - 1) n n A. 1 B. 2 C. 3 D. 2 , 3 E. None 10. Find the limit of the sequence a n = 2 + - 4 5 n : A. 2 B. 6 5 C. - 4 5 D. - 2 E. 4 5 Math 1C: Sample Exam 2 c Jeffrey A. Anderson Page 2 of 10

11. Find the shortest distance from the origin to the surface z 2 = 2 xy + 2 A. 1 √ 2 B. √ 2 C. 1 2 D. 2 E. 1 12. The series ∞ ∑ n =1 1 n α converges if and only if A. 1 < α B. α < 1 C. - 1 < α < 1 D. α ≥ 1 E. - 1 < α 13. Find the minimum value of the function f ( x, y ) = x y subject to the constraint that x 2 + y 2 = 2: A. 1 B. 2 C. - 1 D. 3 2 E. - 3 2 14. Find the values of x for which the series ∞ ∑ n =1 ( x - 1) n : A. - 2 < x < 0 B. 0 < x ≤ 2 C. 0 < x < 2 D. 0 ≤ x ≤ 2 E. - 2 ≤ x < 0 15. Find the directional derivative of the function f ( x, y ) = y 2 · ln( x ) at the point (1 , 2) in the direction of the vector (3 , 4) = 3 i + 4 j : A. 5 16 B. 12 C. 5 12 D. 16 5 E. 12 5 Math 1C: Sample Exam 2 c Jeffrey A. Anderson Page 3 of 10