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Calculus II - Mid Term Exam

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This mid term exam was administered at the Lahore University of Management Sciences for the course Calculus II in 2009-10.
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  MATH 102 Math 102 Calculus II Midterm Exam Problem 1. Find the slope  dy / dx  of the polar curve  r  = 2(1  −  cos θ )  at the pointwith polar coordinates  ( r,θ )=(2 ,π /2) .(A) -2 (B) -1 (C) 0 (D) 1 (E) none of the other choices is correctAnswer: (B)Problem 2. Find the area inside the polar curve  r =3 cos (3 θ ) .(A)  7 π 4  (B)  2 π  (C)  9 π 4  (D)  5 π 2  (E) none of the other choices is correctAnswer: (C)Problem 3. Consider the sequence with terms  a n =( − 1) n  2 n 2 1+ n 2  Which statement belowdescribes the behaviour of the sequence as  n →∞ ?(A)  lim n →∞ a n  exists and equals 0.(B)  lim n →∞ a n  exists and equals 2.(C)  lim n →∞ a n  does not exist because the  a n  grow without bound.(D)  lim n →∞  a n  does not exist because the  a n  oscillate between values very closeto -2 and 2.(E) none of the other choices is correct.Answer: (D)Problem 4. Which statement is true about a series  n =1 ∞ a n ?(A) If  a n → 0  as  n →∞ , then the series  n =1 ∞ a n  converges.(B) If the series  n =1 ∞ a n  does NOT converge, then the sequence  a n  does NOTconverge to zero as  n →∞ .(C) If the sequence  a n  does NOT converge to zero as  n  → ∞ , then the series  n =1 ∞ a n  does NOT converge.(D) If the sequence  a n  converges, then the series  n =1 ∞ a n  converges.(E) none of the other choices is correct.Answer: (C)Problem 5. The series  n =1 ∞ ( − 1) n log ( n ) n 1  (A) is conditionally convergent.(B) is absolutely convergent.(C) is divergent.(D) is conditionally divergent.(E) none of the other choices is correct.Answer: (A)Problem 6. The series  n =0 ∞ ( − 1) n +2 x 2 n +1 is the Taylor series about  x = 0  for thefunction(A)  sin ( x )  (B)  cos ( x )  (C)  x 1+ x 2  (D)  11+ x (E) none of the other choices is correctAnswer: (C)Problem 7. The equation of the plane tangent to  z = xy  at the point (1,2,2) is:(A) z =2 x + y − 4 (B) z =2 x + y − 2 (C) z = − 1 (D) z =3 xy − 2 x − y (E) none of the other choices is correct.Answer: (B)Problem 8. The value of the second-order partial derivative,  f  yz , of the function f  ( x,y,z )=  zy 3 x  + y exp ( z )  at the point (3,2,0) is(A) 4 (B) 1 (C) 5 (D) 2 (E) none of the other choices is correct.Answer: (C)Problem 9. Find non-zero real numbers  α  and  β   such that for all vectors  a  and  b , α ( a +2 b ) − β  a +(4 b − a )= 0 .(A)  α =2 ,β  =1  (B)  α = − 2 ,β   = − 3  (C)  α =1 ,β   =3  (D) α  = − 2 ,β   =3 (E) none of the other choices is correct.Answer: (B)Problem 10. If  u =3 i −  j  + k  and  v =2 i +5  j  + k  then  u × v  is(A)  − 6 i −  j  + 17 k  (B)  − 6 i +  j  + 17 k  (C)  4 i +5  j  + 13 k  (D)  − 4 i −  j  + 15 k (E) none of the other choices is correct.2  Answer: (A)Problem 11.  k =0 ∞ ( − 1) k +2  π 3  k = (A)  11 − π /3  (B)  π /31 − π /3  (C)  33+ π  (D)  π 3+ π  (E) the series does not converge.Answer: (E)Problem 12. Suppose that we try to approximate  sin ( x )  by  S  ( x ) =  x  −  x 3 /3! . Let theerror be  E  ( x )= sin ( x ) − S  ( x ) . Which of the following is true?(A)  | E  ( 0.5 ) |   12 5 · 5!  (B)  | E  ( 0.5 ) | >  12 5 · 5!  (C)  12 5 · 5!  < | E  ( 0.5 ) | <  1 10 (D)  0.1 < | E  ( 0.5 ) | < 0.2  (E) none of the other choices is correct.Answer: (A)Problem 13. What are all the values of  x  for which the series  x  −  x 2 2  +  x 3 3  −  x 4 4   converges?(A)  − 1  x  1  (B)  − 1  x< 1  (C)  − 1  x  1  (D)  − 1 <x< 1 (E) all real numbers.Answer: (D)Problem 14. If  f  (  −  1 ,  3) = 4 ,  f  x (  −  1 ,  3) = 5  and  f  y (  −  1 ,  3) =  −  2 , the linearapproximation to  f  ( − 1.3 , 3.2 )  is(A) 2.1 (B) 4 (C) 5.9 (D) 4.1(E) there is not enough information given to estimate f(-1.3,3.2).Answer: (A)Problem 15. Suppose that  f  ( x, y )  is a function such that  f  (3 ,  4) =  10 ,  f  x (3 ,  4) = 1 ,and  f  y (3 ,  4) = 3 . The equation of the tangent plane to the graph of  f  ( x, y )  at thepoint  (3 , 4)  is(A)  z = x +3 y − 5  (B)  z = x +3 y + 10  (C)  z = − x − 3 y − 5  (D)  z = − x − 3 y − 10 (E) none of the other choices is correct.Answer: (A)Problem 16. In polar coordinates, the equation of the circle  ( x − 1) 2 + y 2 =1  is(A)  r = θ  (B)  r =1  (C)  r =1+ sin θ  (D) r =2 cos θ (E) none of the other choices is correct.3  Answer: (D)Problem 17. The area enclosed by one loop of the lemniscate  r 2 =  sin 2 θ  for  0  θ   π 2  is(A)  14  (B)  34  (C)  12  (D)  35  (E) none of the other choices is correct.Answer: (C)Problem 18. Let  f  ( x,y )= x 3 y 2 . Then  f  xx + f  xy  is equal to(A)  12 x 2 y  (B)  3 x 2 +2 y  (C) x 2 + y  (D)  6 x 2 y +6 xy 2 (E) none of the other choices is correct.Answer: (D)Problem 19. The tangent plane to the surface  x 2 +  y 2 − 2 z 2 = 3  at the point (2,1,1)can be written as(A)  − 2( x − 2)+2( y − 1) − ( z − 1)=0  (B)  2( x − 2)+( y − 1) − 4( z − 1)=0 (C)  ( x − 1)+9( y − 1)+( z − 1)=0  (D)  2 x + y + z =3 (E) none of the other choices is correct.Answer:Problem 20. Given two vectors  u = i +  j − k  and  v = i +  j  + k , the dot product  u · v  is(A)  3 √   (B)  2 i − 2  j  (C) 3 (D) 1 (E) none of the other choices is correct.4
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