Calculus 2        Final Exam    -   Math 231   Spring, 2012

 

 

 

 

1.

 

Mark  A  if the series converges absolutely,  C  if the series converges conditionally,

and  D  if the series diverges.

 

      ∞                                     ∞                              ∞

 a.  ∑(˗1)n (ln n)4 /n2        b.  ∑  (˗1)n / (n+1)1/3      c.  ∑  (˗1)n √n! / 10n

    n=1                                  n=2                           n=1

 

Answers:  a.   A        b.    C      c.  D

 

 

 

2.

Determine whether each integral converges or diverges.

 

            ∞                                 0

a.       ∫ dx/x                     b.  ∫ x e-x dx        Answers:  a. Diverges      b.  Diverges

1                               ˗∞

 

 

 

3.

 

 

 

                                               ∞

Find the value of the series    Σ 1 / n3n              Answer:   ln 3/2

                                              n=1

 

 

 

 

4.

 

 

After making a common trig substitution, the integral   ∫ (1 + x2)3/2  dx  becomes

 

a.      ∫ cos3θ dθ + C     b. ∫ sin3θ dθ + C      c. ∫ cos4θ dθ + C     d. ∫ sec5θ dθ   + C

      e. ∫ sec4θ tanθ dθ  +  C                                Answer:  d

 

 

5.

                                                                                           

Evaluate   ∫ cos2x sin3x dx            Answer:   I  =  (1/5) cos5 x  ˗ (1/3) cos3x  +  C

 

 

 

 

 

 

6.

 

 

 

Find the form of the partial fraction decomposition of   1 / x2 (x+1)(x2 + 4)

 

a.         A/x + B/x2 + C/(x+1) + D/(x2+4)  

b.         A/x2 + B/(x+1) + C/(x-2) + D/(x+2)

c.         A/x2 + B/(x+1) + (Cx +D) / (x2+4)

d.        A/x + B/x2+ C/(x+1) +D/(x-2) + E/(x+2)

e.         A/x + B/x2+ C/(x+1) +(Dx+E)/(x2+4)

 

Answer:  e

 

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