Math 1920, Calculus II

Sample Gateway Exam

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Instructions.  Choose the response which best answers the question. A score of 7 out of 10 is required for successful completion of MATH 1920.

1. Which of the following is the value of the limit

   lim x® ¥  xe-2x

   (a)

   -2

   (b)

   -1

   (c)

   0

   (d)

   1

   (e)

   does not exist

2. Which of the following is the value of the limit

   lim x® 0+  ( 1+x) 1/x

   (a)

   0

   (b)

   1

   (c)

   p

   (d)

   e

   (e)

   e2

3. Which of the following is the value of the limit

   lim x® ¥    ln( x+1) ln( x)

   (a)

   -2

   (b)

   -1

   (c)

   0

   (d)

   1

   (e)

   does not exist

4. What is the derivative of f( x) = ln( x+2)e^-x?

   (a)   -e-x x+2     (b)   e-x x+2 -e-xln( x+2)     (c)  e-x æ ç è   ln(x+2) x+2 ö ÷ ø     (d)   e-x x+2     (e)  ln( 3) -xe-x-1

5. What is the derivative of

   F( x) = ó õ x 1  t1/2  e-t  dt

   (a)  x1/2e-x    (b)  -x1/2e-x    (c)  x1/2e-x-e-1    (d)  2x-1/2e-x    (e)  -2x-1/2e-x

6. Evaluate òe^{sin( x)}   cos( x) dx with a substitution. What is the result?

   (a)  ecos( x) sin( x) +C    (b)  -ecos( x) sin( x) +C    (c)  esin( x) +C    (d)  ecos( x) +C    (e)  esin( x) cos2( x)

7. Evaluate ò tan( x) dx with a substitution. What is the result?

   (a)

   sec2( x) +C

   (b)

   ln| secx| +C

   (c)

   ln| cosx| +C

   (d)

   ln| tan(x) | +C

   (e)

   xtan( x) +C

8. Evaluate òxsin( 2x) dx using integration by parts. What is the result?

   (a)	sin2x - xcos2x + C		(d)	sin2x + xcos2x + C
   (b)	sin2x + cos2x + C		(e)	-sin2x + xcos2x + C
   (c)	-sin2x + xcos2x + C

9. Evaluate òtan^-1( x)  dx using integration by parts. What is the result?

   (a)   1 2 ( tan-1x) 2+C    (b)  xtan-1( x) - 1 2 ln( x2+1) +C    (c)  tan-1( x) -ln( x2+1) +C    (d)  xtan-1( x) - x x2+1 +C    (e)  tan-1( x) -xln( x2+1) +C

10. Evaluate

    ó õ   dx ( x2+1) 3/2

    with a trig substitution. What is the result?
    

    (a)   1 ( x2+1) 1/2 +C    (b)   1 x2+1 +C    (c)   Ö   x2+1   +C    (d)   x ( x2+1) 1/2 +C    (e)  sec( q)+C