Calculus 1  Final Exam    -   Math 221   Fall, 2008

 

 

 

 

1.

 

 

 

Evaluate each of the following limits or state and justify that it does not exist.

 

a.  lim  √{ [( 1 + x – x2 )  - (1 - x)]/2x}          b.  lim [ (x2 – 3x – 4 ) / | x + 1| ] 

    x → 0                                                            x → 3                               

 

Answer for a:  1/2,    Answer for b:  -1

 

 

 

 

 

 

 

 

 

2.

 

Let  f(x) =  2x3  -  9x2  +  12x

 

  1. Find all zeroes, critical points of  f  and the intervals on which the function  f  is

      increasing and those on which it is decreasing

  1. Test critical points for local maxima and local minima of f.
  2. Find absolute maxima and absolute minima of  f  if exist (show your work even

if absolute extrema do not exist)

  1. Find intervals of concavity (up and down) and inflection points of  f.
  2. Sketch the graph    y  =  f(x)

 

Answers:  a.  Zero at  x  =  0,  critical points = 1 and 2,  increasing x < 1,

                decreasing  1 < x < 2,  increasing  x > 2

b.      Local max at x = 1,  local min at x = 2

c.       lim f(x) = ∞       lim  f(x)  =  - ∞

                      x → ∞                x → - ∞

d.      Concave up for  x < 3/2,    concave down for  x > 3/2, inflection point at 3/2

e.       Not shown

 

 

 

 

 

3.

 

 

 

Let     f(x)  =   (cos x )sin x    ,   0  <  x < π/2

 

Evaluate  f ‘ (x)    Hint:  Use logarithmic differentiation.

 

Answer:   f ‘ (x )  =  (cos x )sin x  [ cos x  ln(cos x)  -  sin2x / cos x ]

 

 

 

 

4.

 

 

Find whatever vertical, horizontal, and slant asymptotes for the function

 

     f(x)  =   ( 2x3  -  5x2  +  4x ) / ( x2  - 2x  + 1)

 

Answers:  No vertical asymptotes, Vertical asymptote at  x = 1

                Slant asymptote   y = 2x – 1

 

 

 

5.

 

Use  L’Hopital’s rule to find the limit     lim [ (1 + x2 ) / ( ex – cos x ) ] 

                                                               x → 0    

Answer:  0

 

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