Calculus 1  Hour Exam 1   -   Math 221     Fall, 2011

 

 

 

1.

 

 

 

Answer the questions by looking at the graph given of y = f(x).  Note that the answer

may be a number or ∞ or -∞ or “does not exist”.  Give an approximation if an exact

answer is not possible.  It is not necessary to show your work for this problem.

 

Find:  f ‘(-6),   f ‘(6),   lim f(x)     lim f(x)       lim f(x) 

                                  x→-8         x→6           x→-2+

 

The graphs contain discontinuities including jumps.  (Graphs not shown)

 

 

 

 

 

 

 

 

2.

 

Evaluate each limit as a number, as ∞ or -∞ or “does not exist”.  Show your work or give

a brief explanation as to how you got your answer.  These limits should be done without

use of L’Hopital’s Rule, which we have not covered yet.

 

a.       lim  (x+ 1) / (x + 4)2                             Answer:  - ∞

           x→-4

 

b.      lim (x2 – 1) / |x – 1|                              Answer:  -2

          x→1-             

 

c.       lim ex                                                    Answer:  ∞

    x→∞

 

d.      lim [1/(3+h) – 1/3] / h                          Answer:  -1/9

    x→0

 

 

 

 

 

3.

 

 

Find each derivative  f ‘(x).  You may use any derivative rule including the chain rule.

 

a.       f(x)  =  3x2  -  √x  + ex                                 Answer:  6x  -  ½ x-1/2  + ex

 

b.      f(x)  =  x g(x)  and  g(1)  =  4,  g ‘(1) = 7,  find  f ‘(1)         Answer:  11

 

c.       f(x) =  2 sin x / (3 – tan x)   

 

 Answer:  [2cos x (3 – tan x) – 2sin x (-sec2 x)] / (3 – tan x)2

 

 

 

4.

 

 

a.        State the definition of the derivative   f ‘(a).

 

b.      Use this definition to show that  f ‘(1)  =  ½   for  f(x) = √x

 

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