MATH 111
Fall 2006
Final Exam Name:
Please show all work on this test paper. No credit will be given without proper work to justify your answer.
1. Use the graph of f (x) to answer the following questions: (1 pt. each) a) x lim

2
  f ( x )

___________________________________________________________________ b) lim x

5
 f ( x )

____________________________________________________________________ c) lim x

9 f ( x )

____________________________________________________________________ d) State the value(s) of x for which f has a removable discontinuity: _________________________ e) For what value(s) of x is f not differentiable?___________________________________________ f) List all local extrema for f (label as max or min). g) For what interval(s) of x is f ′(x) > 0?_________________________________________________ h) For what interval(s) of x is f ″(x) < 0?________________________________________________
2.Let the function f be defined as f ( x )



 x
2 x
2 
5

4 a x x

5
,
, x x


5
5
For what values of a , if any, is f continuous at x

5 . Be sure to give your reasons . (4 pts)

3.Evaluate the following limits a. x lim

2
-

4 x


2 x
2 c. x lim

0
1
 x x
2
2 
1
(3 pts each) b. d. lim x

0 lim tan 5 x x
  
4.Use the definition of the derivative to find f

( 3 ) if f ( x )
 x
1

1
.
6 x x x
2

4

1
(4 pts)

5.Differentiate the following functions. a. y

9 x
 x
 tan( x ) b. f ( x )
 x cos( x ) x
4  x

3 c. y

3 sin( x
3
)
DO NOT SIMPLIFY. d. Find d
2 dx
2 y if y

( 3 x

2 )
2
3
6.Find the equation of the tangent line to x y
2  cos(
 y )
 y

7 at (1,2).
(3 pts each)
(5 pts)

7.A small balloon is released at a point 150 feet away from an observer, who is on ground level. If the balloon goes straight up at a rate of 8 feet per second, how fast is the distance from the observer to the balloon increasing when the balloon is 50 feet high? (6 pts)
8. Given that x ≠ 0, lim x

0 f ( x )
 
, x lim

-
 f ( x )

( x

6 ), lim x
  f ( x )

( x

6 ) , and that f (-4) = -13.5; f (2) = 0; f

( x )

( x

2 )
2
( x

4 )
and f

( x )

24 ( x

2 ) x
4
. Find the intervals where f is increasing or decreasing, x
3
CU or CD, and then sketch the graph of f below. The equations of all the asymptotes of f (x) are:
___________________________________ (Do not try to find the function f .) (total 5 pts)

9.Draw the graph of a function that has a critical point at decreasing OR decreasing to increasing at x
 
1 . x
 
1 but does not change from increasing to
(2 pts)
10.
The functions y
 f ( x ) and y
 f

( x ) are graphed below. Clearly label the functions and state 2 different reasons to support your choice. (4 pts)
11.
A box manufacturer wants to construct a box where the length is three times its width. The volume is to be 243 cubic inches. If the material to be used for the top and the bottom costs twice as much as the material for the sides, find the dimensions for the box that will minimize its cost. Use calculus to find and show that your answer is the minimum. (6 pts)

12.
Find f ( x ) if f

( x )

6 x of f is 10 when x

2 .

12 and it is known that
(4 pts) f ( 2 )

5 and the slope of the tangent line for the graph
13.
Find or evaluate the following integrals. a)
 cos( x )
 sec( x ) tan( x )
 
2 dx
(4 pts each) b)
 8
1
3 x

4 x
2 dx c)
 1
0 x ( 4 x 2 
2 ) dx d)



6
3 cos ( 3 x) dx e)

(sin
3 y )(cos y ) dy
14.
If F ( x )
 
x
3 cos (t
2
) dt , find F

( x ) . (4 pts)
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