Math 140 Final Exam Practice Problems
  È

   


a)
b)  c)  d)
e) Does not exist


-----------------------------------------------------------------------------  
2.
Find the derivative of  ab  

  
    
      
a)
b)
 a  b
 a  b
    

c)
d)
e) None of these
  
 a  b
-----------------------------------------------------------------------------      

3.
lim

  
a)  b)  c)  d)  e) 
-----------------------------------------------------------------------------4.
If     2 find  
a)      b)     
c)          d)         
e) None of these
-----------------------------------------------------------------------------
5.
Find the intervals on which the function   
    
is continuous.
a) a   b a b b) a    b a b
c) a   b d) a    b a   b a b
e) None of these
-----------------------------------------------------------------------------6.
An antiderivative of the function      is


a)    b)    c)   


d)     e) None of these
1.
lim
7.
An inflection point on the graph of         
occurs when  
a)   b)   c)  d)  e) None of these
-----------------------------------------------------------------------------8.
If    ab is defined by the equation
      
then find  at the point a b



a)  b) 
c)
d) 
e) None of these



-----------------------------------------------------------------------------
9.
Find the tangent line to the curve  
at the point

a b Express the answer in the form     



a)      b)    c)    





d)    
e) None of these


-----------------------------------------------------------------------------10.


( ˆ  ‰  
a)  a  b  

b)
   

a  b
c)



a  b
  e) None of these
d)

-----------------------------------------------------------------------------11. Solve the initial value problem


    ab  







a)     

b)     





 

 
c)    

d)    
e) None of these





-----------------------------------------------------------------------------12. The curve        is decreasing on the
interval or intervals
a) a    b b) a   b c) a    b
d) a  b e) None of these
-----------------------------------------------------------------------------
13.
Find the total area of the region between the curve and the
x-axis:       ;     
a) 
b) 
c) 
d) 
e) None of these
-----------------------------------------------------------------------------14. Let   a  b a  b . Then  
 a  b
a  b a  b
a)
b)
a  b
a  b
  a  b
a  b a  b
c)
d)
e) None of these
a  b
a  b
-----------------------------------------------------------------------------15. Say ab and ab are related by the equation    where
 is a constant and  is measured in feet,  is measured in
pounds, and  is measured in minutes. Find  given that
 is decreasing at the rate of  feet/minute,    feet
and    pounds
a)  900 pounds/minute
b) 2.25 pounds/minute
c) 4 pounds/minute d) 100 pounds/minute
e) None of these
-----------------------------------------------------------------------------16. The 1st quadrant area bounded by        and the x-axis
is given by the integral
a) (  


b) (  
d) ( ˆ  ‰





c) ( ˆ   ‰


e) None of these

-----------------------------------------------------------------------------17.
Evaluate (  ab ab  
 ab
  ab
a)

b)




 ab

c)   abab  
d) 



e) None of these
-----------------------------------------------------------------------------18. The velocity of a body moving along a coordinate line is
given by       Find an equation    ab for the body's
position at time  if ab   
a)        b)      c)       
d)     
e) None of these
-----------------------------------------------------------------------------19. For the given cost and demand function, find the production
level that will maximize profit

 ab        ab   

Round the answer to the nearest integer.
Use  ab   ab   ab where  ab  ab
-----------------------------------------------------------------------------20. The region bounded by the curve        and the x-axis
is revolved about the y-axis. Find the integral which gives the
volume of the solid that results.
-----------------------------------------------------------------------------21. The area between the curves    and     
is revolved about the x-axis. Find the integral which gives the
volume of the solid that results.
-----------------------------------------------------------------------------22. Consider the curve given by         On what
intervals is the curve increasing? On what intervals is the curve
concave upward?
-----------------------------------------------------------------------------23. The volume of a hot air balloon is increasing at a rate of
210 ft /min. At what rate is the radius of this balloon increasing

at the instant when the radius is  ft? (   

Solutions Practice Final Exam)
1.
2.
  È   È
  




.
     È
a  bˆ  È‰
  È



lim


 The answer is A.
   È

  È
a  b  a  ba  b

a  b
    a      b

a   b
  ab 
    

 The answer is C.
 a  b
3.
      




 
lim
lim
lim

 
 
  
The answer is E.
4.
       
               
          The answer is C.
5.
The function is discontinuous when        
a  ba  b       and    
Therefore the function is discontinuous at    and   and the
function is continuous on the 3 intervals a    b a   b
and a b The answer is D.
6.
Use (            where    
      
(        (    




      The answer is A.

7.
              An inflection point
occurs when           
The answer is B.
8.
                 
          a  b   a  b
 a  b
  

  


At a b  
   The answer is D.


9.
 
10.




( ˆ  ‰   ( ˆ    ‰ 
a  b  ab
    






a   b
a   b
a   b 


At a b  

 

a  b

           Since the tangent line




goes through a b   ab         and





the tangent line is     . The answer is C.


The answer is E.



   


11.
  ( ˆ  ‰  
12.
       a  b   when     







ab
ab   
    
 

ab




          








and     

. The answer is A.



------------   ----------------  ---------------  
  
  
The curve is decreasing from   to  
The answer is A.
13.
The area is shown below.
40
20
0
1
2
3
4
5
6
7
20
The total area is equal to

( ˆ    ‰  ( a    b 




      
The answer is A.
14.
a  b aba  b  a  b
 

a  b
a  b ’a  b  a  b“


a  b
a  b a  b
 The answer is D.
a  b
15.
abab   
16.
The area is shown below with one rectangle.



  or       


 
a  bab



        
 
 


The answer is C.
(9,3)
dy
x=9
    art x  left xb  a   b
The area is (


17.
ˆ   ‰  The answer is C.

Let   ab     ab 


(  abab 
 (  abŒ 

18.
19.


•ab   (   



   ab   The answer is B.



       ( a  b       

ab    ab  ab         and
ab       The answer is A.


 ab  ab   


 ab   
 ˆ     ‰ 


 
      


  Œ
 •   

     

  ab         
  

20.
        a  ba  b   
    The region shown below ranges from    to   
10
2
x
8x
10
12
0 1
2
3
4
5
6
7
8
4 10
When the rectangle shown is rotated about the y-axis,
what results is a cylindrical shell whose volume is
  a      b 
The total volume is ( ˆ      ‰ 


21.
The points of intersection occur when
              a  ba  b  
      
The area ranges from     to    and is shown below with
one rectangle.
10
10
x
x
2
5
6
0
3
3
2
1
0
1
x
2
3
4
5
5
When this rectangle is revolved about the x-axis, what
results is a disk with a hole whose volume is
  a    b  ’a  b   “ 
The total volume is ( ’a  b   “


22
               
  a  b     when    and   
Check the intervals a   b a b and a b using test values.
For example,   ab   ab a  b   
graph is decr. on a b Making other test values, we see that
-------- 0 ---------- 3 ---------incr
incr
decr
Therefore  ab is increasing on the interval a   b 
        a  b   when    or  .
Make test values to check the intervals a   b a b and
a b For example, when      ab   abab  
 curve is concave downward on a b
Making other test values we can see that
--------- 0 --------- 2 -------down
up
down
The graph is concave upward only on the interval a b 
23.
 
 


 
 
     



 




ft/min .
 



ab