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###      NO CLASS WEDNESDAY, December 9, 2009              ###
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###    Test four will still be on the last day             ###
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###   of class which will be Friday, December 11, 2009.    ###
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### The practice test and answers is available on the web. ###
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Instructor Hentzel
Office Phone: 515-294-8141
E-mail:   hentzel@iastate.edu
Math Department Fax:      515-294-5454
http://www.math.iastate.edu/hentzel/class.165.09

Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. 

Wednesday, December  9    REVIEW  


Practice TEST 4     

The actual TEST 4 is Friday, December 11, 2009

During Class 7:30 to 9:00 AM


1.  (a)  Give the definition of y = ln(x).



                                     x
    (b)  Give the definition of y = e



                                        x
    (c)  If (a,b) is on the curve, y = e , 


         what is the derivative of y = ln(x) when x = b.



2.  Express the following as rational numbers.

              2 ln(3/4)
    (a)     e 

        
                       3
    (b)     ln( Sqrt[ e  ] )



    (c)     Log   (27/8) 
               3/2


                       3
    (d)    Log  Sqrt[a   ]
              2  
             a


3.  Find the following derivatives.


    (a)   y = ln(x)



               x
    (b)   y = e





    (c)   y = Sinh[x] 




    (d)   y = Cosh[x] 




    (e)   y = Tanh[x]




    (f)   y = ArcSin[x]




    (g)   y = ArcCos[x]




    (h)   y = ArcTan[x]




    (g)   y = ArcSec[x]



                              /
4.  Derive the solution of   y  = -a y + b    (a >= 0)























5.  A population started with 64 individuals and every 13 years 

the population doubles.  

(a)   Write the equation
















6.  Room temperature is 20 degrees centigrade.  A hot brick starts

at 200 degrees centigrade and cools to 160 degrees centigrade

in 5 minutes.  

(a)  How hot will it be after a total of 30 minutes?


(b)  When will it be 25 degrees centigrade?

















7.  A brick starts at 300 degrees at t = 0.

At t = 1 it is 210 degrees.

At t = 2 it is 192 degrees.

What is the ambient temperature?
















8.  Find the derivative of

               13      15    2
          (x+3)   (x-5)    (x )
    y =  ---------------  e
                 4
           (x+29)

9.  Find the derivative.  Do not simplify.

               2
             (x  + 6 x + 2 )
    (a)     e


                          2
    (b)     ln( 14 x + 3 x  )



    (c)     ArcTan[ 3/x ]


    (d      Sinh[ Sqrt[x] ] 


10.  Evaluate these integrals.

               x/2
              e
(a)   INT   --------    dx
                 x
            1 + e


               Sinh[x]
(b)   INT   -----------  dx
               Cosh[x]


                  1
(c)   INT  ---------------  dx
                _        _
               |     x  2 | 
           Sqrt| 1-(---)  | 
               |_    3   _|




               Sqrt[x]
              e 
(d)  INT  --------------  dx
             Sqrt[x]


11.  Give the formula for interest at r% for n years for these 

     compounding frequencies.  

    (a)  Yearly


    (b)  Monthly

    
    (c)  Daily


    (d)  Instantaneously

                                                x   -x
                                               e - e
12.  (a)  Find the inverse function for y =   --------
                                                 2
                                                                 x   -x
                                                                e - e
      (b)  Find the derivative of the inverse function of y =  --------
                                                                  2 


13.  A tank has 100 gallons of water and contains 5 pounds of

dissolved salt.  Water with 1 pound of salt per gallon flows into

the tank at 2 gal per minute and the well mixed solution drains

out of the tank at 2 gallons  per minute.

  (a)  Write the differential equation for the amount of salt in the

tank at time t.

  (b)  How much salt will be in the tank after 30 minutes.