MAC 2311      CALCULUS I          EXAM 5B                 Dr. Rapalje --  Seminole State College of Florida

Show all work on separate paper.  Turn in ALL worksheets.   Give all irrational answers in exact (radical) form.   When you use the calculator, say so, and explain what you did.

In 1 – 5, find the derivatives and simplify. Show all work!         

         1.                                    2.        

         3.                                                   4.        (Use logarithms)

         5.           (Use log differentiation.  You need NOT find LCD, etc.)

In 6 – 8, evaluate the integrals. Show all work by algebraic techniques!

         6.                           7.                       8.     

         9.         Solve the equations.  Give answer in exact form and also calculate decimal approximations.

                     a)                      b)     

       10.        Derive formulas for the following by using logarithmic and implicit differentiation.  (As in #4.)

    a)      Find the derivative of  , where a is a constant.

        b)      Find the derivative of  , where x (of course!) is the variable.

       11.         Solve the differential equation:  .

       12.         Solve the differential equation:   given the condition that y(1) = 20.

       13  a)   How much money must be invested now at 6% annual rate compounded semiannually, in order to amount to $100,000 in 20 years?  (See formulas below.)

              b)   How much money must be invested now at 6% annual rate compounded continuously, in order to amount to $100,000 in 20 years?

       14.         A certain type of bacteria increases continuously at a rate proportional to the number present.  If there are 500 present at a given time, and 1000 present 2 hours later, how many will there be 5 hours from the initial time given?

        15  a)   What is the underlying mathematical assumption for the exponential growth formula   ?

              b)   Give the major fallacy of population growth and decay solutions.

              c)   Beginning with , show that the solution of this differential equation is .

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