Selected Solutions for Sample Final

8.  h(x)=f(g(x)) so h'(x)=f'(g(x)) g'(x)

    The g and g' functions are on the "inside".  The only values of
    g we know are when x=2.  Thus A and B can be ruled out, because
    x=1 in those.

    h(2)=f(g(2))=f(1)=2
    h'(2)=f'(g(2)) g'(2) = f'(1) 6 = (3)(6) = 18

    Thus only C is correct.

11. Exponential decay decays by the same proportion (or percentage)
    at each point in time.  In our case

    2.10/3.00 = 0.7 = 70%

    Thus (6.10)(0.7) = 4.270 and so the answer is D.

12. If a graph is concave up then f''>0.  This means f' is increasing.
    f' is the slope of the tangent line, so C is correct.

17. Make the substitution
                                       2
                                  u = x  + 4
    Then
                                   du
                                  ---- = 2 x   
                                   dx

    It follows that

                          /                   /
                         |    x              |
                         |  ------ dx = 1/2  |  1/u du
                         |   2               |
                        /   x  + 4          /

    Therefore the answer is C.

18. The formula for future value of an income stream is

                             M
                            /
                           |
                           |   S(t) exp(r(M - t)) dt
                           |
                          /
                            0

    To find the answer we set S(t)=A a constant to solve for,
    M=5 and r=0.07.  Thus

                                5
                               /
                              |
                     50000 =  |   A exp(0.35 - 0.07 t) dt
                              |
                             /
                               0
    and we find that

                            50000
              A = --------------------------  = 8351.875520 $/year 
                     5
                    /
                   |
                   |   exp(0.35 - 0.07 t) dt
                   |
                  /
                    0

    Therefore B is the correct answer.

19A.  The line passing through the point (q,p)=(300,4) with
      slope (-10/.5) is given by

        q - 300  = -(10/.5)(p-4)

      Thus q=380-20p.

19B.  R = pq = p(380-20p) = 380p-20 p^2

19C.  R'(p) = 380-40p

19D.  Critial points when R'(p)=0.  Thus critial when 380-40p=0
      or when p=19/2.

19E.  Since R''(p)=-40 < 0 then this is a maximum by the second
      derivative test.  Thus the price that maximizes revenue
      is p=19/2 = $9.50.

20A. For part A you are approximating the integral

                                    6
                                   /
                                  |
                                  |   r(t) dt
                                  |
                                 /
                                   0


    using a left hand sum with three terms.  Since the interval [0,6]
    is of length 6 then the width of each rectangle is 2.

        A_left = r(0)*2+r(2)*2+r(4)*2 = 90.62741699