Name                                       MATH 106 PRACTICE MIDTERM 1   Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.    PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true, explain why or state the proper theorem. A)  Suppose that f(x) and g(x) are differentiable inverse functions, and f '(2) = 3.  Then g'(2) = 1/3.    B)   If f is a differentiable function such that both f and f ' are positive for all x, then g(x)  =  ln(f(x)) is increasing for all values of x.     PROBLEM 2  Calculate the derivatives of the following functions.A)            d                        (2x)1-x                dx   B)        d                      eln(sin x)             dx   C)         d         23t                                                    dt        t   D)      d                  [ arcsin(1-3x) - (ln x)(arctan x)]          dx   PROBLEM 3 Find the following integrals A)          B)         C)           D)           PROBLEM 4    Let  f(x) = 2x3 + 4x + 5 A.  Prove that f(x) has an inverse.      B.    Find       d                             f -1(11)                     dx      PROBLEM 5    Show that          d                      tanh x   =   sech2 x          dx    Extra Credit:  Write down one thing that your instructor can do to make the class better and one thing that you do not want changed with the class.    Questions, Comments and Suggestions:  Email:  greenl@ltcc.edu