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.      Printable Key    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.      Solution        False. It would be true if we said that g'(f(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.     Solution True,  since                             f '(x)         g '(x)  =                                                   f(x) is positive if both f ' and f are positive PROBLEM 2  Calculate the derivatives of the following functions.A)            d                        (2x)1-x                dx      Solution         (2x)1-x  =  e(1 - x)ln(2x) Now use the chain and product rules.  The derivative of          (1 - x)ln(2x)  is          (1 - x)(1/x) - ln(2x)  =  1/x - 1 - ln(2x) Hence the derivative of           e(1 - x)ln(2x) is          [1/x - 1 - ln(2x)](2x)1-x   B)        d                      eln(sin x)             dx      SolutionFirst use the inverse property of e and ln to get         eln(sin x)  =  sin x Now the derivative is simply          cos x      C)         d         23t                                                    dx        t      Solution We use the quotient rule to get           t(23t)' -23t                                          t2 Now use the chain rule and the fact that          bx '  =  bx ln b to get           3t ln2 (23t) - 23t                                                              t2 D)      d                  [ arcsin(1-3x) - (ln x)(arctan x)]          dx Solution We use the chain rule for the first part and the product rule for the second part to get        PROBLEM 3 Find the following integrals A)            Solution Let          u  =  1 - x    du  =  - dx        x  = 1 - u The substitution produces           B)         Solution  Let          u  =  2 - 3x    du  =  -3 dx The substitution produces           C)           Solution Let          u  =  4 - x2     du  =  -2x dx             D)          Solution   We first complete the square            x2 + 2x + 10  =  x2 + 2x + 1 - 1 + 10         =  (x + 1)2 + 9 Now integrate                 u2  =  (x + 1)2 / 9        u  =  (x + 1) / 3        du  =  1/3 dx We get                   PROBLEM 4    Let  f(x) = 2x3 + 4x + 5 A.  Prove that f(x) has an inverse.      Solution We calculate          f '(x)  =  6x2 + 4 which is always positive, hence f(x) has an inverse. B.    Find       d                             f -1(11)                     dx      Solution   We use the inverse formula              d                                     1                       f -1(11)   =                                                dx                              f '(f -1(11)) Since          f(1)  =  11 We have         f -1(11)  =  1 and          f '(1)  =  6(1)2 + 4  =  10 Hence              d                                     1                         1                       f -1(11)   =                                  =                             dx                              f '(f -1(11))                10   PROBLEM 5    Show that          d                      tanh x   =   sech2 x          dx      Solution             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