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)   (10 points)  The differential equation  is a homogeneous differential equation   B) (10 points)  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)   (10 points)        d                                 (2x)1-x                         dx Solution B)    (10 points)       d                                  eln(sin x)                         dx Solution     C)   (10 points)        d       23t                                                                          dx        t PROBLEM 3 Find the following integrals A)   (10 points)    B)    (10 points)  PROBLEM 4   You have been called as an expert witness in the case involving the recent murder that occurred in room E106.  It is clear that at the time of death the victim was healthy with a temperature of 98.6 degrees.  It is also know that a human body in this situation will cool down to 90 degrees in one hour.  When the body was discovered at 10:00 PM the corpse had a body temperature of 85 degrees.  During the entire day, the temperature of the room was a constant 65 degrees.  A) (10 points) Use the Newton’s Law of Cooling (the rate of change of the temperature of the body is proportional to the difference between the body’s temperature and the ambient temperature) to write down a differential equation for this situation.   B) (10 points) Solve the differential equation from part A. Solution C) (10 points) What was the time of death?   PROBLEM 5    Let  f(x) = 2x3 + 4x + 5 (10 Points)  Prove that f(x) has an inverse. Solution (10 Points)  Find       d                                          f -1(11)                                  dx Solution   PROBLEM 6   Solve the following differential equations A.  (15 Points) y(1 + x2)y'  -  x(1 + y2)  =  0,    y(0)  =  Solution B.  (15 Points)                 x3 + y3                           y'  =                                                              xy2   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