5 A GRAPHING CALCULATOR IS REQUIRED FOR THIS QUESTION You are permitted to use your calculator to solve an equation find the derivative of a function at a point or calculate the value of a definite integral However you must clearly indicate the setup of your question namely the equation function or integral you are using If you use other built in features or programs you must show the mathematical steps necessary to produce your results Your work must be expressed in standard mathematical notation rather than calculator syntax Show all of your work even though the question may not explicitly remind you to do so Clearly label any functions graphs tables or other objects that you use Justifications require that you give mathematical reasons and that you verify the needed conditions under which relevant theorems properties definitions or tests are applied Your work will be scored on the correctness and completeness of your methods as well as your answers Answers without supporting work will usually not receive credit Unless otherwise specified answers numeric or algebraic need not be simplified If your answer is given as a decimal approximation it should be correct to three places after the decimal point Unless otherwise specified the domain of a function f is assumed to be the set of all real numbers for which f x is a real number t hours vp t meters per hour 0 0 3 1 7 2 8 4 0 55 29 55 48 The velocity of a particle P moving along the x axis is given by the differentiable function up where up t is measured in meters per hour and t is measured in hours Selected values of up t Particle P is at the origin at time t 0 are shown in the table above of A a Justify why there must be at least one time t for 0 3 t 2 8 at which up t the acceleration of particle P equals 0 meters per hour per hour b Use a trapezoidal sum with the three subintervals 0 0 3 0 3 1 7 and 1 7 2 8 to approximate the value 2 8 6 vp t dt c A second particle Q also moves along the x axis so that its velocity for 0 t 4 is given by vo t 45 t cos 0 063t meters per hour Find the time interval during which the velocity of particle is at least 60 meters per hour Find the distance traveled by particle Q during the interval when the velocity of particle is at least 60 meters per hour d At time t 0 particle is at position 90 Using the result from part b and the function vo from part c approximate the distance between particles P and Q at time t 2 8