Math 117 Practice Midterm 3 Please work out each of the given problems.  Credit will be based on the steps that you show towards the final solution.  Show your work. Printable Solution Problem 1  Solve the differential Equation         xy' - 2x  =  x2     y(1)  =  5 Solution Problem 2 Determine if                             1         y  =  cos t -       e-t cos t                             5 is a solution to the differential equation         y''' - y'' + y' - y  =  e-t sin t Solution Problem 3  The rate of growth in customers for a small business is proportional to the product of the number of customers who eat at the restaurant divided by the total number of years that the restaurant has been in business.  After one year in business the restaurant averages 20 customers per day, and after two years in business the restaurant averages 25 customers per day.  Approximately how many customers can the restaurant expect to have per day after three years in business? Solution Problem 4 Use the sixth degree Taylor polynomial for         f(x)  =  2 cos x to approximate .  Hint:  Use the fact that         Solution Problem 5 Find the Taylor series center at x  =  c for the given function and find its radius of convergence.                         x2            f(x)  =                          (c  =  0)                       1 + x3  Solution Problem 6 Use Newton's Method to approximate the intersection of the curves         y  =  sin x        and        y  =  1 - x Solution