MATH 222 Final Exam Part I                                       Name____________

Instructions: Answer all the questions in the spaces provided.  You may not use the calculator on this part.  Each question is worth six points.

 

1.  Find  for each of the following:

  a)                                                                                       a)____________

 

 

 

 

 

  b)                                                                               b)____________

 

 

 

 

 

 

2.  Evaluate the following integral:

                                                                                      c)____________

 

 

 

 

 

 

 

3.  Evaluate the following limit:

                                                                                     d)____________

 

 

 

 

 

 

 

4.  Find the sum of the following infinite series (if it exists):

   

                                                                                              e)____________

MATH 222 Final Exam Part II                                     Name____________

Instructions:  Answer any five of the six questions on this part.  You may use the calculator for all questions on this part.  Clearly indicate which question you are not having graded.  Each question is worth 24 points.

1.   Find the Radius and Interval of Convergence of each of the following power series.

 a) 

 

 

 

 

 

 

 

 

 

 b)

 

 

 

 

 

 

 

 

 

 

 

 

  c)


2. a)  Find the equation of the line tangent to the graph of  at x = 1.

 

 

 

 

 

 

 

  b) Find all points on the graph of y = xln(x) at which the tangent lines are horizontal.

 

 

 

 

 

 

 

 

  c) Find the equation of the line tangent to the parametric curve given by  at the point (-2,5)

 

 

 

 

 

 

 

 

 

 

   d) Find all points on the parametric curve given in part c) at which the tangents are horizontal and all the points at which the tangents are vertical.

3.  A sample of a radioactive element arrives at a laboratory.  One day later the test equipment arrives and the lab assistant is able to measure that there is 10.8 kilograms of the element.  Six days later the sample is retested and the element has decayed to 8.1 kilograms.

a) What was the mass of the radioactive element when it arrived?

b) Find an expression for the mass of the radioactive element at time t with t measured in days.

c) What is the half life of the element?

d) When has the sample decayed to one kilogram?

4.  a)  Find the equation in standard form of the parabola with focus at (3,5) and vertex at (5,5).

 

 

 

 

 

 

 

 

 

  b)  Find the equation in standard form of the ellipse with foci at (3,3) and (3,7) and minor axis 4.

 

 

 

 

 

 

 

 

 

 

  c)  Identify the conic section given by the following equation and give its equation in standard form.   x2 + 4y + 4x – 8 = 0.

5.  Determine whether the following infinite series converge. Justify your answers by indicating the tests you use and showing all necessary limits.

a) 

 

 

 

 

 

 

 

 

 

b) 

 

 

 

 

 

 

 

 

 

c)


6.  a)  Using the appropriate substitution, transform the following integral into an integral of the product of trigonometric functions.  Do not evaluate the integral.

 

 

 

 

 

 

 

 

  b)  Evaluate the following integral by integration by parts. 

 

 

 

 

 

 

 

 

 

 

 c)  Evaluate