Final Examination Mathematics 120 Advanced Mathematics with an by broverya75

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									                       Final Examination: Mathematics 120
       Advanced Mathematics with an Introduction to Calculus
  June 2007 and January 2008                                                                                          Version 1.

  Time: 2 hours.                                                                      Formulae: see the last page.
  Scientific calculator required.
  NOT permitted: graphing or programmable calculators.

• Please check that there are 14 pages, including this page.
• There are three parts to the test, Parts A (17 points), B (18 points) and C (25 points).                           Total points: 60.
• Part A appears on pages 2 to 4 and consists of 17 short problems.
  Write your answers clearly in the space provided.
  Answers will be marked right (1 point) or wrong (0 points), with no partial credit.
  Scrap paper is available.
• Part B appears on pages 5 to 7 and consists of six short-response problems, worth 3 points each.
• Part C appears on pages 8 to 12 and consists of six long-response problems, worth 5 points each.
  Work only FIVE of these problems.
• For Parts B and C work must be shown in the space provided.
  Marks will be assigned for clarity of explanation / presentation.
• If you score at least 42/60 on this test and you provide all the information requested below, then you will be
  exempted from writing the Mathematics Placement Test at UNB and other participating universities.


  If your score is at least 42/60, do you want your school to send your name and mark
  to the University of New Brunswick, for release to participating universities?

   Circle your answer:              Yes              No

   Check participating universities you wish to receive the mark:                                     Mount Allison University
                                                                                                      Saint Mary’s University
                                                                                                      University of New Brunswick

   Print Name:             .........................................................................................................
                           Last                             First                            Middle

   Date of birth:          Day: ........                    Month: ........                 Year: ........

   School:                 ..............................................................          Teacher’s initials:        .............

   Signature:              ..............................................................          Mark out of 60:


   email (optional):       ..............................................................
  This examination was prepared at the University of New Brunswick, Fredericton, in consultation with a
  committee of NB teachers.
Name: .....................................................................

                                 PART A: NO partial credit – 17 points.
            Work on scrap paper provided. Put your answer in the box or on the graph, as directed.

                                     One point for each completely correct box or graph.




                         4
    1. Evaluate               (−2j + 1)
                        j=2




    2. An arithmetic sequence satisfies t2 = −3 and t5 = 9.
        Find the common difference.




    3. A sequence is defined by the rule
       t1 = 2, t2 = 5, and tn = tn−2 + tn−1 for n ≥ 3.                               t4 =
        Find t4 .


    4. A geometric sequence has t4 = 3 and t5 = 45.
        Find the common ratio.




    5. Find the exact value of the sum of the infinite geometric series

          1  1   1
            + +    + ... .
          3  6  12


                                                                           n
    6. A sequence is defined by the rule tn = 2 +                              .
                                                                          n+1
        Find lim tn .
                n→∞




                      x2 − 4x + 3
    7. Find       lim             .
                 x→ 3    x−3




                                                                              2
                                                                                           √
 8. Find the range of the function f (x) = 2 −                                              x.




 9. The function f (x) = x3 − x2 − 5x − 3 satisfies f (3) = 0.
                                                                                                 f (x) =
    Write f (x) in factored form.




10. If f (x) = x2 and g(x) = x + 2 then what is the value of
                                                                                                 f (g(0)) =
    f (g(0)) ?




11. The graph shows the tangent to the curve y = f (x) at the point (6, 3) .
    What is the value of f ′ (6)?

                       y
                           8
                           7
                                                                           y = f(x)              f ′ (6) =
                           6
                           5
                           4
                                                              (6, 3)
                           3
                           2
                           1
                                                                                       x
   −3   −2        −1            1        2        3   4       5        6   7       8   9
                       −1
                       −2
                       −3




12. The graph of a function y = f (x) is shown below. Carefully sketch in the graph of y = |f (x)|.

                                             y
                                             5
                                             4
                                             3
                                             2
                                             1
                                                                                       x
   −4        −3            −2       −1                    1            2       3       4
                                             −1
                                             −2
                                             −3
                                             −4
                                             −5
                                             −6




                                                                                             3
                                                         4x2 − 4x + 1
13. What is the value of                                  lim         ?
                                                  x→ − ∞    x2 − 9




14. Sketch the graph of the function y = (x − 1)3 + 2 .

                                          y
                                          6

                                          5

                                          4

                                          3

                                          2

                                          1

                                                                                    x
       −6    −5   −4   −3    −2     −1        0       1    2       3   4    5   6
                                         −1

                                         −2

                                         −3

                                         −4

                                         −5




15. If z1 = 2 + 3i and z2 = 4 + 2i , then what is the value of z1 + z2 ?




16. Represent the complex number w = 2 − 2i as a position vector on the Argand plane.

                                   Im
                                  3i


                                  2i


                                   i

                                                                           Re
        −3        −2        −1      0             1            2       3

                                  −i


                                 − 2i


                                 − 3i




17. What is the value of |2 − 2i| ?




                                                                                        4
              PART B: Short response problems – 18 points.

                   Show your work clearly in the spaces provided.
        Marks will be awarded for method, not for the final answer alone.


                         Each problem in this section is worth 3 points.


1. Algebraic solution required.
  On his first birthday, Robert’s mother put 10 small stones in a corner of the garden. On Robert’s
  second birthday, his mother added twenty stones, and on his third birthday she added thirty stones.
  Each year, the number of stones that Robert’s mother added to the pile was equal to ten times
  Robert’s age.

   (a) Write down a formula for the total number of stones in the pile after Robert’s nth birthday.




   (b) How many stones were in the pile after Robert’s 19th birthday?




2. Algebraic solution required.
                 √
   Solve for x :   x−1 = x−7 .




                                                5
3. (a) Sketch the graphs of each of these functions on the grid provided.

                   1
        f (x) =      + 1 and g(x) = |x − 1|.
                   x

                                              y
                                              6

                                              5

                                              4

                                              3

                                              2

                                              1

                                                                              x
              −6    −5   −4   −3   −2   −1        0   1   2   3   4   5   6
                                             −1

                                             −2

                                             −3

                                             −4

                                             −5




                                                                                  1
   (b) Use your graph to estimate a solution to the equation                        + 1 = |x − 1|.
                                                                                  x

       What is your estimate?




4. Algebraic solution required.
  Solve for x : 2 | x + 3 | − 4 ≤ 1.




                                                                              6
5. Algebraic solution required.
  Find all solutions (real and complex) of the equation x4 + x2 = 12 .




                               (x + 1)(3x + 1)
6. Consider the function y =                   .
                                (x + 1)(x − 3)



   (a) What is the domain of the function?




   (b) Find all vertical asymptotes of the graph of this function.




   (c) Find all horizontal asymptotes of the graph of this function.




                                                   7
               PART C: Long response problems – 25 points.
                         Each problem in this section is worth 5 points.

                 There are six problems, but only five will be marked.
            Circle the five problems you want marked: 1, 2, 3, 4, 5, 6.
      If you do not circle five problem numbers above, then only problems 1 - 5 will be marked.


                    Show your work clearly in the spaces provided.
        Marks will be awarded for method, not for the final answer alone.

1. Algebraic solution required.
  Roberta bought an electric drill. After its first recharge, the battery in the drill lasted 100 minutes.
  Roberta has noticed that, each time she recharges the battery, it lasts 90% as long as it lasted after
  the previous recharge.

   (a) How long did the battery last after its third recharge?
       (Does your answer make sense?)




   (b) Roberta has decided that she will keep recharging the battery until the charge lasts less than
       5 minutes. How many times will Roberta recharge the battery?




                                                 8
2. Algebraic solution required.
  Consider the function f (x) = 3x2 − 4x + 1.

   (a) Use the definition of derivative to prove that f ′ (x) = 6x − 4.




   (b) Find an equation for the tangent line to the graph of y = f (x) at the point (2, 5).




                                                 9
3. Algebraic solution required.
  A rectangular playing area is to be enclosed, and divided into two equal-sized fields. There are 600 m
  of fencing available, to go around the edge, and down the middle, as shown in the diagram.




   (a) Find the dimensions that give the maximal possible total area.

   (b) What is that area?




                                        Did you answer part (b)?



                                                10
4. Algebraic solution required.
  This problem is about the graph of the function
                                    3 2              1
                         y = x3 −     x − 36 x + 54 = (2x − 3) (x − 6) (x + 6) .
                                    2                2
   (a) Find all critical points.




   (b) Find all intervals on which the function increases.




   (c) Mark appropriate scales on the axes below, and sketch the graph, being careful to show all
       intercepts, all critical points and the basic shape of the graph.



                     y




                                         x




                                                11
5. Algebraic solution required.
                                                         x2 − 4
  This problem is about the graph of the equation y =           .
                                                         x2 − 9
   (a) Find all axis intercepts.




   (b) Find the equations of all horizontal and vertical asymptotes.

       Horizontal:




       Vertical:




   (c) Mark appropriate scales on the axes below, and sketch the graph, being careful to show all
       intercepts, all asymptotes and the basic shape of the graph.


                           y




                                             x




                                                 12
  6. Algebraic solution required.

                    √
     Evaluate (1 + i 3)9 .

     You may give your answer in Cartesian or polar form, whichever you prefer.




Please go back to the top of page 8 and circle the five problems from this section which you want marked.

         If you do not circle five problem numbers above, then only problems 1 - 5 will be marked.




                                                   13
Advanced Math 120 – Formula Sheet


             t1 = a


             tn = a + (n − 1) d

                      n
            Sn =        ( 2 a + (n − 1) d )
                      2

                      n
                 =      ( a + tn )
                      2




             tn = a r n−1


                      a ( 1 − rn )
            Sn =
                          1−r


                      a (r n − 1)
                 =
                         r −1


                       a
             S =
                      1−r




( r ( cos θ + i sin θ ) )n = r n ( cos(nθ) + i sin(nθ) )

            ( r cis θ )n = r n cis (nθ)




                          14

								
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