VIEWS: 4 PAGES: 14 CATEGORY: Technology POSTED ON: 1/12/2010
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