MATHEMATICS MATH*1030 Final (Mock) Exam. Student #: Instructor: M. R. Garvie 24 Nov, 2015

Transcription

1 FALL 2015 MATHEMATICS MATH*1030 Final (Mock) Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie 24 Nov, 2015 INSTRUCTIONS: 1. This is a closed book examination. Scientific and/or graphing calculators are allowed. The test is 2 hours long. You may use blank areas of this exam booklet for rough work. 2. The test consists of 40 equally-weighted (independent) multiple choice questions. Answer the multiple choice questions on the computer score sheet (circle your answers on the exam paper). In each question choose the answer that best fits the question. 3. Make sure you have a complete exam booklet (as errors sometimes occur during printing). 4. Fill in the computer score sheet in pencil; make sure you include your name and student ID number, but you don t need a section number or address. Also fill in your name and student number at the top of this exam booklet. 5. The abbreviation d.p. stands for decimal places, e.g. π = 3.14 (2 d.p.) means that π is given to 2 decimal places. 6. Hand in the entire exam booklet and your computer score sheet.

2 1. The solution of x 2 > 1 is (A) x > 1 (B) 1 < x < 1 (C) x > 1 or x < 1 (D) x > ±1 (E) undefined 2. The function 2x 1 3x + 2 has (A) a vertical asymptote x = 2/3, but no horizontal asymptote (B) a horizontal asymptote y = 3/2, but no vertical asymptote (C) a vertical asymptote x = 2/3 and a horizontal asymptote y = 1 (D) a vertical asymptote x = 2/3 and a horizontal asymptote y = 2/3 (E) no asymptotes 2

3 3. The domain of y = 7 2x is (A) [2/7, + ) (B) (, 7/2] (C) (, 7/2) (7/2, + ) (D) [, 7/2] (E) (, 2/7] 4. Let f(x) = 1 and g(x) = x, then the domain of (f + g)(x), i.e. D x f+g is (A) {x R x 0} (B) [0, + ) (C) (, 0) (0, + ) (D) (0, + ) (E) (, 0) 3

4 5. Let f(x) = 1 x and g(x) = x 2 + 1, then (g f)(x) is given by (A) 1 x + 1 (B) 1 x 2 +1 (C) 1 x+1 (D) 1 x (E) undefined 6. The function y = x 2/3 is (A) one-to-many (B) many-to-none (C) one-to-one (D) neither one-to-many nor many-to-one (E) many-to-one 4

5 7. If f(x) = x 1 x+2 then f 1 (x) is (A) 2x 1 x+1 (B) 2x+1 x 1 (C) 2x+1 x 1 (D) 2x+1 x+1 (E) 2x 1 x 1 8. sin(x π) corresponds to (A) a shift of sin(x) in the x-direction by π units (B) a compression of sin(x) in the x-direction by a factor of 1 π (C) cos(x) (D) a shift of sin(x) in the x-direction by +π units (E) a reflection of sin(x) in the line y = π 5

6 9. The function y = [ 1 3 (x + 1)] 2 corresponds to (A) a compression of y = x 2 in the x-direction by a factor of 3 followed by a shift in the x-direction by +1 units (B) a compression of y = x 2 in the x-direction by a factor of 1 followed by a 3 shift in the x-direction by +1 units (C) a stretch of y = x 2 in the x-direction by a factor of 3 followed by a shift in the x-direction by 1 units (D) a compression of y = x 2 in the y-direction by a factor of 3 followed by a shift in the x-direction by 1 units (E) none of the above 10. The function y = x is (A) monotonic increasing (B) monotonic decreasing (C) strictly monotonic decreasing (D) neither strictly monotonic increasing nor strictly monotonic decreasing (E) strictly monotonic increasing 6

7 11. the graph of y = (x 2)(x + 1) is given by (A) (B) (C) (D) (E) 7

8 12. The range and period of f(x) = cos(x) is given by (A) D f = [ 1, 1], P = 2π (B) D f = [ 1, 1], P = π (C) D f = [0, 1], P = 2π (D) D f = [ 1, 0], P = π (E) D f = [0, 1], P = π 13. The function y = x + x is (A) (B) (C) (D) (E) = = = = = { 0 if x 0 2x if x < 0 { 2x if x 0 0 if x < 0 { 2x if x > 0 0 if x < 0 { 0 if x 0 x if x < 0 { 2x if x 0 2x if x < 0 8

9 14. The equation of the straight line passing through the points (1, 2) and (2, 1) is (A) y = 2x + 1 (B) y = x 3 (C) y = x 4 (D) y = x + 3 (E) y = x 15. A firm selling bags of cat food finds that the total cost C(x) in dollars of producing x bags is given by C(x) = 15x 20. The firm charges $20 per bag. What is the profit if 10 bags are sold? (A) $70 (B) $30 (C) $130 (D) $20 (E) $10 9

10 16. The quadratic function y = (x 2) 2 1 has (A) a minimum value of 2 at x = 1 (B) a maximum value of 1 at x = 2 (C) a maximum value of 1 at x = 2 (D) a minimum value of 1 at x = 2 (E) a maximum value of 2 at x = The solution of 5 3x = is (A) 3 (B) +3 (C) +1 (D) 1 (E) +2 10

11 18. The solution of x+3 = 12 is (answer given to 2 d.p.) (A) 0.83 (B) 0.83 (C) 1.07 (D) 1.07 (E) In the Exponential Growth Model (base e) if the growth rate is 2% (units day 1 ) how long does it take for the initial quantity to double? (Answer given to 2 d.p.) (A) days (B) 0.35 days (C) days (D) 0.15 days (E) days 11

12 20. As n gets larger and larger ( n) n gets closer and closer to (A) + (B) π (C) e (D) 0 (E) Consider the recursively defined sequence a 1 = 1, a 2 = 1, a 3 = 3, a n+1 = a n 2 + a n 1 a n, n = 3, 4, 5,... The 5th term is (A) 5/4 (B) 3 (C) 133/39 (D) 13/4 (E) 4/3 12

13 22. The 30th term of the sequence is is given by (A) 61 (B) 60 (C) 63 (D) 107 (E) 29 ( ) 2n 23. lim 3 n 2 +2 n is n 4 +n (A) + (B) (C) 2 (D) 0 (E) 1 {3, 5, 7, 9,...} 13

14 24. Adding 200 terms of the series yields (A) + (B) (C) 250 (D) (E) n=0 2(0.5)n = (A) + (B) 4 (C) 0 (D) 2 (E) 1 S =

15 26. The future value of an investment of $1000 earning 3% p.a. for 100 days is (A) $ (B) $301, (C) $ (D) $ (E) $ $100 is invested at 5% annual interest rate compounded semiannually (twice per year) for 23 months. The future value is (A) $12, (B) $ (C) $9.93 (D) $ (E) $

16 28. The effective interest rate corresponding to 3% compounded monthly is to 2 d.p. (A) 3.01% (B) 3.05% (C) 3.04% (D) 3% (E) 13.55% 29. If f(x) = x2 1 x 1 then lim x 1 f(x) is (A) + (B) 2 (C) (D) 1 (E) 0 16

17 30. If f(x) = 2x3 x x then lim x 1 f(x) is (A) (B) 1 (C) + (D) undefined (E) Consider the graph of a function shown below: Then lim x 1 f(x) is given by (A) 1 (B) 2 (C) does not exist (D) 0 (E) 3 17

18 32. The function f(x) = x 2 + ln(2 x) is (A) continuous for all x (B) discontinuous for all x (C) discontinuous at x = 2 (D) continuous at x = 2 (E) discontinuous at x = The function f(x) = x 2 is (A) continuous from the left at x = 2 (B) continuous everywhere (C) discontinuous everywhere (D) continuous from the right at x = 2 (E) undefined at x = 2 18

19 34. The following data was collected for a road trip: time in hours distance in miles The average speed of the journey during hours 1 to 4 is (A) 42 mph (B) 40 mph (C) 52.5 mph (D) 51 mph (E) 50 mph 35. The difference quotient for the function f(x) = 2x + 1 simplifies to (A) h (B) 2h/h (C) 2 + h (D) 1 (E) 2 19

20 36. Given y = 2x e x then dy dx = (A) 2e x (1 + x) (B) 2x + e x (C) 4xe x (D) 2e x (E) 2xe x 1 + 2e x 37. If y = (1 + x 2 ) 50 then dy dx = (A) 50(1 + x 2 ) 49 (B) 50x(1 + x 2 ) 49 (C) 100x(1 + x 2 ) 49 (D) 100x(1 + 2x) 49 (E) 50x(1 + 2x) 49 20

21 38. Suppose the total cost in dollars to produce x chairs is given by C(x) = 6x 2 50x What is the marginal cost to produce 10 chairs? (A) $100 (B) $60 (C) $70 (D) $76 (E) $ Consider the tabulated data for a function y = f(x): f (0) f (1) f (2) > 0 = 0 < 0 Application of the First Derivative Test yields that we have (A) a point of inflection at x = 1 (B) no extreme (C) neither a local Max nor a local Min (D) a local Max at x = 1 (E) a local Min at x = 1 21

22 40. For a function y = f(x) suppose f (2) = 0 and f (2) < 0. Then application of the Second Derivative Test yields that (A) f(2) is a local Max (B) f(2) is a global Min (C) f(2) is a local Min (D) not enough information to reach a conclusion (E) f(2) is a point of inflection END OF TEST 22