ADVANCED FUNCTIONS (MHF 4U) FINAL EXAMINATION

Transcription

1 Canadian International Matriculation Programme Sunway College (KL) Sdn. Bhd. ADVANCED FUNCTIONS (MHF U) FINAL EXAMINATION Date/Day : December 07, Monday Time : 8.30 am 0.30 am Length : hours Lecturers : Ms. Dzura Mr. Henry Mr. Nithyananthan Ms. Leanne Mr. Welch (Please circle your teacher s name) Student Name : Section/Period : Please read the following instructions carefully before you begin the examination:. This exam paper has 8 printed pages including this cover page and the formula sheet.. The examination is worth 0 percent of your overall mark for this course. 3. The examination consists of THREE parts as follows: Parts Content Number of Questions Marks A MULTIPLE CHOICE 3 B SHORT ANSWERS 3 C PROBLEM SOLVING 5 (choose only ) OVERALL COMMUNICATION 0 TOTAL 00. Please enter your multiple choice answers on the sheet provided on page Solutions for Parts B and C must be written in the space provided.. Answer questions only for Part C. 7. Scientific or Graphing Calculators and paper dictionary are permitted. NO SHARING is allowed. 8. All answers must be written legibly in black/blue pen or pencil. For office use only: K/U T/I COM APPL TOTAL MHF U FINAL EXAMINATION December, 07

2 PART A - Multiple Choice [3 marks] Identify the choice that best completes the statement or answers the question.. Given the functions f ( x) x and g ( x) x, a graph of the combined f ( x) function y most likely resembles: g( x) A. C. B. D.. Express 5 3 as a power with a base of. A. 7 B. C. 3 D. 3. What is the equation of the horizontal asymptote of A. y 0 C. x 0 B. y 5 D. x 5 f ( x)? x 0 MHF U FINAL EXAMINATION December, 07 Page

3 Multiple Choice [3 marks] cont d: Identify the choice that best completes the statement or answers the question.. An equation for a cubic function with zeros, and 3 that passes through the point, is A. y x x x 3 C. y 3 x x x 3 B. y x x x 3 D. y x x x 3 5. The function y log x is 3 A. B. compressed vertically by a factor of stretched horizontally by a factor of 3 C. compressed horizontally by a factor of 3 D. reflected in the y-axis. Over what interval does the graph of the rational function have positive, decreasing slope? A. x C. x B. x D. x 7. Which of the following binomials is a factor of x 3 x x? A. x C. x 7 B. x D. x 3 MHF U FINAL EXAMINATION December, 07 Page 3

4 Multiple Choice [3 marks] cont d: Identify the choice that best completes the statement or answers the question. 8. Use end behaviours, turning points, and zeros to determine which graph represents 3 the polynomial equation : f ( x) 3x 5x x 3. A. y C x 8 0 y x 8 0 B. y D y x Which of the following situations involves an average rate of change? A. The noise is increasing by 0 db as she walks into the arena. B. A car was driven for h C. The temperature dropped C at :00 p.m. D. A child grew cm in three years. MHF U FINAL EXAMINATION December, 07 Page

5 Multiple Choice [3 marks] cont d: Identify the choice that best completes the statement or answers the question. 0. Which graph models walking directly away from a motion sensor at a constant rate? A. d C d t B. d D d t. Given the functions f ( x) x 3 and g ( x) x x 8, determine the domain of the combined function y f ( x) g( x). A. x x 3 C. x x B. x 8 x D. Cannot be determined. Solve the equation : log x 7 x. 5 A. x C. x 7 B. x D. x 8 MHF U FINAL EXAMINATION December, 07 Page 5

6 Multiple Choice [3 marks] cont d: Identify the choice that best completes the statement or answers the question. 3. Which of these is a possible solution for cos 0 5 A. x C. x 7 B. x D. All of the above x in the interval 0, x?. Describe the strategy you would use to solve log x log log 8. A. Express the equation in exponential form, set the exponents equal to each other and solve. B. Use the fact that the logs have the same base to add the expressions on the right side of the equation together. Express the results in exponential form, set the exponents equal to each other and solve. C. Use the product rule to turn the right side of the equation into a single logarithm. Recognize that the resulting value is equal to x. D. Use the fact that since both sides of the equations have logarithms with the same base to set the expressions equal to each other and solve. f ( x) , where x is the number of years after the family purchases the house for $ What is the best estimate for the instantaneous rate of change in the value of the home when the family has owned it for 5 years? 5. The value of a family s home is given by x A. $ 8000 /year C. $ 000 /year B. $ 0000 /year D. $ 000 /year. 3 3 Write the expression 5 79 x y in factored form. A. 8 9xy 8 9xy C. 8 9xy 7xy 8x y B. 8 9xy 7xy 8x y D. 8 9xy 7xy 8x y MHF U FINAL EXAMINATION December, 07 Page

7 MULTIPLE CHOICE ANSWER SHEET NAME : PERIOD: Answer all multiple choice questions on this sheet: MHF U FINAL EXAMINATION December, 07 Page 7

8 PART B - Short Answers [3 marks]. Solve x x = x 3 x+ [K/U: marks]. A bacteria culture doubles every 0 minutes. How many hours will it take a culture of 70 bacteria to grow to a population of 0? [T/I: marks] 3. Let a polynomial be f(x) = 5x 3 kx + 7 where k is a real number. When divided by (x ) the remainder is 3. Solve for k. [T/I: marks] MHF U FINAL EXAMINATION December, 07 Page 8

9 Short Answers [3 marks] cont d:. A satellite moves on a circular orbit which subtends an angle of 5 about the center of the earth. If it is known that it travelled an orbital arc length of 70 km, how far is the satellite from the center of the earth? [APPL: 3 marks] 5. The data table below shows the speed of a sports vehicle over a period of time. Time (seconds) Speed (m/s) [APPL: marks] a) Use quadratic regression to obtain a model of the speed as a function of time (give your answers correct to four decimals) b) Estimate the instantaneous rate of change at the th second (using h = 0.0) using the answer obtained in part a). MHF U FINAL EXAMINATION December, 07 Page 9

10 Short Answers [3 marks] cont d:. Factor the function f(x) = 3x 3 + x + 0x + 8 completely. [APPL: marks] [Note: Show in your working one zero and verify this using the factor theorem. Use any division process to find all of the factors] 7. Solve 3 x = 5 x for x. Round your answer to three decimal places. [T/I: 3 marks] 8. Consider the function f(x) = x + a) Determine the inverse function. b) Sketch both the graph of the function and its inverse on the same axis and then identify one coordinate on each of the graphs. [T/I: marks] MHF U FINAL EXAMINATION December, 07 Page 0

11 Short Answers [3 marks] cont d: 9. The graph shown is the result of transformations applied to y = x. Determine the equation of this transformed equation and write it in the form y = a(x d) + c. Justify your answer. [T/I: marks] y (, 3) x 3 8 (-, -5) 0 (5, -5) 0. The function f(x) = (3x + ) + models the Harrington Paper Company s revenue, where x is the time measured in weeks and f (x) is the revenue measured in thousands of dollars. When will the revenue be more than $0000? [T/I: 3 marks]. Determine the solutions (in radian) to the equation. sin for 0 [T/I: 3 marks] MHF U FINAL EXAMINATION December, 07 Page

12 Problem Solving [ marks] Answer questions only.. The company Shirts Galore has figured its total cost of producing and selling x units of T-shirts per month is given by the equation C x 0.5x x. a) Write a rational function to represent the average cost of producing and selling a shirt. [APPL: marks] b) If the production level is 00 shirts per month, find the average cost of producing and selling a shirt. [APPL: marks] c) If the shirts are sold for $ 7.98 a shirt, what is the profit realized on producing and selling 00 shirts per month? [APPL: marks] MHF U FINAL EXAMINATION December, 07 Page

13 Problem Solving [ marks] cont d: Answer questions only.. a) Draw a distance versus time graph that corresponds to the walk described below. Cathy starts 8.5 m away from the motion sensor and walks in a straight line toward the sensor at a constant rate of m/s for s. She stops for s. Next, she immediately begins walking away from the sensor at a constant rate of 0.5 m/s for 3 s. She stops for s and then walks toward the sensor at a constant rate of.5 m/s for s. b) During which interval of time is the speed the greatest? [APPL: marks] MHF U FINAL EXAMINATION December, 07 Page 3

14 Problem Solving [ marks] cont d: Answer questions only. 3. Carbon- is used by scientists to estimate how long ago a plant or animal lived. The half-life of Carbon- is 5730 years. The mass M in grams of Carbon- left after t years is given by the function M t 00 t 5730 a) State the domain and range for the function M t. [APPL: marks] b) How much Carbon- would remain after 500 years? [APPL: mark] c) Determine the amount of time it will take for the substance to decay to grams. [APPL: 3 marks] MHF U FINAL EXAMINATION December, 07 Page

15 Problem Solving [ marks] cont d: Answer questions only. cos sin. Prove sec tan cos sin [APPL: marks] MHF U FINAL EXAMINATION December, 07 Page 5

16 Problem Solving [ marks] cont d: Answer questions only. 5. The following table of values represents the predicted population in a certain town x years from now: a) Use finite differences to determine the following: the degree of polynomial function, and the sign and value of the leading coefficient; [Note: The differences must be shown completely to earn full marks. You may add/insert more columns below for your answer. Use GC to ease your computation.] [APPL: marks] X Y Degree of the polynomial : Sign and value of the leading coefficient : b) Use the regression feature of the GC to determine the equation of the function that models the given situation and estimate the instantaneous rate of change in the population at the 0 th year. [APPL: marks] ***** END OF QUESTIONS ***** MHF U FINAL EXAMINATION December, 07 Page

17 BLANK PAGE MHF U FINAL EXAMINATION December, 07 Page 7

18 Compound Angle (Addition and Subtraction) Formulae: Addition Formula for Cosine cos(a + b) = cos a cos b - sin a sin b Subtraction Formula for Cosine cos(a b) = cos a cos b + sin a sin b Addition Formula for Sine sin(a + b) = sin a cos b + cos a sin b Subtraction Formula for Sine sin(a - b) = sin a cos b - cos a sin b Addition Formula for Tangent tan a tan b tan( a b) tan a tan b Subtraction Formula for Tangent tan a tan b tan( a b) tan a tan b Double Angle Formulae: sin (x) = sin x cos x cos (x) = cos x - sin x cos (x) = cos x tan x tan( x) cos (x) = sin x tan x Pythagorean Identity: sin x + cos x = sin x = cos x cos x = sin x Quotient Identities: sin x tan x cosx cosx cot x sin x Reciprocal Identities: sec x cosx cot x tan x cscx sin x MHF U FINAL EXAMINATION December, 07 Page 8