Math 143 Practice Final Exam

Transcription

1 Math 143 Practice Final Exam December 11, 2017 Work through these problems in exam conditions; give yourself two hours, and NO CALCULATORS. 1. Solve z + z 6 6 z = 0. A. z = 6 z = 6 7 C. z = ± 6 7 D. z = 0 E. C and D ( 2. Which of the following is not equivalent to A. 9p 8 16(p 2) 2 9p 8 16(p 2 4) 9p 8 16p 2 64p+16 C. ( D. None of A-C are equivalent to ( E. All of A-C are equivalent to 3p 2? 4(p 2)) 4 3p 2. 4(p 2)) 4 3p 2. 4(p 2)) 4 1

2 3. Which of the following expresses the volume of a cube V in terms of its surface area S, given that the cube has side length l? A. V = S2/3 6 2/3 V = S3/2 6 ( 3/2 C. V = D. A and C E. B and C S 6 ) 3 Consider the following figure. 4. Which of these expresses the angle θ in terms of x? A. θ = tan ( ) 1 x 1 x θ = π ( ) 2 tan 1 x 1 x C. θ = sin ( ) 1 x x 1 D. θ = tan ( ) 1 x x 1 E. B and D 5. The expression simplifies to: A. y 11/3 1 y 11/3 C. 1 y 7/3 D. y 7/3 E. 1 y 8/3 (y 1/3 ) 2 (y 4/3 ) 2 (y 1/2 ) 2/3 Page 2

3 6. The length of a rectangle is 150% its width. Let p be the perimeter of the rectangle. An expression for the area of the rectangle in terms of its perimeter is given by... A. 3p 2 50 p 5 C p D. 3p Determine the width of a rectangle with perimeter 5 meters and length 2 meters. A. 1 meter 1 2 meters C. 1 4 meters D. B and C E. There is not enough information to tell. 8. A closed box (rectangular prism) has a square bottom with side length 4 inches and surface area 64 square inches. What is the height of the box? A. 4 in 3 in C. 2 in D. 3 in 2 E. A box with these dimensions cannot be defined. 9. Which of the following expresses the area A of an equilateral triangle in terms of its side length x? A. A = 1 2 xh A = x2 3 4 C. A = 1 x(x sin(π/3)) 2 D. A = x2 3 2 E. B and C Page 3

4 10. The expression is equivalent to A. ( x 2 y) 2 C. D. 1 x 2y x+2y x y x+y E. 1 x y 2 y x What is the domain of the function f(x) = x+2 x 3? A. All real numbers x 3 C. x 2 D. x 2 and x 3 E. x > 2 and x What is the range of the function f(x) = 1 x? A. All real numbers x 0 C. x 0 D. x > Given f(1) = 12 and f(4) = 6, what is the average rate of change of f as x increases from 1 to 4? A. 2 6 C. 6 D. 2 E Dog is 20 feet from his driveway when he begins to pull away at a constant speed of 30 ft/sec. Let d denote Dog s distance from his driveway in feet since he attained a speed of 30 ft/sec and t represent the time in seconds that have elapsed since that moment. Which of the following are valid in all possible situations? A. d = 30t d = 30 t Page 4

5 C. d t = 30 D. B and C E. All of the above Let f and g be functions defined on the domain {0, 1, 2, 3, 4} (this is the set containing the numbers 0, 1, 2, 3 and 4, not an interval of any kind). Questions are based on the table below, which displays the output values of f and g. 15. Evaluate f(g(4)). A. 1 0 C. 2 D. 3 E Evaluate g(f 1 (3)) A. 0 1 C. 2 D. 3 E. We cannot evaluate f 1 (3) 17. Evaluate f 1 (g 1 (2)). A. 4 3 C. 2 D. 1 E. We cannot evaluate f 1 (g 1 (2)). x f(x) g(x) An ice cube (which is in fact a perfect cube) originally has a side length of 6 cm. As it melts, the side length decreases at a rate of 0.5 cm/minute. Which of the following defines the volume of the ice cube (in cm 3 ) as a function of the number of minutes since the cube began melting? Page 5

6 A. g(t) = t t3 g(t) = (6 0.5t) 3 C. g(t) = 216 (0.5t) 3 D. A and B E. All of the above An object falls off of a suspended platform. The following graph depicts the functional relationship d = f(t) between the number of meters d the object has fallen away from the platform, and t the number of seconds that have passed since the object fell. Questions are based on this graph. 19. Evaluate f(1) and explain what this quantity represents in the context of the falling object. A. f(1) = 5; one second after the object began to fall, it had fallen 5 meters. f(1) = 5; five seconds after the object began to fall, it had fallen 1 meter. C. f(1) = 0.5; one second after the object began to fall, it had fallen.5 meters. D. f(1) = 0.5; a half a second after the object began to fall, it had fallen 1 meter. E. f(1) = 5, but neither A nor B provide the correct context. 20. Which option best describes the quantity f(2) f(1) in the context of the falling object? A. After 1 second, the change in distance of the object is 15 meters. The average speed of the object from t = 1 to t = 2 seconds since it fell from the platform C. After 2 seconds, the object has fallen 20 meters. D. The time elapsed as the object fell from 5 meters to 20 meters away from the platform E. The change in distance from the platform by the object between t = 1 and t = 2 seconds Page 6

7 21. Which of the following statements are true about the graph of f(t)? A. It is concave down because the distance from the ground is decreasing at an increasing rate. It is concave up because the distance of the object from the platform is increasing at a decreasing rate. C. It is concave up because the distance of the object from the platform is increasing at an increasing rate. D. It is neither concave up nor concave down because the object falls at a constant speed. E. None of these statements are true. 22. Which of the following function rules describe the graph of f(t)? A. d = f(t) = 5t d = f(t) = 15t 10 C. d = f(t) = 4(5 t ) D. d = f(t) = 5t As word spreads about Ilsa s diner, she finds that the number of people eating at her restaurant is growing exponentially. More specifically, the number of customers C she serves each day is a function of n, the number of days since October 1st, 2016, and the rule with which she estimates this relationship is C = f(n) = 10(1.1) n. Which of the following statements is false? A. On October 1st, 2016, Ilsa had 10 customers. On October 2nd, 2016, Ilsa had 11 customers. C. Each day, the number of customers visiting the diner is 1% greater than the day before. D. The growth factor associated with the function f(n) is 1.1. E. None of these statements are false. 24. On October 1st, 2017, Ilsa had 200 customers in her diner. But it was at this point that she began to notice she was losing customers, as word began to spread that Ilsa s was no longer the cool place to be seen. In fact, each day after Oct. 1st, 2017, the number of customers C in the diner was 20% less than it was the day before. Which function f(n) expresses the number of customers in Ilsa s diner n days after October 1st, 2017? A. C = f(n) = n C = f(n) = 200(0.8) n C. C = f(n) = 10(0.8) n D. C = f(n) = 200(0.2) n E. C = f(n) = n Page 7

8 25. The following is a graph of a function that represents Judy s distance from a wall as a function of the number of seconds since Judy started walking. The horizontal axis represents the number of seconds since she started and the vertical axis represents her distance from the wall in feet. Which of the following statements is false? A. Judy spends most of the time between 0 seconds and 12 seconds not moving at a constant speed. As the time increases from 4 to 8 seconds, Judy s distance increases at an increasing rate. C. Between 0 and 8 seconds, Judy s distance from the wall increases. D. As time increases from 8 to 12 seconds, Judy s distance from the wall decreases at a constant rate. E. As the time increases from 0 to 4 seconds, Judy s distance from the wall increases at an increasing rate. Page 8

9 Questions 26 and 27 are based on the diagram below. 26. According to the above diagram, cos θ appears to be equal to which angle? A. A B C. C D. D E. None of the above 27. According to the above diagram, tan θ appears to be equal to which of the following? A. A B C. C D. D E. None of the above. 28. Consider the function f(x) = 1 x 2. As values of the input x approach 2 from the left, which of the following statements best describes the behavior of the corresponding outputs f(x)? A. The output values of f(x) decrease and get very very large. The output values of f(x) increase and get very very large. C. The output values of f(x) decrease and get very very negative. D. The output values of f(x) increase and get very very negative. E. None of the above Page 9

10 29. The square mileage of prairie land left in a particular region is modeled by the function A(m) = 3m m 2 + 2, the inputs of which correspond to the number of months that have passed since Sukup initiated a grain genetics project in the area. After a very large number of months, the square mileage of prairie gets very close to which of the following values? A. 3/5 5 C. A very large number D. 0 E Suppose that a population of bacteria triples every day. What is the 3-day growth factor of the population? A C. 3 1/3 D. 3 E. None of the above Questions are based on the figure. The y-scale is hard to discern (sorry), but it does match up with the x-scale. 31. Which of the following gives the best approximation of sin θ? A C D. 0.8 E Which of the following gives the best approximation of 3 sin θ? Page 10

11 A C D. 0.8 E Which of the following statements are true? A. sin θ > cos θ θ < π 2 C. sin θ > 3 cos θ D. A and C E. All of the above 34. Which of the following is the best approximation of sin 1 (0.25)? A. π 6 π 12 C. π 2 D. π 4 E. π Which of the following is equivalent to sin ( 1 sin ( 7π 8 π A. 8 7π 8 C. csc ( sin ( )) 7π 8 D. None of the above E. All of A through C ))? Page 11

12 36. As a fan turns counterclockwise, a bug sits 1.8 feet from the center of rotation on one of its blades. The bug is at the 3 o clock position on the fan when it begins to turn. If the fan makes less than one full rotation, determine the distance the bug has traveled along its arc from the 3 o clock position when its final position is 0.7 feet above the horizontal diameter for the second time. A. sin 1 (0.7/1.8) 1.8 sin 1 (0.7/1.8) C. π sin 1 (0.7/1.8) D. 1.8(π sin 1 (0.7/1.8)) E. None of the above 37. Suppose ( u = log 2 (6), and v = log 2 (3). Which of the following expressions is equivalent to log 18 ) 2 4? A. u + v 2v 1 C. 2u 3 D. u + v 2 E. B, C, and D are equivalent to log 2 ( 18 4 ). Page 12

13 Questions are based on the setup pictured below. One end of a straight wire is attached to an 8-foot tall pole. The other end of the wire is attached to the ground, forming an angle θ. A system within the pole lengthens the wire, while a track on the ground keeps the wire in place so as to maintain a triangular configuration. 38. At the moment the ground-end of the wire is 6 feet from the pole s base, which statement expresses the length of the wire? A. 6 tan θ feet 6 cos θ feet C. 10 feet D. A, B, and C E. B and C 39. After the the moment the ground-end of the wire is 6 feet from the pole s base, the distance between the wire and the base of the pole increases at a rate of 3 feet per minute. Which expresses the angle θ as a function of t, the minutes passed since the wire was six feet from the base of the pole? A. θ = f(t) = tan 1 ( 8 3t+6 θ = f(t) = 8 3t C. θ = f(t) = tan ( ) 1 8 3t D. θ = f(t) = 10 tan ( ) 1 8 3t E. There is not enough information to tell. ) Page 13

14 40. Consider the function of question 39. As the number of minutes that have passed since the ground-end of the wire was 6 feet from the pole gets very large, what can be said about the corresponding values of θ? (You do not need to know the correct answer to #39 in order to answer this question.) A. They approach 0. They approach. C. They approach π 2. D. They approach tan 1 ( 8 6 ). E. There is not enough information to tell. Page 14