MATH15062022PRACTICEFINALEXAM

Consider the following traffic flow diagram, and use it to answer questions 1, 2, 3, 4, and 5. 1. In the augmented matrix of the system of linear equations that describes this traffic flow diagram, what is the line that represents the flow of traffic through intersection D? A) (1 − 1 | 300) B) (1 1 | 300) C) (1 0 0 0 1 0 0 0 | 300) D) ((1 0 0 0 − 1 0 0 0 | 300)) E) None of the above. 2. What is the maximum number of cars that can pass along x 6 ? A) 150 B) 250 C) 300 D) 50 E) None of the above. 3. Suppose we know that at least 170 cars per minute enter the intersection B from the north. Determine the minimum number of cars that will be leaving the intersection E towards south. A) 10 B) 20 C) 30 D) 40 E) None of the above. Copyright 2022 A. McEachern 1

4. Would the system of linear equations describing the traffic flow diagram be square, overdetermined, underdetermined, or none of the above? A) Square B) Overdetermined C) Underdetermined D) None of the above 5. How many free variables are in this system? A) 0 B) 1 C) 2 D) 3 E) None of the above. 6. Consider the following system of linear equations: 2 x + 2 y + 3 z = − 8 x − y + z = 0 What is the solution to this system? A) ( − 2 − 5 4 t, − 2 − 1 4 t, t ) , −∞ < t < ∞ B) (2 − 5 4 t, 2 − 5 4 t, t ) , −∞ < t < ∞ C) (2 − 5 4 t, − 2 − 5 4 t, t ) , −∞ < t < ∞ D) ( − 2 − 5 4 t, 2 − 5 4 t, t ) , −∞ < t < ∞ E) None of the above. 7. The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese? A) Hot dog = 1060, Cottage cheese = 800 B) Hot dog = -860, Cottage cheese = -1000 C) Hot dog = 1000, Cottage cheese = 860 D) Hot dog = 860, Cottage cheese = 1000 E) None of the above. Copyright 2022 A. McEachern 2

12. What is the lim x → 0 − p ( x )? A) 0 B) π 2 C) -1 D) 3 − sin(3) E) None of the above. 13. What is the lim x → 3 + p ( x )? A) 0 B) π 2 C) -1 D) 3 − sin(3) E) None of the above. 14. What is the lim x →∞ p ( x )? A) e 3 B) π 2 − 1 C) ∞ D) 3 − sin(3) E) None of the above. Copyright 2022 A. McEachern 4

Consider the following picture and use it to answer questions 15 and 16. Let the red graph be f ( x ) and the blue graph be g ( x ). 15. If h ( x ) = f ( x ) g ( x ), what is h ′ (3)? A) -15 B) -9 C) -6 D) 12 E) None of the above. 16. If h ( x ) = f ( x ) − g ( x ), what is h ′ (0)? A) -15 B) -9 C) -6 D) 15 E) None of the above. Copyright 2022 A. McEachern 5

Use the following information to answer questions 20, 21, and 22: The sigmoidal kinetics function is given by v = Kc 2 k 2 n + c 2 where c > 0 20. What is the first derivative of v ? A) v ′ = − Kk n c ( k 2 n + c 2 ) B) v ′ = 2 Kk n c − K 2 c 2 ( k 2 n + c 2 ) 2 C) v ′ = 2 Kk n c ( k 2 n + c 2 ) 2 D) v ′ = Kk n c ( k 2 n + c 2 ) 3 E) None of the above. 21. Let 0 < c < q k 2 n 3 . As c increases, what can we say about the rate of change of v ? A) As c increases, the rate of change of v is negative. B) As c increases, the rate of change of v decreases. C) As c increases, the rate of change of v increases. D) As c increases, the rate of change of v neither increases nor decreases. E) None of the above. 22. Select the statement that is true . A) v can output a value greater than K . B) v has no critical points. C) v is always concave down. D) v has one inflection point. E) None of the above. Copyright 2022 A. McEachern 7

Use the following description to answer questions 23 and 24: The vertical displacement y of a boat that is rocking up and down on a lake, with y measured relative to the bottom of the lake. It has a maximum displacement of 12 meters and a minimum of 8 meters, a period of 3 seconds, and an initial displacement of 11 meters when measurement was first started (i.e., t = 0). Assume the boat is rising at this time. 23. Which of the following trigonometric functions could be used to approximate this cycle? A) y = − 2 cos( 2 π 3 ( t − 1 2 )) B) y = 10 + 2 cos( 2 π 3 ( t − 1 2 )) C) y = 8 + cos( 2 π 3 t ) D) y = 10 + 2 cos( 3 2 π ( t + 1 2 )) E) None of the above. 24. What was the predicted rate of change of the number of y ( t ) when t = 4? A) − 4 π 3 sin( 7 π 3 ) B) − 2 π 3 sin( 14 π 3 ) C) 4 π 3 sin( 7 π 6 ) D) π 3 sin( π 3 ) E) None of the above. Use the following information to answer questions 25 and 26: A mass suspended from a spring is raised a distance of 8 m above its resting position. The mass is released at time t=0 and allowed to oscillate. After 2 seconds, it is observed that the mass returns to its highest position, which was 6 cm above its resting position. 25. Which of the following functions could be used to approximate this phenomena? A) 8(0 . 75) t 2 cos( πt ) B) 6(0 . 75) t 2 sin( πt ) C) 8(0 . 75) 2 t cos( πt ) D) 8 e t 2 cos( πt ) + 8 E) None of the above. Copyright 2022 A. McEachern 8

30. Given the Folium of Descartes x 3 + y 3 = 2 axy, a > 0 Where on the graph of the Folium is the tangent line equal to 0? A) x = − q 3 ay 2 and x = q 3 ay 2 B) x = − q 2 ay 3 and x = q 2 ay 3 C) x = 3 y 2 2 a D) The tangent line is never equal to 0. E) None of the above. 31. Given the equation x 3 + y 3 = 6 xy What is the slope of the tangent line, assuming y is the independent variable? A) dx dy = y 2 − 2 x 2 y − x 2 B) dy dx = 2 y − x 2 y 2 − 3 x C) dx dy = 3 y 2 − 6 x 6 y − 3 x 2 D) dy dx = 6 y − 3 x 2 3 y 2 − 6 x E) None of the above. Copyright 2022 A. McEachern 10

32. Given the equation x 3 + y 3 = 2 xy At how many points is the slope of the tangent line zero, assuming y is the independent variable? A) 0 B) 1 C) 2 D) 3 E) None of the above. Use the following information to answer questions 33 and 34: A population of mice grows by four-thirds every 3 months. Initially there are 9 mice present. 33. What is the equation describing the number of mice after t months? A) A ( t ) = 9 ( 3 4 ) t 3 B) A ( t ) = 9 ( 4 3 ) t 3 C) A ( t ) = ( 1 3 ) 3 t D) A ( t ) = 9(4 . 3) t E) None of the above. 34. What is the rate of change of the number of mice after 1 year? A) 9 ( 4 3 ) t 3 ln( 4 3 ) B) 3 ( 4 3 ) 1 3 ln( 4 3 ) C) 3 ( 1 3 ) − 2 3 ln( 4 3 ) D) 9 ( 7 3 ) 4 3 ln( 4 3 ) E) None of the above. Copyright 2022 A. McEachern 11

40. Given the following diagram What is L (4 . 5)? A) 4.4 B) 4.5 C) 4.6 D) 4.7 E) None of the above. 41. What is T 2 ( x ) if f ( x ) = e 2 x and the center is a = 3? A) T 2 ( x ) = e 3 + 2 e 3 ( x − 3) + 2 e 3 ( x − 3) 2 B) T 2 ( x ) = e 6 + 2 e 6 ( x − 3) + 2 e 6 ( x − 3) 2 C) T 2 ( x ) = e 6 + 2 e 6 x + 2 e 6 x 2 D) T 2 ( x ) = − e 3 − 2 e 3 ( x − 3) − 2 e 3 ( x − 3) 2 E) None of the above. Copyright 2022 A. McEachern 13

Use the following image of f ′ ( x ) to answer questions 42, 43, 44, and 45. Assume f ( x ) is defined on [-1.5, 1.5]. 42. Where is f ( x ) increasing? A) ( − 1 . 5 , 1 . 5) B) ( − 1 , 1 . 5) C) ( − 1 . 5 , 1) ∪ (1 , 1 . 5) D) ( − 1 . 5 , − 1) ∪ ( − 1 , 0) ∪ (1 , 1 . 5) E) None of the above. 43. Where is f ( x ) concave down? A) ( − 1 . 5 , 1 . 5) B) ( − 1 , − 1 . 5) C) ( − 1 . 5 , 1) ∪ (1 , 1 . 5) D) ( − 1 . 5 , − 1) ∪ ( − 1 , 0) ∪ (1 , 1 . 5) E) None of the above. Copyright 2022 A. McEachern 14

48. What is the constraint function? A) 1000 = 6 x + 10 y B) 1000 = x + y C) 1000 = 10 x + 6 y D) 1000 = 6 x + 6 y E) None of the above. 49. What will the objective function look like if we solve for y using the constraint and plug that into the objective function? A) A ( x ) = x 500 − 5 x 3 B) A ( x ) = 2 x ( 500 − 5 x 3 ) C) A ( x ) = x + ( 500 − 5 x 3 ) D) A ( x ) = x ( 500 − 5 x 3 ) E) None of the above. 50. What are the dimensions that maximize the area of the pen? A) x = 100 , y = 0 B) x = 50 , y = 250 3 C) x = 40 , y = 60 D) x = 30 , y = 50 E) None of the above. 51. Every optimization problem requires us to classify the critical points when searching for global extrema. Select the reasoning that justifies the previous sentence. A) The Maximum Value Theorem states that every function on a closed domain is guaranteed to have a maximum at the critical points on the closed interval. B) The Maximum Value Theorem states that every function on a closed domain is guaranteed to have a maximum at the endpoints of the interval. C) The Extreme Value Theorem states that every continuous function on a closed domain must have a global maximum and a global minimum. D) The Intermediate Value Theorem states that every continuous function on a closed domain must have a global maximum and a global minimum. E) The first sentence is incorrect. Copyright 2022 A. McEachern 16

52. Select the choice that makes the entire statement correct. The slope of the tangent line at the point ( x 0 , f ( x 0 )), where f ( x ) is differentiable at x 0 . A) is equivalent to the super fast rate of change of the function f ( x ) at x 0 . B) is equivalent to the average rate of change of the function f ( x ) at x 0 . C) is equivalent to the average-ish rate of change of the function f ( x ) as x increases by h at x 0 . D) is equivalent to the kind of slow rate of change of the function f ( x ) as x increases by x 0 + h at x 0 . E) None of the above. 53. Select the choice that makes the entire statement correct. The slope of the secant line between the points ( x, f ( x )) and ( x + h, f ( x + h )) on a function f ( x ) A) is equivalent to the instantaneous rate of change of the function f ( x ). B) is equivalent to the average rate of change of the function f ( x ). C) is equivalent to the average rate of change of the function f ( x ) as x increases by h . D) is equivalent to the instantaneous rate of change of the function f ( x ) as x increases by h . E) None of the above. 54. Decide if the following statement is true or false, and the correct reasoning: According to the Intermediate Value Theorem, If f ( x ) = 3 x − x 3 then f(x) = 0 has a solution over the interval [ − 1 , 1]. A) False: f(x) is not continuous on the interval [-1, 1], so the Intermediate Value Theorem does not apply here. B) True: since f(x) is continuous on the interval [-1, 1], f ( − 1) < 0 and f (1) > 0, there exists a value z on [-1, 1] such that f ( z ) = 0. C) True: since f(x) is continuous on the interval [-1, 1], f ( − 1) > 0 and f (1) < 0, there exists a value z on [-1, 1] such that f ( z ) = 0. D) False: since f(x) is continuous on the interval [-1, 1], f ( − 1) > 0 and f (1) > 0, we can’t know if f ( x ) has a root on [-1, 1]. E) False: since f(x) is continuous on the interval [-1, 1], f ( − 1) < 0 and f (1) < 0, we can’t know if f(x) has a root on [-1, 1]. Copyright 2022 A. McEachern 17

57. The First Derivative Test is able to classify a critical point as a local maximum when A) the first derivative is negative on the interval to the left of the critical point, and then positive on the interval to the right of the critical point. B) the first derivative is positive on the interval to the left of the critical point, and then positive on the interval to the right of the critical point. C) the first derivative is positive on the interval to the left of the critical point, and then negative on the interval to the right of the critical point. D) the first derivative is negative on the interval to the left of the critical point, and then positive on the interval to the right of the critical point. E) None of the above. 58. In a population of size 500 at time t = 0, if there are 50 births per year and 25 deaths per year, what is the solution to the differential equation that describes the change in population per year? A) N ( t ) = 0 . 05 e 0 . 5 t B) N ( t ) = e 25 t C) N ( t ) = 500 e − 0 . 01 t D) N ( t ) = 500 e 0 . 05 t E) None of the above. 59. Given the differential equation with the initial condition y ′ ( t ) = 0 . 9 y, y 0 = 44 What is the solution to this initial value problem? A) y ( t ) = 44 e 0 . 9 t B) y ( t ) = e − 0 . 9 t C) y ( t ) = e 44 t D) y ( t ) = 0 . 9 44 e t E) None of the above. 60. Given the following differential equation dy dt + 3 y = 2 t − 1 What value of C causes y = 2 t 3 − C + e − 3 t to satisfy it? A) 5 9 B) 5 3 C) 3 2 D) − 3 2 E) None of the above. THIS IS THE LAST PAGE OF THE FINAL EXAM. Copyright 2022 A. McEachern 19