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Computational Tools EM - September 4th, 2019
Surname Name Student ID
Write the solutions using two decimal digits.
(1) Consider the numbers 3.493427e+19 and 2.392993e+23. Then say whether both numbers are small and the first is larger then the second. (2) Draw the line and the parabola y = −x + 1 , y = x^2 + x + 2 over the interval [− 2 , 3 ] and find the ordinate of the lowest point belonging to the graphs. (3) Consider the expression abs(-1:2)+log(1:2). Which of the following stataments holds? (a) The first recycled number was 2 (b) The first recycled number was 0 (c) There was no need to recycle any argument. (d) The first recycled number was 1 (4) Which of the following statements is correct? (a) Simulation is based on several optimization runs to gain insight. (b) The terms ’distribution’ and ’simulation’ are equivalent. (c) A distribution is a function taking values in [ 0 , 1 ] to be maximized by simulation. (d) Simulation can answer the question ’What’s the average outcome of my experiment?’. (5) Type set.seed(107) and press enter. Then immediately store in a vector 100 normal pseudo-random numbers with mean 5 and standard deviation 2. Compute the sum of 8 com- ponents of the vector, starting from position 51. (6) Minimize, if possible, the following function: f (x) = x^4 − x^3 − 3 x^2 − 4 x + 4. Write the objec- tive function at the optimum or -999 if there is no solution. (7) In the smartphone market, the number of customers (in billions) of Samsung and Apple can be modelled by the functions f (x) = x/ 100 + 0. 2 e^0.^09 x, and g(x) = 0. 33 + 0. 025 x, where x > 0 denotes year 2000+x. When is Samsung reaching one billion customers for the first time? (8) How many solutions do you have in the system of non-linear equations: x^2 + 2 xy + y^2 − x = 0 , (x − 1 )^2 + (y + 1 )^2 = 1? (9) Assume that t = 0 corresponds to May 1st 2014. Consider the BTP denominated, according to the standard conventions, BTP3.5% 01Oct2015. Compute the gross return of the BTP if the cost is 101.21. Use 3 decimal digits for this exercise !!! (10) Enter the following commands: set.seed(102);x <- 1:10;y <- rnorm(10) and plot the points whose coordinates are x and y. How many points are inside the triangle formed by the three lines y = − 0. 5 x + 2 .5, y = x − 7 and y = 2. a) There is no point inside the triangle. b) One point is inside the triangle. c) Two points are inside the triangle. d) Three or more points are inside the triangle. (11) Let f be the function x^2 + xy + y^2 − y 9
2
(^2) −y . Plot the levels of f and find its global minimum. (12) The production of a firm is
- 25 l^0.^7 k^0.^3 , 1
where l is labour and k is capital. The unit costs of labour and capital are 0.5 and 1.1, respec- tively. Which is the maximum production if the cost of the factors cannot exceed 2? (13) Profits depend on price p and can be described as f (p) = 3 e^1.^5 p^ − 0. 3 e^4 p. The actual price is 0.58. Which of the following is true? (a) The profit-maximizing price is about 0.53. (b) The optimal profit is about 5.016. (c) Profit would be maximized decreasing the price by about 7.1%. (d) Increasing the price can result in profit’s increase. (14) Consider the assets ( 118 , 99 , 109 )′, ( 96 , 101 , 92 )′^ and ( 84. 6 , 80 , 81. 4 )′^ whose prices are 100.16, 87.17, 74.546, respectively. Is the third asset replicable? Do you have an arbitrage opportunity on the third asset? (a) No, it’s not replicable. Hence, there is an arbitrage opportunity. (b) No, it’s not replicable and there is NO arbitrage opportunity. (c) Yes, it’s replicable and there is an arbitrage opportunity. (d) Yes, it’s replicable but there is NO arbitrage opportunity. (15) Solve the system (^)
x y z
Write the sum of the components of the solution or -999, if the system has no solution. (16) The quality of a new item is assessed using 2 numeric measures. Assume they are uniform random numbers. Estimate how likely it is that the maximum of the measures exceeds 0.5. a) About 0.22; b) About 0.75; c) About 0.9; d) About 0.39.
(12) The production of a firm is
- 4 l^0.^8 k^0.^2 , where l is labour and k is capital. The unit costs of labour and capital are 1 and 1.1, respec- tively. Which is the maximum production if the cost of the factors cannot exceed 2? (13) Profits depend on price p and can be described as f (p) = 5 e^2 p^ − 0. 3 e^3 p. The actual price is 2.84. Which of the following is true? (a) The profit-maximizing price is about 2.41. (b) The profit at price 2.84 is -43.42. (c) The optimal profit is about 248.974. (d) The profit-maximizing price is about 2.05. (14) Consider the assets ( 104 , 104 , 104 )′, ( 106 , 95 , 105 )′^ and ( 84 , 79. 6 , 83. 6 )′^ whose prices are 93.60, 90.57, 63.669, respectively. Is the third asset replicable? Do you have an arbitrage opportunity on the third asset? (a) No, it’s not replicable. Hence, there is an arbitrage opportunity. (b) No, it’s not replicable and there is NO arbitrage opportunity. (c) Yes, it’s replicable and there is an arbitrage opportunity. (d) Yes, it’s replicable but there is NO arbitrage opportunity. (15) Solve the system (^)
x y z
Write the sum of the components of the solution or -999, if the system has no solution. (16) Run a simulation to estimate the probability that a standard random uniform is bigger than another uniform random number in [ 0. 4 , 0. 9 ]. a) About 0.5; b) About 0.45; c) About 0.4; d) About 0.35.
Computational Tools EM - September 4th, 2019
Surname Name Student ID
Write the solutions using two decimal digits.
(1) Consider the numbers 2.658881e+22 and 1.1259e+15. Then say whether both numbers are small and the first is larger then the second. (2) Draw the line and the parabola y = x + 2 , y = −x^2 − x + 4 over the interval [− 2 , 1 ] and find the ordinate of the lowest point belonging to the graphs. (3) Consider the expression 2*(-1:2)+log(1:4). Which of the following stataments holds? (a) The last recycled number was 0. (b) There was no need to recycle any argument. (c) The last recycled number was 0 (d) The last recycled number was 0. (4) Let x <- seq(0.4,3,len=11) and y <- 10+sin(10x). Consider the functions g(x) = ex and h(x) = − 12 x+15. Plot both the functions and the points defined by x and y on the interval [ 0 , 3 ]. Which of the following sentence is true? (a) The curve is mostly above the line and 3 points are below the graph of the exponential. (b) The curve is mostly above the line and 2 points are below the graph of the exponential. (c) The curve is mostly below the line and 2 points are below the graph of the exponential. (d) The curve is mostly above the line and 5 points are below the graph of the exponential. (5) Type set.seed(184) and press enter. Then immediately create a 4x3 matrix, filled by columns with (standard) random uniform numbers. Let u be the product of the matrix times the vector ( 1 , 2 , − 1 ). Finally, provide the sum of the components of u. (6) Minimize, if possible, the following function over the interval [ 0 , 2 ]: f (x) = x^4 + 3 x^3 + 2 x^2 + 4 x + 4. Write the objective function at the optimum or -999 if there is no solution. (7) In the smartphone market, the number of customers (in billions) of Samsung and LG can be modelled by the functions f (x) = x/ 100 + 0. 21 e^0.^079 x, and g(x) = 0. 3 + 0. 023 x, where x > 0 denotes year 2000+x. When is Samsung reaching one billion customers for the first time? (8) Solve the non-linear system of equations 5x + 2 y = exp(−x^2 ) + 5, 3x − 4 y = exp(−y^2 ) + 5, and write the y coordinate of the solution. (9) Assume that t = 0 corresponds to May 1st 2014. Consider the BTP denominated, according to the standard conventions, BTP4% 01Sep2015. Compute the gross return of the BTP if the cost is 101.72. Use 3 decimal digits for this exercise !!! (10) Enter the following commands: set.seed(143);x <- 1:10;y <- rnorm(10) and plot the points whose coordinates are x and y. How many points are inside the triangle formed by the three lines y = − 0. 5 x + 4 .5, y = 0. 5 x − 4 .5 and y = 1. a) There is no point inside the triangle. b) One point is inside the triangle. c) Two points are inside the triangle. d) Three or more points are inside the triangle. (11) Let f be the function x^2 + xy + y^2 − y 2
(^2) −x
(^2) − 2 y .
Plot the levels of f and find its global minimum.
Computational Tools EM - September 4th, 2019
Surname Name Student ID
Write the solutions using two decimal digits.
(1) Consider the numbers 1.562882e-18 and 1.215767e+19. Say whether one number is large, the other is small and 1.562882e-18 is smaller than 1.215767e+19. (2) Draw the line and the parabola y = −x − 2 , y = x^2 + 2 x + 4 over the interval [− 2 , 3 ] and find the ordinate of the lowest point belonging to the graphs. (3) Consider the expression 2**(0:2)+1:3. Which of the following stataments holds? (a) There was no need to recycle any argument. (b) The last recycled number was 1 (c) The last recycled number was 2 (d) The last recycled number was 0. (4) Which of the following statements is correct? (a) Simulation is based on several optimization runs to gain insight. (b) The terms ’distribution’ and ’simulation’ are equivalent. (c) Simulation can answer the question ’How likely is that a random outcome is acceptable?’. (d) Simulation can answer the question ’What’s the most appropriate outcome for an exper- iment?’. (5) Type set.seed(182) and press enter. Then immediately store in a vector 100 normal pseudo-random numbers with mean 5 and standard deviation 3. Compute the sum of 10 components of the vector, starting from position 11. (6) Maximize, if possible, the following function over the interval [− 3 , 1 ]: f (x) = −x^4 − 2 x^3 + 2 x^2 + 3 x + 4. Write the objective function at the optimum or -999 if there is no solution. (7) Find the leftmost root of the equation
−x^3 + 2 x^2 + x − 4 = 0.
(8) How many solutions do you have in the system of non-linear equations: x^2 + 2 xy + y^2 + x = 0 , (x + 1 )^2 + (y − 1 )^2 = 1? (9) Assume that t = 0 corresponds to May 1st 2014. Consider the BTP denominated, according to the standard conventions, BTP4.5% 01Jun2015. Compute the gross return of the BTP if the cost is 101.65. Use 3 decimal digits for this exercise !!! (10) Enter the following commands: set.seed(183);x <- 1:10;y <- rnorm(10) and plot the points whose coordinates are x and y. How many points are inside the triangle formed by the three lines y = − 0. 5 x + 3 .5, y = 0. 5 x − 2 .5 and y = −1. a) There is no point inside the triangle. b) One point is inside the triangle. c) Two points are inside the triangle. d) Three or more points are inside the triangle. (11) Let f be the function
x^2 + xy + y^2 − y 2
(^2) − 2 x
(^2) −y .
Plot the levels of f and find its global minimum.
(12) The production of a firm is
- 45 l^0.^2 k^0.^8 , where l is labour and k is capital. The unit costs of labour and capital are 1 and 0.5, respec- tively. Which is the maximum production if the cost of the factors cannot exceed 4? (13) Profits depend on price p and can be described as f (p) = 3. 5 e^2.^5 p^ − 0. 7 e^3 p. The actual price is 2.57. Which of the following is true? (a) The profit-maximizing price is about 2.85. (b) The profit at price 2.57 is 658.24. (c) The profit-maximizing price is about 2.43. (d) The profit at price 2.57 is 538.56. (14) Consider the assets ( 120 , 111 , 116 )′, ( 97 , 93 , 112 )′^ and ( 126. 9 , 120. 6 , 137. 4 )′^ whose prices are 104.79, 94.50, 118.800, respectively. Is the third asset replicable? Do you have an arbitrage opportunity on the third asset? (a) No, it’s not replicable. Hence, there is an arbitrage opportunity. (b) No, it’s not replicable and there is NO arbitrage opportunity. (c) Yes, it’s replicable and there is an arbitrage opportunity. (d) Yes, it’s replicable but there is NO arbitrage opportunity. (15) Solve the system (^)
x y z
Write the sum of the components of the solution or -999, if the system has no solution. (16) The quality of a new item is assessed using 4 numeric measures. Assume they are normal random numbers. Estimate how likely it is that the maximum of the measures exceeds 0.5. a) About 0.04; b) About 0.94; c) About 0.77; d) About 0.24.
(12) Minimize, the following function:
f (x, y) = 2 x^2 − 2 xy + y^2 − 5 x − 5 y + 4 , under the constraints x ≤ 5 , y ≤ 4 , x + y ≥ 4 and write the objective function at the optimum. (13) Profits depend on price p and can be described as f (p) = 3. 5 e^2.^5 p^ − 0. 7 e^3 p. The actual price is 2.57. Which of the following is true? (a) Profit would be maximized increasing the price by about 8.3%. (b) Decreasing the price can result in profit’s increase. (c) The profit-maximizing price is about 3.14. (d) Profit would be maximized increasing the price by about 11.1%. (14) Consider the assets ( 107 , 116 , 98 )′, ( 119 , 107 , 92 )′^ and ( 67. 8 , 66. 9 , 57 )′^ whose prices are 94.28, 96.64, 67.274, respectively. Is the third asset replicable? Do you have an arbitrage opportunity on the third asset? (a) No, it’s not replicable. Hence, there is an arbitrage opportunity. (b) No, it’s not replicable and there is NO arbitrage opportunity. (c) Yes, it’s replicable and there is an arbitrage opportunity. (d) Yes, it’s replicable but there is NO arbitrage opportunity. (15) Solve the system (^)
x y z
Write the sum of the components of the solution or -999, if the system has no solution. (16) The quality of a new item is assessed using 2 numeric measures. Assume they are uniform random numbers. Estimate how likely it is that the maximum of the measures exceeds 0.3. a) About 0.66; b) About 0.21; c) About 0.8; d) About 0.91.
Computational Tools EM - September 4th, 2019
Surname Name Student ID
Write the solutions using two decimal digits.
(1) Consider the numbers 3.046394e-21 and 3.552714e-15. Say whether both numbers are small and 3.046394e-21 is larger than 3.552714e-15. (2) Draw the line and the parabola y = − 2 x + 2 , y = x^2 − 2 x + 4 over the interval [− 2 , 1 ] and find the ordinate of the lowest point belonging to the graphs. (3) Consider the expression abs(0:2)+log(1:5). Which of the following stataments holds? (a) The first recycled number was 1. (b) The first recycled number was 1. (c) The first recycled number was 0 (d) The first recycled number was 0. (4) Let x <- seq(0.4,3,len=11) and y <- 10+sin(10*x). Consider the functions g(x) = e^0.^9 x^ and h(x) = − 6 x + 5. Plot both the functions and the points defined by x and y on the interval [ 0 , 3 ]. Which of the following sentence is true? (a) The curve is mostly below the line and 2 points are below the graph of the exponential. (b) The curve is mostly above the line and 3 points are below the graph of the exponential. (c) The curve is mostly above the line and 2 points are below the graph of the exponential. (d) The curve is mostly below the line and 4 points are below the graph of the exponential. (5) Type set.seed(192) and press enter. Then immediately create a 2x3 matrix, filled by rows with (standard) random uniform numbers. Let u be the product of the matrix times the vector ( 2 , 1 , − 1 ). Finally, provide the sum of the components of u. (6) Minimize, if possible, the following function: f (x) = −x^4 − x^3 − 2 x^2 + x + 4. Write the ob- jective function at the optimum or -999 if there is no solution. (7) Find the righmost root of the equation x^3 − 4 x^2 + 3 x + 4 = 0. (8) How many solutions do you have in the system of non-linear equations: x^2 + 2 xy + y^2 + 3 x = 0 , (x − 1 )^2 + (y − 1 )^2 = 4? (9) Assume that t = 0 corresponds to May 1st 2014. Consider the BTP denominated, according to the standard conventions, BTP5% 01Oct2015. Compute the gross return of the BTP if the cost is 101.43. Use 3 decimal digits for this exercise !!! (10) Enter the following commands: set.seed(121);x <- 1:10;y <- rnorm(10) and plot the points whose coordinates are x and y. How many points are inside the triangle formed by the three lines y = − 0. 5 x + 0 .5, y = 0. 5 x − 3 .5 and y = 0. a) There is no point inside the triangle. b) One point is inside the triangle. c) Two points are inside the triangle. d) Three or more points are inside the triangle. (11) Let f be the function x^2 + xy + y^2 − y 2
(^2) +x
(^2) −y .
Plot the levels of f and find its global minimum.
Computational Tools EM - September 4th, 2019
Surname Name Student ID
Write the solutions using two decimal digits.
(1) Consider the numbers 1.407375e+14 and 9.847709e+20. Then say whether both numbers are small and the first is larger then the second. (2) Draw the line and the parabola y = x + 1 , y = x^2 + x − 5 over the interval [− 2 , 1 ] and find the ordinate of the lowest point belonging to the graphs. (3) Consider the expression exp(-1:2)+log(1:6). Which of the following stataments holds? (a) The last recycled number was 0. (b) The last recycled number was 1. (c) The last recycled number was 1. (d) The last recycled number was 1 (4) Let x <- seq(0.4,3,len=11) and y <- 10+sin(10*x). Consider the functions g(x) = e^0.^9 x^ and h(x) = − 13 x + 15. Plot both the functions and the points defined by x and y on the interval [ 0 , 3 ]. Which of the following sentence is true? (a) The curve is mostly below the line and 2 points are below the graph of the exponential. (b) The curve is mostly above the line and 2 points are below the graph of the exponential. (c) The curve is mostly below the line and one point is below the graph of the exponential. (d) The curve is mostly above the line and 4 points are below the graph of the exponential. (5) Type set.seed(134) and press enter. Then immediately create a 3x3 matrix, filled by rows with (standard) random uniform numbers. Let u be the product of the matrix times the vector ( 0 , 2 , − 1 ). Finally, provide the sum of the components of u. (6) Minimize, if possible, the following function: f (x) = −x^4 − 4 x^3 − 4 x^2 + 4 x + 1. Write the objective function at the optimum or -999 if there is no solution. (7) In the smartphone market, the number of customers (in billions) of Apple and LG can be modelled by the functions f (x) = x/ 100 + 0. 17 e^0.^079 x, and g(x) = 0. 32 + 0. 023 x, where x > 0 denotes year 2000+x. When is Apple reaching one billion customers for the first time? (8) Solve the non-linear system of equations 2x + 3 y = exp(−x^2 ) + 6, 5x − 4 y = exp(−y^2 ) + 6, and write the x coordinate of the solution. (9) Assume that t = 0 corresponds to May 1st 2014. Consider the BTP denominated, according to the standard conventions, BTP4% 01Jul2015. Compute the gross return of the BTP if the cost is 101.41. Use 3 decimal digits for this exercise !!! (10) Enter the following commands: set.seed(123);x <- 1:10;y <- rnorm(10) and plot the points whose coordinates are x and y. How many points are inside the triangle formed by the three lines y = − 0. 5 x + 3 .5, y = 2 x − 11 and y = 2. a) There is no point inside the triangle. b) One point is inside the triangle. c) Two points are inside the triangle. d) Three or more points are inside the triangle. (11) Let f be the function x^2 + xy + y^2 − y 3
(^2) + 2 x
2 .
Plot the levels of f and find its global minimum.
(12) Minimize, the following function:
f (x, y) = −x^2 − 2 xy + 5 y^2 + x + y + 4 , under the constraints x ≤ 4 , y ≤ 5 , 3 x + y ≥ 4 and write the objective function at the optimum. (13) Profits depend on price p and can be described as f (p) = 4. 5 e^2 p^ − 0. 6 e^3 p. The actual price is 1.45. Which of the following is true? (a) The profit at price 1.45 is 38.83. (b) The profit at price 1.45 is 31.77. (c) The profit-maximizing price is about 1.61. (d) Profit would be maximized increasing the price by about 8.2%. (14) Consider the assets ( 113 , 105 , 116 )′, ( 105 , 107 , 117 )′^ and ( 109 , 106 , 116. 5 )′^ whose prices are 98.60, 99.10, 108.850, respectively. Is the third asset replicable? Do you have an arbitrage opportunity on the third asset? (a) No, it’s not replicable. Hence, there is an arbitrage opportunity. (b) No, it’s not replicable and there is NO arbitrage opportunity. (c) Yes, it’s replicable and there is an arbitrage opportunity. (d) Yes, it’s replicable but there is NO arbitrage opportunity. (15) Solve the system (^)
x y z
Write the sum of the components of the solution or -999, if the system has no solution. (16) The quality of a new item is assessed using 4 numeric measures. Assume they are normal random numbers. Estimate how likely it is that the maximum of the measures exceeds 0.5. a) About 0.26; b) About 0.21; c) About 0.77; d) About 0.1.
Plot the levels of f and find its global minimum. (12) Minimize, the following function:
f (x, y) = x^2 − 5 xy + 4 y^2 + x + 4 y + 5 , under the constraints x ≤ 5 , y ≤ 4 , x + y ≥ 5 and write the objective function at the optimum. (13) Profits depend on price p and can be described as f (p) = 5 e^2.^5 p^ − 0. 6 e^4 p. The actual price is 0.92. Which of the following is true? (a) The profit-maximizing price is about 1.1. (b) Profit would be maximized increasing the price by about 14.7%. (c) The profit-maximizing price is about 0.94. (d) The optimal profit is about 35.508. (14) Consider the assets ( 115 , 111 , 117 )′, ( 105 , 97 , 106 )′^ and ( 100. 5 , 97. 1 , 104. 1 )′^ whose prices are 102.24, 91.71, 89.550, respectively. Is the third asset replicable? Do you have an arbitrage opportunity on the third asset? (a) No, it’s not replicable and there is NO arbitrage opportunity. (b) No, it’s not replicable. Hence, there is an arbitrage opportunity. (c) Yes, it’s replicable and there is an arbitrage opportunity. (d) Yes, it’s replicable but there is NO arbitrage opportunity. (15) Solve the system (^)
x y z
Write the sum of the components of the solution or -999, if the system has no solution. (16) The quality of an organizational process is assessed using 3 numeric measures. Assume they are uniform random numbers. Estimate how likely it is that the minimum of the measures exceeds 0.5. a) About 0.88; b) About 0.62; c) About 0.34; d) About 0.12.
Computational Tools EM - September 4th, 2019
Surname Name Student ID
Write the solutions using two decimal digits.
(1) Consider the numbers 1.215767e+19 and 4.398047e+12. Then say whether both numbers are small and the first is larger then the second. (2) Draw the line and the parabola y = 2 x − 2 , y = x^2 − 2 x − 3 over the interval [− 1 , 2 ] and find the ordinate of the lowest point belonging to the graphs. (3) Consider the expression exp(1:2)+log(1:6). Which of the following stataments holds? (a) There was no need to recycle any argument. (b) The last recycled number was 0 (c) The last recycled number was 7. (d) The last recycled number was 0. (4) Which of the following statements is correct? (a) Simulation can answer the question ’What’s the correct outcome of my experiment?’. (b) Simulation can be used to estimate how probable is an event. (c) Simulation is based on several optimization runs to gain insight. (d) Simulation has to do with the insight that be be gained on similar problems. (5) Type set.seed(149) and press enter. Then immediately store in a vector 100 normal pseudo-random numbers with mean -5 and standard deviation 1. Compute the sum of 14 components of the vector, starting from position 41. (6) Maximize, if possible, the following function over the interval [− 3 , − 1 ]: f (x) = −x^4 + 4 x^3 + 2 x^2 + 3 x − 4. Write the objective function at the optimum or -999 if there is no solution. (7) In the smartphone market, the number of customers (in billions) of Apple and LG can be modelled by the functions f (x) = x/ 100 + 0. 18 e^0.^085 x, and g(x) = 0. 31 + 0. 021 x, where x > 0 denotes year 2000+x. When is Apple reaching one billion customers for the first time? (8) How many solutions do you have in the system of non-linear equations: x^2 − 2 xy + y^2 + 3 x = 0 , (x + 1 )^2 + (y − 1 )^2 = 1? (9) Assume that t = 0 corresponds to May 1st 2014. Consider the BTP denominated, according to the standard conventions, BTP3.5% 01Oct2015. Compute the gross return of the BTP if the cost is 101.41. Use 3 decimal digits for this exercise !!! (10) Enter the following commands: set.seed(160);x <- 1:10;y <- rnorm(10) and plot the points whose coordinates are x and y. How many points are inside the triangle formed by the three lines y = − 0. 5 x + 4 .5, y = x − 4 and y = −1. a) There is no point inside the triangle. b) One point is inside the triangle. c) Two points are inside the triangle. d) Three or more points are inside the triangle. (11) Let f be the function
x^2 + xy + y^2 − y 4
(^2) +x
(^2) −y .
Plot the levels of f and find its global minimum.
Computational Tools EM - September 4th, 2019
Surname Name Student ID
Write the solutions using two decimal digits.
(1) Consider the numbers 8.225263e-20 and 1.1259e+15. Say whether one number is large, the other is small and 8.225263e-20 is smaller than 1.1259e+15. (2) Draw the line and the parabola y = x + 1 , y = x^2 − x − 5 over the interval [− 2 , 3 ] and find the ordinate of the lowest point belonging to the graphs. (3) Consider the expression exp(-1:2)+log(1:4). Which of the following stataments holds? (a) The last recycled number was 0. (b) The last recycled number was 1. (c) The last recycled number was 7. (d) There was no need to recycle any argument. (4) Let x <- seq(0.4,3,len=11) and y <- 10+sin(10*x). Consider the functions g(x) = ex and h(x) = 2. 8 x + 5. Plot both the functions and the points defined by x and y on the interval [ 0 , 3 ]. Which of the following sentence is true? (a) The curve is mostly above the line and 5 points are below the graph of the exponential. (b) The curve is mostly below the line and 4 points are below the graph of the exponential. (c) The curve is mostly below the line and 3 points are below the graph of the exponential. (d) The curve is mostly above the line and 2 points are below the graph of the exponential. (5) Type set.seed(147) and press enter. Then immediately store in a vector 100 normal pseudo-random numbers with mean 5 and standard deviation 1. Compute the sum of 8 com- ponents of the vector, starting from position 51. (6) Minimize, if possible, the following function: f (x) = −x^4 + 2 x^3 + 3 x^2 + 2 x − 3. Write the objective function at the optimum or -999 if there is no solution. (7) Find the leftmost root of the equation −x^3 − 2 x^2 + 3 x − 1 = 0. (8) Solve the non-linear system of equations 4x + 5 y = exp(−x^2 ) + 5, 3x − 2 y = exp(−y^2 ) + 5, and write the y coordinate of the solution. (9) Assume that t = 0 corresponds to May 1st 2014. Consider the BTP denominated, according to the standard conventions, BTP3% 01Oct2015. Compute the gross return of the BTP if the cost is 101.12. Use 3 decimal digits for this exercise !!! (10) Enter the following commands: set.seed(191);x <- 1:10;y <- rnorm(10) and plot the points whose coordinates are x and y. How many points are inside the triangle formed by the three lines y = −x + 7, y = x − 4 and y = 0. a) There is no point inside the triangle. b) One point is inside the triangle. c) Two points are inside the triangle. d) Three or more points are inside the triangle. (11) Let f be the function x^2 + xy + y^2 − y 9
(^2) − 2 x
(^2) + 2 y .
Plot the levels of f and find its global minimum.
(12) Maximize, the following function:
f (x, y) = 4 x^2 + 4 xy + 2 y^2 + 4 x − 5 y + 2 , under the constraints x ≤ 3 , y ≤ 4 , 3 x + y ≥ 5 and write the objective function at the optimum. (13) Profits depend on price p and can be described as f (p) = 3. 5 e^1.^5 p^ − 0. 3 e^3 p. The actual price is 1.4. Which of the following is true? (a) Profit would be maximized decreasing the price by about 12.8%. (b) The profit at price 1.4 is 7.72. (c) The optimal profit is about 12.353. (d) Profit at price 1.4 is 8.58. (14) Consider the assets ( 119 , 100 , 118 )′, ( 99 , 93 , 116 )′^ and ( 113. 9 , 104. 4 , 129. 2 )′^ whose prices are 97.30, 91.20, 102.350, respectively. Is the third asset replicable? Do you have an arbitrage opportunity on the third asset? (a) No, it’s not replicable and there is NO arbitrage opportunity. (b) No, it’s not replicable. Hence, there is an arbitrage opportunity. (c) Yes, it’s replicable and there is an arbitrage opportunity. (d) Yes, it’s replicable but there is NO arbitrage opportunity. (15) Solve the system (^)
x y z
Write the sum of the components of the solution or -999, if the system has no solution. (16) The quality of an organizational process is assessed using 2 numeric measures. Assume they are uniform random numbers. Estimate how likely it is that the average of the measures exceeds 0.6. a) About 0.56; b) About 0.32; c) About 0.13; d) About 0.89.
