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1. Consider the multi-variable function g(x, y) = x²y+ln(xy) – y3+V1 – 2-sin(my) (a) (4 marks)...
Question 4 a) Differentiate with respect to x, i. y = sin 2x ii. y =...
Question 4 a) Differentiate with respect to x, i. y = sin 2x ii. y = x In(5x + 2) b) Show that if y = cotx, dy dx -cosec? x c) Show that if y = tan x, then dy dx 1 1+xal Question 5 Use calculus to find any turning points of the function A(t) = te-020 and determine their nature (maximum, minimum or inflexion) using any method. Question 6 a) Find tan” x dx b) Use integration...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
1 point) Sketch the graph of the function y = x(4-2) - 103 In x. Indicate...
1 point) Sketch the graph of the function y = x(4-2) - 103 In x. Indicate the transition points (local extrema and points of inflection) (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list of e-coordinates. Enter NULL in answer field if there is no such point.) Local maximum at = help (fractions) Local minimum at I= Inflection at I= (1 point) Sketch the graph of the function y = 81x +...
Question 3 [20 marks] A measure of skewness is defined by Y such that: Note that...
Question 3 [20 marks] A measure of skewness is defined by Y such that: Note that when a distribution is symmetrical about the mean, the skewness is equal to zero. If it is skewed to the right, the measure of skewness will be positive; if it is skewed to the left, the measure of skewness will be negative. Let X be random variable, and a function f(x) is defined such that 21, 2, 3, 4, 5, 6, f(x)- (a) Create...
Suppose an individual’s utility function for two goods X and Y is givenby U(X,Y) = X^(3/4)Y^(1/4)...
Suppose an individual’s utility function for two goods X and Y is givenby U(X,Y) = X^(3/4)Y^(1/4) Denote the price of good X by Px, price of good Y by Py and the income of the consumer by I. a) (2 points) Write down the budget constraint for the individual. b) (4 points) Derive the marginal utilities of X and Y. c) (3 points) Derive the expression for the marginal rate of substitution of X for Y. Write down the tangency...
y 1) (4 marks) Algebraically invert the Lorentz transformation, I' (x- ut) (1) V (2) (3)...
y 1) (4 marks) Algebraically invert the Lorentz transformation, I' (x- ut) (1) V (2) (3) 7(t - ux/c) where y = (1 - u2/)-1/2. In other words, solve for x, y, z and t in terms of X',V, :' and t'. Note: I expect you to simplify your expressions appropriately. 2
Q6 [10+1+3=14 Marks] Let F be a force field given by F(x, y) = y2 sin(xy?)...
Q6 [10+1+3=14 Marks] Let F be a force field given by F(x, y) = y2 sin(xy?) i + 2xy sin(xy?)j. (a) Show that F. dr is exact by finding a potential function f. (b) Is I = S, y2 sin(xy2) dx + 2xy sin(xy?) dy independent of path C? Justify your answer. (c) Use I to find the work done by the force field F that moves a body along any curve from (0,0) to (5,1).
D X-EC2010-1 M. 1. An individual consumer with Cobb-Douglas preferences over two products, x and y,...
D X-EC2010-1 M. 1. An individual consumer with Cobb-Douglas preferences over two products, x and y, maximises utility, U(x,y) = x1910, subject to the constraint that all income, M, is spent on x and/or y. Products x and y are priced at px and Py, respectively. (a) Set up the appropriate lagrangian for this maximisation problem, find the appropriate first-order conditions for this lagrangian and solve for X and y in terms of Px. Py and M. (40 marks) (b)...
4. This question is about using the composite Simpson's Rule to estimate the integral 1 =...
4. This question is about using the composite Simpson's Rule to estimate the integral 1 = (exp() dr to ten decimal places. (a) Enter and save the following Matlab function function y = f(x) y =exp(x/2); end [O marks) (b) Now complete the following Matlab function function y = compSR (a,b,N) end The function is to return the estimate of I found by applying Simpson's Rule N times. The Matlab function from the previous part of the question should be...
X-EC2010-1 1. An individual consumer with Cobb-Douglas preferences over two products, x and y, maximises utility,...
X-EC2010-1 1. An individual consumer with Cobb-Douglas preferences over two products, x and y, maximises utility, U(X.y) = x10y10, subject to the constraint that all income, M, is spent on x and/or y. Products x and y are priced at Px and Py, respectively. (a) Set up the appropriate lagrangian for this maximisation problem, find the appropriate first-order conditions for this lagrangian and solve for x and y in terms of px, Py and M. (40 marks) (6) For product...
Question 4 a) Differentiate with respect to x, i. y = sin 2x ii. y = x In(5x + 2) b) Show that if y = cotx, dy dx -cosec? x c) Show that if y = tan x, then dy dx 1 1+xal Question 5 Use calculus to find any turning points of the function A(t) = te-020 and determine their nature (maximum, minimum or inflexion) using any method. Question 6 a) Find tan” x dx b) Use integration...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
1 point) Sketch the graph of the function y = x(4-2) - 103 In x. Indicate the transition points (local extrema and points of inflection) (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list of e-coordinates. Enter NULL in answer field if there is no such point.) Local maximum at = help (fractions) Local minimum at I= Inflection at I= (1 point) Sketch the graph of the function y = 81x +...
Question 3 [20 marks] A measure of skewness is defined by Y such that: Note that when a distribution is symmetrical about the mean, the skewness is equal to zero. If it is skewed to the right, the measure of skewness will be positive; if it is skewed to the left, the measure of skewness will be negative. Let X be random variable, and a function f(x) is defined such that 21, 2, 3, 4, 5, 6, f(x)- (a) Create...
y 1) (4 marks) Algebraically invert the Lorentz transformation, I' (x- ut) (1) V (2) (3) 7(t - ux/c) where y = (1 - u2/)-1/2. In other words, solve for x, y, z and t in terms of X',V, :' and t'. Note: I expect you to simplify your expressions appropriately. 2
Q6 [10+1+3=14 Marks] Let F be a force field given by F(x, y) = y2 sin(xy?) i + 2xy sin(xy?)j. (a) Show that F. dr is exact by finding a potential function f. (b) Is I = S, y2 sin(xy2) dx + 2xy sin(xy?) dy independent of path C? Justify your answer. (c) Use I to find the work done by the force field F that moves a body along any curve from (0,0) to (5,1).
D X-EC2010-1 M. 1. An individual consumer with Cobb-Douglas preferences over two products, x and y, maximises utility, U(x,y) = x1910, subject to the constraint that all income, M, is spent on x and/or y. Products x and y are priced at px and Py, respectively. (a) Set up the appropriate lagrangian for this maximisation problem, find the appropriate first-order conditions for this lagrangian and solve for X and y in terms of Px. Py and M. (40 marks) (b)...
4. This question is about using the composite Simpson's Rule to estimate the integral 1 = (exp() dr to ten decimal places. (a) Enter and save the following Matlab function function y = f(x) y =exp(x/2); end [O marks) (b) Now complete the following Matlab function function y = compSR (a,b,N) end The function is to return the estimate of I found by applying Simpson's Rule N times. The Matlab function from the previous part of the question should be...
X-EC2010-1 1. An individual consumer with Cobb-Douglas preferences over two products, x and y, maximises utility, U(X.y) = x10y10, subject to the constraint that all income, M, is spent on x and/or y. Products x and y are priced at Px and Py, respectively. (a) Set up the appropriate lagrangian for this maximisation problem, find the appropriate first-order conditions for this lagrangian and solve for x and y in terms of px, Py and M. (40 marks) (6) For product...
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