y''' + 2y'' − y' − 2y = sin(4t),  y(0) = 0,  y'(0) = 0,  y''(0) = 1

1. Explain the Differences Between Subjective and Objective Measures.

Solve the following initial value problem over the interval from t = 0 to 1 where y(0) = 1 using the following methods with a step size of 0.25.

dy/dt=(1+4t)*sqrt(y)

a) Analytical method

b) Euler's method

c) Heun's method without iteration

d) Ralston's method

e) Fourth-order Runge-Kutta method

f) Display all your results obtained above on the same graph

Solve the following initial value problem over the interval from t = 0 to 2 where y(0) = 1 using the following methods.

dy/dt=y*t^2−1.1y

a) Analytical method

b) Euler's method with h=0.5 at t=2

c) Euler's method with h=0.25 at t=2

d) Midpoint method with h=0.5 at t=2

e) Fourth-order Runge-Kutta method with h=0.5 at t=2

f) Display all yor results obtained above on the same graph

Consider the subgroup H={R0,R180} of D6. List the elements of the factor group D6/H.

Use undetermined coefficients to find the particular solution to

y''−9y'+20y=−3936sin(4t)

Researchers collected data on the numbers of hospital admissions resulting from motor vehicle​ crashes, and results are given below for Fridays on the 6th of a month and Fridays on the following 13th of the same month. Use a 0.05 significance level to test the claim that when the 13th day of a month falls on a​ Friday, the numbers of hospital admissions from motor vehicle crashes are not affected. Friday the​ 6th: 10  5 11 11  4 4 Friday the​ 13th: 14 11 14 10 6 12 .

Identify the critical​ value(s). The critical values are t= ±

2.0152.015.

Introduction In the popular computer game League of Legends, a player’s Effective Health, E, when defending against physical damage is given by E = H(1 + 0.02A) where H and A denote the player’s chosen units of health and armor, respectively. Furthermore, a unit of health costs 4 gold pieces and a unit of armor costs 24 gold pieces. If you have 3,600 gold pieces to spend and you need to maximise the effectiveness E of your health and armor to survive as long as possible against enemy attacks, how much of each should you buy?

A3. Thirty minutes into the game, you have 2,000 health and 40 armor. You have 1,800 gold pieces to spend, and again health costs 4 gold per unit and armor costs 24 gold per unit. If the goal is still to maximise the effectiveness E of your resulting health and armor (after purchase), how much of each should you buy with your 1,800 gold pieces (show your work).

Write a 300-word essay of the importance of inequalities in real life. Share a task-personal of professional that could be modeled mathematically thorough inequalities. Explain how this might be used in making good decisions.

Introduction In the popular computer game League of Legends, a player’s Effective Health, E, when defending against physical damage is given by E = H(1 + 0.02A) where H and A denote the player’s chosen units of health and armor, respectively. Furthermore, a unit of health costs 4 gold pieces and a unit of armor costs 24 gold pieces. If you have 3,600 gold pieces to spend and you need to maximise the effectiveness E of your health and armor to survive as long as possible against enemy attacks, how much of each should you buy?

A1. (1 mark) Write down the Lagrange function associated with this optimisation problem.

A2. Solve the problem using the Lagrange Multiplier method (show your work).

A3. Thirty minutes into the game, you have 2,000 health and 40 armor. You have 1,800 gold pieces to spend, and again health costs 4 gold per unit and armor costs 24 gold per unit. If the goal is still to maximise the effectiveness E of your resulting health and armor (after purchase), how much of each should you buy with your 1,800 gold pieces (show your work).

Describe some advantages and disadvantages of Value at Risk (VaR) as a risk measure relative to other risk measures

The Seaberg Library is interested in the relationship between the inches of rain in a month and the number of patrons visiting the library in a month. They collect data for seven months below.

a) State your null formally and in lay terms for a simple regression (10)

b) Calculate r and the regression line (y = a + bx) and reject/accept at a=.05. You must state the estimated regression line even if Ho is accepted. (10)

c) Explain your findings in lay terms using r-square, r, b as appropriate (20)

d) Calculate a 95% confidence interval for the slope IF NECESSARY and if not explain WHY NOT. Provide the formal interval and explain in layterms. (5)