Sec 2.2:  Analyze the DE dx/dt = x(2-x) - h, where h is the rate of harvesting.   For each h>0, what are the critical points of the DE?   Which of them are stable?   What is the bifurcation point?   Draw a bifurcation diagram indicating which equilibrium points are stable and which are unstable.  (Use a phase line diagram to aid in your analysis.)

Write Matlab programs implementing the algorithms based on bisection,Newton, and secant method for numerical solution of scalar nonlinear equa-tions. Use these programs to compute approximations to real roots of the

following equations:

exp(x)−3x^2=0, (1)

x^3=x^2+x+1, (2)

exp(x) =1/(0.1 +x^2), (3)

and

x= 1 + 0.3 cos(x). (4)

Use an error tolerance tol=10^(−12). Display the obtained approximations to the roots of these equations, and compare the number of iterations, starting with the same initial values x0 for bisection and Newton methods, and with x0 and x1, where x1 was computed by bisection starting with x0, for secant method.

A brine solution of salt flows at a constant rate of 8 L/min into a large tank that initially held 100 L of brine solution in which was dissolved 0.15 kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is 0.03 ​kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.01 kg/L?

Develop a three-period moving average forecast for April 2019 through January 2020. Calculate the MFE, MAD, and MAPE values for April through December 2019. LOADING... Click the icon to view the time series data. Develop a three-period moving average and fill-in the table below (enter your responses rounded to one decimal place). Month Demand Forecast January 2019 123123 February 7474 March 114114 April 7777 103.66103.66 May 9898 88.3388.33 June 136136 96.3396.33 July 103103 103.66103.66 August 106106 112.33112.33 September 100100 103103 October 9393 99.6699.66 November 9090 94.9394.93 December 8989 90.6690.66 The forecast for January 2020 is nothing. (Enter your response rounded to one decimal place.) The MFE is nothing. (Enter your response rounded to one decimal place and include a minus sign if necessary.) The MAD is nothing. (Enter your response rounded to one decimal place.) The MAPE is nothing %. (Enter your response rounded to one decimal place.) Enter your answer in each of the answer boxes.

Implement the Lorenz-63 model in MATLAB, and solve numerically (using MATLAB’s ode45 or other built-in solver) for any random, non-zero initial conditions to reproduce, qualitatively, Figures 3 and 4. Note that, as usual, you should label your axes and make the plots as “pretty” as possible.

The model equations are

dx/ dt = σ(y − x),

dy /dt= x(ρ−z)−y,

dz/dt = xy − βz.
Use parameter values σ = 10, ρ = 28, and β = 8/3.

A toboggan with two people on it weighs 300 lb. It starts from rest down a slope, 1/4 mile long, from a height 200 ft above horizontal level. The coefficient of sliding friction is 3/100 and the force of the wind resistance is proportional to the square of the velocity. When the velocity is 30 ft/sec, this force is 6 lb.

(a) Find the velocity of the toboggan as a function of the distance and of the time.

(b) With what velocity will the toboggan reach the bottom of the slide?

(c) When will it reach the bottom?

(d) What would its terminal velocity be if the slide were infinite in length?

Answers:

(a) v= 74.1 (e^(0,105t)-1)/(e^(0.105t)+1), v^2=5484(1-e^(-0.0014s)

(b) 68 ft/sec

(c) 30 sec, approx.

(d) 74.1 ft/sec

I'm having trouble solving for v originally. Any help would be much appreciated.

y(4)+18y''+81y=0

y(0)=2,  y'(0)=8,  y''(0)=0,  y'''(0)=−108

Note; y(4) is the 4th derivative of y

Solve the initial value problem y(t)= ?

To find a root of a polynomial equation, we can use an iterative process.
We start with an initial guess for the value of the root, x 0 , plug it in to the iterative formula and
solve for x 1 . Then we plug x 1 back into the iterative formula and solve for x 2 . We continue this
process until x n+1 and x n are equal to a specified number of decimal places. When this happens,
this is our approximate solution to the polynomial equation.
We will be solving for a root of a cubic equation:
f(x n ) = c3 x n 3 + c2 x n 2 + c1 x n + c0
where c3, c2, c1 and c0 are the coefficients of each polynomial term.
The iterative formula we will use is:
x n+1 = x n - ( f(x n ) / f '(x n ) )
where f '(x n )is the derivative of f(x n )

Define a public static method named cubicRoot that accepts the coefficients of the cubic
equation and an initial guess for the root . This method computes and returns a root of the cubic
equation by using the iterative process described below (you must use a while loop):
1. Start with the guess for the root passed to the method as x n

2. Compute x n+1 using the formula above Note: you can write the equation for the
derivative in terms of the coefficients, exponents and x terms.
3. Compare x n+1 and x n
i. if these are equal within 4 decimal places, then return the value
ii. If not, x n should be updated - repeat Step 2

in java code

A parachutist whose mass is 80 kg drops from a helicopter hovering 2000 m above the ground and falls toward the ground under the influence of gravity. Assume that the force due to air resistance is proportional to the velocity of the​ parachutist, with the proportionality constant b1=20 ​N-sec/m when the chute is closed and b2=100 ​N-sec/m when the chute is open. If the chute does not open until the velocity of the parachutist reaches 35 ​m/sec, after how many seconds will the parachutist reach the​ ground? Assume that the acceleration due to gravity is 9.81 m/s^2.

a) verify that y1 and y2 are fundamental solutions

b) find the general solution for the given differential equation

c) find a particular solution that satisfies the specified initial conditions for the given differential equation

1. y'' + y' = 0; y1 = 1 y2 = e^-x; y(0) = -2 y'(0) = 8

2. x^2y'' - xy' + y = 0; y1 = x y2 = xlnx; y(1) = 7 y'(1) = 2

Determine the form of a particular solution to the following differential equations (do not evaluate coefficients).

(a)y′′ −4y′ = x+1+ xe^(2x) + e^(4x) + e^(4x)sin4x

Confidence Interval Worksheet with Sample Data MTH 160: Statistics

Read the following scenario and complete each of the problems below:

A flashlight company claims that the new bulb in its heavy-duty flashlight will average 246 hours of light. A statistics student decides that he/she wants to test this claim at a 5% level of significance to determine if there is evidence to support the claim. The student randomly selects and tests 15 flashlight bulbs and records how long the bulb lasts until it burns out. Assume the life of a bulb is normally distributed. T

he data is in the table below

Trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Hrs: 246 224 231 242 237 240 243 236 239 255 256 239 247 231 253

A. The standard deviation of the population is 7.4 hours. Construct a 95% confidence interval for this study.

B. Same scenario, but the population’s standard deviation is not known. Construct a 95% confidence interval for this study.

C. Write a statement comparing and contrasting the two confidence intervals and results.

Read the following scenario and complete each of the problems below:

A new car manufacturing company has emerged and has claimed that its new hybrid car, the Pusho, gets a better gas mileage than the highest ranked Toyota Prius. Consumer Reports Magazine decides to test this claim at a 5% level of significance. Consumer Reports randomly selects 10 of each type of car, calculates the miles per gallon for each car in the study, and records the data in the table below. Assume miles per gallon of the cars is normally distributed.

Pusho 54.1 52.4 55.7 49.7 50.6 48.9 51.8 54.5 56.9 49.8

Prius 53.2 54.3 49.8 50.1 50.5 56.1 47.8 53.4 56.8 48.7

A. Construct a 90% confidence interval for the difference in the gas mileage of Pusho and Prius.

B. Construct a 95% confidence interval for the difference in the gas mileage of Pusho and Prius.

C. Write a statement comparing and contrasting the two confidence intervals and the significance of having 0 in both of those intervals.

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is a vector space over R. A polynomial f(x, y) is called degree d homogenous polynomial if the combined degree in x and y of each term is d. Let Vd be the set of degree d homogenous polynomials from R[x, y]. Is Vd a subspace of R[x, y]? Prove your answer.

By constructing a suitable bijection, show that the number of subsets of an n-set of odd size is equal to the number of subsets of an n-set of even size.

Let us divide the odd positive integers into two arithmetic progressions; the red numbers are 1, 5, 9, 13, 17, 21, ... The blue numbers are 3, 7, 11, 15, 19, 23,....

(a) Prove that the product of two red numbers is red, and that the product of two blue numbers is red.

(b) Prove that every blue number has a blue prime factor.

(c) Prove that there are infinitely many blue prime numbers. Hint: Follow Euclid’s proof, but multiply a list together, multiply the result by four, then subtract one.