Using the wave functions

for the potential energy step, apply the boundary conditions of ψ and

dψ/dx

to find B' and C' in terms of A', for the potential step when particles are incident from the negative x direction. Evaluate the reflection and transmission coefficients

R=

and

T=

.

( k₀=42 and k₁=12)

Boundary-Value Problems in Other Coordinate Systems

1. Solve ∆u = 0 in a disk x^2 + y^2 ≤ 25, where u(5, θ) = 7 sin 3θ − 6 sin 8θ and u is bounded when   r = 0.
2. Solve ∆u = 0 in an annulus 1 ≤ x^2+y^2 ≤ 4, where u(1, θ) = 75 sin θ, u(2, θ) = 60 cos θ.
3. Find the steady-state temperature distribution in a disk of radius 1 if the upper half of the circumference is kept at 100◦ and the lower half is kept at 0◦ .

Quantity take-off problem: A capital improvement project requires the installation of a property line fence along the 250 -ft northern boundary line. The decorative aluminum fence is constructed of posts spaced at 10-ft center to center and an ornate picket infill panel between two posts. Given the material costs below, the cost for the construction of the fence is most nearly:

a. $65,771.25

b. $66,461.60

c. $68,402.10

d. $71, 065.58

Material costs:

Aluminum posts: $645.35 each

Ironworker: $78/hr

Picket infill panel: $1,985.50 each

Placed concrete $498.00/cy

1) Several central banks have used negative official interest rates in recent years, which until recently seemed unimaginable. Indeed, economists used to talk about a zero interest-rate boundary (ZIRB). How is it possible for a central bank to cut its official interest rate below zero?

2) Japan has the world's highest level of government debt. The interest rate (or yield) on 10-year Japanese government debt has been approximately zero for the past three years. What has made this possible? How is it related to quantitative easing?

The refractive index of a transparent material can be determined by measuring the critical angle when the solid is in air. If θ_c= 41.5° what is the index of refraction of the material?

A light ray strikes this material (from air) at an angle of 37.1° with respect to the normal of the surface. Calculate the angle of the reflected ray (in degrees).

Calculate the angle of the refracted ray (in degrees).

Assume now that the light ray exits the material. It strikes the material-air boundary at an angle of 37.1° with respect to the normal. What is the angle of the refracted ray?

The fully-developed, steady,Poiseuilleflow of a liquidina horizontal, circular cylinderwith no swirl (i.e., Vθ= 0)under certain simplifying assumptions is given by:

Vz = (−∆???)4?(R2 –r2)Vr = 0Vθ= 0Here, Ris the inner radius of the pipe and r, θand zare the (cylindrical) coordinates of any point inside the pipe.Apply the continuity equation in cylindrical coordinates (Appendix B.8 of the text), given by()()()0zr1rrr1t=+++zrVVVto this flow (giving reasons why you drop each term) and obtain an ordinary differential equation for Vr. Integrate your solution and apply appropriate boundary conditions to obtain Vr.

Q 5.

Reagan has accumulated $3200 in savings and wishes to invest this money sensibly. The types of investments and their corresponding percentages, recommended by a financial advisor, are shown in the following pie chart. Find the amount of money Reagan should invest in bonds. Round your answer to the nearest whole dollar

Recommended Categories of Investment

Bonds 38.28%

Stocks 15.74%

Real Estate 10.22%

Mutual Funds 10.54%

Annuities 25.22%

Q 6.

Consider the following frequency table representing the distribution of cost of a paperback book (in dollars).

Step 1 of 2:

Determine the cumulative frequency for the second class.

Step 2 of 2:

Determine the cumulative frequency for the fourth class.

Q 7.

Consider the following frequency table representing the scores on a test.

Copy Data

Step 1 of 5:

Determine the lower class boundary for the fifth class.

Step 2 of 5:

Determine the upper class boundary for the second class.

Step 3 of 5:

Determine the class width of each class.

Step 4 of 5:

Choose the interval that contains the score, 57.9

Step 5 of 5:

Determine the number of scores between 39.5 and 55.5

2. The growth rate of a population of bacteria is directly proportional to the population p(t) (measured in millions) at time t (measured in hours).

(a) Model this situation using a differential equation.

(b) Find the general solution to the differential equation.

(c) If the number of bacteria in the culture grew from p(0) = 200 to p(24) = 800 in 24 hours, what was the population after the first 12 hours?

3. Find the particular solution y(x) to the following boundary-value problem, y ′′ + 9y = 0, with y(0) = −1 and y(π/6) = 1.

4. Consider the differential equation y ′ + 3t 2 y = e −t 3 with initial condition y(0) = 1. Find the particular solution of the differential equation.

6. Consider a rod: 0 < x < L with insulated sides. The temperature at the sides is fixed at 10 degrees C. At time t = t0, the rod is given an initial temperature distribution of f(x) degrees C, for 0 < x < L. Let u(x, t) denote the temperature at the point x on the rod at time t. The heat flow is modeled by the heat equation, ut − 2uxx = 0.

(a) Write the initial condition(s) of the problem in terms of u(x, t).

(b) Write the boundary condition(s) of the problem in terms of u(x, t).

(c) What does the solution of this problem represent for the rod?

(d) Suppose you are not interested in solving the problem but in finding the steady-state solution. What differential equation would you solve instead?

Given the data on scores of students final grade in statistics (in percent) determine the following statistics.

43 45 48 51 53 54 57 59 60 60 60 60 61 70 70 71 71 72 72 72 75 76 76 79 81 81 83 85 87 88 88 89 89 91 92 93 96 98 98 99 100 101 101

Assume students are only allowed to transfer the class if they receive a grade of 70 % or above. Use this fact to create a binomial distribution for students that are able to transfer and students not able to transfer the class. Do this by finding the proportion of students that receive a grade of 70 or above (this will be the value p and then q = 1 - p).

a. Determine the mean and standard deviation using the binomial distribution formulas.

b. Determine the range of usual value by finding the values that are significantly low and significantly high.

c. Use a normal continuous distribution to APPROMATE the binomial discrete probability distribution to determine the probability that at least 30 students score at a 70 or more. (Be sure to use the boundary to get the more accurate/correct answer.) Show an approximation box to verify your boundaries.

d. Use a normal continuous distribution to APPROMATE the binomial discrete probability distribution to determine the probability that exactly 30 students score at a 70 or more. (Be sure to use the boundary to get the more accurate/correct answer. Show an approximation box to verify your boundaries.

4. Gradient descent. Gradient descent is one of the most popular algorithms in data science and by far the most common way to optimise neural networks. A function is minimised by iteratively moving a little bit in the direction of negative gradient. For the two-dimensional case, the step of iteration is given by the formula xn+1 , yn+1 = xn, yn − ε ∇f(xn, yn). In general, ε does not have to be a constant, but in this question, for demonstrative purposes, we set ε = 0.1. Let f(x, y) = 3.5x 2 − 4xy + 6.5y 2 and x0 and y0 be any real numbers. (a) For all x, y ∈ R compute ∇f(x, y) and find a matrix A such that [3] A x y = x y − ε ∇f(x, y). Write an expression for xn yn in terms of x0 and y0 and powers of A. (b) Find the eigenvalues of A. [1] (c) Find one eigenvector corresponding to each eigenvalue. [2] (d) Find matrices P and D such that D is diagonal and A = P DP −1 . [1] (e) Find matrices Dn , P −1 and An . Find formulas for xn and yn. [4] (f) Suppose x0 = y0 = 1. Find the smallest N ∈ N such that xN yN ≤ 0.05. [3] (g) Sketch the region R consisting of those (x0, y0) such that xN ≥ 0, yN ≥ 0 and [4] xN yN ≤ 0.05, xN−1 yN−1 > 0.05, where N is the number found in part (f). Write an equation for the boundary of R. Which points of the boundary belongs to R and which do not?