a rectangular plate OPQR is bounded by the lines x=0, y=0 x=4, y=4 determine the potential distribution u(x,y) over the rectangular using the laplace equation uxx+uyy=0 boundary conditions are u(4,y), u(x,0)=0, u(x,0)=x(4-x) usimg separation of variables

The McBurger Corporation developed a new vegan burger and introduced the product to several test markets. The burger comes with the non-meat based patty and the usual accoutrements such as lettuce, tomato, ketchup, mustard, pickle, onion, special sauce, and a bun. The company is interested in customer response, and surveyed customers about satisfaction with the burger. A data analyst proposed a multiple regression model with an interval level dependent variable of repurchase (measured by a ratings thermometer from 0 (will definitely not repurchase) to 100 (will repurchase). The data analyst is not sure on what independent variables will best predict repurchase, and needs your help. What independent variables would you include in the multiple regression model? Explain your reasoning for including each independent variable. Which variable would be your best predictor of repurchase?

Identify at least four independent variables.

You are in a land inhabited by people who either always tell the truth or always tell falsehoods. You come to a fork in the road and you need to know which fork leads to the capital. There is a local resident there, but he has time only to reply to one yes-or-no question. What one question should you ask so as to learn which fork to take? Suggestion: Make a table.

For each of the following determine whether ∗ is a binary operation on R. If so, determine whether or not ∗ is associative, commutative, has an identity element, and has inverse elements.

(a) a ∗ b = (ab) / (a+b+1)

(b) a ∗ b = a + b + k where k ∈ Z

(c) a ln(b) on {x ∈ R | x > 0}

When already given tool values, how do I create cut-off values from scratch for a screening tool?

Fill in the blank with “all”, “no”, or “some” to make the following statements true. Note that “some” means one or more instances, but not all. • If your answer is “all,” then give a brief explanation as to why. • If your answer is “no,” then give an example and a brief explanation as to why. • If your answer is “some,” then give two specific examples that illustrate why your answer it not “all” or “no.” Be sure to explain your two examples. An example must include either a graph or a specific function.

(a) For functions f, if f′′(0) = 0, there is an inflection point at x = 0. (

b) For functions f, if f′(p) = 0, then f has a local minimum or maximum at x = p.

(c) For functions f, a local minimum of a function f occurs at a critical point of f.

(d) For functions f, if f′ is continuous and f has no critical points, then f is everywhere increasing or everywhere decreasing.

4. Prove that the universal quantifier distributes over conjunction, using constructive logic,

(∀x : A, P x ∧ Qx) ⇐⇒ (∀x : A, P x) ∧ (∀x : A, Qx) .

6. We would like to prove the following statement by contraposition, For all natural numbers x and y, if x + y is odd, then x is odd or y is odd.

a. Translate the statement into a statement of predicate logic.

b. Provide the antecedent required for a proof by contraposition for the given statement.

c. Provide the consequent for a proof by contraposition for the given statement.

d. Prove the contrapositive statement is true, from which you can conclude that the original statement is true. You may use either Coq or the informal proof shown in the text.

a) Prove that an isolated point of set A is a boundary point of A (where A is a subset of real numbers).

b) Prove that a set is closed if and only if it contains all its boundary points

Prove that 1+ cos theta + cos 2theta + .... cos ntheta = 1/2 + (sin(n+1/2)theta)/2sin(theta/2)

olin is due to pay $9000 in five years. If she makes three equal payments, in 20 months, 30 months, and 5 years from today, what is the size of the equal payments if money is worth 5.16% compounded monthly?

Are the following statements true or false? In each case, prove your answer.

1. There is a strictly decreasing function f from N to N with f(0) = 100.
2. Let f(x) and g(x) be strictly increasing functions from R to R. Then (f + g)(x) is also strictly increasing.
3. Once again, let f(x) and g(x) be strictly increasing functions from R to R. Then (f×g)(x) is also strictly increasing.

Locate the bifurcation values for the one-parameter family and draw the phase lines for values of the parameter slightly smaller than, slightly larger than, and at the bifurcation values. Explain steps.
1.) dy/dt = y^2 + a

Let H be the hemisphere x2 + y2 + z2 = 66, z ≥ 0, and suppose f is a continuous function with f(4, 5, 5) = 5, f(4, −5, 5) = 11, f(−4, 5, 5) = 12, and f(−4, −5, 5) = 15. By dividing H into four patches, estimate the value below. (Round your answer to the nearest whole number.) H f(x, y, z) dS

2. Do the Maclaurin series expansion of 1 √(1−?) out to 3 terms

3. Find the Taylor series for the two functions:

i. ?(?) = ? −6? about ? = −4

ii. ?(?) = 7 ? 4 about ? = −3