Consider the system modeled by the differential equation

                              dy/dt - y = t    with initial condition y(0) = 1

the exact solution is given by y(t) = 2et − t − 1

Note, the differential equation dy/dt - y =t can be written as

                                              dy/dt = t + y

using Euler’s approximation of dy/dt = (y(t + Dt) – y(t))/ Dt

                              (y(t + Dt) – y(t))/ Dt = (t + y)

                               y(t + Dt) = (t + y)Dt + y(t)

                              New Value = change + current value

1. Using R implement Euler’s method directly to numerically solve the equation and construct a Table as below – list data to four digits past the decimal point. Submit your R session

               time     ∆t = 0.1           ∆t = 0.0001        Exact Value       %Relative                  %Relative

                                                                                                             Error ∆t = 0.1       Error ∆t = 0.0001                    

                  0

                 0.1

                 0.2

                 0.3

                 0.4

                 0.5

                 0.6

                 0.7

                 0.8

                 0.9

                 1.0

Solve equations

1.) y'-y=t/y

2.) y'-(1/t)y=y²sin(t)

3.) y'+y=y²cos(t)

4.) y'-2y=cos(t)/(y^1/2)

6- Write a C++ program that determines the amount of memory used by char, short int, unsigned short int, float, and double types and display the information on the screen. You program should also display the typical range for each data type.

Where is the function ln(sin x ) defined and continuous?

3-24 Today’s Electronics specializes in manufacturing modern electronic components. It also builds the equipment that produces the components. Phyllis Weinberger, who is responsible for advising the president of Today’s Electronics on electronic manufacturing equipment, has developed the following table concerning a proposed facility:

a. develop an opportunity loss table.

b. what is the minimax regret decision?

The minimax regret decision rule states the best alternative is the medium-sized facility.

Just need help with making an excel spreadsheet showing formulas for these answers .

Find the general solution of the following equations:

y′′ −4y′ +4y=0; y′′ −5y′ +6y=0; y′′ − 2y′ = 0

Solve the differential equation y'' − y' − 2y = 9e^2t , with initial conditions y(0) = 3, y' (0) = −2, using two different methods. Indicate clearly which methods you are using. First method:

Second method:

Obtain the general solution of the following equation 4Uxx+12Uxy+9Uyy-9U = 9

Let f : V mapped to W be a continuous function between two topological spaces V and W, so that (by definition) the preimage under f of every open set in W is open in V : Y is open in W implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that the preimage under f of every closed set in W is closed in V . Feel free to take V = W = R^n to simplify things. Hint: show that the “preimage of” operation plays nice with set-complements, and then use the fact that every closed set is the complement of some open set. Note that R^n is both open and closed as a subset of itself.

A solution of hydrochloric acid accidentally leaks into a spring fed freshwater lake, which in turn feeds a single stream. the acid leaks into the lake at a rate of 2 gallons per hour and has a concentration if 3 pounds per gallon. The lake initially contains 4000 gallons of water. Water flows into the lake from the spring at a rate of 100 gallons per hour and fluid flows out of the lake into the stream at a rate of 100 gallons per hour.
a) Set up the initial value problem that models the dynamics if acid in the lake.
b) Solve the initial value problem from part a.
c) compute the amount of acid in the lake after 1 hour

By computing both sides, show that for an m × n matrix A, vectors u and v ∈ Rn , and a scalar s ∈ R, we have (a) A(sv) = s(Av); (b) A(u + v) = Au + Av; (c) A(0) = 0. Here 0 denotes the zero vector. Is the meaning of 0 on the two sides identical? Why or why not? Hint: Let x = (x1, . . . , xn) and y = (y1, . . . , yn) be vectors in Rn . The definition of vector equality is that x = y if and only if, for each i between 1 and n, we have xi = yi . So verify this property for the vectors in (a), (b), (c).

How do you determine convexity/concavity of a function f(x,y)? How about a function f(x,y,z)?

1.Determine whether S spans V . Justify your answers.

V = C ^0 [−1, 1] (the vector space of continuous functions on [−1, 1]) and S = {1, t, t2 , t3 , . . . }.

2.Let S be a set in a vector space V and v any vector. Prove that span(S) = span(S ∪ {v}) if and only if v ∈ span(S).

PROOFS:

1. State the prove The Density Theorem for Rational Numbers

2. Prove that irrational numbers are dense in the set of real numbers

3. Prove that rational numbers are countable

4. Prove that real numbers are uncountable

5. Prove that square root of 2 is irrational