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Functional Analysis

Exponential Function V(t)

A fax machine is purchased for $5,800. Its value each year is about 80% of the value of the preceding year. So after t years the value, in dollars, of the fax machine, V(t), is given by the exponential function V(t) = 5800 (0.8)t. a. Give a sketch of the graph of the function V(t). Your graph can be a "rough draft".

Real Valued Function

The lemma is in the attachment along with further questions. The lemma defines a way to rewrite f(X)-f(Xo)

First Order Non-linear Differential Equation

The dependence of the velocity v of a particle upon time t obeys the differential equation: dv/dt = -av^2 - bv where a>0, b>0 are constants. The initial condition is v(0) = V0 where V0 is the velocity at time t=0. a. What is the order of this equation? Is it linear or non-linear? b. Find the general solution of the equation

Functional changing variables

Changing variables in functionals See attachment for full equation and answer the following questions: a. Obtain the Euler-Lagrange equation corresponding to this functional. b. By changing to a new independent variable u = exp(-2x), show that the functional S[y] is equivalent to the functional (see attachment)... c. Det

Equation of motion of a bead on a rotating parabola

A bead of mass m moves on a parabolic wire with equation z=(1/2*x^2), where z measures the height of the bead and x is the horizontal distance in the plane of the wire. The plane in which the wire lies rotates about a fixed vertical axis passing through x = 0, so that its angle relative to a fixed plane is ø. (See attachment fo

Steady state diffusion through three layers system

Consider a system that consists of a flat layer of material of thicness L that creates radioactive particles surrounded on each side by a very thic layer of material that absorbs the partices that are produced by the radioactive layer.... see attached

Proving the parallelogram law and polarization identity

If x,y are elements  of a Hilbert space H, then prove that:                          [norm of (x + y)]^2 + [norm of (x - y)]^2 = 2(norm of x)^2 + (norm of y)^2                                                Alternatively, prove that in any Hilbert space H,

Derivitaves + Story Problems

Please show all work. 1. Consider the function a) Determine the critical points. (Apply the quotient rule carefully to find derivative.) b) For what intervals in the domain of f is the function increasing? c) For what intervals in the domain of f is the function decreasing? 2. Find all maximum and minimum values o

Chain Rule & Demand Function

Please show all the steps to calculate the answers to the attached questions about using the chain rule on an American population function and a demand function for desk lamp prices.

Finding the Slope and Intercept in Linear Equations

1. A server purchased at a cost of $55,000 in 2002 has a scrap value of $10,000 at the end of 5 years. If the straight-line method of depreciation is used. (Please show work) a) Find the rate of depreciation. Include your units. b) Find the linear equation expressing the server's book value at the end of t years. Use proper fu

Setting and solving linear programming question

A furniture manufacturer produces sofas, tables, and chairs. The profits per item are, respectively, $70, $120, and $80. The pieces of furniture require the following labor-hours for their manufacture. Carpentry Upholstery Finishing Sofas 3 5 1 Tables 8 0 2 Chairs 6 2 1 The following amounts of labor-hours are availabl

Gauss-Jordan Elimination - Airlines

In 2001, 400 seat Boeing 747s were priced at $200 million each, 300 seat Boeing 777s were priced at $160 million, and 200 passenger Airbus A321s were priced at $60 million. Suppose you were the purchasing manager of an airline company and had a $2100 million budget to purchase new aircraft to seat a total of 4500 passengers. You

Setting matrix for assembly and packaging times

A company manufactures tables and chairs. Each chair requires 2.5 hours of assembly and 0.5 hours of packaging. Each table requires 5 hours of assembly and 0.65 hours of packaging(Please show all work) a) Write a matrix A that represents the required time for assembly and packaging of chairs and tables. Label your rows and co

Matrix multiplication and its interpretation

A flu epidemic hits a large city. Each resident of the city is either sick, well, or a carrier. The proportion of people in each of these categories by age is given by the matrix:(Please show all work) A = AGE 0-10 10-30 Over 30 WELL .65 .58 .60 SICK .10 .20 .30 CARRIER .25 .22 .10 The population of the city is

Manufacturing Company - Determine Objective Function -

A manufacturing company makes three types of office chairs: Model A, Model B, and Model C. The construction of each chair requires a process involving three departmental steps; material workup, assembly, and packaging. The three departments have a maximum of 30, 53, and 47 hours available each week, respectively. The information

Laplace Transform and the Driven Oscillator

Please provide help on this assignment with a clear explanation. 1. A mechanical system is described by the differential equation : y''+(w^2)y=f(t)1, 0 < t < a f(t) = { y0=y'0=0 0, otherwise (1/w^2) (1-cos(wt)) t < a show that : y = { (1/w^2) [(cos(w)(t-a) - cos(wt)] t > a 2. Sketch the m

Numerical Integration of Irradiance Spectrum

Calculate the total power per unit area generated by a xenon lamp determined at 50 cm from the lamp by integrating the irradiance spectrum over the specified wavelength range of 250 - 400 nm. As the irradiance spectrum cannot be described analytically, convert the integral to a summation expression, and divide up the specified w

Convergence and divergence of infinite products

** Please see the attached file for the complete problem description ** Please provide a detailed solution to the following questions: Discuss the convergence of (please see the attached file) (1+ c_n) and that of (please see the attached file). a) c_2n-1 = 1/(sqrt)n, c_2n = -1/(sprt)n; b) c_2n-1 = -1/(sprt)n, c_2n =

Mittag-Leffler Theorem problem

Problem: Obtain a partial fraction expansion of 1 / [(e^z) - 1]. I was told that the solution should be (1/z) - (1/2) + 2z summ inf;n=1 [1/ {(4n^2)(pi^2)+(z^2)}] z not = 0, +-2ipi,+-4ipi, I need to get this solution with the Mittag-Leffler theorem, convergenc, and evaluating the constant. I also know that 1 / [(e^z

well question for work

Question 5 A bucket that weighs 6 lb and a rope of negligible weight are used to draw water from a well that is 90 ft deep. The bucket is filled with 42 lb of water and is pulled up at a rate of 2.5 ft/s, but water leaks out of a hole in the bucket at a rate of 0.25 lb/s. Find the work done in pulling the bucket to the top of t

Determine which subsets given below are subspace of R^3 1) {x l x_1 greater or equal to 0} 2) {x l x_1=0} 3) {x l x_1+x_2+x_3=1} 4) {x l x_1+x_2+x_3=0}

The doubling period is attained.

The count in a bacteria culture was 900 after 20 minutes and 1300 after 35 minutes. What was the initial size of the culture? Find the doubling period. Find the population after 80 minutes. When will the population reach 13000.

Volume of the large box

2 boxes of brazils are mathematically similar, their lengths are 12cm and 24cm. The surface area and volume of the smaller box are 250cm2 and 240cm2 respectively. What are the corresponding figures for the large box/ Give answer in Area & Volume (details explanation)

Union of Two Segments

Exercise # 24. In Exercises 21-32, use the figure to find the following Exercise 9.3 # 16 In Exercises 15-18, determine (a) the area and (b) the circumference of the circle. Use the key on your calculator and round your answer to the nearest hundredth _____________________________________________________

Uncountable or countable functions

Please help with the following problem. A) Let P be the set of all functions f : N -----> N such that for some M in N and all n>M, f(n + 1) = f (n). In other words, f is in P provided that after some point, f is constant. Show that P is countable. b) Let E be the set of all strictly increasing functions f : N ----> N. Show

Find the limit using the given theorem

Theorem: Suppose that lim as z-->z0 of f(z)=w0 and lim as z-->z0 of F(z)=W0 then lim as z-->z0 of [f(z)+F(z)]=w0 + W0, lim as z-->z0 of [f(z)F(z)]=w0*W0 and if W0 can not equal 0, then lim as z-->z0 of f(z)/F(z)=w0/W0. Find lim as z--> i of ((iz^3)-1) / (z+i).

Determining Average Speed

Assume that a watermelon dropped from a tall building falls y=16t^2 in t sec. Find the watermelon's average speed during the first 3 sec of fall and the speed at the instant t=3sec. a. 48 ft/sec; 96 ft/sec b. 49 ft/sec; 98 ft/sec c. 24 ft/sec; 48 ft/sec d. 96 ft/sec; 49 ft/sec

Some statics problems

1. A horizontal circular plate is suspended as shown from three wires that are attached to a support at D and form 30° angles with the vertical. Knowing that the z component of the force exerted by wire BD on the plate is -32.14 N, determine ( a) the tension in wire BD 2. A 300- N force is applied at A as shown. Determine (

Find the area of the surfaces.

1.)The part of the hyperbolic paraboloid z= y^2 - x^2 that lies between the cylinders x^2 + y^2 = 1 and x^2 + y^2 = 4 2.)The part of the sphere x^2 + y^2 + z^2 = a^2 that lies within the cylinder x^2 + y^2 = ax and above the xy-plane