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Functional Analysis

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

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

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

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

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

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 (

Compact Subset of R^m with Convergent Sequences

Please help with the following problem. Let A be a proper subset of R^m. A is compact, x in A, (x_n) sequence in A, every convergent subsequence of (x_n) converges to x. (a) Prove the sequence (x_n) converges. Is this because all the subsequences converge to the same limit? (b) If A is not compact, show that (a

Transfer Function Analysis and Models

Please see the attached file for the fully formatted problems. Two identical stirred tanks with a recycle stream are connected as shown in the diagram below: Fr CAi,Fi CA1

Asymptotes

Find all vertical and horizontal asymptotes of f(x) = (x^2+ 4)/(x^2- 4x-12) Please show steps & graph if applicable!

Asymptotes

Explain and contrast the types of asymptotes considered for rational functions.

Vectors : Divergence, Gradients and Curls

1) V(x, y, z) = (x + y + z)2 i + (x + y)2 j + x2 k. Find div V(3, 2, 4) ≡ ∇? V (3, 2, 4) 2) F (x, y) = xe2y i + y/(x + y) j. Find ∇ ? F (4, 0) 3) F (x, y, z) = -yz i + xz j - xy k. Find curl F (1, 2, 5) = ∇×F ( 1, 2, 5)

Green's Theorem Enclosed Curves

Use Green's Thereom to find the area enclosed by the curve: {abs(x)}^(1/2) + {abs(y)}^(1/2) = 1. It is important that you solve this problem using Green's Method because that is how my professor prefers it. I know that you have to make the above statement true and switch to a different set of coordinates. Something al

Equation of ellipse and hyperbola

1 Find an equation of the ellipse with the center (0,0) , vertical major axis 14 and minor axis 10. 2 Find an equation for the hyperbola with the focus (11,12) and asymptotes 4x-3y=18 and 4x+3y=30. 3 Find the arc length of the curve given by x = sin t - cos t, y = sin t + cos t, pi/1 <= t <= 3pi/4

Polar and Rectangular Coordinates

1 Express the polar equation r^2 = 2cos2&#920; in rectangular form. 2 Find the total area enclosed by the graph of the polar equation r = 1 + cos2&#920;

Finding Area of a Region Bounded by a Line and a Curve

Sketch the region bounded by the graph of the functions and find the area of the region 1) f(x) = - x^2+ 4x + 2, g(x) = x + 2 2) f(y) = y(2 - y), g(y) = -y 3) f(x) = 3^x, g(x) = 2x + 1 keywords: integration, integrates, integrals, integrating, double, triple, multiple

Concave Up

On what interval does the function f(x) = -x^3 + 3x^2 + 24x + 5 concave up? keywords: concave-up

Finding Absolute Minimum

Determine the absolute minimum of the function f(x) = x^3 - 3x - 1 on the interval [0, 4]. Make sure to show all work that is involved.

Number Lines

________________(____)________ -5 -4 -3 -2 -1 0 1 2 3 4 5 How would I write this in equality form and/or in interval form? x + 2 =+2x is what I came up with. The graph indicates that the solution is between 0 and 2, where 2 is included in the solution and 0 is not.

Ellipse

Given an ellipse, could you please explain AND show on a piece of paper exactly why the distance between the origin and vertex equals a? Please don't just show me how to derive the equation for an ellipse. I have that. And please don't just draw a picture without explaining it. Thanks!