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Calculus and Analysis

Partial Differential Equation : Diffusion Equation and Explicit Series

Consider the diffusion equation ut = ku.xx for 0 < < pi and t > 0 with the boundary conditions ux(0, t) = 0 and u(pi, t) = 0 and the initial condition u(x,0) = 1. (a) Find the separated solutions satisfying the differential equation and boundary conditions. (b) Use these solutions to write an explicit series solution to t

Word problem/sketch involving integrals

A high-tech company purchases a new computing system whose initial value is V. The system will depreciate at the rate f=f(t) and will accumulate maintenance costs at the rate g=g(t), where t is the time measured in months. The company wants to determine the optimal time to replace the system. a) Let C(t)=1/t the integral fr

Polar Forms : Cardinoid, Rose Curve, Lemniscate and Limacon

Questions: 4,6,10,12,18,22,26,30,34,36,42,44,46,48,50 on page 6.3. 2,8,22,14,18,12,24 on page 393 4. Identify each of the curves as a cardinoid, rose curve (state number of petals), lemniscate, limacon, circle, line of none of the above. Please see attached for all questions.

LaPlace Transformations with some Initial Value Problems

Problem 9.1 (Prob. 29. P. 252) Two particles each of mass m moves in the plane with co-ordinates (x(t), y(t)) under the influence of a force that is directed toward the origin and had magnitude k/(x2 + y2) an inverse-square central force field. Show that mx''=-kx/(r^3) and my''= -ky/(r^3) where r = sqrt(x2 + y2) Problem 9.2

Polar Coordinates; Laplace's Equation; Boundary Conditions

I need some clues on figuring out these questions. Please see attachment for complete problems (regarding the below: "..." indicates an equation to be found in the attachment. Thanks!) (a) Using polar coordinates, find all the separated solutions of Laplace's equation satisfying the attached boundary conditions in the "wedge

Brachistochrone Curves

What is it's origin? State the problem with picture. Explain how the cycloid relates to the solution.

Average and Instantaneous Rate of Change and Application to Revenue

Find the average rate of change of the function over the indicated interval. Than compare the average rate of change to the instantaneous rate of change at the end points of the interval. f(x)= x^2 + 3x - 4; (0,1) The annual revenue R (in millions of $ per year) of Wm. Wrigley Jr. Company for the years 1994-2000 can

Capacitator, Negative Exponential, Capacitance & Charge Time

Background Information: The paddles of a defibrillator in an ambulance or the emergency room of a hospital are actually the two plates of an electronic device called a capacitor. A capacitor is built from two conducting plates that are attached to a voltage source. Due to the voltage source, electrical charges move through

Rates of Change: Cost and Marginal Cost

3.) A company determines that the cost in dollars to the manufacture x cases of the dvd "math caught in embarrassing moments" is given by C(x) = 100 + 15x - x^2, (0< or = x < or = 7) a.) Find the average rate of change of the cost per case for the manufacturing between 1 and 5 cases [ review ex. 2a] So on the average, t

Fundamental Solution of N-Dimensional Laplace equation

Please see the attached. The fundamental solution of the n-dimensional Laplace equation solves , (1) where is the n-dimensional delta function. a. Show that if , the solution of the above equation (1) for is , where is a constant. b. Use the n-dimensional Gauss theorem to evaluate the

Calculus...

Hi 1) A right triangular plate of base 8.0 m and height 4.0 m is submerged vertically, find the force on one side of the plate. (W = 9800N/m^3)? 2) The cable of a bridge can be described by the equation y = 0.06x^(3/2) from x = 0 to x = 200 ft. find the length of the cable? 3) A conical tank is resting on its apex. Th

Recursive Sequence Given Recursive Formula and Seed

Please see the attached file for the fully formatted problems. Please show me how to solve question 1 part d of lecture two in the "non elegant" way. I'd like you to "work backwards from what you want to prove until you arrive at a true formula" l like in part C of question 1. I've provided the solutions so you can see

Vector Analysis : Construction of a Plane Curve

Please see the attached file for the fully formatted problems. Problem: Assume you are given a non-negative function K(s). We would like to construct a plane curve B(s) with curvature K(s)..... HINT: Use the Fundamental Theorem of Calculus to show B has unit speed and then compute dT/ds. Problem: I. If K(s) = ... usin

PDE with Time-Dependent Domain

Please see the attached file for the fully formatted problems. Consider the diffusion equation: on the time-dependent domain where a is a constant. We wish to solve the initial and boundary value problem having for and a prescribed . Thus, u is prescribed as a function of time on the left boundary that moves at

Differential Equations : Population Growth

9. The rate of change of the population of a town in Pennsylvania at any time t is proportional to the population at that time. Four years ago, the population was 25,000. Now, the population is 36,000. Calculate what the population will be six years from now. A. 43,200 B. 52,500 C. 62,208 D. 77,760 E. 89,580

Normal Modes : Second Order Simultaneous Equations

This question is concerned with finding the solutions of the second order simultaneous equations where a = 38, b = -9, c = 378, d = -79 (i) Find the particular solutions to the differential equations which satisfy the initial conditions x = -10 and y = 7 at t = 0 together with the condition at t = 0.. For this part

Heat Diffusion Equation and Standard Heat Equation

A) Let the temperature u inside a solid sphere be a function only of radial distance r from the center and time t. Show that the equation for heat diffusion is now: {see attachment}. This is not an exercise in doing a polar coordinate transformation. First you should derive an integral form for the equation by integrating over a

Business Calculus Functions

Given the following table...(a) Is y a function of x? Explain your answer. (b)Is x a function of y? Explain your answer. (See attachment for full question) Hi, I am having trouble doing these problems listed below. Please show me how to solve these problems for future reference. Thank you very much. I would like for you

Temperature Distribution Exposed to a Hot Gas Stream

See the attached file. You solution can be similar, but IT CANNOT BE IDENTICAL OR LOOK ANYTHING CLOSE TO IDENTICAL. Please see the attached file for the fully formatted problem. L .M. Chiappetta and D.R. Sobel ("Temperature distribution within a hemisphere exposed to a hot gas stream," SIAM Review 26, 1984, p. 575?577)

Laplace transform : Solve an Initial Value Problem

Please see the attached file for the fully formatted problem. Suppose there was an IVP such as the following: where Where and how do you begin to set this problem up to be solved using the Laplace transform? The value y(4)(t) is the fourth derivative of function y(t).

Inverse Laplace Transforms and partial fractions

I have a transform F(s) of which I need the inverse transform for. The form of the transform is not of a common form and I am having trouble reducing it to a workable form. I am looking at a problem that requires the inverse laplace transform of f(t) to be found using the following transform: F(s) = (s*e^(-s/2))/(s^2 + p

Parabolic Curve : Application to Buoyancy

You have been hired as a special consultant by u.s coast guards to evaluate some proposed new design for navigational aids buoys. The buoys are floating cans that need to be visible from some distance away without rising too far out of the water. Each buoy has a circular cross-section (viewed from below) and will be lifted with

Variation of parameters and Undetermined coefficients

1. Write a short paragraph comparing and contrasting the method of undetermined coefficients and variation of parameters. How are they similar, how they are different? If you had your choice, which method would you use? 2. Consider the differential equation: my"+cy'+ky=mg+sqrt(t) Why would the method of undetermined

Laplace equation in cylindrical coordinates

In this problem, you will find the electrostatic potential inside an infinitely long, grounded, metal cylinder of unit radius whose axis coincides with the z-axis (See figure below). In cylindrical coordinates, the potential, V(r, theta, z), satisfies Laplace's equation... <i>Please see attached</i>... Let us assume that the po

Linear dependency, Wronskian and Bessel's Equation

Three problems regarding the Wronskian and solutions of a second order differential equation. Example of a question 1. Determine whether the following sets of functions are linearly dependant or independent... Please see attached. 2. Bessel's equation x²y" + xy' + (x² - n²)y = 0 where n is a constant, i