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# Calculus and Analysis

### Find the Limit : L'Hopital's Rule

Lim n-->&#8734; 2^(-n) ln n

### 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

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

### The problems are from Boundary Value Problems

The problems are from Boundary Value Problems. Undergrad 400 level course. Mainly uses partial differential skills. Some problems might require using MATLAB. Please explain each step of your solutions. Thank you very much.

### The problems are from Boundary Value Problems

The problems are from Boundary Value Problems. Undergrad 400 level course. Mainly uses partial differential skills. Some problems might require using MATLAB. Please explain each step of your solutions. Thank you very much.

### Rate of Change of Height of Cone

Gravel is dumped from a conveyor belt at the rate of 30 cubic feet per minute and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and heght are always equal. How fast is the height of the pile increasing when the pile is 10 feet high? include the correct units in your answer. Let h=the heig

### Boundary Conditions for Transport Phenomena

In solving transport phenomena problems, we use "boundary conditions" and "initial conditions". In mathematical terms, (that is, types of variables), what are "boundary conditions" and what are "initial conditions"

### Non-empty bounded subset of R.... then prove a function

Pls provide step by step proof with explaination. Thanks.

### Differential Equation into Transfer Function

Transform the attached differential equation into a transfer function and find the homogeneous solution to the differential equation from the transfer function. See the attached file.

### Advanced Calculus Analysis : Cauchy and Convergent Sequences

We proved the following theorem in the class: "If a>0 and if a sequence... is covergent, then the sequence... is convergent." In proving this theorem, we proved that... is Cauchy instead of proving it converges directly. Why did we have to do that? Please see attachment for full question.

### Advanced Calculus Analysis : Simplify using Set and Interval Notation

Let f: R &#8594; R defined by f(x) = x² for all xER. Use set notations (for example, &#8745;, U, &#8722;) and interval notations to simplify the sets f(f-¹ [-1, 3]) and f-¹(f[-1, 3]).

### Advanced Calculus : Negate the Definition of a Function Being Onto

Convention: Let X and Y be non-empty sets of real numbers. Let g: X &#8594; Y be a function. 1. Negate the definition of the function g: X &#8594; Y being onto. (8 points)

### Solve the Differential Equation

Solve: ((e^x)(sin(y))-2ysin(x))dx+((e^x)(cos(y))+2cos(x))dy= 0

### Solve the Differential Equation

Solve: (cos x + lny)dx + ((x/y)+ e^y)dy = 0

### Solve the Differential Equation

(x^2 +1)y' + 3xy = 6x

### Question about Solve the Differential Equation

Question about Solve the Differential Equation (6xy-y^3)dx + (4y+3x^2-3xy^2)dy = 0

### Find the general solution to the driven differential equation.

Find the general solution to the driven differential equation attached. The solution is detailed and well presented. The solution received a rating of "5" from the student who posted the question.

### 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)

### Find the Laplace Transform of a Product

Simple Laplace transform of a product of two trig identities of which I cannot remember how to integrate. See the attached file.

### Laplace Transform to Solve IVP

Use Laplace transform to solve an IVP style problem. Where y^n(t)+a^2y(t)=0 and a is not equal to zero.

### 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

### Laplace Equation Inside Quarter-Circle/Disk

Solve the Laplace equation inside the quarter-circle if radius {see attachment} is subject to boundary condition: {see attachment} Thank you.

### Laplace Expansion - Exponential Product

My problem deals with the Laplace expansion property applied to a exponential product function. I am unsure as to how the problem should be solved based on the given property formula and the Differential Equation books that I have used have been of no help. If familiar, please try to use the DiPrima notation.

### Find the Inverse Laplace Transform

The following is a sample test question involving the use convolution to find the inverse Laplace transform of the below equation. I have thus so far not been able to break the initial equation down into the two separate equations F(s) and G(s). Any help would be appreciated as it has been a while since I have used partial fract

### 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

### Dynamics and Hamiltonians: Hamilton's Equations

A dynamical system, with one degree of freedom has Hamiltonian (see attachment for equation) ? Write down the Hamilton's equations governing the motion of this system, and deduce that H remains constant during the motion. ? Solve Hamilton's equations with initial conditions (see attachment) and show that q(t) and p(t) both t

### Antiderivative Integral Problems

I've included the equation in the attachment. I realize that it is something larger than w, but I don't know how to get the antiderivative of f(x)f(1/x). Whoever helps me on this, please include more than just the answer in your reply. Thank you very much.

### Rate of change of water level in a frustum of a cone

A water bucket is shaped like the frustum of a cone with height 24 inches, base radius of 6 inches and top radius of 12 inches. Water is leaking from the bucket at 10 cubic inches per minute. At what rate is the water level falling when the depth of the water in the bucket is 12 inches?

### Computing an Integral Using Residues

Use residue theory to compute the integral: Use residue theory to compute the integral: I know that the singular points are 4i and -4i, and that I need to integrate from -R to R and then around the half circle from 0 to pi. Only 4i is in this half-circle, so I need to calculate the residue for 4i. This is where I get stu