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Relating Transform of a Function and Transform of the Derivative

Please see the attached file for the fully formatted problems. Problem statement: What really makes Laplace transforms work for differential equations is the relationship between the transform of a function and the transform of the derivative of that function. Therefore, the formula you will prove below is key to all that

Integral Domains and Fields: Embedding Theorem

Problem: Note: C is set containment If R is an integral domain, show that the field of quotients Q in the Embedding Theorem is the smallest field containing R in the following sense: If R C F, where F is a field, show that F has a sub-field K such that R C K and K is isomorphic to Q.

Integral Domains, Fields and Subfields

Problem: Note: Q is rational numbers, R is real numbers , sqrt() means square root Show that Q(sqrt(2)) is the smallest subfield of R that contains sqrt(2).

Wave Packet: Orthonormal Functions and Complete Set

1. Consider the set of functions ("wave packets") (see attached) where e is a fixed positive constant. a) SHOW that these wave packets are orthonormal. b) SHOW that these wave packets form a complete set.

Representation of the Dirac Delta Function

Please see the attached file for the fully formatted problems. Show that is a representation of the Dirac S-function. Discussion: Let and let f(x) be a function which is piecewise continuous on [?a, a], in particular, (Dirac delta function) one must show that One way of doing this is to follow the approach u

Integrating Periodic Functions

Please see the attached file for the fully formatted problems. Suppose that f(x + 2pi) = f(x) is an integrable functionof period 2pi. Show that S f(x) dx 2pi + a ---> a = S f(x) dx 2pi ----> 0 where a is any real number.

Integration By Trigonometric Substitution: Solving Trickier Problems?

Hello! I'm having trouble using Trigonometric Substitution to find the anti-derivative of non-simple integrands. For details on my situation, please consult my missive, which I've included as an attachment in MS Word '95 (WordPad compatible) and Adobe PDF (ver 3+) files. (The files contain identical information; if you can re

Curve Sketching, Integration, Stationary Points and Asymptotes

Please see the attachment for the full question. I require full, detailed, step by step workings for all sections of this problem Coursework 2 Question 2 a) For the curve with the equation y = x^3 + 3x^2 - 2 i) Find the position and nature of any stationary points. ii) Make up tables of signs for y, y' and y''. Us

Multivariable Calculus : Iterated Integral

( f ^n_r means that n is on the top of the f and r is on the bottom) Evaluate the iterated integral: f ^(pi/2)_0 f ^(pi/2)_0 cos x sin y dy dx f: is the integral symbol

Multivariable Calculus: Triple Integral - Cylindrical Coordinates

Question: Solve by triple integration in cylindrical coordinates. Assume that each solid has unit density unless another density function is specified: Find the volume of the region bounded above by the spherical surface x^2 + y^2 + z^2 = 2 and below by the paraboloid z = x^2 + y^2.

Multivariable Calculus : Triple Integral

Compute the value of the triple integral   _T f(x, y, z) dV: f(x, y, z) = xyz; T lies below the surface z = 1 - x^2 and above the rectangle -1*x*1, 0*y*2 in the xy-plane. : is the integral symbol

Multivariable Calculus : Double Integral - Polar Coordinates

(  ^n_r means that n is on the top of the  and r is on the bottom) Evaluate the given integral by first converting to polar coordinates:  ^1_0  ^(square root of 1 - x^2)_0 (1/(square root of 4 - x^2 - y^2)) dy dx : is the integral symbol

Multivariable Calculus : Integral Explanation

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you. (  ^n_r means that n is on the top of the  and r is on the bottom) Sketch the region of integration, reverse the order of integration, and evaluate the resulting integral:  ^1_0  ^1_y

Volume Using Cylindrical Shells

The region bounded by the curve and the lines and is revolved around the x-axis to form a solid S. The volume of S by the method of washers is given by the integral . Find the volume using the method of cylindrical shells. Please show the complete solution (particularly how to find the cross-sectional area), including al

Integral Solution Evaluation

Please see the attached file for full problem description. Can somebody help me to evaluate the following integral,

Calc II/Setting up integral to calculate volume of rotation

a) Take the region bounded by y = cos x and y = 1 - x for 0 ≤ x ≤ and rotate about the line x = . Set up the integral which calculates the volume of rotation (using one variable only). b) Complete the integration without using integral formulas or calculator estimates. (Please show work on this so that I can see

Differential Equation

Given dy/dx= -xy/(ln y), where y>0 find the general solution of the differential equation What solution satisfies the condition that y=e^2 when x=0... express in y=f(x) Why is x=2 not in the domain found from that?

Integrals and Continuity Calculus

20) If the function f is continuous for all real numbers and lim as h approaches 0 of f(a+h) - f(a)/ h = 7 then which statement is true? a) f(a) = 7 b) f is differentiable at x=a. c) f is differentiable for all real numbers. d) f is increasing for x>0. e) f is increasing for all real differentiable ans is B. Explain

Arc Length

Evaluate the integral in the attached file "Arc Length.doc" for arc length (L). The intent is to solve for a numerical answer and the values for a, b, and t are all constant.