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    Calculus and Analysis

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    General Solution & Phase Portrait

    A. Find the general solution to the attached system of differential equations. x' = y y' = 2x b. Draw the phase portrait for this system and describe in words what happens to the solution for all initial conditions: x(0) = x0 and y(0) = y0. See the attached file.

    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

    Functions: Domain, Increasing/Decreasing, Image Set and Inverse

    This question concerns the function,f (x) = ln (x + 3). a) State the domain of the function f, justifying your answer briefly. b) State whether the function f is: increasing, decreasing, neither increasing nor decreasing, one-one or many-one. c) Write down the image set of f. d) Explain briefly why the function f has

    Particular Solutions, Homogeneous and Non-Homogeneous

    See attachment for complete questions. 1 and 2. Find the general solutions to the attached equations. 3. Non-homogeneous equation, associated homogeneous equation - find solution satisfying the given initial conditions. 4. Find the particular solution to the attached equation.

    Calculus: Area and volume

    Please see the attached file for full problem description. --- 1. A solid is bounded by two bases in the horizontal planes z = h/2 and z = -1/2, and by such a surface that the area of every section in a horizontal plane is given by a formula of the sort Area = a0 z3 + a1 z2 + a2 z + a3 (where as special cases some of

    Rate of Change and Instantaneous Rate of Change

    The effectiveness of E (on a scale from 0-1) of a pain killing drug t hours after entering the bloodstream is given by E= 1/27(9t+3t^2-t^3) o< or = t < or = to 4.5 Find the average rate of change E on the indicated intervals and compare this rate with the instantaneous rates of change at the end points of the intervals a.) [

    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

    Vector Calculus - Verify Divergence Theorem

    Hi, This problem is from a book I have, and has three parts. I have attached both the problem (p23.jpg) and the solution (a23.jpg). I understand in general how the theorem works, but I can't come up with the correct answer no matter what I try. I think I'm not setting up the problem correctly. I'd appreciate someone sho

    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

    Numerical Methods for Differential Equations

    In this problem we will use Euler's method to find an approximation for e. Consider the differential equation f' = f with initial data f(0) = 1. We know the solution is f(t) = e', therefore f(1) = e. Using as step size h = 0.5, and fo = 1, use Euler's method to obtain f20. Your answer is an approximation to f(t10) = f(1).

    Limit Points of a Bounded Set of Real Numbers

    Question: Construct a bounded set of real numbers with exactly three limit points (put the limit points at 0, 1 and 2005). (Please explain in your own words how the proof works. If you use a theorem, please state what it is and if possible, where you got it).

    Rate of Change of Volume of a Cone..

    The radius of a right circular cone is increasing at a rate of 2 inches per second and its height is decreasing at a rate of 2 inches per second. At what rate is the volume of the cone changing when the radius is 30 inches and the height is 20 inches? ___ cubic inches per second.

    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

    Tension on a Cord in a Mobile

    A mobile is hanging from the ceiling with two metal pieces strung by cord, one under another. I only have the masses for each figure. How do I determine the tension on the top cord and then how do I determine the tension on the bottom cord?

    A novice inventor has invented an exciting new toy for dogs.

    A novice inventor has invented an exciting new toy for dogs. He believes it will cost him $.95 per toy to produce these doggy marvels. Unfortunately, to get mass-producing these items, he has had to spend $6000 of his hard earned money and countless hours observing animals. He plans on selling the toys for $1.69 each. a.

    Differentials

    The dimensions of a closed rectangular box are measured as 100 centimeters, 60 centimeters, and 60 centimeters, respectively, with the error in each measurement at most .2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box.

    Principal Value and Principal Branch of an Integrand

    Use parametric representation in exercise 10 for the oriented circle C0 there to show that....where a is any real number other than zero and where the principal branch of the integrand and where the principal value of R^G are taken. Please see the attached file for the fully formatted problems.

    Calculus analysis problem

    1. Show that u (x, y) is harmonic in some domain and find a harmonic conjugate v (x, y) when (a) u (x, y) = 2x (1 - y) (b) u (x, y) = 2x - x3 + 3xy2 (c) u (x, y) = sinh x?sin y (d) u (x, y) = y / (x2 + y2) (Question is also included in attachment)

    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