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    Calculus and Analysis

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    Inductance, Resistance and Voltage

    Find the equation for the current i(t) versus time in a series circuit with inductance L=0.1H, resistance R=8 ohms, and voltage V=12V. The initial current i(0)=2A.

    Equilibrium Points and Initial Conditions

    Consider the system: dx/dt = x+2y+1 dy/dt = 3y (a) Derive a general solution (b) Find equilibrium points of the system (c) Find the solution that satisfies the initial condition x(0) = -1, y(0) = 3. See attached file for full problem description.

    Differential Equations : Spring with Damping Force

    A body that weighs 16 lb. is attached to the end of a spring which is stretched 2 ft. by a force of 100 lb. It is set in motion from a position of ½ foot from the equilibrium position of the spring with an initial velocity of -10 ft/sec. Assume the motion of this body is subject to a damping force that provides 6 lb of resistan

    Differential Equations : Masses and Springs

    A body that weighs 16 lb. is attached to the end of a spring which is stretched 2 ft. by a force of 100 lb. It is set in motion from a position ½ foot from the equilibrium position of the spring with an initial velocity of -10 ft/sec. (a) Find the differential equation that governs the position of the body over time relative

    Solving Linear Homogeneous Differential Equations

    Consider the differential equation y''-2y'+2y = cost (a) Find a general solution to this equation using techniques for solving linear homogeneous differential equations with constant coefficients in combination with a variation of parameters. (b) Use Laplace Transforms to solve this differential equation with the initial c

    Business Calculus : Time Value of Money

    There are 3 sheets in the file with approx 21 questions. Most of them are mathematical, but there are also some that are subjective type questions. I have also attached the tutorials with the problem sets in the same file. In the problem sets on sheet one, I will answer number 9. Chapter 3: Project 3 Mortgages: Princi

    Business Calculus : Maximizing Profits, Cost and Price-Demand Functions

    A company manufactures and sells x air-conditioners per month. The monthly cost and price-demand equations are C(x) = 180x + 20,0(X) p = 220 - 0.001x 0 =< x <=100,000 (A) How many air-conditioners should the company manufacture each month to maximize its monthly profit? What is the maximum monthly profit, and what should

    Laplace Transforms and Block Diagrams

    Represent this block diagram as an equation in the Laplace domain. The equation should be arranged so that a dependent variable is expressed as the sum of independent variables, each of these multiplied by some transfer function. (Note in the diagram that y is subtracted from xsp.) If you could choose Gf to be anything you want

    Parametric Equations and Equation of an Ellipsoid

    (1) Find parametric equations for the line through the point (0, 1,2) that is parallele to the plane x+y+z = 2 and perpendicular to the line x = 1 +t, y = 1 ?t,z = 2t. (2) An ellipsoid is created by rotating the ellipse 4x2 + y2 = 16 about the x-axis. Find an equation of the ellipsoid. See the attached file.

    Calculus

    Evaluate the integral along the parametric curve C: / l (x^(2) - y^(2)) dx + x dy l / C C: x = t

    Write Equation in Cylindrical Coordinates

    Find a parametric representation of the surface in terms of the parameters r and theta , where (r, theta, z) are the cylindrical coordinates of a point on the surface: z = 2xy

    Calculus III

    Evaluate the iterated integral: / 2 / y^(2) / z l l l y z dxdzdy l l l / 0 / -1 / -1

    Differential Equations : Steady State Conditions and Heat Transfer Functions

    Problem 1 Consider the following transfer function .... (a) What is the steady state gain and time constant? (b) If U(s) =2/s, what is the value of the output when t&#8594;&#8734; (c) For the same U(s) what is the value of the output when t = 10? (d) If U(s) is a unit rectangular pulse what is the output when t&#8594;&#873

    Differential Equations and Laplace Transforms

    See the attached file. Problem 1 For the following function a) Plot y(t) versus time b) Find the Laplace transform Y(s) Problem 2 Find the solution of the following differential equation using Laplace transforms. The initial conditions for y(t) and its derivatives are zero and the forcing function is the uni

    Systems of Differential Equations : Modeling a Battle between Two Armies

    When one models a pair of conventional forces in combat, the following system arises x1' -a -b x1 p x2' = -c -d x2 + q The unknown functions x1(t) and x2(t) represent the strengths of opposing forces at time t. The terms -ax1 and -dx2 represent operational loss rates and -cx1 and -bx2 represent combat loss rates. The c

    Use Laplace transforms to solve an initial value problem.

    Use Laplace transforms to solve the initial value problem y'' + ty' - 2y = 1, y(0) = 0, y' (0) =0. Because this equation does not have constant coefficients may need to use the frequency differentiation property of Laplace transforms ( L[(t^n)f(t)](s) = ((-1)^n)F^(n)(s) and the fact that if y(t) is a solution to this differenti

    Absolute Minimum and Maximum Value of a Function on a Set

    Find the absolute maximum and minimum values of f on the set D. 28. f(x, y) = 3 + xy ? x ? 2y, D is the closed triangular region with vertices (1, 0), (5, 0), and (1, 4) 24. Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral.