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Calculus and Analysis

Elementary Differential Equations : Power Series Methods

Attached are two problems, one with an answer that I don't understand how it was derived and one problem without the answer that I would like to see how it is solved. Power Series Methods - Introduction and Review of Power Series 14. Find two linearly independent power series solutions of the given differential equation.

Solving Differential Equations with Substitution and Bernoulli

56. Suppose that n does not equal to zero and n does not equal to one. Show that the substitution v = y1-n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation dv/dx + (1-n)P(x)v(x) = (1-n)Q(x). 63. The equation dy/dx = A(x)y2 + B(x)y + C(x) is called a Riccati equation. Suppose that one parti

Solving Differential Equations

57. Show that the substitution v = lny transforms the differential equation dy/dx + P(x)y = Q(x)(ylny) into the linear equation dv/dx + P(x) = Q(x)v(x) 58. Use the idea in Problem 57 to solve the equation x (dy/dx) - 4x2y + 2ylny = 0 59. Solve the differential equation dy/dx = (x-y-1)/(x+y+3) by finding h and k so t

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

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

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

Calculus Questions - Automotive Examples

See the attached file for full description. 40) The value of a certain automobile purchased in 1997 can be approximated by the function v(t)=25(0.85)^t , where t is the time in years, from the date of purchase, and v is the value, in thousands of dollars. (a) Evaluate and interpret v(4). (b) Find an expression for v1(t) inclu

Partial Differential Equations : Heat Equations

1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function. 2) The wave equation (c^2 &#8706;^2 u / &#8706; x^2 = &#8706;^2 u / &#8706; t^2) and the heat equation (c &#8706;^2 u / &#8706;

Ordinary Differential Equations Fourth Order Runge Kutta Method

Question Use Runge-Kutta method of order four to approximate the solution to the given initial value problem and compare the results to the actual values. y'=e^(t-y) , 0 <=t <=1 , y(0)=1 with h = 0.5(Interval) Actual solution is y(t)= In((e^t+e-1). For full description of the problem, please see the attached question

Four problems are solved in this posting. One is involved Taylor Series expantion about x - 0, the second is involved finding Partial Derivatives, the third is involved Double Integral and the fourth is involved finding Divergence and Curl of a given vector field. For complete description of the questions, please see the posted questions.

Question (1) Write the Taylor series with center zero for the function f(x) = In(1 + x^2 ) Question (2) Compute the first-order partial derivatives of f(x, y) = 2x/(x-y) Question (3) Evaluate the double integral (1 to 3)(0 to 1) of (2x-3y)dx dy Question (4) Calculate the divergence and curl of the vector field F(x,

Solving Inseparable Differential Equations

Consider the following very simple model of blood cholesterol levels based on the fact that cholesterol is manufactured by the body for use in the construction of cell walls and is absorbed from foods containing cholesterol: Let C(t) be the amount (in milligrams per deciliter) of cholesterol in the blood of a particular person a

Random Variables and Limit State Functions

Consider the following two collections of data that represent realizations of two random variables X1 and X2: X1: 18.9 21.1 17.8 20.2 16.0 19.0 20.9 19.1 22.5 18.7 15.:3 17.5 22.1 19.8 20.76 X2: 2:3.9 17.8 20.7 20.6 20.0 21.6 25.0 21.9 21.5 20.6 22.0 20.4 2:3.2 21.5 2:3.0 2:3.:3 21.8 2:3.8 26.6 2:3.0 22.0 2:3.8 22.1 (a) Es

Modeling Data With Linear functions, Trends and Forecasting

1. America creates more garbage than any other nation. According to Denis Hayes, president of Seattle's nonprofit Bullitt Foundations and a founder of Earth Day, "We need to be an Heirloom Society instead of a Throw-Away Society." The EPA estimates that, on average, we each produce 4.4 pounds of garbage daily (Source: Take out

Lagrange Multipliers

Use the method of Lagrange multipliers to find the extreme valus of 3x - 4y +12z on the spherical surface with equation x^2+y^2+z^2=1.

Finding Zeros of Functions

Find the zero of the linear function f(x)=3x-12 Find the zeros of f(x)=x^2-2x-3 Find the vertex of f(x)=x^2-2x+4 Find the axis of symmetry of f(x)=x^2-2x+4 Find the zeros and state the multiplicity of each for f(x)=x^2(x+3)(x+1)^4 Find the zeros of f(x)=x^2-8x+12

Calculus - Vectors - Area of Triangle, Volume of Parallepipe

Question (1) a = (3 , 1 , 2 ) , b = ( - 1 , 1 , 0 ) , c = ( 0 , 0 , - 4 ) , then show that a × ( b × c ) &#8800; (a × b) × c Question(2) Given P ( 2 , 1 , 5 ), Q = ( - 1 , 3 , 4 ) and R = ( 3 , 0 , 6 ), then find (a) a vector orthogonal to the plane through the points P,Q and R (b) Find the area of the triangle PQR

Determine whether the given vectors are orthogonal, parallel or neither

Determine whether the given vectors are orthogonal, parallel or neither. <-5,3,7> and <6,-1,2> <4,6> and <-3,2> -i + 2j + 5k and 3i + 4j - k 2i + 6j - 4k and -3i -9j +6k Find a unit vector that is orthogonal to both i+j and i+k. If a = <3,0,-1>, find a vector b such that comp_a b = 2 (component of b in the a direction

Eigenvalues, Eigenfunctions and Sturm-Liouville Problems

1. y'' + k*y = 0 BC: y'(0) = 0 y'(L) = 0 2. y'' + k*y = 0 BC: y(0) = y(&#61552;) y'(0) = y'(&#61552;) 3. y'' + k*y = 0 BC: y(0) = 0 y(&#61552;) +2*y'(&#61552;) = 0 4. y'' + 2*y' + (1+k)*y = 0 BC: y(0) = y(1) =0 Please see the attached file for

Inverse Function

4. A retailer you spoke with in New York City's fashion district imports haute couture from European designers. One of the accommodations which must be considered when importing fashion from other countries is the difference in the size charts. A function that will convert dress sizes in the United States to those in Italy is

Finding X, Y and Z Intercepts of 3-D Graphs

How do you find the x, y, and z-intercepts of a 3-Dimensional graph step by step? I am trying to interpret some graphs and would like this information to assist me in the interpretation. keywords: 3D, 3Dimensional

Simple harmonic oscillation driven by an external force

2. A spring with a 4-kg mass has natural length 1 m and is maintained stretched to a length of 1.3 m by a force of 24.3 N. If the spring is compressed to a length of 0.8 m and then released with zero velocity, find the position of the mass at any time t. 10. As in Exercise 9, consider a spring with mass m, spring constant k,