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Calculus and Analysis

limit theorems and series

By using partial fractions show that a. the sumation from 0 to infinity of 1/(n+1)(n+2)=1 b. the sumation from 0 to infinity of 1/(alpha+n)(alpha+n+1)=1/alpha >0 if alpha>0 c. the sumation from 0 to infinity of 1/n(n+1)(n+2) =1/4 apply the theorem: let (Xsubn) be a sequence of positive real numbers such that L := lim(Xsubn

Finding Limits for Equations

Please see file attached. Find Limit f'(x) as x--->0 Dear TA, I don' understand the answer where limit as x0- and lim as x0+ don't equal each other since one equals -∞ and the other +∞. At f'(0) the function is undefined. Do I need to use the def of limit and find the limit from the right side and left side.

Volume

A)The region bounded by y = e^2x, y = e^x, x = 0, and x = 2 is rotated around the x-axis. Find the volume. b)Consider the region bounded by y = x^2, y = 3, and the y-axis. Find the volume of the solid obtained by rotating the region around the y-axis.

Calculus - Derivatives and Numerical Values

Need help with calculus homework problems. In Exercises 41-48, find dy/dx 44. 5x 4/5 + 10y 6/5 = 15 ____________________________________________________________________ 54. a. By differentiating x 2 - y2 =1 implicitly, show that dy/dx = x/y b. Then show that d2y/dx2 = - 1/y3. ____________________

Accounting : Profit Margins, Turnover and Return on Investment

The Valve Division of Bendix, Inc, produces a small valve that is used by various companies as a component part in their production. Bendix, Inc., operates its divisions as autonomous units, giving its divisional managers great descretion in pricing and other decisions. Each division is expected to generate a mimimum required

Differential Equation Given the Roots of the Auxiliary Equation

Please see the attached file for the fully formatted problems. 1. The roots of the auxiliary equation corresponding to a certain 12th order homogeneous linear equation with constant coefficients are: 2, 2, 2, 2, 2, 2, 1, 4, 3+4i, 3-4i, 2+5i, 2-5i Write the general solution 2. Find the general solution to y"- y= x^3

Line Integrals of Vector Fields

Please see the attached file for the fully formatted problems. Let gamma be given on 0<u<1 by (x,y)=(1-u,u) The vector field (F1,F2) is a gradient field. Compute: Integral over gamma of Fdx+F2dy Where: F1 = x*y^2 F2 = x^2*y

Boundary Value Problem

Hello, I need a detailed solution to the attached problem. Using an appropriate transform solve the boundary value problem... ** Please see the attached file for the complete problem description **

Calculus - Differentiation

A Ladder 25 feet long is leaning against the wall of a house ... (Please see the attachment for the figure and questions)

Second Order Homogenous Differential Equation

Show that y=t and y=1/t are solutions of t^2*y" + ty' - y = 0 Determine the following solutions of this equation, or explain why none exist. y(1)=0, y'(1)=2 all solutions satisfying y(t)=0 as t approaches zero all solutions satisfying y(t)=0 as t approaches infinity.

Example of Finding a Taylor Series

How do I find the Taylor series for this function: (1/x) Integral(x to 0) (e^t - e^-t)/2t dt. ...up to 4 terms? And the approximate value at x = 1.

An application of the Laplace transform

Define the unit ramp function by ... 1. Determine the Laplace transform of H(t) 2. Use the Laplace transform to solve the ODE ..... Questions: 1. To determine the Laplace transform, do we use ..... &#8747; (note the difference in integration limits). I know both integrals will give the same answer. But I am confused bec

Laplace Transform of Unit Ramp Function

The unit ramp function H(t) is defined by H(t) = 0 t < 0 t t > 0 a) Determine the Laplace transform of H(t) b) Solve the ODE y''+y = H(t −1) y(0) = 0 y'(0) = 0 
using
 Laplace
 transforms.
 Please see attached file for proper formatting.

Differential equation calculated

------ Consider&#8233; the&#8233; differential &#8233;equation&#8233; &#8233; y'' +(1/x)y'-(1+9/(4x^2))y=0 1. Determine the location and type of all singular points. 2. For regular singular points, determine the indicial equation that gives the leading exponents of a Frobenius expansion. 3. Find a series solution ab

Sampling to Conduct Survey Attitudes

Suppose you want to conduct a survey of the attitude of psychology graduate students studying clinical psychology toward psychoanalytic methods of psychotherapy. One approach would be to contact every psychology graduate student you know and ask them to fill out a questionnaire about it. a) What kind of sampling method is this?

Existence an duniqueness

Using the existence-uniqueness theorem show that the following IVP has a solution but it is not unique: y''+y'/x-9y/x^2=0 y(0) bounded general solution: y=Ax^3+Bx^{-3}

Differential Equation and Unit Step: Analytic Solution

Consider the differential equation (d^2y/dt^2) + 3(dy/dt) + 2y = u where y(0)=dy(0)/dt and u(t) is a unit step. 1. Determine the solution y(t) analytically. Please Show all steps and explanation of each step Thank You

Tangent line problem

Explore the following function. You are to decide which of threee lines given are tangent to the graph of f(x) at given points. The function to define is F(x)=2x^3-4x^2+3x-5 The possible tangent lines are: y= x-5 y= 2x-5 y= 3x-5 a) what is/are the zero(es) for this function? In other words, what is the solu

Solutions to Legendre's Differential Equation

The attached file has all the information about this problem. On this problem, we know from a previous problem that the Legendre polynomials satisfy the DE. It is a second order DE. Usually these have two linearly independent solutions. Are these the only polynomials that satisfy the DE, or is there another set, linearly ind

Differential Equation and Compound Interest

Attached INVESTMENT: Let A be the amount in a fund earning interest at the annual rate of r, compounded continuously. If the continuous cash flow of P dollars per year is withdrawn from the fund, then the rate of decrease of A is given by the differential equation: dA/dt = rA - P where A = A_0, when t = 0 a) solve t

Rate of change

See attachment Section 11.3 1) The quarterly profit of Cunningham Realty depends on the amount of money x spent on advertising/quarter according to the rule: where and x are measured in thousands of dollars. What is Cunningham's profit when its quarterly advertising budget is $35,000.00? 2) Suppose

Inverse Laplace Transformation and Partial Fraction Expansion

Please help me with these problems. section 7.4 6,16,20,24 Samples of these questions appear below. Please see the attached files for the fully formatted problems. Find the inverse Laplace transformation. Determine Partial Fraction Expansions for the given rational function. Determine L^-1{F}.

Laplace transform

Hi, Please help on these problems Please show all steps Section 7.3 # 4,8,14,20 See attached determine the Laplace transform of the given function.

Business calculus

Problem attached Revenue: A company sells a seasonal product. The revenue R (in dollars per year) generate by sales of the product can be modeled by: R = 410.5t^(2)e^(-t/30) +25,000 0 < t < 365 where t represents the day (a) Find the average daily revenue during the first quarter which is given by 0 < t < 90 (b) Find t

Average Daily Revenue of the Seasonal Product

Problem attached Revenue: A company sells a seasonal product that generates a daily revenue R (in dollars per year) modeled by: R = 0.06t^2 (365 - t)^1/2 + 1250 0 < t < 365 where t represents the day (a) Find the average daily revenue over a period of one year (b) Describe a seasonal product whose seasonal sales pattern re

Business Calculus: Revenue

Please help with the following questions regarding calculus and analysis. Revenue: Two models, R1 and R2, are given for the revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2004-2008, with t = 4 corresponding to the year 2004. Which model is projecting the

Business Calculus re: Mortgage Debt

Mortgage Debt. The rate of change of mortgage debt outstanding for one to four family homes in the United States from 1993 to 2002 can be modeled by: dM/dt = 5.4399t^2 + 6603.7e^-t where M is the mortgage debt outstanding (in billions of dollars) and t = 3 corresponds to 1993. In 1993, the outstanding mortgage in the US was

Extreme values

1.)f(x,y)= 5 - 3x + 4y, D is the closed triangular region with vertices (0,0), (4,0), and (4,5) 2.)f(x,y)= x^2 + 2xy + 3y^2, D is the closed triangular region with vertices (-1,1), (2,1), and (-1,-2) 3.)f(x,y)= 1 + xy - x - y, D is the region bounded by the parabola y=x^2 and the line y=4