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Calculus and Analysis

Differential Equations : Solving Boundary-Value Problems

3. Solve the boundary-value problem, if possible. a. y''-6y'+9y =0, y(0) =1 and y(1) = 0 b. 9y''-18y'+10y = 0 , y(0) =0 and y(pie) = 1 4. If a, b and c are all positive constants and y(x) is a solution of the differential equation ay''+by'+cy = 0, show that lim x->infinity y(x) = 0

Inverse LaPlace Transform two different ways

Find the Inverse LaPlace Transform using different methods described in the attachment. To see the description of the problem in its true format, please download the attached question file.

solve differential equation using Laplace transform

1. Solve y''+4y'+4y=g(t) using Laplace transforms.... y(0) = 2; y'(0)=-3 Express the answer in terms of a convolution integral 2. Find the inverse of F(s) = 1/[(s^4)(s^2+1)] the answer can be left as a convolution integral 3. Solve y''''+5y''+4y=g(t) using laplace transforms.... y(0) = 1; y'(0)=y''(0)=y'''(0)=0 E

Find the Laplace transform of the following signal examples

See attached file for full problem description. 1. cos(200*pi*t) 2. sin(200*pi*t) 3. sqrt(2) & cos(200*pi*t - pi/4) 4. If the signal above was written like this, can a time delay be applied??

Newton's Law of Cooling

Case: At 9:00 PM a coroner arrived at a hotel room of a murder victim. The temp. of the room was 70 degrees F. It was assumed that the victim had a body temp. of 98.6 degrees F (AT THE TIME OF DEATH)(not at 9:00PM). The coroner took the victim's temp. at 9:15 PM at which it was 83.6 degrees F and again at 10:00 PM at which it wa

Applications of Cost Functions

4. A hospital has seven large identical heat pumps. If one of the heat pumps malfunctions, it could seriously impact hospital operations. If preventive maintenance is performed weekly, it costs \$1200 to perform preventive maintenance (PM) on all seven of them. If one of the heat pumps breaks down between PM inspections, the av

Solving Differential Equations, Volumes of Solids and Taylor Series

See attached file for full problem description. 1. First find a general solution of the differential equation dy 3y dx = . Then find a particular solution that satisfies the initial condition that y(1) = 4. 2. Solve the initial value problem dy y3 dx = , y(0) = 1. 3. Nyobia had a population of 3 million in 1985. Assum

Differential Equations and Springs

1. Solve the initial-value problems and graph the solutions on the same set of axes. y'' + 4y' + 2y = 0 y(0) = 5; y'(0) = 0 y'' + 4y' + 2y = 0 y(0) = 0; y'(0) = 5 2. Repeat problem 1 for the equation: y'' + 2y' + 5y = 0 y(0) = 5; y'(0) = 0 y'' + 2y' + 5y = 0 y(0) = 0; y'(0) = 5

Solving Inequalities, Limits and Derivatives

Please see the attached file for the fully formatted problems. Includes: Solve the following inequalities/equations expressing the solution. Show all work and box the final answer. Write an equation for the line described below. Graph the following function. What symmetries, if any, does the graph have? Find the v

Harmonic Functions

A)Let a be less than b and set M(z)=(z-ia)/(z-ib). Define the lines L1={z:F(z)=b}, L2={z:F(z)=a} and L3={z:R(z)=0}. The three lines split the complex plane into 6 regions. Determine the image of them in the complex plane. b) Let log be principal branch of the logarithm. Show that log(M(z)) is defined for all z in C with the

Applications of Derivatives Word Problems : What dimensions will maximize the enclosed area?

A farmer has 480 meters of fencing. He wishes to enclose a rectangular plot of land and to divide the plot into three equal rectangles with two parallel lengths of fence down the middle. What dimensions will maximize the enclosed area? Be sure to verify that you have found the maximum enclosed area.

Function Calculation Process

See attached file for full problem description. Find a function f(x) = x^k and a function g such that f(g(x)) = h(x) = sqrt(3x+ x^2)

Paraboloid of revolution

(Parabaloid of revolution) Determine the shape assumed by the surface of a liquid being spun in a circular bowl at constant angular velocity, W. Hint: consider a particle of liquid located at (x, y) on the surface of the liquid. The forces acting on the particle are m*W^2*x in the x direction and -m*g in the y direction.

Differential Equations

1. A tank with a capacity of 500 gal. originally contains 200 gal. of water with 100 lb. of salt in solution. Water containing 1 lb. of salt per gallon is entering at the rate of 3 gal./min. and the mixture flows out at a rate of 2 gal./min. (i) Write a differential equation for the concentration in the tank before the tank o

Tangent line

Finding tangent line for f(x) that pass through the given point ( and the point isn't on the curve) part 1 f(X) = 4x-x^2; (2,5) Part 2 f(x) =x^2; (1,-3)

Differential Equations : Chain Falling off a Table

A 24 foot chain, weighing Y lbs/foot of length, is stretched out on a very tall frictionless table with 6 feet hanging over the edge of the table. If the chain is released from rest , in the configuration described above, find the following: a) How long before the chain falls off the table? b) What is the velocity of the cha

Find the present value and future value of an income stream of \$1000 a year, for a period of 5 years, if the interest rate is 8%

Determine concentration of wine watered in a vat - There is a vat of wine with a tap at the bottom and a water faucet at the top. The tap is open and draining wine at a certain rate. The water faucet is adding water at the same rate, keeping the volume the same. How do I determine the concentration of the wine for any moment?

There is a vat of wine with a tap at the bottom and a water faucet at the top. The tap is open and draining wine at a certain rate. The water faucet is adding water at the same rate, keeping the volume the same. How do I determine the concentration of the wine for any moment?

Solving Differential Equations by Separation of Variables

Consider a right cylinderical hot tub. Radius = 5 feet; Height = 4 feet; placed on one of its circular ends. water is draining from the tub through a circular hole in the base of the tub 5/8inches in diameter. k = .6; using Torricelli's Law v = [2*g*h(t)]^1/2 and the equation dV/dt = -kAv where A is the are

Find value of b - integral

(See attached file for full problem description) --- If - &#8747;3b 3x2 dx = 37 Then find the value of b. Note - The b is supposed to be directly over the 3.

First order differential equations-mixing problem

Consider two tanks, labeled Tank A and Tank B. Tank A contains 100 gallons of solution in which is dissolved 20 lbs of salt. Tank B contains 200 gallons of solution in which is dissolved 40 lbs of salt. Pure water flows into tank A at a rate of 5 gal/s. There is a drain at the bottom of tank A. The solution leaves tank A via thi

Length of graph

Please show me how to calculate the length of a curve. See attached file for full problem description.

Finding limits using Squeeze theorem and L' Hospital theorem

1. Compute the following limit: lim (x->2) [sqrt(6-x) - 2]/[sqrt(3-x) - 1] 2. Prove using the squeeze theorem lim (x->0) x^4 cos(2/x ) = 0 3. Show by means of an example that lim x-> a (f(x) + g(x)) may exist even though neither lim x-> a f(x) nor l

Standard Decay Age of Sample Using Calculus

8. a. Let y(t) be the amount of a radiactive material with relative decay rate k. Let Q(t) be the decay rate. Use the differential equation for y (not the solution formula) to show that the quantity Q also undergoes exponential decay with rate constant k. b. It is sometimes easier to measure the rate of radioactive decay t

Calculus Finding Volume of the solid of revolution obtained by rotating a region.

Question (1) What is the volume of the solid of revolution obtained by rotating the region bounded by y = 1 and y = 5 - x^2 around the X-axis. Question (2) Find the volume of the solid of revolution obtained by rotating the region bounded by y = 1 and y = Tan x about the x-axis from x = 0 to x = pi/4. See attached file

Nonhomogeneous Differential Equations : Method of Undetermined Coefficients

Using the method of undetermined coefficients to find the particular solution of the nonhomogeneous equation, find the solution of the following d.e. satisfying the given initial conditions y"+4y=x^2+3e^x y(0)=0 y'(0)=2

Use reduction of order to find a second solution to a differential equation.

Given that y1(x)=e^-x is a solution of y'' + 2y' + y =0, find second solution using the method of reduction of order. I substituted y =ve^-x so y' = v'e^-x - v^-x; y''= v''e^-x -v'e^-x +v e^-x I came out of that with v''e^-x + v'e^-x =0 I think the next step is to seperate variables. Looking for help to finish the pro

Using implicit differentiation to determine equation of the line

Use implicit differentiation to find an equation of the line tangent to the curve x^3+2xy+y^3 = 13 at the point (1,2).

Limit using epsilon delta defintion

Show that: lim (x+y)=o as x and y approach zero; using the epsilon-delta definition Also, show that: lim f(x)=1 as x approaches zero; using the epsilon-delta definition. **Note: x is a vector in this case, with a right arrow going over it. It is not just "x".

Modeling with Higher Order Differential Equations

Two springs are attached in series as shown in Figure 5.42. If the mass of each spring is ignored, show that the effective spring constant k ot the system is defined by I/k = I/k + I/k2. A mass weighing W pounds stretches a spring 1/2 foot and stretches a different spring 1/4 foot. The two springs are attached, and the mass is