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Calculus and Analysis

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

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)

Average Rate of Change Problem

A car is moving from left to right along a road which lies along the straight line 3x + 4y = 12 in the xy-plane. (The x and y coordinates are measured in miles.) The following short table gives the x-coordinate of the car at several times: t 3:00pm 3:10pm 3:15pm 3:30pm 4:00pm x -28 -24

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)

Business Calculus

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%

Radius of Convergence

Find the radius of convergence of About x= (-1/3) Thanks! Please see the attached file for the fully formatted problem.

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 - ∫3b 3x2 dx = 37 Then find the value of b. Note - The b is supposed to be directly over the 3.

Lennard-Jones Potentials

The Lennard-Jones potential for the interaction of two molecules separated by distance R is U(R)= A/R^12 - B/R^6 where A and B are constants. the equilibrium separation Re is that value of R at which u(R) is a minimum and the binding energy is De = -U(Re). Express: (a) A and B in terms of Re and De. (b) U(R) in terms of R,

Solving Differential Equations by Variation of Parameters

Determine the particular solution of the following nonhomogeneous differential equation using the method of variation of parameters y" + 4y' + 4y = x^-2 e^-2x ; x>0 homogeneous equation is y =C1e^-2x +C2xe^-2x y= u1e^-2x +u2xe^-2x after differentiating and letting u'1e^-2x +u'2xe^-2x =0, we have _2u'e^-2x -2'u2xe^-2

Monthly payment calculus question

A $100,000 mortgage is to be paid over 15 years at 7.4% per annum, compounded semi-annually, with monthly payments. What are the monthly payments (to the nearest dollar)?

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.

Four calculus questions on finding the limits

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

Differential Equations : Variation of Parameters

Determine the particular solution for the following nonhomogeneous differential equation using the method of variation of parameters Y" + y =tan(x); 0< x < pi/2 I got characteristic equation as y = U1 sin(x) + U2 cos (x) I was able to get thru to set the original D.E. to u'1 (x) cos (x) - u'2 (x) sin (x) = tan (x)

Applications of Derivatives and Differential Equations

The relative decay rate of naturally occurring uranium U-238 was given in Example 4. Scientists estimate the age of the earth to be about 4.55 billion yr. Determine the fraction of the U-238 present when the earth was formed that is still present as U-238. EXAMPLE 4: A 1-G sample of pure uranium produces about 740,000 de

Applications of Derivatives and Differential Equations

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