Explore BrainMass

Calculus and Analysis

Work, Force, Consumer Surplus, Cost Functions and Maximum Profit (9 Problems)

36. A holding tank has the shape of a rectangular parallelepiped 20ft by 30ft by 10 ft. a) How much work is done in pumping all the water to the top of the tank? b) How much work is done in pumping all the water out of the tank to a height of 2ft above the top of the tank? Please see the attached file for all nine questions

Partial Differential Equation : Diffusion Equation and Explicit Series Solution

Consider the diffusion equation ut = ku.xx for 0 < < pi and t > 0 with the boundary conditions ux(0, t) = 0 and u(pi, t) = 0 and the initial condition u(x,0) = 1. (a) Find the separated solutions satisfying the differential equation and boundary conditions. (b) Use these solutions to write an explicit series solution to t

Word problem/sketch involving integrals

A high-tech company purchases a new computing system whose initial value is V. The system will depreciate at the rate f=f(t) and will accumulate maintenance costs at the rate g=g(t), where t is the time measured in months. The company wants to determine the optimal time to replace the system. a) Let C(t)=1/t the integral fr

Differentials : Rate of Change, &#916;x and dx

Let x=1 and delta x= 0.01, find delta y. f(x)=5x^2 - 1 f(x)=sq rt 3x compare the values of dy and delta y y=x^3 x=1 delta x=dx=0.1 y=x^4+1 x=-1 delta x=dx=-0.1 Part 2 Use differentials to approximate the change in cost , revenue, or profit corresponding to an increase in sales of one

Polar Forms : Cardinoid, Rose Curve, Lemniscate and Limacon

Questions: 4,6,10,12,18,22,26,30,34,36,42,44,46,48,50 on page 6.3. 2,8,22,14,18,12,24 on page 393 4. Identify each of the curves as a cardinoid, rose curve (state number of petals), lemniscate, limacon, circle, line of none of the above. Please see attached for all questions.


Problems: 2, 6, 8, 12, 14,16,18,20,and 22, 28 done. Page 372: 2, 6,10,14,16,20,22,24,26,28,32,34 done. 28. Find the area of the region that contains the origin and is bounded by the lines 2y = 11 - x and y = 7x + 13 and the curve y = x² - 5. Please see attached for full question.

LaPlace Transformations with some Initial Value Problems (8 Problems)

Problem 9.1 (Prob. 29. P. 252) Two particles each of mass m moves in the plane with co-ordinates (x(t), y(t)) under the influence of a force that is directed toward the origin and had magnitude k/(x2 + y2) an inverse-square central force field. Show that mx''=-kx/(r^3) and my''= -ky/(r^3) where r = sqrt(x2 + y2) Problem 9.2

Work Done by Force Field Along a Helix

Find the work done by the force field F(x,y,z)=... on a particle that moves along the helix... Please see the attached file for the fully formatted problems.

Derivatives : Rate-of-Change Word Problem

A motorist, in a desert 5 km from point A, which is the nearest point on a long straight road, wishes to get to point B on the road. If the car can travel 30 km/ hr and 80 km/hr on the road, find the point where the motorist must meet the road to get to point B in the shortest possible time if point B is 5 km from point A. Use c

Polar Coordinates; Laplace's Equation; Boundary Conditions

I need some clues on figuring out these questions. Please see attachment for complete problems (regarding the below: "..." indicates an equation to be found in the attachment. Thanks!) (a) Using polar coordinates, find all the separated solutions of Laplace's equation satisfying the attached boundary conditions in the "wedge

Brachistochrone Curves

What is it's origin? State the problem with picture. Explain how the cycloid relates to the solution.

Vector Calculus - Maxwell's Equation

Hi, I'm having trouble figuring out how to solve this. I think I figured out some of it, but I don't understand it in general. I attached the problem as a jpeg. I'd appreciate seeing how to show the answer. You can ignore the pencil markings, they are notes to myself in trying to figure out the problem. Thanks.

Average and Instantaneous Rate of Change and Application to Revenue

Find the average rate of change of the function over the indicated interval. Than compare the average rate of change to the instantaneous rate of change at the end points of the interval. f(x)= x^2 + 3x - 4; (0,1) The annual revenue R (in millions of $ per year) of Wm. Wrigley Jr. Company for the years 1994-2000 can

Differential Equation

Solve the differential equation for absolutely integrable (on R) solutions (See attached for equation)

Rate of Change and Instantaneous Rate of Change

The effectiveness of E (on a scale from 0-1) of a pain killing drug t hours after entering the bloodstream is given by E= 1/27(9t+3t^2-t^3) o< or = t < or = to 4.5 Find the average rate of change E on the indicated intervals and compare this rate with the instantaneous rates of change at the end points of the intervals a.) [

Rates of Change : Cost and Marginal Cost

3.) A company determines that the cost in dollars to the manufacture x cases of the dvd "math caught in embarrassing moments" is given by C(x) = 100 + 15x - x^2, (0< or = x < or = 7) a.) Find the average rate of change of the cost per case for the manufacturing between 1 and 5 cases [ review ex. 2a] So on the average,

Fundamental Solution of N-Dimensional Laplace equation

Please see the attached. The fundamental solution of the n-dimensional Laplace equation solves , (1) where is the n-dimensional delta function. a. Show that if , the solution of the above equation (1) for is , where is a constant. b. Use the n-dimensional Gauss theorem to evaluate the