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Calculus and Analysis

Linear PDE, Order, Homogeneous, Non-Homogeneous : Boundary Value Problems

A linear PDE can be written in differential operator notation L(u) = f. where L is the linear differential operator, u is the unknown function, and f is the right-hand side function. For each of the following PDEs, determine the linear operator and the right-hand side function, the order of the PDE, and whether the PDE is homog

Solving the Differential Equation

Please give me a step-by-step solution to this attached ODE. x'' + x = d2(t) + u2(t) x(0) = 0 x'(0) = 0 Please see the attached file for the fully formatted problems.

Differential Equation: Torricelli's Law

Please help with the following problem. Provide step by step calculations for each. (1) (a) Let h(t) and V (t) be the height and volume of water in a cylindrical tank at time t. If water leaks through a circular hole with area a at the bottom of the tank, Torricelli's law says that the rate of change of volume is given by the

Work, Force, Consumer Surplus, Cost Functions and Maximum Profit

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

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 : Minimizing Fencing

The math department is planning to build a park for calculus students along the riverbank. The park is to be rectangular with an area of 512 square yards and is to be fenced off on the three sides not adjacent to the river (draw a picture). a.) What is the least amount of fencing required for this job? b.) How long and w

Finding the Second Derivative of a Function

How did the book get from here: -2X(1+x^2)^2 - 4X(1-x^2)(1+x^2) ______________________________ (1 + x^2)^4 to here? -2x (1+ x^2)[1+x^2+2(1-x^2] ___________________________ (1+x^2)^4 I have worked and reworked and I cannot get that numerator to become that. What is the deal? (This

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, Δx 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.

Volume generated by rotation of area

52. The base of the solid is an isosceles right triangle whose legs are each 4 units long. Each cross section perpendicular to a side is a semicircle. Please see attached for all questions.

Find the Volumes of Bounded Regions

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.

Velocity & Acceleration

In science, we calculate the displacement of an object (how far it travels) and the velocity of an object (how fast it is moving) using the displacement equation and the velocity equation. Solving using the Method of substitution.

Find Derivatives using the Fundamental Theorem of Calculus

Use part I of the Fundamental Theorem of Calculus to find the derivatives of the following functions; answers must use correct variable. a. f(x)=the integral as pi goes to x of (1+cos[t])dt; f'(x)=___ b. f(u)=the integral as -1 goes to u of [1/(x+4x^2)]dx; f'(u)=___

LaPlace Transformations with some Initial Value 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.