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# Calculus and Analysis

### Find the dimensions of the rectangle of area A sq. units that has the smallest perimeter.

I am in Freshmen Level Calculus. We are in section 4.5 of the Salas, Hille, and Etgen book, "Calculus: One and Several Variables" 9th edition. The name of the chapter is The Mean value theorem and its applications, But from my understanding of the examples, It does not use the MVT. The previous sections were about local ex

### Vertex and intercepts

Find the vertex and intercepts for the quadratic function and sketch its graph y=x^2+4x

### Differential Equations : Variation of Parameters

Use variation of parameters to solve the differential equation y'''+4y'=cot2t.

### 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

### 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

### 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

### 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

### 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.

### 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

### Differential Equations Variation of Parameters and Undetermined Coefficients and IVP

1. Find a Particular Solution using undetermined coefficients, then find a general solution y³ + y" - 6y' = 3e²&#8319;. 2. Find a particular solution using the method of variation of parameters then find a general solution y" + 9y = cos(3x). 3. Solve the Initial Value Problem x" + 4x = 6sin(3t). x(0)=4, x'(0) = 0. Ple

### 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.

### 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

### Limit of a falling object using L'Hopital's rule

This problem has been particularly confusing to me: "If an object with mass m is dropped from rest, one model for its speed v after t seconds, taking air resistance into account, is v=[(mg)/(c)][1-e^([-ct]/m)] where g is the acceleration due to gravity and c is a positive constant. (a)Calculate the limit as t approache

### Capacitator, Negative Exponential, Capacitance & Charge Time

Background Information: The paddles of a defibrillator in an ambulance or the emergency room of a hospital are actually the two plates of an electronic device called a capacitor. A capacitor is built from two conducting plates that are attached to a voltage source. Due to the voltage source, electrical charges move through

### Functions: Domain, Increasing/Decreasing, Image Set and Inverse

This question concerns the function,f (x) = ln (x + 3). a) State the domain of the function f, justifying your answer briefly. b) State whether the function f is: increasing, decreasing, neither increasing nor decreasing, one-one or many-one. c) Write down the image set of f. d) Explain briefly why the function f has

### Particular Solutions, Homogeneous and Non-Homogeneous

See attachment for complete questions. 1 and 2. Find the general solutions to the attached equations. 3. Non-homogeneous equation, associated homogeneous equation - find solution satisfying the given initial conditions. 4. Find the particular solution to the attached equation.

### 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, t

### 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

### Calculus...

Hi 1) A right triangular plate of base 8.0 m and height 4.0 m is submerged vertically, find the force on one side of the plate. (W = 9800N/m^3)? 2) The cable of a bridge can be described by the equation y = 0.06x^(3/2) from x = 0 to x = 200 ft. find the length of the cable? 3) A conical tank is resting on its apex. Th

### Limit Points of a Bounded Set of Real Numbers

Question: Construct a bounded set of real numbers with exactly three limit points (put the limit points at 0, 1 and 2005). (Please explain in your own words how the proof works. If you use a theorem, please state what it is and if possible, where you got it).

### Rate of Change of Volume of a Cone..

The radius of a right circular cone is increasing at a rate of 2 inches per second and its height is decreasing at a rate of 2 inches per second. At what rate is the volume of the cone changing when the radius is 30 inches and the height is 20 inches? ___ cubic inches per second.

### Recursive Sequence Given Recursive Formula and Seed

Please see the attached file for the fully formatted problems. Please show me how to solve question 1 part d of lecture two in the "non elegant" way. I'd like you to "work backwards from what you want to prove until you arrive at a true formula" l like in part C of question 1. I've provided the solutions so you can see

### Calculus analysis problem

1. Show that u (x, y) is harmonic in some domain and find a harmonic conjugate v (x, y) when (a) u (x, y) = 2x (1 - y) (b) u (x, y) = 2x - x3 + 3xy2 (c) u (x, y) = sinh x?sin y (d) u (x, y) = y / (x2 + y2) (Question is also included in attachment)

### Vector Analysis : Construction of a Plane Curve

Please see the attached file for the fully formatted problems. Problem: Assume you are given a non-negative function K(s). We would like to construct a plane curve B(s) with curvature K(s)..... HINT: Use the Fundamental Theorem of Calculus to show B has unit speed and then compute dT/ds. Problem: I. If K(s) = ... usin

### PDE with Time-Dependent Domain

Please see the attached file for the fully formatted problems. Consider the diffusion equation: on the time-dependent domain where a is a constant. We wish to solve the initial and boundary value problem having for and a prescribed . Thus, u is prescribed as a function of time on the left boundary that moves at

### Differential Equations : Particle Position at Time

2. A particle moves along a straight line so that its acceleration at time t seconds is (t + 1)2 cm/sec2. The particle's position at time t = 0 is at the origin, and its initial velocity is 1 cm/sec. What is the position of the particle, in cm. at time t seconds? A.((t+1)4/12)+(2/3)t-1/12 B.((t+1)4/12)+(2/3)t+1/12 C.((t+1

### Differential Equations : Population Growth

9. The rate of change of the population of a town in Pennsylvania at any time t is proportional to the population at that time. Four years ago, the population was 25,000. Now, the population is 36,000. Calculate what the population will be six years from now. A. 43,200 B. 52,500 C. 62,208 D. 77,760 E. 89,580