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# Calculus and Analysis

### 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

### Differential Equations : Particular Solutions, Variation of Parameters and Resonance Frequency

Please see the attached files for the fully formatted problems.

### 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.

### Find the slope of a line tangent to a curve.

Find the slope of the line tangent to the curve y=x^(x-1) at x=3.

### 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

### General Solution & Phase Portrait

A. Find the general solution to the attached system of differential equations. x' = y y' = 2x b. Draw the phase portrait for this system and describe in words what happens to the solution for all initial conditions: x(0) = x0 and y(0) = y0. See the attached file.

### Solve the Differential Equation Subject to Initial Conditions

Solve, subject to the given initial conditions: x(0) = x'(0) = 0 solve x'' + x = sec t t ε (-pi/2, pi/2). See the attached file.

### Prove one of maxwell's equations using supplied info

Hi, I have worked this problem, but want to know if I did it correctly. If I did not do it correctly I would like to see how to do it. Attached is the problem and my answer. Thanks

### 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.

### Differential Equation for Absolute Integrable Solutions

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

### Calculus: Area and volume

Please see the attached file for full problem description. --- 1. A solid is bounded by two bases in the horizontal planes z = h/2 and z = -1/2, and by such a surface that the area of every section in a horizontal plane is given by a formula of the sort Area = a0 z3 + a1 z2 + a2 z + a3 (where as special cases some of

### 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, t

### Determine the Equation of Parabola

Analyze and then plot the following curve: y = x^2 + 2x - 3.

### Vector Calculus - Verify Divergence Theorem

Hi, This problem is from a book I have, and has three parts. I have attached both the problem (p23.jpg) and the solution (a23.jpg). I understand in general how the theorem works, but I can't come up with the correct answer no matter what I try. I think I'm not setting up the problem correctly. I'd appreciate someone sho

### 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

### Goat on a Rope : Length of rope so it can graze half of the field?

A goat is tethered to a stake at the edge of a circular field with radius 1 unit. Use a computer or graphing calculator to determine how long the goat's rope should be so that the goat can graze over half of the field.

### 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

### Numerical Methods for Differential Equations: Recursive Formula

5. Knowing that the recursive formula was obtained from a differential equation, try to determine the original equation.

### Numerical Methods for Differential Equations

In this problem we will use Euler's method to find an approximation for e. Consider the differential equation f' = f with initial data f(0) = 1. We know the solution is f(t) = e', therefore f(1) = e. Using as step size h = 0.5, and fo = 1, use Euler's method to obtain f20. Your answer is an approximation to f(t10) = f(1).

### 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

### Tension on a Cord in a Mobile

A mobile is hanging from the ceiling with two metal pieces strung by cord, one under another. I only have the masses for each figure. How do I determine the tension on the top cord and then how do I determine the tension on the bottom cord?

### Laplace Equation with Neumann Boundary Condition on a Circle

Solve by using series representation and summing the series to get a Poisson integral formula for this Neumann boundary condition problem. Please see the attached file for the fully formatted problems.

### Limit Response Functions

Lim f(x) where f(x)={2-x, x<1 x->1- {2x-x^2, x>1