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# Calculus and Analysis

### 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

### A novice inventor has invented an exciting new toy for dogs.

A novice inventor has invented an exciting new toy for dogs. He believes it will cost him \$.95 per toy to produce these doggy marvels. Unfortunately, to get mass-producing these items, he has had to spend \$6000 of his hard earned money and countless hours observing animals. He plans on selling the toys for \$1.69 each. a.

### Differentials

The dimensions of a closed rectangular box are measured as 100 centimeters, 60 centimeters, and 60 centimeters, respectively, with the error in each measurement at most .2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box.

### Find the Radius and Center of a Circle by Completing the Square

Complete the square to determine the radius and center of the circle. x^2+y^2+2x-8y+12=0 and sketch its graph.

### Principal Value and Principal Branch of an Integrand

Use parametric representation in exercise 10 for the oriented circle C0 there to show that....where a is any real number other than zero and where the principal branch of the integrand and where the principal value of R^G are taken. Please see the attached file for the fully formatted problems.

### 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

### Derive Source Solution using a Laplace Transform

Please see the attached file for the fully formatted problems.

### Limits : L'Hopital's Rule (6 Problems)

Please see the attached file for the fully formatted problems.

### X-Coordinate of Centroid of Closed Region

What is the x-coordinate of the centroid of the closed region Please see attachment for function and region description. Please explain why the answer is correct. Thank you!

∫(2-x)^6dx

### 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

### Solve a Differential Equation

8. What is the solution of the differential equation dy/dt = 4y2t3, subject to the condition y(1) = 1? A. y = 1/(2-t4) B. y = 2 - 1/t4 C. y = t4 D. y = 1/t4 E. y = e^(1-t2)

### Differential Equations : Rate of Population Growth

6. A population grows exponentially. At 10 years, the population is 1,000. At 20 years, it is 2,000. What was the approximate population at 5 years? A. 140 B. 250 C. 500 D. 700 E. 750

### Differential Equations : Separation of Variables

5. Let y(x) be the solution to the differential equation (x2+1)1/2dy - (x/y)dx = 0 satisfying y(31/2) = 3. Then [y(81/2)]2 = A. 6 B. 8 C. 10 D. 11 E. 13

### Differential Equations : Radioactive Decay

The rate of decay of a radioactive substance is proportional to the amount of the substance present. Two years ago there were 5 grams of substance. Now there are 4 grams. How many grams will there be 4 years from now? A. 16/25 B. 2 C. 64/25 D. 16/5 E. 25/4

### Differential Equation: Increase at a Rate Equal to Product of Time Elapsed

The amount of a chemical increases at a rate equal to the product of elapsed time (in minutes) and the amount of the chemical. If the initial amount of the chemical is 10 units, what is the number of units at 4 minutes? A. 14 B. 10 + e8 C. 10 + e16 D. 10e8 E. 10e16

### Solve the Differential Equation

Which of the following is a solution to the differential equation: ylny + xy' = 0 for x > 0 ? A. xlny = 1 B. xylny = 1 C. (lny)2 = 2 D. -y(lny)(lnx) = 1 E. lny + (x2/2)y = 1

### How do I solve this problem of second - degree equations?

Sketch the graphs of each pair of circles to determine the number of points of intersection. If the circles are tangent or fail to intersect, say so. Then solve the system. 2 2 2 2 x + y - 4y= 0, x = y - 2x = 4 How do I go about solving?

### Normal Modes : Second Order Simultaneous Equations

This question is concerned with finding the solutions of the second order simultaneous equations where a = 38, b = -9, c = 378, d = -79 (i) Find the particular solutions to the differential equations which satisfy the initial conditions x = -10 and y = 7 at t = 0 together with the condition at t = 0.. For this part

### Heat Equation : Moving Source - Dirac Impulse Function

Please see the attached file for the fully formatted problem. Use superposition to solve: with boundary conditions: and initial condition

### Prove Using the Chain Rule

Please see the attached file for the fully formatted problems. Let x,y denote Cartesian coordinates and denote polar coordinates, and let . Using the chain rule show that (you may assume that ). Hence show that