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Calculus and Analysis

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.

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

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!

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 : 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

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

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

Rate of Change of Volume Given Rate of Change of Height

The height h of a right circular cone is 20 cm and is decreasing at the rate of 4 cm/sec. At the same time, the radius r is 10 cm and is increasing at the rate of 2 cm/sec. What is the rate of change of the volume in cm3/sec? (Note: The volume of a right circular cone is V = 1/3p r^2h.)

Find the Limit of Equation

Please see the attached file for the fully formatted problem. If lim x --> 0 f(x) = 3 then lim x--> 0 (e^(2x)-1)/(x f(x)) = ?

Heat Diffusion Equation and Standard Heat Equation

A) Let the temperature u inside a solid sphere be a function only of radial distance r from the center and time t. Show that the equation for heat diffusion is now: {see attachment}. This is not an exercise in doing a polar coordinate transformation. First you should derive an integral form for the equation by integrating over a