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Calculus and Analysis

optimal coughing radius

When a foreign object gets stuck in your windpipe you cough. The radius of your windpipe contracts so as to increase the air velocity and dislodge the obstruction. It can be shown that air velocity is related to windpipe radius by v(r) = (r0 - r)r^2 ,where r0 is the usual 'rest radius' of the windpipe. Physical constraints preve

Deriving the volume element in spherical coordinates

A differential of volume is given by: dV = r^2*sin(theta)*dr*d(theta)*d(phi) can you derive this for me i.e. using a differential of volume and spherical coordinates show how this equation is arrived at? r is a radius, theta is the angle in the x-y plane and psi is the the z-y plane Please show a diagram

Linear Approximations and Differentials

2.8 Exercises Find the linearization L(x) of the function at a. 2. f(x) = 1/sqrt(2 + x), a = 0 Explain, in terms of linear approximations or differentials, why the approximation is reasonable. 16. (1.01)^6 ~ 1.06 24. Use differentials to estimate the amount of paint needed to apply a coat of pain 0.05 cm thick to

Use differential equations to find the salt in the tank

Please show all steps to solution. A tank with a capacity of 500 gal originally contains 200 gal of water with 100 lb of salt in solution. Water containing a salt concentration of 1 lb/gal enters the tank at a rate of 3 gal/min, and the mixture is allowed to flow out of the tank at the rate of 2 gal/min. (a) Find the amount

Equilibrium of Bodies: Tension in the ropes of a decoration.

Ropes 3 m and 5 m in length are fastened to a holiday decoration that is suspended over a town square. The decoration has a mass of 5 kg. The ropes, fastened at different heights, make angles of 52 degrees and 40 degrees with the horizontal. Find the tension in each wire and the magnitude of each tension.

Evaluate Functions

How to Evaluate Functions Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x. 1. f(x) = 5x - 3 2. f(x) = x2 - 3x + 2 3. f(x) = 2x3 + 7x2 - x - 1 4. f(x) = 10x 5. f(x) = ln x

Stochastic Differential Equations

Question on Stochastic Differential Equations. Show that G=exp(t+aexp(X(t))) is a solution of the stochastic differential equation... Stochastic Calculus (a) Show that Is a solution of the stochastic differential equation (b) By considering the form show that where should be determined. Similar form que

Parabola Application

A new bridge is to be constructed over a big river. The space between the supports must be 1000 feet; the height at the center of the arch needs to be 320 feet. The support could be in the shape of a parabola or a semi-ellipse. An empty tanker needs a 250 foot clearance to pass beneath the bridge. The width of the channel for ea

Find the maximum profit

Total cost of producing goods. See attached for full description. 4. The total cost of producing q units of product is given by C(q) = q^3 - 60q^2 +1400q + 1000 for 0 < q <= 50; the product sells for $788 per unit. What production level maximizes profit? Find the total cost, total revenue, and total profit at this production

What happens to concavity when functions are added

#4 What happens to concavity when functions are added? a) If f(x) and g(x) are concave up for all x, is f(x) + g(x) concave up for all x? Yes b) If f(x) is concave up for all x and g(x) is concave down for all x, what can you say about the concavity of f(x) + g(x)? For example, what happens if f(x) and g(x) are both polyn

Calculus - Exponential Distribution

The following table gives the percent of the US Population living in urban Areas as a function of year2. ... 5. (a) Estimate f'(2) using the values of f in the table. ... 5. The thickness, P, in mm, of pelican eggshell depends ... 11. The quantity, Q mg, of nicotine in the body t minutes after a cigarette is smoked ... [Pl

Calculus to find volume

1. direction: Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. y=2/x, y=-x+3 ( note: the 2 is over a fraction bar and the x is the denominator) 2. Solve the problem. An auxiliary fuel tank for a helicopter is shaped l

Series and Sequences of Parametric Equations

Following are the instructions from my teacher for the final review. please follow directions precisely and show all steps by hand. Answers must be exact unless otherwise indicated. 1) Consider the parametric equations x =2t+1/t and y = 1-t. (i) Using a table sketch the curve represented by the parametric equations. Writ

Elementary Differential Equations

Please show the steps in the attached problems so that I can better understand what I am learning. Q1 is a Bernoulli equation, primes denote derivatives with respect to x. Please only complete questions 1 and 2. 1. Find the general solution to 2xy' + y^3 * exp(-2x) = 2xy. 2. Find the general solution of reducible sec

Provide a plot of temperature versus distance east.

Linear equation that relates Celsius to Fahrenheit : Your company was very impressed with your results from the recent study. They have asked that you investigate another location for them and present this new promising location at the upcoming sales meeting. Again, it is your job to check out the various areas and the frien

Model the Data for Temperature of Water and the Heat Supplied

Water is the most important substance on Earth. One reason for its usefulness is that it exists as a liquid over a wide range of temperatures. In its liquid range water absorbs or releases heat directly in proportion ot its change in temperature. Consider the following data that shows temperature of a 1,000 g sample of water at

ACME Construction Company is building a suspension bridge over the Miami River. They need to know how much material will be required to construct the main support cables and what sort of cable they need to buy. The support cables will be attached at either end to the top of 100 meter tall concrete pillars. The two concrete pillars are 200 meters apart. The cable should hang down 50 meters at its lowest point. Gottfried Leibniz and Christian Huygens in 1691 determined that any cable hanging under the force of gravity must have the shape of the graph This shape is known as a catenary. The parameter "a" is the ratio of cable tension to cable density and . The only use of the parameter b is to provide a vertical shift, if necessary. ACME would like to hire your group to find two things for them. First, what values must a and b have in order for the catenary to fit the constraints imposed by the placement of the concrete pillars and the low point of the cable? They are especially interested in the parameter a since this tells them what tension the cable will be under. Second, what length cable do they need? You should try to give a formula for the cable length in terms of the cable function y(x). That way, ACME can use your result for other cable shapes as well. Following are several hints for solving this problem. When you write your report for ACME, you must explain to them, step by step, how you solved the problems. You will need to use a combination of graphs, equations and text. You must try to convince them that your results are correct. It's no use to them if they have to solve the problem themselves in order to verify your results.

ACME Construction Company is building a suspension bridge over the Miami River. They need to know how much material will be required to construct the main support cables and what sort of cable they need to buy. The support cables will be attached at either end to the top of 100 meter tall concrete pillars. The two concrete pilla

acceleration, velocity and position of an object

An object moves along the x-axis with initial position x(0)=2. The velocity of the object at time t is greater than or equal to 0 is given by v(t)=sin((pi/3)t). a.) What is the acceleration of the object at time t=4? b.) Consider the following two statements. Statement I: For 3<t<4.5, the velocity of the object is increa

Schwarzschild Radius for Sun

The Schwarzschild radius describes the critical value to which the radius of a massive body must be reduced for it to become a black hole. R = 2 G M / c 2 where G = gravitational constant 6.7x10 -11 M= mass of the object C = speed of light 3x10 8 The sun has M = 2x10 30 . What is the Schwarzschild radius

Differential Equations Boundary Conditions

Consider the following problem for u = u(x,t): , , a) Seeking a solution of this problem of the form , show that f and g satisfy the coupled system ; where and ; , ; , . b) Eliminating g between the differential equations in a), show that f satis

Deriving the equation of motion of a projectile shot vertically upward considering the effects of air resistance and solving the first order differential equation obtained.

Consider a projectile of mass m which is shot vertically upward from the surface of the earth with initial velocity V. Assume that the gravitational force acts downward at a constant acceleration g while the force of air resistance has a magnitude proportional to the square of the velocity with proportionality constant k>0 and a

First find a general solution of the differential equation.

Please see the attached file. dy/dx = 3/y First find a general solution of the differential equation. Then find a particular solution that satisfies the initial condition y(0) = 5. ******************* A bacteria population is increasing according to the natural growth formula and numbers 100 at 12 noon and 156

Interval where function is increasing and decreasing

Find the interval where the function is increasing and the interval where it is decreasing. (If you need to enter - or , type -INFINITY or INFINITY. If there is no interval where the function is increasing/decreasing, enter NONE in those blanks.) ( , ) (increasing) ( , ) (decreasing) 2. [TanApCalc7 4.1.0

Two standard proofs

Text Book: - Taylor & Menon 1.) Prove that a compact set is bounded. 2.) Prove that if the sequence {xn} converges, than the sequence {IxnI} also converges. Is the converse true as well?

Fundamental Set of Solutions to an ODE

Suppose that p and q are continuous on some open interval I, and suppose that y1 and y2 are solutions of the ODE y'' + p(t)y' + q(t)y = 0 on I. (a) Suppose that y1, y2 is a fundamental set of solutions. Prove that z1, z2, given by z1 = y1 + y2, z2 = y1 &#8722; y2, is also a fundamental set of solutions. (b) Prove that i