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Calculus and Analysis

Second Order Ordinary Linear Differential Equations with Constant Coefficients

1). Given the differential equation for 1. L[y]= y''+2by'+b2y = exp(-bx)/x2, x>0 ; a) Find the complementary solution of (1) by solving L[y] = 0. b) Solve (1) by introducing the transformation y[x]= exp(-bx) v(x). into (1) and obtaining and solving completely a differential equation for v(x) . Now identify the part

Writing Equations from Word Problems

Investments Morgan has $50,000 to invest and wants to receive $5000 interest the first year. He puts part in CD's earning 5.75%APY, part in bonds earning 8.75%APY and the rest in a growth fund earning 14.6% APY. How much should he invest at each rate if he puts the least amount possible in the growth fund? Mixing Acid Solu

First Order Differential Equations

1. In the following, represent the situation pictorially and deduce the governing 1st order differential equations and initial condition for the quantity in question. a) A tank, containing 300 gallons of pure water initially, is emptied out in the following fashion. A salt solution of concentration ½ lb of salt per gallon is

Surface area, volume, and work

Find the area of the surface obtained when the graph of y = x2, 0 ≤ x ≤ 1, is rotated around the y-axis. Find the volume of the solid that is generated by rotating the region formed by the graphs of y = x2, y = 2, and x = 0 about the y-axis. A 100-ft length of steel chain weighing 15 lb/ft is hanging

Solid of Revolution and Limits

Evaluate &#8747;(log3x / 2x) (dx) R is bounded below by the x-axis and above by the curve y = 2cosx, Figure 11.1. Find the volume of the solid generated by revolving R around the y-axis by the method of cylindrical shells 0(< or =) x(< or =) pi/2 Find limx&#8594;&#8734; ( 3x-2/3x+2 )^x

R is the region that lies between the curve

R is the region that lies between the curve (Figure 15.1) and the x-axis from x = -3 to x = -1. Find: (a) the area of R, (b) the volume of the solid generated by revolving R around the y-axis. (c) the volume of the solid generated by revolving R around the x-axis. y=1 / x^2+4x+5

Integrals, Differential Equations and Limits

Please see the attached file for the fully formatted problems. Question 1 Find &#8747; x3+4 ________________________________________x2 dx Question 2 Solve the initial value problem: dy ________________________________________dx = x^/¯(9+x2) ; y(-4) = 0 Question 3 Figure 3.1 f(x) = x2+3 Figure 3.2

Roots, rate of change, and maximum and minimum

Find the maximum and minimum values attained by the function on the interval [0, 2]. h(x)=x-1/x+1 The equation has three distinct real roots. Approximate their locations by evaluating f at -2, -1, 0, 1, and 2. Then use Newton's method to approximate each of the three roots to four-place accuracy f(x)= x^3- 3x+ 1

Rotating a System

Write the equation in terms of a rotated x'y'-system using q, the angle of rotation. Write the equation involving x' and y' in standard form. x2 + 2xy + y2 - 8x + 8y = 0; q = 45° x'2 = -4sqrt2y'2 x'2 = -4sprt2y' 3x'2 - 4sqrt2x'y' + y'2 = 0 2x'2 - sqrt2x'y' + 2y'2 = 0

Minimizing Perimeter of a Fence and Finding the Nearest Point on a Line

1. A rancher wishes to fence in a rectangular corral enclosing 1300 sq yards and must divide it in half with a fence down the middle. If the perimeter fence costs $5 per yard and the fence down the middle costs $3 per yard, determine the dimensions of the corral so that the fencing cost will be as small as possible. 2. Find t

Find a value for c so that f(x) is continuous for all x.

Please show all work. Find a value for c so that f(x) is continuous for all x. c2-x2 if x<0 f(x)={ _______________ ccosx if x>0 use the four-step process to find a slopepredictor function m(x). Then write an equation for the line tangent to the curve at the point x = 8. 4

Estimating Area under a Graph

If you have not seen it yet, consider flying with Professor Goetz over Rio hills. His GPS recorded the this graph of the velocity function v(t) . Based on this graph estimate the total distance traveled during the glider flight from the take off to the landing on the beach. Explain in words how you do this estimate. Please s

Differential Equations : Predator / Prey Models

Part a) Given the following predator prey model where x(t) is the predator population and y(t) is the prey population: dx/dt = - ax + bxy + (z1)*x dy/dt = cy - gxy +(z2)*y Here both z1 and z2 can be positive or negative; parameters a, b, c, g are all defined to be positive. Parameters z1 and z2 can r

Differential Equations : Spring Compression and Automobile Suspension Systems

36. An automobile's suspension system consists essentially of large springs with damping. When the car hits a bump, the springs are compressed. It is reasonable to use a harmonic oscillator to model the up-and-down motion, where y(t) measures the amount the springs are stretched or compressed and v(t) is the vertical velocity of

Differential Equations and Harmonic Oscillators

In Exercises 21?28, consider harmonic oscillators with mass in, spring constant k, and damping coefficient b. (The values of these parameters match up with those in Exercises 13?20). For the values specified, (a) find the general solution of the second-order equation that models the motion of the oscillator; (b) find the parti

Applications of Differential Equations: Mechanics

See the attached file. A perfectly flexible cable hangs over a frictionless peg as shown, with 8 feet of cable on one side of the peg and 12 feet on the other. The goal of this problem is to determine how long it takes the cable to slide off the peg, starting from rest. (a) At time t 0 what proportion of the whole cable is

Solving word problems using differential equations and their solutions.

Question 5 Suppose Anytown, USA has a fixed population of 200,000. On March 1, 3000 people have the flu. On June 1, 6000 people have it. If the rate of increase of the number N(t) who have the flu is proportional to the number who don't have it, how many will have the disease on September 1? Question 7 Suppose th

Converting Parametric and Rectangular Equations

Eliminate the parameter. Find a retangular equation for the plane curve defined by the parametric equations. X=3t, y=t+7 Find a set of parametric equations for the rectangular equation. Y=2x-2

Differential Equations and Rate of Change

A tank initially contains 100 gallons of a solution that holds 10 pounds of a chemical. A solution containing 1 pound of the chemical runs into the tank at a rate of 4 gallons per minute, and the well-mixed mixture runs out of the tank at a rate of 6 gallons per minute. a. How much chemical is in the tank after 25 minutes? b

Differential Equations : Phase Lines and Bifurcation Diagrams

Please see the attached file for the fully formatted problems. 22. (a) Use PhaseLines to investigate the bifurcation diagram for the differential equation .... where a is a parameter. Describe the different types of phase lines that occur. (b) What are the bifurcation values for the one-parameter family in part (a)? (c) U

Acceleration Rate of Change of Homer Simpson

Homer Simpson lies directly in the path of the flame-spewing juggernaut, with only the meager acceleration of the family station wagon standing between him and utter destruction. Assume Homer's velocity (in feet per second) is given by the equation: V(t)=t^3-4t^2-t-x+1 , where t is measured in seconds and . Answer the f

Marginal Revenue and Maximizing Profit

Please choose the correct answer: 10. Acme estimates marginal revenue on a product to be 200q^-1/3 dollars per unit when the level of production is q units. The corresponding marginal cost is 2q dollars per unit. Suppose the profit is $250 when the level of production is 1 unit. What is Acme's profit when 8 units are produced

Important Information About Calculus Questions

Please choose the correct answer and show the process: 1. Find the equation of the tangent line to y = 2 ln x at the point where x = 8. y = x/2 - 1 + ln 2 y = x/2 - 1 + 2 ln 2 y = x/4 - 1 + ln 2 y = x/4 - 2 + 6 ln 2 y = x/8 - 1 + ln 2 y = x/8 - 1 + 2 ln 2 y = x/8 - 1 +

Calculus

Introduce slack variables as necessary, and then write the initial simplex tableau for each linear programming problem. 1). Find x1 &#8805; 0 and x2 &#8805; 0 such that X1 + x2 &#8804; 10 5x1 + 3x2 &#8804; 75 and z = 4x1 + 2x2 is maximized 2. Production -Knives The Cut-Right Company sells set of kitchens knives. Th

Differential Equations and Determinants

Question 1: ----------- Quoting from the book: ----------------------------------------------------------------- Example 2. Form a differential equation by eliminating the constants c1 and c2 from the equation x 2x y = c1*e + c2*e Since there are two constants to eliminate, three equatio

Linear Cost Function

1. Write a linear cost function for each situation. Identify all variable used. A parking garage charges 50 cents plus 35 cent per half-hour 2. Find the cost function in each case. Marginal cost: $90; 150 items cost $16,000 to produce. 3. Supply and Demand: Let the supply and demand functions for butter pecan ice cream b

Elementary Differential Equations : Power Series Methods

Attached are two problems, one with an answer that I don't understand how it was derived and one problem without the answer that I would like to see how it is solved. Power Series Methods - Introduction and Review of Power Series 14. Find two linearly independent power series solutions of the given differential equation.