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# Calculus and Analysis

### Probability

Find the indicated probability. 1) Two fair dice are rolled. What is the probability that a sum of 6 or 11 is obtained? 1) _______ A) 7/6 B) 17/36 C) 1/66 D) 7/36 Find the simple interest. Assume a 360-day year. Round results to the nearest cent. 2) \$ 49,417 at 7.7% for 17 months 2) _______ A) \$

### To establish non-existence of multivariable limit

We examine the above function and consider its limit as (x,y)-> (0,0).We take two different paths in the x-y plane for approaching the point (0,0),and find that f(x,y) approaches two different values .This enables us to conclude that the given limit does not exist.For a detailed discussion see the solution given.

### Calculus - Derivatives and Numerical Values

Need help with calculus homework problems. In Exercises 41-48, find dy/dx 44. 5x 4/5 + 10y 6/5 = 15 ____________________________________________________________________ 54. a. By differentiating x 2 - y2 =1 implicitly, show that dy/dx = x/y b. Then show that d2y/dx2 = - 1/y3. ____________________

### Accounting : Profit Margins, Turnover and Return on Investment

The Valve Division of Bendix, Inc, produces a small valve that is used by various companies as a component part in their production. Bendix, Inc., operates its divisions as autonomous units, giving its divisional managers great descretion in pricing and other decisions. Each division is expected to generate a mimimum required

### An application of the Laplace transform

Define the unit ramp function by ... 1. Determine the Laplace transform of H(t) 2. Use the Laplace transform to solve the ODE ..... Questions: 1. To determine the Laplace transform, do we use ..... &#8747; (note the difference in integration limits). I know both integrals will give the same answer. But I am confused bec

### Tangent line problem

Explore the following function. You are to decide which of threee lines given are tangent to the graph of f(x) at given points. The function to define is F(x)=2x^3-4x^2+3x-5 The possible tangent lines are: y= x-5 y= 2x-5 y= 3x-5 a) what is/are the zero(es) for this function? In other words, what is the solu

### Solutions to Legendre's Differential Equation

The attached file has all the information about this problem. On this problem, we know from a previous problem that the Legendre polynomials satisfy the DE. It is a second order DE. Usually these have two linearly independent solutions. Are these the only polynomials that satisfy the DE, or is there another set, linearly ind

### Rate of change

See attachment Section 11.3 1) The quarterly profit of Cunningham Realty depends on the amount of money x spent on advertising/quarter according to the rule: where and x are measured in thousands of dollars. What is Cunningham's profit when its quarterly advertising budget is \$35,000.00? 2) Suppose

### Inverse Laplace Transformation and Partial Fraction Expansion

Please help me with these problems. section 7.4 6,16,20,24 Samples of these questions appear below. Please see the attached files for the fully formatted problems. Find the inverse Laplace transformation. Determine Partial Fraction Expansions for the given rational function. Determine L^-1{F}.

### Laplace transform

Hi, Please help on these problems Please show all steps Section 7.3 # 4,8,14,20 See attached determine the Laplace transform of the given function.

Please help with the following questions regarding calculus and analysis. Revenue: Two models, R1 and R2, are given for the revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2004-2008, with t = 4 corresponding to the year 2004. Which model is projecting the

### (1) Find a general solution to the following non homogeneous differential equation: y'' + 5y' + 6y = 6x^2 + 10x + 2 + 12 e^x; yp(x) = e^x + x^2 (2) Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equations (a) y'' - y' + y = (e^t + t)^2 (b) y'' + y' + y = e^t + 7 (c) y'' - 2y' + 3y = cosh t (3) Find general solution to the following differential equations: y'' + 4y = sin q - cos q y'' + y' - 12 y = e^t + e2t - 1 y'' - 4y' + 4y = t^2 e^t - e^(2t) (4) Find a particular solution to the given higher order differential equation: y^(4) - 5y'' + 4y = 10 cos t - 20 sin t

Please help working on these attached problems section 4.5 #6,10,12,14,20,28,36,38 Thanks (6) Find a general solution to the following non homogeneous differential equation: y'' + 5y' + 6y = 6x^2 + 10x + 2 + 12 e^x; yp(x) = e^x + x^2 Decide whether the method of undetermined coefficients together with superposition

### Finding Extreme Values for Functions of Two Variables

1.)f(x,y)= 2x^2 + xy^2 + 5x^2 + y^2 2.)f(x,y)= ysquareroot x - y^2 - x + 6y.

### 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

### Calculus Problem Set - Let p(x) = x3 -ax, where a is constant a) If a<0, show that p(x) is always increasing. ubstituting in values ... [See the attached Question File.]

#5 Let p(x) = x3 -ax, where a is constant a) If a<0, show that p(x) is always increasing. substituting in values b) If a>0, show that p(x) has a local maximum and a local minimum. c) Sketch and label typical graphs for the cases when a<0 and when a>0. Find the value(s) of x for which: a) f(x) has a local maximum

### 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

### 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

### 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

### 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

### Projectile motion and differential equations

Please follow all instructions and answer accordingly. See the attached file. Consider a projectile of mass m

### 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

### Find a Cartesian equation for the curve and identify it.

r = 10sin(theta) + 10cos(theta) which one is it ? circle ellipse parabola line

### 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

### A solution of the 2d Laplace equation

Solve Laplace of u = 0 subject to the conditions: u(x,0) = f1(x) u(0,y) = 0 u(x,b) = 0 u(a,y) = 0 0<x<a 0<y<b (The question attachment contains a slightly different question. The question is restated correctly in the solution attachment)

### Upper and lower control limits for /x- and R-charts

Find the upper and lower control limits for /x- and R-charts for the width of a chair seat when the sample grand mean (based on 30 samples of 6 observations each) is 27.0 inches, and the average range is 0.35 inches.