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# Calculus and Analysis

### 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.

### Use one-sided limits to find the limit or determine that the limit does not exist.

Use one-sided limits to find the limit or determine that the limit does not exist: limx => 4 16-x^2 / 24-x The problem might be easier to read like this: limit 16 - x^2 x --> 4 24 - x

### Graphing and Critical Points

3. Use calculus to sketch the graph of the function . Your sketch should address: a. An analysis of : it's domain, intercepts, end behavior, and any asymtotes. b. An analysis of : the critical points, the extrema and classification, and each region of increase or decrease. c. An analysis of : the points of inflection (

### 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

### Modeling using Differential Equations

One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor: a) Write a differential equation that is satisfied by y ( use k for the constant of proportionality) dy/dt = b) Solve t

### 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 &#8804; x &#8804; 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, initial value problems, limits

See the attachment. Solve the initial value problem.... Evaluate by interpreting the limit of Riemann sums... Find the integral....

### Euler's method problem

The function of y(x) satisfies the differential equation and the initial condition y(1)=1. Firstly solve the equation to get an exact value then use Euler's method to obtain the value of y(2). Compare this value with the analytical value and discuss how the approximate value obtained by Euler's method may be improved. P

### A ball thrown vertically upward at time t = 0 (s) with initial velocity 80 ft/s and with initial height 96 ft has height function

A ball thrown vertically upward at time t = 0 (s) with initial velocity 80 ft/s and with initial height 96 ft has height function: a) What is the maximum height attained by the ball? b) When and with what impact speed does the ball hit the ground? y(t)= -16t^2+ 80t+ 96 Find the interval on which the function is

### 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

### Differential Equations: Rate of Change Word Problem

Set up but do not solve a differential equation that models the amount of salt in the tank for the following: A tank, having a capacity of 700 liters, initially contains 3 kilograms of salt dissolved in 100 liters of water. At time t=0, a solution containing 0.4 kilograms of salt per liter flows into the tank at a rate of 3 lite

### 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

### Transfer Functions, Inverse Laplace Transforms and State-Space Representations

Consider the system (see attached file). Determine the transfer function of the system in terms of Laplace transform. Expand the transfer function by means of partial fractions and find your new state equations by means of inverse Laplace transform. Obtain the state-space representation of the system.

### Differential Equations : Harmonic Oscillator with Damping Coefficient

22. Consider a harmonic oscillator with mass m = 1, spring constant k = 1, and damping coefficient b = 4. For the initial position y(O) = 2, find the initial velocity for which y(t) > 0 for all t and y(t) reaches 0.1 most quickly. [Hint: It helps to look at the phase plane first.] Differential Equations From Phase plane for

### Instantaneous Rate of Change

5. An object is moving along the straight line as follows. It starts at and then it moves to the right to . Then the objects moves to the left to , and finally to the right to stop at . Sketch a possible graph of the position function . Sketch a possible graph of the velocity (the instantaneous rate of change of ).

### Center of Mass of a 4D Pyramid Using Calculus

Find the center of mass of a 4D pyramid (cube base) using calculus. Explanations, step-by-step procedure, diagram if possible.