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Calculus and Analysis

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 ∫ 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

Estimating the Area under the Curve

Please see the attached file for the fully formatted problems. 1. Graph . Using the formula for the area of a rectangle to find the function: . What is ? 2. Graph . Using the formula for the area of triangles (or trapezoids) to find the function: . (assume that ) What is ? 3. Based on the graph sketched to

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

Maximum Profit

An apartment complex has 240 units. When the monthly rent for each unit is $360, all units are occupied. Experience indicates that for each $16 per month increase in rent, 3 units will become vacant. Each rented apartment costs the owner of the complex $46 per month to maintain. What monthly rent should be charged to maximize pr

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

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 on the left side of the p

Critical Points

The graph of f(x) = ln(x^2) has a. neither a relative minimum nor a point of inflection at x = 0 b. a relative minimum that is not an inflection point at x = 0 c. a relative maximum that is not an inflection point at x = 0 d. an inflection point that is not a relative minimum at x = 0

Equations of Tangent Lines

What is the equation of the tangent line to f(x) = bar e^(x^2) at x = 2? Please see the attached file for the fully formatted problems.

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

Limits and Uniform Continuous Mappings

Suppose that A = R^2 with {(0,0)} removed and that f :A&#8594; R is a uniform continuous mapping on A. a)Prove that there exists L an element of R so that lim f (x,y) = L [(x,y) &#8594; (0,0), (x,y) element of A]. b)Using L from part (a) prove that F(x,y) = { f(x,y) when (x,y) &#8800; (0,0) and L when (x,y) = (0,0)}

Differential Equations, Chain Rule, and Rate of Change

1. A ladder 10 feet long rests against a vertical wall. If the top of the ladder slides down at a rate of 1 ft/sec how fast is the bottom of the ladder sliding away when the bottom of the ladder is 8 ft away from the wall? 2. Two people start from the same point. One runs west at 13 km/hr and the other walks north at 2 km/hr

Differentials, Derivatives and Differentiability

Text Book: - Advance Calculus Author: - Taylor & Menon In page number 199 : - following questions to be answered : 1, 2, 4 In page Number 206: - Following questions to be answered : 1, 2, 3 & 7 Please mention each and ever step.

Proof Involving Existence of Limit for Piecewise Function

Please see attached file. Let f(x) = x, if x is a rational number, and f(x) = x^2 if x is an irrational number. For what values of a, if any, does lim(f(x)) as x --> a exist? Justify your answer. I know that the answer is 0 and 1, but why? Please explain. Thank you.