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Calculus and Analysis

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→ 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) → (0,0), (x,y) element of A]. b)Using L from part (a) prove that F(x,y) = { f(x,y) when (x,y) ≠ (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.

Systems of Differential Equations

From First Order Systems--Modeling via Systems. Please explain each step of your solution and double check your final answers. Thank you.

Vectors, Gradients and Rate of Change

8. The Intergalactic Ship Zora is hanging motionless at (5, 1, 10) (Universal Galatic Coordinates) when the crew spots an interesting object at (7, 5, 6). The temperature in that part of the galaxy is given by T(x,y,z) = x2 + y/x + z3. As the crew starts to move the Zora directly toward the unknown object, what is the rate of ch

Limits and Continuity

What is the domain of the function f(x) = square root of x-1/square root of x2? Is this function continuous everywhere where it is defined? Explain.

Limits and Continuity

Sketch by hand the graph of a function f that satisfies (a) f(2)=3 and (B) lim as x approaches 2 of f(x)=4. Is the function f(x) continuous at x=2? explain.

Secant Lines

Let f(x) = cos(x). Let p=(pi,-1) be a point on the graph y = f(x). a) calculate the slope of the secant line passing through P and F(pi+0.2), (pi+0.2)) b) calculate the slope of the secant passing through P and (pi-0.1),f(Pi-0.1)) what would your guess on the value of slopes of secant lines passing through P and other p

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

Calculus Problem

Let f(x)=3+x^2+tan(pie*x/2), where -1<x<1. a. find f-1(inverse) of (3) b. find f(f-1(5))