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# Calculus and Analysis

### 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.

### Limits and Piecewise Functions

Please help with the following problems regarding limits and piecewise functions. 1) Let f(x) {o if x is a natural odd number {1 otherwise Does f(x) have a limit as x approaches infinity? Explain you answer. 2) Let f(x) {1 if x is a natural odd number { 1-1/x

### Differential Equations: LRE and CRE Circuits

A 20 ohm resistor, a .05 farad capacitor, and an alternating power source equal to 40 cos(t) are placed in series. If the initial charge on the capacitor is 3 coulombs, find a general formula for the charge at any time t.

### Differential Equations: Electromotive Force and Maximum Current

An electromotive force of 100 volts is in series with a 2 henry inductor and a 50m ohm resistor. a. Determine the current i at any time t after the switch is closed b. Determine the maximum current that can be obtained in the simple circuit described. Please see the attached file for the fully formatted problems.

### 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 and Rate of Change: Example Problems

A tank contains 54 gallons of pure water. A salt water solution with 2 pounds of salt per gallong enters the tank at a rate of 3 gallons per minute, and the well-stirred mixture leaves the tank at the same rate (3 gallons per minute). a. How much salt is in the tank at any time t? b. When, to the nearest minute, will the wat

### Applying Differential Equations and Logistic Equations

The number of bacteria in a Petri dish was initially determined to be 200. After one hour, the number had increased to 500 and after another hour to 1,000. Assume that the rate of bacterial growth in the dish at any time t can be calculated using the logistic equation dB/dt = B(a-bB), which I know to use the formula B = aBo/(bBo

### Nonlinear Differential Equations: Chemical Reactions Problem

Question: A Chemical X is produced from a reaction involving chemicals A and B. The rate of production of X varies directly as the product of the instantaneous amounts of A and B present. The formation of X requires 6 pounds of A for every 4 pounds of B. 30 pounds of A and 20 pounds of B are present initially and 10 pounds of

### 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 +

### Differential Equations : Damped Springs

The differential equation y" = -ky - cy' is used to model a motion of a mass on a spring with damping, where k is the spring constant and c is the damping coefficient. a) Show the function y = e^-t cos 3t satisfies the differential equation y" = -10y - 2y'. b) Show the graph of y(t) (^ means exponent)

### 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.

### Solving Differential Equations with Substitution and Bernoulli

56. Suppose that n does not equal to zero and n does not equal to one. Show that the substitution v = y1-n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation dv/dx + (1-n)P(x)v(x) = (1-n)Q(x). 63. The equation dy/dx = A(x)y2 + B(x)y + C(x) is called a Riccati equation. Suppose that one parti

### Solving Differential Equations

57. Show that the substitution v = lny transforms the differential equation dy/dx + P(x)y = Q(x)(ylny) into the linear equation dv/dx + P(x) = Q(x)v(x) 58. Use the idea in Problem 57 to solve the equation x (dy/dx) - 4x2y + 2ylny = 0 59. Solve the differential equation dy/dx = (x-y-1)/(x+y+3) by finding h and k so t

### Elementary Differential Equations : Substitution Methods and Exact Equations

Substitution Methods and Exact Equations Homogeneous Equations: Dy/dx = F(y/x) v = y/x, y = vx, dy/dx = v + x(dv/dx) x(dv/dx) = F(v) - v Bernoulli Equations: dv/dx + (1-n) P(x)v = (1-n) Q(x) Criterion for Exactness: F(x,y) = &#8747; M(x, y) dx + g(y) Verify that the given differential equation is exa

### Differential Equations : Spring with Damping Force

A body that weighs 16 lb. is attached to the end of a spring which is stretched 2 ft. by a force of 100 lb. It is set in motion from a position of ½ foot from the equilibrium position of the spring with an initial velocity of -10 ft/sec. Assume the motion of this body is subject to a damping force that provides 6 lb of resistan

### Differential Equations : Masses and Springs

A body that weighs 16 lb. is attached to the end of a spring which is stretched 2 ft. by a force of 100 lb. It is set in motion from a position ½ foot from the equilibrium position of the spring with an initial velocity of -10 ft/sec. (a) Find the differential equation that governs the position of the body over time relative

### Managerial Accounting : Break-Even Point, Internal Rate of Return, Net Present Value and Capital Budgeting

Please see the attached file for the fully formatted problems. 42. A company paid off a \$10,000 long-term note by issuing common stock to the creditor. This transaction would be reflected on the company's statement of cash flows as: a. an addition of \$10,000 and a deduction of \$10,000 under investing activities. b. an addit

### Business Calculus : Time Value of Money

There are 3 sheets in the file with approx 21 questions. Most of them are mathematical, but there are also some that are subjective type questions. I have also attached the tutorials with the problem sets in the same file. In the problem sets on sheet one, I will answer number 9. Chapter 3: Project 3 Mortgages: Princi

### Business Calculus : Maximizing Profits, Cost and Price-Demand Functions

A company manufactures and sells x air-conditioners per month. The monthly cost and price-demand equations are C(x) = 180x + 20,0(X) p = 220 - 0.001x 0 =< x <=100,000 (A) How many air-conditioners should the company manufacture each month to maximize its monthly profit? What is the maximum monthly profit, and what should

### Differential Equations : Steady State Conditions and Heat Transfer Functions

Problem 1 Consider the following transfer function .... (a) What is the steady state gain and time constant? (b) If U(s) =2/s, what is the value of the output when t&#8594;&#8734; (c) For the same U(s) what is the value of the output when t = 10? (d) If U(s) is a unit rectangular pulse what is the output when t&#8594;&#873

### Laplace Equation in a Unit Disk

Find the solution of the Laplace equation: (See attachment) in the unit disk given by Fourier method (separation of variables). Then compare your answer with the one given by the Poisson Integral Formula to compute the definite integral. (see attachment)

### Relative maxima, relative minima, and saddle points

Find the relative maxima, relative minima, and saddle points of the function (x^2)*y - 6*(y^2) - 3*(x^2).

### Calculus Questions - Automotive Examples

See the attached file for full description. 40) The value of a certain automobile purchased in 1997 can be approximated by the function v(t)=25(0.85)^t , where t is the time in years, from the date of purchase, and v is the value, in thousands of dollars. (a) Evaluate and interpret v(4). (b) Find an expression for v1(t) inclu

### Find the mass M (in grams) of a rod coinciding with the interval [0,4] which has the density function p(x)=5sin(pi/4)x

Find the mass M (in grams) of a rod coinciding with the interval [0,4] which has the density function p(x)=5sin(pi/4)x See attached file for full problem description.