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Calculus and Analysis

Properties of the equilibrium price for a demand function

The demand function for a certain good is D(p,m) and the supply function is S(p). For a given m, the equilibrium price p* is given by p* = f(m). a) Show that f'(m) = [dD(p*,m)/dm] / [S'(p*) - dD(p*,m)/dp*] b) Verify this result when S(p) = 2p and D(p,m) = 6m^2p^(-1) + m c) By using the implicit function rule, or oth

Production Function, returns to scale, Taylor expansion, Euler's theorem

Consider a production function of the form F(K,L)=(K^(-a)+L^(-a))^(-1/a). (a) Is this function homogeneous? (b) Does it display increasing, constant or decreasing returns to scale? (c) Let G be a differentiable function. Find an expression for G(K+g,L+h) by taking a first-order Taylor expansion of G about (K,L). (d)

Step by Step Explanation of Differential Equations

I am asking for the step-by-step workings for all of the attached problems. ** Please see the attached file for complete problem description ** 1st problems. Please find the general solution of: (1) dy/dx = y/sin(y) - x (2) dy/dx = y + cos(x)y^2010 In the process of finding the solutions for the problems make use of both

Concentration and Differential Equations

A polluted river with a nutrient concentration of 110g/m^3 is flowing at a rate of 130m^3/day into an estuary of volume 1000m^3. At the same time, water from the estruary is flowing into the ocean at 120m^3/day. The initial nutrient concentration in the estuary is 35g/m^3. (i) Let N(t) be the amount of nutrient (in grams) in

What is the value of given endowment in today's dollars?

A wealthy philanthropist has established the following endowment for a hospital. The details are as follows: a cash deposit of $ 8 M one year from now; an annual cash deposit of $3M per year for the next five years. The first $3M will start today; at the end of 5 years, the hospital will also receive a lump sum payment of $18M.

Business Calculus and Average Cost Function

1. Average Cost. A company manufacturing snowboards has fixed cost of $200 per day and total cost of $3800 per day at a daily outputs of 20 boards. a) Assuming that the total cost per day c(x) is linearly related to the total output per day x, write an equation for the cost function. b) The average Cost per board for an out

find out the stationary point, saddle point, min and max

1) Let f(x,y)=25e^(-1/5x^2)-y^5+5y+3 a) Find all stationary points of the function f(x,y) and enter their coordinates by "" with at least 3 decimal places. b)Let (xs,ys) be the saddle point of the function f(x,y). Calculate the following expression: f(xs,ys)-(xs+ys) and enter the value with at least 3 dp. c) what is(are) t

Concentration of a Certain Drug

The following figure shows the devastating effect the opening of a new discount department store had on an established department store in a small town. The revenue of the discount store at time t (in months) is given by f(t) million dollars, whereas the revenue of the established department store at time t is given by g(t) mill

Solutions to First-Order Ordinary Differential Equations

Question 1: (1 points) Solve the following equations for given initial condition. Please represent your answer in the implicit form ( ). Use y(x) instead of y Question 2: (1 points) Integrate the following equations and write a solution for given initial condition. Please represent your answer in the implicit form

Calculus- Revenue/Future Value

Please help with the following problems. Homework Set 17: Problem 4 Section 6.3, Problem 5 Section 6.3, Problem 6 Section 6.3 Section 6.3: Problem 4 pg. 288 13. Harley-Davidson Inc. manufactures motorcycles. During the years following 2003 (the company's 100th anniversary), the company's net revenue came be approxima

Estimate the total solar PV production between 2000 and 2010.

Homework Set 15: Problem 10 Section 5.4, Problem 11 Section 5.4 Section 5.4: Problem 10 pg. 257 10. World annual natural gas consumption, N, in millions of metric tons of oil equivalent, is approximated by N= 1770 + 53t, where t is in years since 1990. A. How much natural gas was consumed in 1990? In 2010? B. Esti

Estimate total oil consumption during this 25-year period

Homework Set 13: Problem 8 Section 5.1, Problem 12 Section 5.1 Section 5.1: Problem 8 pg. 239 8. The following table gives world oil consumption, in billions of barrels per year. Estimate total oil consumption during this 25-year period. Year 1980 1985 1990 1995 2000 2005 Oil (bn barrels/year) 22.3 21.3 23.9 24.9

Particle velocity and harmonic motion

You are in your Physics lab and your professor has an experiment set up to help you learn about harmonic motion. The set up is that you have 112.5/pi^2 (approxmately 11.39 kg) on a frictionless surface that is attached to a spring constant 12.5 N/m. You pull the block to 3/pi meters (approximately .955 meters) such that x(0) = 3

Integral Approximation and Nim Strategy

** Please see the attached file for the complete solution response ** I need a little assistance with a couple of problems. Please include all the work with each problem. Thank you for your assistance. Question 1: Two players take turns removing 1, 2, 3, or 4 objects from a set of 16 identical objects (without replaci

Maximization of Profit for Canoe Rentals

Imagine that you run a consulting business that helps small businesses with process improvement. Justin runs a small canoe-rental business somewhere in Florida and currently charge $10.50 per canoe rental. He has an average of 36 rentals per day. With the rising cost he wants to increase the charge for the rental however he want

Calculus: Retirement Plans

A friend wants to know how much money she will have at retirement if she invests $10,000 compounded continuously in a security that earns interest at ten percent per year. Identify the functional relationship of her balance on-account over time. a) Compute the retirement balance if my friend is currently 50 yrs old and retire

Mr. Mick Mouse has a trap company.

Mr. Mick Mouse has a trap company with fixed costs of $846, variable (i.e. operating) costs of $2 per trap and a demand curve of: traps=116-2(price). Since Mick does not understand economics, find the revenue and cost functions in terms of price for his business. a) Describe the graph of these two functions over the price do

Differential equations

Please see the attached file for full description. 1. Find both the first and second order differentials (y' and y") for the following functions: 2. Use integrating factor to convert the following equation into "exact ODE" form and solve for y. 2xy' = (y -x) (y + x)/ y 3. Solve the differential equation, y is a functi

Solve: Looking at Retirement Balances

A colleague wants to know how much money she will have at retirement if she invests $25,000 in a bank account that yields six percent compounded annually. a) List some possible retirement balances if my friend is currently 50 yrs old. (hint: some folks retire at 60 and some at 62). b) Graph the balance on account against

A simple beginning to differential equations containing full explanations and reasoning along with step-by-step directions for completing an example first order ordinary differential equation.

(e^y + 1)^2 * e^-y dx + (e^x + 1) * e^-x dy = 0 An ordinary differential equation (ODE) is an equation that involves derivatives, but no partial derivatives. The "dx" and "dy" found in the equation denote the derivatives involved. We read these notations as "with respect to x" (dx) and "with respect to y" (dy). Their inclu

Examine the carrying capacity.

The predator prey system below has a term for the carrying capacity of the prey species. Initially the prey population N(0) = 330 and the predator population P(0) = 270. dN/dt = 0.07N(700-N) - 0.05NP dP/dt = 0.04PN - 4P (a) In the absence of predators, what is the carrying capacity of the prey's enviornment? (b) Use a

Initial value problem in Differential Equations

The following problem was given as an example by the professor but I can't seem to come up with the same answer as he did. The answer he got was C1 = -2 and C2 = 1/4 I keep getting C1 = - 1/4 and C2 = - 1/2 There are three possibilities 1) He's wrong 2) I'm wrong or 3) we are both wrong. I need to know which it is. Can you

Dynamic Programming

For a computer to work properly, three subsystems of the computer must all function properly. To increase the reliability of the computer, spare units may be added to each system. It costs $100 to add a spare unit to system 1, $300 to system 2, and $200 to system 3. As a function of the number of added spares (a maximum of two s

Find the Point of Intersection & Radius of Curvature

(a) Find the point of intersection of the line containing the points (3; 2; 1) and (4; 3; 3), and the plane containing the points (5; 4; 2), (3; 1; 6) and (6; 5; 3). (b) Find the radius of curvature at (1; 0; 1) on the three dimensional curve given by r(t) = cos ti + sin tj + e^tk. Sketch this curve, and briefly describe its

Exercises on Functions: Evaluating Linear Functions

1. Evaluate the function f(x) = 4x + 6 for x=4. 2. Evaluate the function f(x) = 9x - 6 for x=0. 3. Take a look at the following table: x -2 -1 0 1 2 f(x) -5 -2 1 4 7 a. Write out an equation for f(x). Assume the function is linear b. What is the slope? Is it negative or positive?