Mathematical Analysis II- Group Project

Transcription

1 Mathematical Analysis II- Group Project Student #1: Last Name First Name Student #2: Last Name First Name Functions used for the project: Price Function: Problem 1 Cost Function: Revenue Function: Problem 2 Cost Function: Prepared by Ioannis Souldatos in February 2017 for MTH Page 1

2 General Instructions Every student will be given a unique set of numbers and functions. These numbers and functions are posted on Webwork under Functions for Project. You have to answer the following questions using your own set of numbers and functions. Provide your answers on a piece of paper following the order of the questions below and showing all your work. For all that follows x is the number of units produced/sold. Unless you are instructed otherwise, you can assume that x is in the interval [0, ). If you are asked to provide the graph of a function, you will have to find the graph of this function online, print it and provide it alongside your other work. There are many websites you can use for this purpose. For instance, is one of them that allows you to save the graph as an image. For every graph you provide make sure that the functions are displayed on the appropriate domain. Problem 1 Throughout Problem 1 use the same functions p(x), C(x) which are provided on Webwork Functions for Project Problem 1. The function p(x) is the price function of a certain product and the function C(x) is the cost function of the same product. Price-Demand and Elasticity 1. What is the input and output of p(x)? 2. Is p(x) increasing for all x, decreasing for all x, or neither of the two? 3. Find the inverse function of p(x) and its domain. What is the input and the output of the inverse function? The inverse function is called the demand function. 4. Is the demand function increasing for all x, decreasing for all x, or neither of the two? 5. Provide a graph of the price function and a graph of the demand function (two different graphs). 6. Find the price elasticity of demand function η(x) (read eta of x ) and its domain. 7. Explain the meaning of η(x). 8. Find when the demand is elastic, inelastic and when it has unit elasticity. Prepared by Ioannis Souldatos in February 2017 for MTH Page 2

3 9. Provide a graph that contains both η(x) and the constant function 1. Explain how we can use this graph to identify when the demand is elastic, inelastic and when it has unit elasticity. Cost and Average Cost Recall that C(x) is the cost function of a certain product. 1. Find the fixed cost. 2. Explain the meaning of the fixed cost. 3. Is C(x) increasing for all x, decreasing for all x, or neither of the two? 4. Find the average cost function C(x). 5. What is the meaning of C(x)? 6. Is C(x) increasing for all x, decreasing for all x, or neither of the two? 7. Find the derivative of C(x). 8. Find the critical points of C(x) and the intervals of increase/decrease. Identify any local maximum or minimum values (if any). 9. Find the value x ac that will minimize the average cost? What is the minimum average cost? 10. Provide a graph that contains both the cost function and the average cost function. Identify the fixed cost on this graph. 11. Is it true that C(x) always has a vertical asymptote at x = 0? If so, what is the reason? Profit and Revenue Recall that you are given two functions p(x), C(x). The function p(x) is the price function of a certain product and the function C(x) is the cost function of the same product. 1. Find the revenue function R(x) and the profit function P (x), and their domains. All these should be functions of x. 2. Find the cost function C(p), the revenue function R(p) and the profit function P (p); this time all functions must be functions of p, not x. 3. What is the difference between R(x) and R(p)? 4. Find the value of R(x) when x = 0. Is it true that R(x) always takes the same value at x = 0? What is the reason? 5. Find the value of P (x) when x = 0. What is significant about this value? Prepared by Ioannis Souldatos in February 2017 for MTH Page 3

4 6. Provide a graph of R(x) and identify on the graph when the demand is elastic and when it is inelastic. Does your answer here agree with your previous answer on elasticity? If not, why not? 7. Find the break-even point and explain its meaning. 8. Provide a graph the profit function P (x) and identify the break-even point and the fixed cost on the graph. 9. Provide a graph the profit function P (p) (this is different than before; the independent variable is price p now versus x before). Explain the meaning of the p-intercept(s) of P (p). Problem 2 Throughout Problem 2 use the same functions R(x), C(x), P (x) which are provided on Webwork Functions for Project Problem 2. The function R(x) is the revenue function, the function C(x) is the cost function and the function P (x) is the profit function. All functions refer to the same product. These functions maybe different than the functions of Problem 1 and should not be confused with them. Marginal Cost, Marginal Revenue and Marginal Profit 1. Find the price function p(x) and its domain. 2. Find the marginal revenue, the marginal profit and the marginal cost. How do these three functions relate to each other? 3. Find the critical points of the cost, the revenue and the profit function (if any). 4. Find the intervals of increase/decrease for the revenue function. Identify any local maximum or minimum values. 5. Which of the following do you expect to be true for the revenue function in general? (a) The revenue function has a maximum value. (b) The revenue function has a minimum value. (c) The revenue function increases to infinity, as x increases to infinity. (d) The revenue function decreases to -infinity, as x increases to infinity. Explain your answer. 6. Find the intervals of increase/decrease for the profit function. Identify any local maximum or minimum values. Prepared by Ioannis Souldatos in February 2017 for MTH Page 4

5 7. Which of the following do you expect to be true for the profit function in general? (a) The profit function has a maximum value. (b) The profit function has a minimum value. (c) The profit function increases to infinity, as x increases to infinity. (d) The profit function decreases to -infinity, as x increases to infinity. Explain your answer. 8. If applicable, identify the x-value x r that will maximize the revenue? What is the maximum revenue in this case? What is the price p r that will result in maximum revenue? 9. If applicable, identify the x-value x p that will maximize the profit? What is the maximum profit in this case? What is the price p p that will result in maximum profit? 10. Are the values p r, p p equal? Do you expect p r and p p to be equal in all cases? Explain. 11. Find the equation of the tangent line to R(x) at the point x = x p (be careful; R(x) is the revenue function, but x p is the point that maximizes profit, not revenue). What is the slope? 12. Find the equation of the tangent line to C(x) at the point x = x p (that s the same point as in the previous question). What is the slope? 13. Compare your answers in the previous two questions. What do you observe? Is this always true? If so, what is the reason? 14. Provide a graph that contains both R(x) and C(x) as well as the tangent lines to both functions at the point x = x p. What do you observe? 15. Provide a graph that contains all three functions: marginal cost, marginal revenue and marginal profit. Identify the points x = x r and x = x p on this graph. Find the point where the graph of the marginal cost intersects the graph of the marginal revenue. What do you observe? Prepared by Ioannis Souldatos in February 2017 for MTH Page 5