"Pricing is both an art and a science." Suppose you're relatively new in business and want to launch your product in the market. You have a great deal of flexibility in how you set your prices, you may want to consider pricing for optimum market penetration. This means that you initially sell your product at a low introductory price P. (say) to attract new customers, then raise prices once you've secured your share in the market. Determining the most appropriate pricing model for your business is tricky and takes considerable research. If it is known that change in price P depends upon the demand D and Supply S of your product, where both D and S are linearly related to price P. (a) Write the differential equation the describes the change in price, (b) Describe the pattern of change of price for different phases. (e) For what values of parameter, you have equilibrium price (Hint: Recall equilibrium solution of differential equations) (d) Whether your equilibrium price represents the stability of the market of your product. (e) By taking different values of your parameters. Use MATLAB to draw various graphs on single plot. (f) Discuss in detail whether your plot in part (e) supports your discussions done in (b)-(e). (g) Discuss whether your proposed model is ideal for all market situations, or there are some factors that are not considered by you. If so, then give a brief description on what would be their effect on the model. (h) Summarize your work in such a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, marketing strategies, a short description of your methodology, concise analysis, and your main conclusions. Assume the reader is familiar with the basics of calculus and differential equations, so there is no need to walk through every step of your solution process or include equations. However, you should still describe the processes and mathematical techniques you used to reach your conclusions and explain why you used them. Project II Suppose you want to model the retirement plan of a Bank Manager Mr. XYZ, using differential equations, who has Pin a retirement account earning an interest of r% compounded continuously, (A) Write the initial value problem, (b) If Mr. XYZ. plans to retire in a years, how much amount he can expect (c) If he still plans to make contributions to his retirement fund during his next years of employment. His annual contribution will be thousands, which his employer will generously match. If we assume that these contributions will spread out evenly over the course of the year, How you will incorporate this information into your original initial value problem? (d) After a years, when he will need to begin withdrawing money from his account. He estimates that he will need to withdraw thousands a year for living expenses. Of course, whatever remains in his account at any given time will still collect interest. Modify his retirement model for the years, after t = (c) Use MATLAB to plot the solutions of previous parts. By taking the fixed values of parameters in the way if last four digits of roll of one of your group members are alxd then r =% (choose such roll number .t c0) a = ab years and be thousands (1) Explain your graphs obtained in part (e) supports your discussion (a)-(e). "Pricing is both an art and a science." Suppose you're relatively new in business and want to launch your product in the market. You have a great deal of flexibility in how you set your prices, you may want to consider pricing for optimum market penetration. This means that you initially sell your product at a low introductory price P. (say) to attract new customers, then raise prices once you've secured your share in the market. Determining the most appropriate pricing model for your business is tricky and takes considerable research. If it is known that change in price P depends upon the demand D and Supply S of your product, where both D and S are linearly related to price P. (a) Write the differential equation the describes the change in price, (b) Describe the pattern of change of price for different phases. (e) For what values of parameter, you have equilibrium price (Hint: Recall equilibrium solution of differential equations) (d) Whether your equilibrium price represents the stability of the market of your product. (e) By taking different values of your parameters. Use MATLAB to draw various graphs on single plot. (f) Discuss in detail whether your plot in part (e) supports your discussions done in (b)-(e). (g) Discuss whether your proposed model is ideal for all market situations, or there are some factors that are not considered by you. If so, then give a brief description on what would be their effect on the model. (h) Summarize your work in such a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, marketing strategies, a short description of your methodology, concise analysis, and your main conclusions. Assume the reader is familiar with the basics of calculus and differential equations, so there is no need to walk through every step of your solution process or include equations. However, you should still describe the processes and mathematical techniques you used to reach your conclusions and explain why you used them. Project II Suppose you want to model the retirement plan of a Bank Manager Mr. XYZ, using differential equations, who has Pin a retirement account earning an interest of r% compounded continuously, (A) Write the initial value problem, (b) If Mr. XYZ. plans to retire in a years, how much amount he can expect (c) If he still plans to make contributions to his retirement fund during his next years of employment. His annual contribution will be thousands, which his employer will generously match. If we assume that these contributions will spread out evenly over the course of the year, How you will incorporate this information into your original initial value problem? (d) After a years, when he will need to begin withdrawing money from his account. He estimates that he will need to withdraw thousands a year for living expenses. Of course, whatever remains in his account at any given time will still collect interest. Modify his retirement model for the years, after t = (c) Use MATLAB to plot the solutions of previous parts. By taking the fixed values of parameters in the way if last four digits of roll of one of your group members are alxd then r =% (choose such roll number .t c0) a = ab years and be thousands (1) Explain your graphs obtained in part (e) supports your discussion (a)-(e)