Ingredients of Multivariable Change: Models, Graphs, Rates

Project 9.1: Competitive and Complementary Products



Competitive and complementary products play an important role in today's business markets. Two commodities are called competitive if an increase in demand for one results in a decrease in demand for the other. Two different brands of the same item are competitive. For example, Pepsi and Coke are competitive. In this case, when the demand for Pepsi increases, the demand for Coke decreases, and when the demand for Coke increases, the demand for Pepsi decreases. It is important to note that the driving force in the change in the demands for competitive products is often the increase (or decrease) in the prices of the products. That is, when the price of Pepsi increases, the demand for Pepsi will decrease and the demand for Coke will increase. Likewise, when the price of Coke increases, the demand for Coke will decrease and the demand for Pepsi will increase.

Sometimes products are used together because they complement each other (coffee and cream, computers and floppy disks, VCRs and tapes).Two products are called complementary when each loses sales if one raises prices.


1. Let f(x, y) be the demand function for a certain automobile sold for x dollars by Ford, and let h(x, y) be the demand function for a similar automobile sold for y dollars by Honda. If the total number of these two types of automobiles remains constant, explain why the automobiles are competitive if


2. Give four examples of commodities you believe would be competitive in today's market.

3. Let D (a, b) be the demand function for product A which sells for a dollars, and let F(a, b) be the demand function for product B which sells for b dollars. If products A and B are complementary, restate the definition of complementary products in terms of their demand functions using partial derivative notation.

4. Give four examples of products you believe would be complementary in today's market.


Prepare a written report containing detailed answers to the preceding tasks.


Project 9.2: Expert Witness



Investigators at accident scenes often wish to determine the speed at which a vehicle was traveling before the driver applied the brakes. Normally, all an investigator has to go on is the length of the skid marks and the weight of the vehicle that caused the skid marks. The investigator for an involuntary manslaughter lawsuit has called you as an expert witness to testify regarding the probable speed of the vehicle at the time of the accident.


1. Under normal driving conditions, if a person driving a vehicle on an asphalt road slams on the brakes and skids to a stop, then the length of the skid marks can be modeled by S(v, w) 5 0.000013wv2 feet, where the vehicle was traveling v miles per hour when the driver slammed on the brakes. The vehicle weighs w pounds.

Suppose the investigator measures the skid marks and weighs the vehicle that made them. He finds that the skid marks are 112 feet long and the vehicle weighs 3145 pounds. Determine the velocity at which the vehicle must have been traveling if it was empty.

2. It is possible that when it was weighed, the vehicle did not contain everything that was in it when the skid marks were created. For instance, it is extremely unlikely that the driver and passengers were still in the vehicle. Therefore, the investigator's ability to predict accurately the velocity of the vehicle is affected by the rate of change of velocity with respect to weight. Find a formula for the rate of change of the velocity of a vehicle with respect to its weight. How quickly will the velocity of the 3145-pound vehicle that made skid marks 112 feet long be changing when its weight increases?

3. The investigator knows that the vehicle at the time of the accident contained one male driver 6 feet 1 inch tall and of average build, one female passenger 5 feet 5 inches tall and of average build, one 2-year-old child described as "chubby," and a child safety seat. Discuss whether the weight of the driver and passengers would significantly affect the calculation of the velocity of the vehicle in the case under investigation. Note that the accident occurred in a 50-mph zone.


1. Prepare a document summarizing your analysis of the vehicle's velocity to be admitted as court evidence. This report should be technical in nature with your conclusions clearly stated.

2. Prepare a witness statement to be given under oath. You will be cross-examined by the defense attorney and should be prepared to justify your assumptions and conclusions.


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