Chapter 3 Project

Demand Curves and Revenue

The graph of the relationship between the price p of a given product and the quantity x of that product sold is called the demand curve for the product. It typically reflects the fact that an increase in price results in lower sales and a decrease in price results in higher sales. The revenue realized by selling quantity x of the product at price p is then given by the function $R\left(x\right)=xp$.

1. One common relationship between p and x is an equation of the form

   $p+bx=a$,

   where a and b are positive constants.

   1. With the horizontal axis representing x and the vertical axis representing p, sketch the general shape of the demand curve associated with this relationship. (Hint: It may help to initially pick specific values for a and b in order to identify the behavior.)

   2. In words, how would you describe the dependence of p on x?

   3. What value of x corresponds to the largest feasible value for p?

   4. What value of p corresponds to the largest feasible value for x?

   5. Find a formula for the revenue function $R\left(x\right)$ as a function of x alone

   6. What class of function is $R\left(x\right)$?

   7. How would you describe the graph of $R\left(x\right)$?

2. Another common relationship between p and x is an equation of the form

   $px=a$,

   where again a is a positive constant.

   1. With the horizontal axis representing x and the vertical axis representing p, sketch the general shape of the demand curve associated with this relationship. (It may again help to initially pick a specific value for a in order to identify the behavior.)

   2. What is the smallest feasible value for x in this relationship?

   3. What value of p corresponds to the smallest feasible value for x?

   4. Is there a smallest feasible value for p in this relationship?

   5. Find a formula for the revenue function $R\left(x\right)$.

   6. What class of function is $R\left(x\right)$?

   7. How would you describe the graph of $R\left(x\right)$?