Chapter 2.1, Problem 96E

A company sells custom embroidered apparel and promotional products. Table 1 shows the volume discounts offered by the company, where x is the volume of a purchase in dollars. Problems 95 and 96 deal with two different interpretations of this discount method.

Table 1 Volume Discount (Excluding Tax)

Volume ($x)	Discount Amount
$300 ≤ x < $1,000	3%
$1,000 ≤ x < $3,000	5%
$3,000 ≤ x < $5,000	1%
$5,000 ≤ x	10%

96. Volume discount. Assume that the volume discounts in Table 1 apply only to that portion of the volume in each interval. That is, the discounted price for a $4,000 purchase would be computed as follows:

  $300+0.97(700)+0.95(2,000)+0.93(1,000)=3,809$

1. (A)   If x is the volume of a purchase before the discount is applied, then write a piecewise definition for the discounted price P(x) of this purchase.
2. (B)   Use one-sided limits to investigate the limit of P(x) as x approaches $1,000. As x approaches $3,000.
3. (C)   Compare this discount method with the one in Problem 95. Does one always produce a lower price than the other? Discuss.