The Piecewise Example:

Suppose you plan to buy many blank compact disks. You check price lists and find out that if you buy a 1000 CD’s or less you pay $0.74 each. However if you buy between 1000 and 2000 CD’s the price drops to $0.69 each for the second thousand. Also, for any purchase of more than 2000, the price for the CD’s drops again to $0.64 for each after the 2000^th.

The solutions:

You should have created the piecewise function below.

• The first line is easy. At $0.74 each the rule is just 0.74 n.
• However, for the other lines remember that we need to charge the correct amount for the first 1000 and for the number above 2000.
• For the second line remember to include the cost of the first 1000! That's 1000($0.74) = $740. Also you need to adjust for that amount by adding on only what you need for each additional CD above the first 1000. Then variable cost is 0.60(n-1000)
• For the third line, the cost for the first 2000 is $1430. You adjust in the same manner as for the the second line. Then variable cost is 0.64(n-2000).

When you graph these equations, you should enter the following data into your calculator in the Graph menu:

Y1: 0.74 x,[0,1000]

Y2: 0.69( x-1000)+740,[1000,2000]

Y3: 0.64( x-2000)+1430,[2000,5000]

This creates the graph below. The graph is very subtly bent at x = 1000 and x = 2000.

Oh by the way... The graph in your calculator is continuous. In the real world, this graph is a series of discrete point! Ever bought half a CD?

To evaluate for n = 250, use Y1. For n = 1000, you should also use Y1 although Y2 returns the correct value. This is because the function is continuous at n = 1000.

For the n = 2000 calculation use Y2 and for n = 3500 use Y3.

You should obtain these values:

p(250) = 185 dollars

p(1000) = 740 dollars

p(2000) = 1430 dollars

p(3500) = 2092.40 dollars