calculus question?

f(200) = 2200

at a price of $200, 2200 items are sold

f'(175) = -12.5

at a price of $175 the rate of change of the number of items sold is -12.5. ie. if you were to change the price from 175 --> 176 you would sell 12.5 less items. Note this is an approximation because I just changed the interval from an infinitely small to one (hence calculus) but you get the idea.

Note, in economics this is also called the elasticity and is an indication of how sensitive a particular item is to changes in the price.

f(200) = 2200 means that if you set the price at $200 per item, then you will be able to sell 2200 of them.

f'(175) = -12.5 This is more complicated. Let's say that if you price the item at $175 you will be able to sell x items (we don't know what x is). Then this equation says that for each dollar change in price, the change in the number you can sell will be -12.5. So if you increase the price to $176, you will be able to sell x - 12.5 items (approx). If you decrease the price to $174, you will be able to sell x+12.5 items (approx). If you increase the price to $177, you will be able to sell x - 25 items (approx). In general, the change in the number of items you can sell will be -12.5 times the change in price (approx) (decrease means negative change). Note that all these equations involving change are only approximate, not exact. The smaller the change is, the better your approximation will be.

The first means that, at a price of 200, 2200 items are sold.

The second means that at a price of 175, raising the price by 1 will reduce the number sold by about 13.

f(200)=2200

At a price of 200 dollars 2200 units of an item are sold

f'(175)=-12.5

At a price of 175 dollars the rate of change of items sold is -12.5. This means that as you increase the price of the item by one dollar, you will sell -12.5 less units