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14.1 - Section 15.1 Functions of Several Variables Dening...

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Section 15.1 Functions of Several Variables “Defining and Graphing Functions of more than one Variable” Except for a couple of special examples (cylindrical surfaces, planes), we have not yet really touched upon the idea of functions of more than one variable (instead we have focused on curves in 3-space). In this section, we shall consider the basic properties of a function of two variables. 1. Functions of Two Variables - Basic Definitions We start with a formal definition of a function of two variables. Definition 1.1. A function of two variables f is a rule that assigns to an ordered pair of real numbers ( x, y ) a unique real number denoted by f ( x, y ). The domain of f ( x, y ) is the set of all ordered pairs ( x, y ) for which f can be evaluated, and the range is all the values that f takes. There are many different ways to represent a function of two variables. Just like single variable functions, one way is to list the entries in a table. The only problem with this method is that just a couple of data points requires a large table. We illustrate. Example 1.2. Market research suggests that the number of people who use the train to commute between Salem and Portland every morn- ing is a function of the cost of a ticket and the time it takes to get from Salem to Portland, so N ( T, P ) where N = the number of people, T is time and P is price. Research indicates the following results: Cost P ($) Time T (Hrs) 5 10 15 20 25 1 35 33 28 20 15 1.5 34 30 27 20 12 2 20 15 9 7 1 2.5 12 10 7 1 0 ( i ) How many people take the train if it take 1(1/2) hours and costs $ 15? We have N (1 . 5 , 15) = 27 ( ii ) In order to maximize revenue, what is the best time and price to choose? The revenue will be equal to the number of people who take the train multiplied by the cost for a ticket. Therefore, we can construct a new function for revenue depending upon cost P 1
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2 and time T in hours by simply multiplying each column in the previous table by the price of the ticket: Cost ($) Time (Hrs) 5 10 15 20 25 1 175 330 420 400 375 1.5 170 300 405 400 300 2 100 150 135 140 25 2.5 60 100 105 20 0 We can see from the table that the most revenue is gathered when P = $15 and T = 1. As the last example suggests, using tables to represent functions of two variables is inefficient - in general, we need a lot more information to make even simple conclusions about a function. Therefore, as with sin- gle variable, we usually use graphs and equations to represent functions and study them. For algebraic representations, many of the results re- garding functions will be identical to the single variable case. With graphs however, it is more complicated. We explore some algebraic expressions first.
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