Page 16 - How To Solve Word Problems In Calculus

Because of the geometric or physical nature of a prob-
lem, many word problems arising from everyday situations
involve functions with restricted domains. For example, the
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squaring function f (x) = x discussed in Example 2 allows all
real x, but if f (x) represents the area of a square of side x, then
negative values make no sense. The domain would be the set
of all nonnegative real numbers. (We shall see later that it is
sometimes desirable to allow 0 as the dimension of a geomet-
ric figure, even though a square or rectangle whose side is 0 is
difficult to visualize).
Finally, please note that when dealing with problems in
elementary calculus, such as those discussed in this book, only
functions of a single variable are considered. We may write
A = xy to represent the area of a rectangle of width x and
length y, but A is not a function of a single variable unless
it is expressed in terms of only one variable. Techniques for
accomplishing this are discussed in the pages that follow.

Strategy for Extracting Functions
The most important part of obtaining the function is to read
and understand the problem. Once the problem is understood,
and it is clear what is to be found, there are three steps to
determining the function.

Step1
Draw a diagram (if appropriate). Label all quantities,
known and unknown, that are relevant.

Step2
Write an equation representing the quantity to be ex-
pressed as a function. This quantity will usually be repre-
sented in terms of two or more variables.
Step3
Use any constraints specified in the problem to elimi-
nate all but one independent variable. A constraint defines
a relationship between variables in the problem. The pro-
cedure is not complete until only one independent variable
remains.

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