Mathematics for Teaching Calculus Variables, Constants and Parameters

Variables, Constants and Parameters

Whether a mathematical notation is a variable,  parameter, or constant depends on what you mean by it.

  • If you intend to represent the value of a quantity whose measure varies within a situation, then you are using that notation as a variable.
  • If you intend to represent the value of a quantity whose measure is the same within all situations (e.g., pi), then you are using that notation as a constant.
  • If you intend to represent the value of a quantity that is constant in a particular situation, but which can vary from one situation to another, then you are using that notation as a parameter.

Examine the cylinders depicted in the figure below. They are partially filled with water. As they sit, it appears that nothing about the cylinders or water in them varies. Now imagine that each cylinder is being filled from a pipe at its bottom. Then the water’s height in each cylinder varies.

If we let h1, h2 and h3  represent the water’s height in the respective cylinders and imagine that each height varies, then the values of h1, h2 and h3 vary as the water’s height varies within its cylinder.

 

Let h1, h2, and h3 represent the height of the water in the three cylinders, respectively. Each variable’s value varies as the water’s height varies. Any tank’s height and radius do not vary.

Some quantities do not vary within a cylinder but vary across cylinders. Any cylinder’s radius does not vary, nor does its height, nor does the area of its water’s exposed surface. For each cylinder, its water’s height and volume do vary. So if we let r represent a cylinder’s radius, the value of r, for each cylinder, does not vary. But the value of varies across cylinders. Similarly, if we let H represent a cylinder’s height, the value of H, in each cylinder, does not vary. But the value of H varies across cylinders.

Again, when we say that a notation has the meaning of a variable, we mean that it represents the value of a quantity whose value varies within a situation. When we use a notation to represent a value that is constant in a situation but which can vary from one situation to another, we are using the notation as a parameter. We mean that the notation represents the value of a quantity that is constant within a situation, but the quantity could have different values in different situations.

By convention, the term domain is used to refer to the set of values that a variable can have. If in the figure we use b to represent the initial height of water in a cylinder, then the domain of hi for any of i = 1, 2, or 3, is b ≤ hi ≤ H.The letters br, and H are used as parameters. The letters , and are used as variables.

We also can speak of a variable’s value varying in the absence of a specific quantity. When we say something like, “The value of x varies between 2 and 5”, we are actually saying that there is some yet-to-be-known quantity lurking in the background whose measure varies from 2 to 5, passing through all real numbers between 2 and 5.

Values that do not vary are called constants. Some constants are universal, like π and e. Their values do not depend on a particular context. Other constants are constant only with respect to a situation. In the statement, “Billy has 5 toys. His father gave him 2 more. He has 5+2 toys altogether”, the number of toys Bill started with and the number of toys he received are constants.

In the statement, “Billy has m toys. His father gave him n more. He has m+n toys altogether”, the number of toys he started with and received are constants within the situation. But they are expressed generally. This means that the statement is given with the intention that we are describing many potential situations. The statement “m+n” says what to do with those numbers once they are known. The letters “m” and “n” are used as parameters.

You can find this kind of explanation above from the textbook Newton meets Technology and it’s free. I like how the precalculus concepts were developed in the book. It’s useful resource even for those teaching or learning algebra or advanced algebra.

 

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