Parameter vs. Constant
Date: 12/08/2002 at 23:52:40 From: Chatchai Watcha Subject: Parameter and constant Dear Sirs, I am confused about parameters and constants. I have always understood that parameter has the same meaning as constant. Later, I found that there are a lot of equations which depending on a parameter can be differentiated with respect to parameter. So, I think that I misunderstood something. Can you explain the difference between parameter and constant? Thank you so much. Chatchai
Date: 12/09/2002 at 09:20:59 From: Doctor Rick Subject: Re: Parameter and constant Hi, Chatchai. That's a good question! A parameter is a variable that is held constant when the function is used. Varying a parameter gives you a family of functions. For instance, if we have a function f(x) = 3x + b it is x that varies when you evaluate the function (it is the input to the function); b is treated as a constant. But if you change b (for instance, let b = 0, 1, 2, and 3 in turn), and graph the function (y versus x) for each value of b, you see a family of parallel lines, each of slope 3, and each with a different y-intercept (b). If you know that a function is a member of this family and you know that, for instance, f(2) = 8, then you can find the value of b that identifies f(x) as a particular member of the family of functions. To do this, you plug in x=2 and f(x) = 8: 8 = 3*2 + b b = 8 - 3*2 = 2 Thus f(x) = 3x + 2. Notice that in order to determine the value of the parameter b, we treated b as a variable (and replaced x and y by particular constants). This is because, in the problem of determining b, b is the unknown. This problem (identifying one member of a family of functions) is a distinct problem from using the function (identifying the value of the function for a particular value of the variable). Something similar happens in the kind of problems you are asking about. You may differentiate with respect to the parameter when the purpose is to find the value of the parameter, or more generally whenever you have the entire family of functions in view, not just a single function characterized by a particular value of the parameter. We could write the family of functions that I have used as an example this way: f(x;b) = 3x + b Variables go before the semicolon and parameters go after the semicolon. This makes more explicit that the function depends on the parameter b as much as it does on the variable x. When the function is used as a function, only x is varied; but for other purposes it is just as valid to vary b as it is to vary x. Does this help? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 12/11/2002 at 11:03:03 From: Chatchai Watcha Subject: Thank you (Parameter and constant) Thank you so much. It is really nice to know there are people like you who are expert in maths and are willing to help other people. Thank you again, Dr. Rick. Chatchai
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