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### 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
```
Associated Topics:
High School Functions

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