Calculus
for Physics 50 series
–
an 8-page guide to concepts important for these courses.
By Todd Sauke
We start with the important concept of a
Function
.
A function is a mathematical
relation for which, if given an input value (the
"argument"), the result (the "output" value of
the function for that argument) is given by
exactly one number.
In the same way that we
can refer generally to an arbitrary or unknown
quantity as "x", we can refer generally to some
arbitrary or unknown function using an
abstraction like "F(x)".
A function can be
thought of as a mathematical
operation
(or
"
rule
") that produces a single output number
for any input number in the domain of the
function.
The specific relation, F(x) = x
2
+ 1
is a function; for
every
value of the input, x,
you can compute
exactly one
output value:
F(3) = 3
2
+ 1 = 10, for example.
The input
value of a function is also called an
"
independent variable
"
.
f
(x
)
=
x
½
(the
square root of x) is
not
a function, since the
result, for example, for x=4 gives
plus or
minus
2 (
two
possible results, rather than
exactly one
result.
A function can be plotted
as a graph showing a curve of
any
shape, as
long as there is exactly one and only one value
corresponding to each input value. We can
also have functions with
multiple
arguments
(or multiple independent variables), say a
function of x, y, and z such that there is
exactly one output result for any combination
of input values x, y and z.
For example, F(x, y, z) = 3 + 2xy
2
– z
4
is a
function of x, y and z.
For x=1, y=3 and z=2,
F(1, 3, 2) = 3 + 18 – 16 = 5,
exactly one
output number corresponding to the specific
input values.
Sometimes a curve on a graph,
say the circle in the x,y plane given by the
equation x
2
+ y
2
= R
2
, does not represent a
function y(x).
However, this same curve
could be considered a function r(
θ
) by
changing from
Cartesian coordinates
to
polar coordinates
.
In that case, each value of