In science and engineering, you often approaimate complicated functions
which are easier to calculate with, and which show the relations between the variables more
course, the approximation must be close enough
to give you reasonable accuracy.
For this reason, approximation is a skill, one your other teachers will expect you to have.
This is a good place to start acquiring it.
Throughout, we will use the symbol a to mean "approximately equal to"; this is a bit
vague, but making approximations in engineering is more
art than science.
The simplest way to approximate
for values of
is to use a linear
The linear function we shall use is the one whose
graph is the tangent line
tangent line at (a,
a good approximation to the graph of
at (a, f(a))-
into calculus, we need the equation for the tangent line. Since the line goes
f(a)) and has slope
its equation is
can be expressed as
that for x near
can be approximated
the linear function
on the right of (2). This function
the one whose graph is the tangent line
The appraoimation (2) is often written in an equivalent form
that you should become
it makes use of a dependent variable. Writing
the approximation (2) takes the
In this form, the quantity on the
represents the change in height of
taent line, while the left measures the change in height of the eraph.
Here are some examples of linear approximations.
this being the most important
can be found
(2)above and calculating
derivative. You should verify each of them, and memorize the approximation.