In many cases, you can compute
by plugging a in for x:
For example,
This situation arises often enough that it has a name.
Definition. A function
is continuous at a if
This definition really comprises three things, each of which you need to check to
show
Example. The Folium of Descrates is given by the equation
The graph consists of all points
on the graph, because
,
. Picture:
which satisfy the equation. For example,
is
, satisfies the equation.
Observe, however, that the graph is not the graph of a func
Functions
for all
and
and
does not mean "
Example.
are inverses if
. If f has an inverse, it is often denoted
. However,
"!
and
Notice that the inverse of
are inverses, since for all x,
is not
!
Example. Functions which are inverses "undo" one another. Th
Given a function
at the point
, how do you find the slope of the tangent line to the graph
?
(I'm thinking of the tangent line as a line that just skims the graph at
,
without going through the graph at that point. This is a vague description, but it will
If you graph
and
indistinguishable near
, you see that the graphs become almost
:
That is, as
,
. This approximation is often used in applications - e.g.
analyzing the motion of a simple pendulum for small displacements. I'll use it later on
to derive the
The Chain Rule computes the derivative of the composite of two functions.
The composite
is just "g inside f" - that is,
(Note that this is not multiplication!)
Here are some examples:
Here's a more complicated example:
One way to tell which function is "i
Here are the formulas for the derivatives of
and
:
I'll derive them at the end. First, I'll give some examples to show how they're used.
Example. Compute
.
Using the Product Rule, I get
Note: Don't write "
Either put the "
" for the second term, since tha
There are many rules for computing limits. I'll list the most important ones. There are
analogous results for left and right-hand limits; just replace "
" with "
" or "
".
I mentioned the first rule earlier:
If
is a polynomial, then
Remember that a polyno
In some cases, you let x approach the number a from the left or the right, rather than
"both sides at once" as usual.
means: Compute the limit of
as x approaches a from the right.
means: Compute the limit of
as x approaches a from the left.
The left- and
Here is the graph of
:
The graph approaches the horizontal line
write:
as it goes out to the left and right. You
In general, to say that
means that the graph of
approaches
larger positive values for x.
as you plug in larger and
means that the graph of
app