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
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.
Obse
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
!
Examp
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
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
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 co
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 wr
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 e
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 l
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