There are several definitions from calculus presented, including the
definition of a limit, definition of continuous function (at
a point and on an interval), limit of a sequence, differentiable function (at
a point and on an interval), and the Riemann integral.
In conjunction with the Taylor series, the following are also defined:
-
Taylor polynomial 
and its associated remainder term 
(or truncation error); - Maclaurin series
Theorems exposed are
-
the equivalence of continuity at
and limits of convergent sequences at ; - differentiability implies continuity;
- Rolle's Theorem;
- Mean Value Theorem (a generalization of Rolle's, and then the official....)
- Generalized Rolle's Theorem (which just goes to show that there's more than one way to generalize a result!);
- Extreme Value Theorem;
- Weighted Mean Value Theorem for integrals (a generalization of the Mean Value Theorem for integrals);
-
Intermediate Value Theorem (note: in their statement, they implicitly specify
their understanding of the word ``between'': they mean ``strictly between'',
otherwise the theorem fails for
when a=-1 and b=1); and - Taylor's Theorem.
Errata: having implied that when they say ``between'' they mean ``strictly between'', they operate in #26 as though ``between'' means ``or equal to''.
A lot of our time in this course will be spent with our old friend calculus, as we deal with issues that we handled in other ways back in the day.... Now we will ask questions such as how to integrate functions which are not integrable using elementary functions; how to talk about derivatives of functions defined only by data points; how to find roots of functions; etc.
As examples of intractable functions that we glossed over in calculus class, consider the following problems:
-
Compute
- Find the roots of a given cubic polynomial.
Both of these problems are intractable using methods from your calculus; however, we can use the calculus to get an answer to either to whatever precision required - that's the job of numerical analysis!