In addition to learning the customary computational skills in
calculus, the student will be exposed to mathematics as
mathematicians view it.  Students will learn to write proofs and
think more rigorously about mathematics, and will come to grips
with challenging concepts and problems.  In that regard, this
course is an excellent preparation for students considering a
career in law, medicine, or the sciences.

1. Upper and lower sums, the definition of the integral, the
convenient criterion for integrability; examples and counterexamples.
2. Properties of the integral.
3. The Fundamental Theorem of Calculus.
4. Applications: areas under curves, volumes, arclength, work.
5. The logarithm and exponential functions.
6. Methods of integration: integration by substitution, by parts,
by partial fractions (including sketch of proof), by
trigonometric substitution. Improper integrals.
7. Taylor polynomials; in-depth treatment of the algebra and
calculus of Taylor polynomials. Taylor's Theorem with remainder,
 with applications to indeterminate forms and approximate
integration.
8. Sequences and series. Bolzano-Weierstrass Theorem, Cauchy
sequences.  Comparison and limit comparison tests, ratio test,
integral test.  Conditional convergence and rearrangement. 
9. Sequences and series of functions. Pointwise and uniform
convergence.  Applications to power series and computation of
explicit numerical series.
10. Complex numbers and power series. Explanation of
singularities on the circle of convergence.