Last revised 28 OCT 2010, by JMB
Maybe you really want the index page instead.
All these notes are available in both DVI and PDF formats.
Notes for 110.108 Calculus I
Derivatives (2 pages) Differentiability of a function of one
variable is defined as the existence of a good linear approximation.
Proofs in this context of the standard rules for derivatives are given.
Available in your choice of:
General Exponential Functions (2 pages) The general
exponential function, a to the power x, is introduced
axiomatically, and its algebraic properties are deduced. When we attempt
to differentiate it, we find the natural logarithm of a and deduce
its properties. Available in your choice of:
The Natural Exponential Function (1 page) This is presented
as a special case of the general exponential function. Its properties
follow easily. Available in your choice of:
The Riemann Zeta Function (2 pages) The p-series
for the Riemann zeta function is discussed as an example of a series that
converges too slowly for practical computation. Available in your choice of:
Projections and Components (1 page) An arbitrary vector
is decomposed as a sum of a vector that is parallel to a given nonzero
vector and one that is orthogonal (perpendicular) to it. This leads to
a short proof of Schwarz's inequality. Available in your choice of:
The Tangent Vector to a Curve (1 page) The derivative of
a vector-valued function is identified with the geometric tangent vector
to the curve traced out. Available in your choice of:
Derivatives and Differentials (2 pages) The directional
derivative of a scalar function f(x) of a vector variable x
is introduced first, and used to define the differential
df. Partial derivatives are treated as a special case. The coordinate
differentials are introduced last. Available in your choice of:
Differentiability and the Tangent Plane (2 pages) The
differentiability of a function of two variables is discussed, both
analytically and geometrically, in terms of the tangent plane.
Available in your choice of:
Local Maxima and Minima (2 pages) The various possible
behaviors of a scalar function f(r) of a 2-dimensional vector
r near a stationary point are classified in terms of the quadratic
form Q(h). Available in your choice of:
Step Functions in Two Dimensions (2 pages) The elementary
properties of step functions in 2 dimensions are discussed, as a prelude
to the 2-dimensional Riemann integral. Available in your choice of:
The Riemann Integral in Two Dimensions (2 pages) The
Riemann integral is defined in terms of step functions and the
order-preserving property. This leads to a short proof of Fubini's Theorem,
with no mention of continuity. Available in your choice of:
Evaluation of Double Integrals (2 pages) The main result
is a Fubini-type theorem, based on the Riemann integral defined as the
limit of Riemann sums, with no mention of continuity. A short sketch proof
is given. Also listed are two auxiliary theorems that are needed to make
the result useful for applications. Available in your choice of:
A Fubini Counterexample (1 page) This example shows that
the order of integration in a double integral cannot be reversed if the
integrand has a bad discontinuity, even if all the single integrals that
occur involve only continuous functions. Available in your choice of: