See
handout for administrative information.
This quarter we will cover four main topics: Trigonometry, Calculus with
Trigonometry, Integrals,
and Transcendental functions.
Most of the time
will be spent on integration, the opposite of differentiation.
Trigonometry is about the ratios of the sides of right triangles.
The mnemonic SOHCAHTOA
is useful in remembering which function
corresponds to which ratio. It stands for
sine over hypoteneuse,
cosine adjacent over hypoteneuse, and tangent opposite over adjacent.
One point of view of calculus is that it is about rates of change.
In principle, knowing the position of a car allows
one to know its
speed. Conversely, knowing the speed and the initial location
allows one to know its position.
Qualitatively, a function which
increases rapidly has a large positive derivative. One which
grows slowly has a small positive derivative.
Similar statements hold
for decreasing functions. A slowly decreasing function has a small
negative derivative.
A rapidly decreasing function has a large
negative derivative.
Last quarter, we used the derivative to make these observations
about the rates of change
with numbers,
that is, quantitatively.
This quarter we will go from the rates of change to the original function
quantitatively using the integral.
One application of the integral is to find the area of a region in the plane.
There are various tricks one may try, which are
successful in finding the
area of a triangle. The sure way is with the integral.
Naturally,
we will have to develop some mathematical machinery to define the integral.
This is just an introduction to the many exciting topics
we will cover this quarter.
What better time, than
late spring to discover the fundamental theorem of calculus?