Applied Calculus Math 215 - University of Hawaii

4 CHAPTER 0. A PREVIEW
In due time we will explain all of this in more detail. You noticed that
we need the idea of a line. When you look at (2) and see the square of the
variable you can imagine that we need parabolas. So we review and elaborate
on lines and parabolas in Chapter 1. We also introduce the, possibly,
two most important functions in life science applications, the exponential
function and the logarithm function.
Chapter 2 is devoted to the precise definition of the derivative and the
exploration of related ideas. Relying only on the definition, we calculate the
derivative for some basic functions. Then we establish the major rules of
differentiation, which allow us to differentiate many more functions.
Chapter 3 is devoted to applications. We investigate the ideas of monotonicity
and concavity and discuss the 1st and 2nd derivative tests for finding
extrema of functions. In many applications of calculus one proceeds
as follows. One finds a mathematical formulation for a problem which one
encounters in some other context. One formulates the problem so that its
solution corresponds to an extremum of its mathematical formulation. Then
one resorts to mathematical tools for finding the extrema. Having found the
solution for the mathematically formulated problem one draws conclusions
about the problem one started out with.
E.g., look at a drop of mercury. Physical principles dictate that the
surface area be minimized. You can derive mathematically that the shape
of a body which minimizes the surface area, given a fixed volume, is a ball.
This is roughly what you see. There is a slight perturbation due to the effect
of gravity. This effect is much greater if you take a drop of water, for which
the internal forced are not as strong as the ones in a drop of mercury.
Often calculus is used to solve differential equations. These are equations
in which a relation between a function and its rate of change is given 2 .The
unknown in the equation is the function. E.g., for some simple population
models the equation (Malthusian Law)
P ′ (t) =aP (t)
is asserted. The rate at which the population changes (P ′ (t)) is proportional
to the size of the population (P (t)). We solve this and some other population
related differential equations. We will use both, analytical and numerical
means.
The second principal concept is the one of the integral. Suppose you need
to take a certain medication. Your doctor prescribes you a skin patch. Let
2
In more generality, the relation may also involve the independent variable and higher
derivatives.