Basic concepts; differentiation, differential equations
and integ ration with applications
directed primarily to the life sciences.
Pre: C– or better in 140 or assessment exam.
0. (Re)introduction to function theory (1 week)
A. We review polynomial ( especially linear ) functions,
logarithms, exponentials and to
some extent trigonometric functions . Special attention will be paid to the
graphs of these
functions.
B. The concept of composition of functions and inverse
function is also considered again.
C. New topics introduced here may include boundedness,
continuity, limits at infinity,
monotonicity, local and absolute extremes and concavity of functions.
1. Differentiation (8 weeks)
A. We define and give several inter pretations to the idea
of differentiation. Differentials
are used to approximate the change in the value of a function .
B. The derivative of most of the functions considered in
Section 0 are introduced. Some
of these derivatives will be computed using the definition; others will simply
be given and
these formulas will be supported by computer experimentation. If subsequently a
more
rigorous justification of any of these latter formulas is possible then it will
be given.
C. The standard rules for differentiating sums, products ,
quotients and composition of
functions are presented. The chain rule is used to obtain the derivative to the
inverse of a
differentiable function .
D. The Mean Value Theorem is presented and used. In
particular, we will show that
antiderivates are determined up to an additive constant.
E. Applications of differentiation are considered. In
particular, properties of the graphs of
functions and optimization are studied.
F. The study of differential equations is begun; we
examine exponential growth and decay,
the logistic differential equation and perhaps separation of variables . Also
Euler’ s method
can be introduced and used as the basis for numerical studies.
G. Partial differentiation could be introduced here.
2. Integration (5 weeks)
A. We define and give several interpretations to the idea
of integration. Numerical methods
of integration are considered.
B. The Fundamental Theorem is presented and used.
C. The standard integration rules are studied, in
particular, substitution and integration
by parts.
D. Applications of integration are introduced.
E. Improper integrals are defined and computed.
3. Introduction to calculus of several variables (2
weeks)
A. Partial derivatives are introduced.
B. As an application we could consider optimization
problems in 2 variables.
Various topics in this course will be supplemented by work
in a computer lab that will
meet once each week. Obvious examples are numerical integration and solving
differential
equations using Euler’s method. Also the students will be introduced to a
symbolic
manipulation package by means of which they will be able to algebraically (or
formally)
differentiate and integrate functions.
