We began our study of scalar-valued functions, which are just the usual fucntions that you studied in algebra and precalculus courses. They are called scalar-valued because in our context single numbers are called scalars. We began by presenting a certain perspective on scalar-valued functions that is useful to keep in mind. One can think of these functions as describing a relationship between two quantities: an independent variable, which can take on a certain range of values (the domain of the function), and the dependent variable, whose values are determined (depend on) the values of the independent variable. In calculus, one of the main objectives is to determine the effect on the dependent variable of a small change in the independent variable--this is one interpretation of the derivative, and a very useful one. We looked at several examples of such functions, and also noted that many times variables are related by an equation, and so any dependence is only expressed implicitly. Which variable is independent depends on your point of view.
Determining tangents to the graphs of such functions is easier than
for the vector-valued functions we have been doing, and can be considered
as a special case of what we have already sone. Recall that a curve which
is the graph of a function y=f(x) can be represented parametrically
by the pair x=t, y=f(t). Then dx/dt is always 1, and dy/dt=f
' (t), so putting everything together gives dy/dx=f ' (t) = f '
(x). Thus, the derivative of the original function directly gives the
slope of the curve. Other than that, everything is as before: once we have
the slope, we can find equations of tangent lines just like we did before.
We went through a couple of examples doing this, using the Maple file class0629.mws;
we also looked at one or two examples where we interpreted the derivative
as relating rates of change between two quantities. Remember, units are
important, and we can often figure out what a derivative represents in
a particular case by paying attention to units of measurement.
