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Some of the material in this unit should be familiar to you. Some of it will almost certainly be new to you. All of it is essential to a thorough understanding of calculus and of the other scientific concepts in this course.
In many branches of physical and social science, researchers gather lots of data, plot the data, and then try to figure out a formula that describes the data. The idea behind this practice is that by measuring the different values of some quantity while changing the values of another variable under the scientist's control, you can predict the value of the quantity for any other value of the control variable. Depending on the situation, the scientist may or may not have an a priori idea of what kind of formula to expect. The research must also take into account the fact that measurement errors and other uncertainties in the data may occur. But we do our best.
The simplest, and probably the most familiar, kind of formulas and graphs to consider are the linear ones. As everybody knows, what we mean when we say that the variable y depends linearly on the control variable x is that the mathematical relationship between y and x is
y = mx + b
where m and b
are parameters which specify the slope and y-intercept
of the graph of the line, respectively. In class, you will see a Maple
animation depicting just how changing m and b
affects the graph of the line.
The most important characteristic of lines is the fact that they have a
well-defined value of the slope. Recall that the slope of a line is
defined as the change in y for a given change in x.
The formula for slope, given two points (x_1, y_1) and
(x_2, y_2) on the line, is
The most striking and important thing about this formula is that the same
value of m results whichever two points on the line are chosen
to be (x_1, y_1) and (x_2, y_2). Lines (i.e. linear functions)
are the only graphs (i.e. functions) for which this is true.
To get a little practice with Maple and linear functions, go through exercises 1-1, 1-2, and 1-3 in your assignment pack.