Precalculus

Calculus is the mathematical study of change. It is useful in tracking the growth of investments; the growth or decline of animal populations; the paths of the planets, missiles, and the space shuttle; and much more. Calculus opens a doorway to the quantitative analysis needed for informed decision making. More about calculus can be found on my calculus pages.

Precalculus opens the door to calculus.

The material in precalculus is mostly familiar to high school students who have taken four years of mathematics. But the approach to problem-solving is new to many college students, and the pace is much faster. Students are asked to take more responsibility for their own learning, to do more work outside of class.

The major tool used in the analysis of change is the function: functions relate the various quantities that change. For example, we can think of the value of a savings account as a function of time. The amount of a chemical product in a reaction is a function of the amount of raw materials used. The area of a rectangle is a function of its height and width. It is important to understand what a function is, and that can be done from at least four different perspectives.

Descriptively, a function is defined by words:

"The distance traveled by a falling object is proportional to the square of the time it has been falling, with the constant of proportionality equal to half the acceleration of gravity."

Algebraically, a function is a rule which relates the variables:

D = 16t2

Numerically, a function is a collection of paired numbers. Thought of in this way, a function is often represented as a table:

Time Distance
0 0
1 16
2 64
3 144
etc. etc.

Graphically, a function is a geometric picture showing the relation. More precisely, the function is the set of ordered pairs in the tD-plane:

We sometimes call this four-pronged approach to understanding functions DANG. Precalculus students, presumably overwhelmed by enthusiasm, are often heard to utter this acronym.