Module 3 - Introduction
Although the SMRs are organized into discrete content domains (e.g., algebra or calculus), prospective teachers should know that these domains cannot be rigidly separated. For example, the importance of the exponential function (SMR 1.3c) stems primarily from the fact that it is the unique solution of the differential equation f'(x) = f(x) with the initial condition f(0) = 1 (SMR 5.3f). It should be noted that because of this differential equation, the exponential function ex shows up in the growth and decay problems of algebra textbooks.
The fundamental difference between polynomial functions and both exponential and logarithmic functions should be emphasized (SMR 1.3b, c). The overriding concern with a polynomial is to locate its roots and the roots of its derivative (to get the x-intercepts as well as the "peaks" and "valleys" of its graph).
Please read carefully the following summary description of this topic. Don't worry if you don't yet understand it completely, or, if you have already studied this material, check your understanding with the practice problems at the end and move on to the next topic.
For exponential and logarithmic functions, however, such a concern does not exist because log x has exactly one root whereas exp x has no root at all. Moreover, both are strictly increasing functions; their graphs have no "peaks" or "valleys."
>Therefore our interests in the latter functions are different in kind. Our interests in the exponential and logarithmic functions are that log x converts multiplication into addition
while exp x does the opposite
— exp (a+b) = (exp a)(exp b) —
and the fact that they are inverses to each other
log (exp x) = x for all x and
exp(log y) = y for all positive y.
The algebraic properties of log x account for its historical importance as a computational aid (logarithm tables). Analytically, it is the fact that exp x is the solution of f'(x) = f(x), as discussed above, and that log x is the function that has derivative 1/x and satisfies
log 1 = 0.
The trigonometric functions are important for yet a different reason: periodicity (SMR 5.1c). Many natural phenomena are periodic, and their modeling would require the trigonometric functions. Such a conceptual understanding of these three classes of functions is indispensable to helping teachers make sense of the functions they see almost daily in algebra classes.
(Commission on Teacher Credentialing)
|
|
|