GRADES 9-12: Standard 13 - Conceptual Underpinnings of The Calculus

In grades 9-12, the mathematics curriculum should include the informal exploration of calculus concepts from both a graphical and a numerical perspective so that all students can--

• determine maximum and minimum points of a graph and interpret the results in problem situations;
• investigate limiting processes by examining infinite sequences and series and areas under curves;

and so that, in addition, college-intending students can--

• understand the conceptual foundations of limit, the area under a curve, the rate of change, and the slope of a tangent line, and their applications in other disciplines;
• analyze the graphs of polynomial, rational, radical, and transcendental functions.
##### Focus

This standard does not advocate the formal study of calculus in high school for all students or even for college-intending students. Rather, it calls for opportunities for students to systematically, but informally, investigate the central ideas of calculus--limit, the area under a curve, the rate of change, and the slope of a tangent line--that contribute to a deepening of their understanding of function and its utility in representing and answering questions about real-world phenomena.

Most of the mathematics described in the other 9-12 standards involve finite processes, such as determining a sequence of transformations that maps a figure onto a congruent figure or approximating a zero of a polynominal function using an iterative technique. In contrast, the concept of limit and its connection with the other mathematical topics in this standard is based on infinite processes. Thus, explorations of the topics proposed here not only extend students' knowledge of function characteristics but also introduce them to another mode of mathematical thinking.

Instruction should be highly exploratory and based on numerical and geometric experiences that capitalize on both calculator and computer technology. Instructional activities should be aimed at providing students with firm conceptual underpinnings of calculus rather than at developing manipulative techniques.

##### Discussion

The development of the calculus represents one of the great intellectual accomplishments in human history; perhaps the greatest achievement in the application of mathematics is the use of calculus in physics during the first third of this century. Today, methods of calculus are applied increasingly in the social and biological sciences, and in business as well. As students explore the topics proposed in this standard, it is important that they develop an awareness of, and appreciation for, the historical origins and the cultural contributions of the calculus.

The topics proposed for investigation in this standard should be developed as natural extensions of ideas students have previously encountered. The study of finite sequences and series recommended in the discrete-mathematics standard leads naturally to consideration of the corresponding infinite cases and concepts associated with limiting processes. Considerations of infinite sequences and series can occur at many different levels of abstraction and formalism. Consider, for example, the series

At a very concrete level, this series can be summed in parts, as shown in figure 13.1.

Fig. 13.1. An informal approach to infinite series

At a somewhat higher but still intuitive level, the series can be summed by investigating the limit of the sequence of partial sums (either by using a calculator or a simple looping computer program or by applying the formula for the sum of a finite geometric series). An understanding of the concept of limit in contexts such as this should in turn contribute to the meaningful development of the remaining topics in this standard.

Similarly, it is important that all students recognize how the concept of the area under a curve builds on and extends their previous experiences with areas of geometric figures. College-intending students also should recognize how the rate of change builds on and extends their experiences with uniform motion and associated rates in algebra and trigonometry, and how the slope of a tangent to a curve generalizes the notion of the slope of a line as developed in algebra.

Many of these concepts can be approached through informal activities that focus on the understanding of interrelationships. For example, the concepts of slope, derivative, velocity, and acceleration could be addressed through experiences such as the following:

Given the velocity-time graph shown in figure 13.2, construct reasonable distance-time and acceleration-time graphs for the same system.

Fig. 13.2. A velocity-time graph

Possible solutions are shown in figure 13.3.

Fig. 13.3. Related distance-time and acceleration-time graphs

Note that correct responses may vary; for example, the distance-time graph could be translated vertically.

Computing technology makes the fundamental concepts and applications of calculus accessible to all students. The area under a finite portion of a curve, for example, can be approximated geometrically by partitioning the region into rectangles with bases of equal length and heights given by function values at an endpoint of each base. Using a calculator or a computer-based algorithm, students can easily sum the areas to obtain a numerical approximation of the desired area. This approximation can then be sharpened by using sequences of rectangles whose bases are made to decrease toward zero. Project work might require students to investigate other ways of partitioning the region so as to obtain a more precise estimate. (A description of a process using trapezoids can be found in the standard on mathematical connections.)

All students could use a graphing utility to investigate and solve optimization problems, including the maximum-minimum problems traditionally associated with the first college-level course in calculus, without computing a derivative. (For an example, see Examples of Content Differentiation,) A great deal of mathematical understanding is reinforced in the context of solving these problems: data analysis, problem formulation, mathematical modeling, geometric topics, translation across multiple representations, and validation.

Using interactive graphing utilities, college-intending students could examine other characteristics of the graphs of functions, including continuity, asymptotes, end behavior (i.e., behavior as |x| ), and concavity. Moreover, such analysis can be applied with equal ease to the graphs of a variety of complicated functions and to curves specified by polar and parametric equations.

Computing technology also permits the foreshadowing of analytic ideas for college-intending students. From a computer-graphics perspective, for example, a differentiable function can be viewed as a function having the property that a small portion of its graph, when highly magnified, approximates a line segment. The "zoom in" feature permits students to magnify the graph of a function over a small interval to view the approximate segment representation and to compute the gradient (slope). Using the computer to plot successive gradients for a series of short intervals (see fig. 13.4) suggests a process that derives functions from functions.

Fig. 13.4. Using computer graphics to see that the derivative of y = sin x is y = cos x

Although developing a foundation for the future study of calculus remains a goal of the 9-12 curriculum for college-intending students, equally important is the development of prerequisite understandings for further study of statistics, probability, and discrete mathematics. This standard calls for a new balance of skills, concepts, and applications in that portion of the curriculum traditionally associated with preparation for calculus. Instead of devoting large blocks of time to developing a mastery of paper-and-pencil manipulative skills, more time and effort should be spent on developing a conceptual understanding of key ideas and their applications. All students should have the benefit of a computer-enhanced introduction to some of the types of problems for which calculus was developed.