BACKGROUND OF THE INVENTION 
     The present invention relates to a teaching aid for use in understanding and visualizing calculus concepts and specifically to a teaching aid adapted to aid students in understanding and visualizing the mathematical concept of the first derivative in calculus. 
     In calculus, the derivative is a measurement of how a function changes when the values of its inputs change. The derivative of a function at a chosen input value describes the best linear approximation of the function near that particular input value. For a real-valued function of a single real variable, the derivative of a point equals the slope of the tangent line to the graph of the function at that point. 
     Differentiation is a method used to compute the rate at which a quantity, y, changes with respect to the change in another quantity, x, upon which it is dependent. This rate of change is called the derivative of y with respect to x. The dependency of the variable y on x means that it is a function of x, which is represented by the variable f(x). Therefore, if x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of tangent lines to this graph at each point. 
     In teaching mathematical concepts, it is often helpful for students to utilize a visual model in order to better understand these concepts. For example, to better visualize the concept of derivatives, it is often helpful to graphically display the curve of a simple function. It has been found that students often have difficulty visualizing such concepts, and therefore have difficulty in understanding and solving such problems. Accordingly, the graphical representation of such concepts aids in the students learning. 
     It is therefore an object of the present invention to provide a calculus teaching aid for modeling the determination of a first derivative in order to assist students in visualizing the concept of the first derivative. It is a further object of the present invention to provide such an aid that can be adapted to show a variety of different models. Yet another object is to provide an aid comprising a plurality of different indicia configured to represent the various characteristics of the model. A further object of the present invention is to provide such an aid that is relatively simple in its construction yet which greatly enhances a student&#39;s ability to understand and visualize the concepts necessary to understand the various aspects of the first derivative. 
     SUMMARY OF THE INVENTION 
     In accordance with the invention, a calculus teaching aid for demonstrating the characteristics of the first derivative includes a substrate, a plurality of parts configured to represent a graph of the first derivative, and a plurality of symbolic labels having various indicia written on them for identifying various features of the graph. The parts and labels are generally selectively attachable to a surface of the substrate. The substrate is preferably substantially planar defining a rectangle or square and constructed of a plastic, metal, wood or other suitable material. Further, the substrate may be configured for use with parts and labels to ensure that the pieces and markers remain secured to the substrate, e.g. the substrate may be metal and the parts and labels may be magnetic. 
     The graphical parts and symbolic labels are adapted to be selectively secured to a surface of the substrate to visually represent a graph of a function having an input. The graphical pieces may include a plurality of curvilinear pieces for representing the actual graph as well as a plurality of circular pieces for representing particular points along the graph. The symbolic labels preferably include written indicia on an upper surface thereof. However, it is understood that the symbolic labels may be configured to be user-definable so that a variety of different information may be presented thereon. The symbolic labels may be used by the user to label the various information depicted by the graph defined by the graphical pieces to thereby make it easier to understand the graph. 
     Various other features, objects and advantages of the present invention will be made apparent from the following detailed description and the drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings illustrate one preferred embodiment presently contemplated for carrying out the invention. 
       In the drawings: 
         FIG. 1  is a top plan view of a calculus teaching aid constructed in accordance with the present invention; 
         FIG. 2  is a top plan view of the calculus teaching aid of  FIG. 1  demonstrating the determination of the average rate of change between two points of a function; and 
         FIG. 3  is a top plan view of the calculus teaching aid of  FIGS. 1 and 2  depicting the calculation of the derivative of the function represented. 
     
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     Referring to the drawings, and initially to  FIG. 1 , a calculus teaching aid  10  in accordance with the present invention is illustrated. Calculus teaching aid  10  includes a substantially flat substrate  12  defining a rectangle or square, however, other suitable shapes may be utilized in practicing the present invention. Substrate  12  is preferably constructed of a magnetically attractive material, such as a ferrous metal or other satisfactory material. However, it is understood that other materials may be used for construction of substrate  12  such as, for example, Velcro, plastic and other non-metallic materials. 
     Calculus teaching aid  10  further comprises a plurality of labels  14  configured to be selectively attached and arranged on a surface of substrate  12 . The labels  14  are preferably constructed of any satisfactory material, and may be magnetized for secure attachment to the magnetically attractive substrate  12 , however, it is understood that labels  14  may be formed of any other suitable material. 
     Labels  14  preferably comprise a number of labels having different indicia written on surfaces thereof to visually represent various parts of the graph depicted in teaching aid  10 . For example, as shown in  FIGS. 1-3 , labels  14  may comprise a label  16  including the text “x,” a label  18  including the text “f(x),” a label  20  including the text “x+Δx,” a label  22  including the text “(x, f(x)),” a label  24  including the text “f(x+Δx),” a label  26  including the text “((x+Δx), f(x+Δx)),” and a label  28  including the text “←Δx→.” Although the labels  14  shown here have predetermined indicia written on them, it is understood that labels  14  may be configured for user customization. 
     Calculus teaching aid  10  further comprises a plurality of parts  30  configured to represent various aspects of a graph. Similar to labels  14 , parts  30  are preferably constructed from a magnetized metal or the like but may alternatively be constructed of any material suitable for practicing the present invention. 
     Parts  30  preferably include at least one curvilinear member  32  and a pair of vertical strings  34  having attachment means  36  at one end thereof. While elements  34  are described as flexible members in the form of strings, it is also understood that the elements  34  may also be in the form of non-flexible members such as rods or the like. Attachment means  36  may comprise any suitable attachment means such as a clip, magnet, or other such suitable connector. Further, a movable rod  38  configured to be slidably coupled to attachment means  36  at a right angle to vertical string  34  is preferably provided. Attachment means  36  is also preferably selectively slidably attachable to curvilinear members  32  thereby enabling the movement of movable rod  38  to different positions on the graph. 
     A pair of circular members  40   a ,  40   b  may also be provided for representing two discrete points on the graph and may be selectively attachable to attachment means  36 . A plurality of rectangular markers  42  are also preferably provided for indicating the position of circular members  40   a ,  40   b  along either of the axes of the graph. Finally, a generally L-shaped member  44  may be provided for representing the axes of the graph of the present invention. Alternatively, rather than utilizing a single L-shaped member  44 , the present invention may utilize a two-piece structure adapted to be configured as the axes of the graph. 
     Referring now to  FIGS. 2 and 3 , and in particular  FIG. 2 , the calculus teaching aid  10  of the present invention is shown in operation. L-shaped member  44  is generally centrally positioned on substrate  12  to represent the x axis  46  and y axis  48  of a graph  50 . A pair of curvilinear members  32  are centrally positioned to represent a curve  52  of the function shown in graph  50 . Circular members  40   a ,  40   b  are positioned at two distinct points along curve  52  to represent two distinct points along the curve  52 . Circular member  40   a  represents the point (x, f(x)) and is labeled by label  18  to convey this information to the user. Similarly, circular member  40   b  represents the point ((x+Δx), f(x+Δx)) and label  26  is positioned nearby to indicate this. Vertical strings  34  are shown with attachment means  36  coupled to circular members  40   a  and  40   b  such that vertical strings  34  extend in the y-direction of the graph. Rectangular markers  42  are placed at locations along the x-axis  46  to identify the positions of the circular markers  40   a  and  40   b . Label  28  is positioned on the substrate  12  between the lengths of the vertical strings  34  to identify that the distance between circular member  40   a  and  40   b  is represented by the value Δx. Further, two additional rectangular markers  42  are placed along the y-axis of the graph to represent the positions f(x) and f(x+Δx) and labels  18  and  24  are located nearby to identify each position. 
     In  FIG. 2 , movable rod  38  is configured to define a secant line extending between circular markers  40   a  and  40   b . A secant line is a straight line that intersects a curve at two or more discrete points along the curve. The slope of the secant line represents the average rate of change and is defined by Δy/Δx. As such, the slope of the secant line shown in  FIG. 2  is equal to the (f(x+Δx)−f(x))/x+Δx−x), or (f(x+Δx)−f(x))/Δx). 
     Now turning to  FIG. 3 , the calculus teaching aid  10  of the present invention in operation is shown being used to graphically demonstrate the calculation of the first derivative of the function. 
     In mathematics, the first derivative represents the instantaneous rate of change, or the rate of change at a particular point along the curve of a function. The slope of the tangent line through the point is used to determine the instantaneous rate of change. The slope of the tangent line shown in  FIG. 3 , or first derivative, is represented by the following equation: 
               lim       Δ   ⁢           ⁢   x     -&gt;   0       ⁢       (       f   ⁡     (     x   +     Δ   ⁢           ⁢   x       )       -     f   ⁡     (   x   )         )     /     (     Δ   ⁢           ⁢   x     )             
Thus, solving for the above limit will give the slope of the tangent line. However, because students often have difficulty understanding the concept of limits, the present invention aids by graphically demonstrating the concept.
 
     The mean value theorem guarantees that there is at lease one point there where the slope of the tangent line equals the slope of the secant line provided the function is continuous and differentiable. As such, the present invention may demonstrate the concept of the mean value theorem through the visual depiction thereof as is described further below. 
     Specifically, calculus teaching aid  10  graphically demonstrates the above-noted principles by allowing the user to selectively slide attachment means  36  along curvilinear members  32  to a position where movable rod  38  acts as a tangent line through a single point on curve  52 . As such, the vertical strings  34  become positioned nearer to one another with respect to the x-axis, thereby decreasing the value represented by the Δx or label  28 . As such, the concept of using limits to calculate the slope of tangent line is graphically demonstrated. Accordingly, calculus teaching aid  10  illustrates the complicated principle of the first derivative through a rather simple mechanism. 
     The present invention has been described in terms of the preferred embodiment, and it is recognized that equivalents, alternatives, and modifications, aside from those expressly stated, are possible and within the scope of the appending claims.
A calculus teaching aid includes a substrate, a plurality of visual components configured to define a two-dimensional graphical model of the first derivative, and a plurality of symbolic labels configured to define a variety of characteristics of the graphical model. The substrate, visual components and labels are preferably magnetized for secure attachment to one another. The calculus teaching aid may be configured in a plurality of different ways to demonstrate the various aspects of the first derivative.