Proceedings of 2014 Zone 1 Conference of the American Society for Engineering Education (ASEE Zone 1) Visual Differential Calculus Andrew Grossfield, Ph.D., P.E., Life Member, ASEE, IEEE Abstract— This expository paper is intended to provide = (y2 – y1) / (x2 – x1) = = m = tan(α) Equation 1 engineering and technology students with a purely visual and intuitive approach to differential calculus. The plan is that where α is the angle of inclination of the line with the students who see intuitively the benefits of the strategies of horizontal. Since the direction of a straight line is constant at calculus will be encouraged to master the algebraic form changing techniques such as solving, factoring and completing every point, so too will be the angle of inclination, the slope, the square. Differential calculus will be treated as a continuation m, of the line and the difference quotient between any pair of of the study of branches11 of continuous and smooth curves points. In the case of a straight line vertical changes, Δy, are described by equations which was initiated in a pre-calculus or always the same multiple, m, of the corresponding horizontal advanced algebra course. Functions are defined as the single changes, Δx, whether or not the changes are small. valued expressions which describe the branches of the curves. However for curves which are not straight lines, the Derivatives are secondary functions derived from the just mentioned functions in order to obtain the slopes of the lines situation is not as simple. Select two pairs of points at random tangent to the curves. The derivatives of the derivatives are on a curve and the corresponding straight lines through these related to the turning of the tangent lines. The concepts involved pairs will in general have neither the same direction, nor the in the study of continuous curves are not difficult to comprehend. same slope nor the same ratio of incremental differences. This But parsing complexity arises from the many combinations of can be observed for the circle in Figure 1 forms, kinds and operations on functions. The paper will focus on the properties of continuous and smooth curves while y maintaining an organized topic structure. Subsequently a 4 student, enabled with the goals and structure of a course in P2 differential calculus, can refer to conventional texts to fill in and 3 y = 0.5x + 3 expand on subordinate details. Q2 2 y = 0.2x + 2 Index Terms—branches of curves, visual derivative, curvature Q1 1 P1 x I. INTRODUCTION −4 −3 −2 −1 1 2 3 4 5 −1 IFFERENTIAL calculus is primarily concerned with effects of small changes in related variables and the direction of D −2 curves. If two variables x and y are functionally related then the relationship can be imagined as a branch of a curve, −3 say y = f(x). If the curve is continuous and smooth then it would be expected that small changes in the variable, x, would −4 produce small changes in the variable, y. Figure 1 Pairs of points on a curve may not have the same A simple example of the functions being studied in calculus slope is a straight line of the form, y = mx + b. Here we are concerned with how small changes in the horizontal variable, If the curve is smooth then at a fixed point, say P, the ratio x relate to the corresponding changes in the vertical variable, of incremental differences for the tangent line will be close to y. The change in x, x – x , is called the run and is represented 2 1 the ratio of incremental differences on the curve near P. This by the symbol Δx. The change in y, y2 – y1, is called the rise and is represented by the symbol Δy. The significance for a fact, that is, that on the curve is approximately equal to straight line of the ratio of the ‘change in y’ to the ‘change in the slope, m, of the tangent line is the basis of differential x’ which is called the difference quotient can be observed in calculus. Equation 1. The following graph, Figure 2, illustrates this principle. A circle of radius 5 is drawn with a line, T, tangent to the circle at the point P(3,4). The equation of the tangent line, T, is Manuscript received February 7, 2014. Andrew Grossfield is a member of the faculty in the Basic Science Department of Vaughn College of Aeronautics and Technology, Flushing, New York. NY 11369 phone: 718-429-6600 ext. 262 fax: 718-429-6339; (y – 4) = – (x – 3) or y = –0.75 x + 6.25. e-mail: [email protected]). 978-1-4799-5233-5/14/$31.00 ©2014 IEEE y R order to refer to these concepts. Differential calculus was y = -0.75x + 6.25 5 crystallized by Isaac Newton in England and by G.W. Leibniz Q P(3,4) in Germany. Each devised his own notation. Currently 4 xx+yy = 25 y = -0.5x + 5.5 Newton’s notation has evolved to writing y(x) to represent the 3 original function; y′(x) to represent the derivative function which was previously called m(x) and y′′(x) to represent the 2 T second derivative. This notation is clean and easy to read in circumstances where there is only one independent variable. 1 On the other hand the notation used by Leibniz appears x more cluttered but better describes the meaning and −5 −4 −3 −2 −1 1 2 3 4 5 application of the derivative and enabled the concepts of −1 calculus to develop and spread faster in Europe than in England. In the notation of Leibniz, the derivative was −2 represented by the symbol, dy/dx and the second derivative −3 by the symbol, d²y/dx² . These symbols, lacking the name and parentheses do not convey the functional aspect of the −4 derivatives. However the symbol dy/dx represents a number −5 which magnifies or compresses small changes in x to produce closely the corresponding changes in y which can be written Figure 2 The rotation of a secant line until it coincides with as dy = dy/dx dx. In the notation of Leibniz, the derivative the tangent appears as a quotient of the differential variables dx and dy. This equation naturally extends to a formula, called the A secant line, R, is drawn through the point P which chain rule, for the derivatives of multiple compositions of intersects the circle at the point Q. If the point of intersection, functions. If y = f(u) and u = g(w) and w = h(x) then Q, is moved along the circle toward the point P, the secant line, R, will be forced to rotate clockwise. When Q coincides dy = dx . Stated in words, if we are considering with P the secant line will become tangent to the circle and coincide with the tangent line, T, and acquire the same slope functions of functions of functions, that is, chains of functions, as T. This value, the slope of the tangent line at a given point, the effect of making small changes in the initiating variable, P, is a measure of the direction of the curve at P. This value is dx, on the terminal variable, dy, can be easily computed by called the derivative of the describing function at the point P. multiplying dx by the derivatives of each of the functions in An animation3 of a rotating secant line approaching a tangent the chain. line can be found on the Internet. Some additional notes on notation will be needed. It is seen that for a curve the direction of the tangent line Individual points are described using an equal sign e.g. x = 6 changes from point to point, implying that the slope of the or x = 7. Intervals whose endpoints are a and b are described tangent line is a function of position on the curve. This using inequalities, e.g., a < x < b, which is most clearly read function which is called the derivative function, m(x) is as ‘x is between a and b’. Therefore, it will be seen below that related at each point to, 1) the direction of the tangent line, 2) the zeros, extreme points and points of inflection will be angle of inclination of the tangent line and 3) the ratio of described with equalities. Additionally inequalities will be incremental differences. needed to describe the alternating intervals where the function Another concern of differential calculus regards the turning is positive or negative, the function is rising or falling and of the curve. If at a point, the curve lies above its tangent line; where the function is turning up or down. that is, the curve turns upward, then the slopes of the nearby tangent lines increase as x increases and if the curve lies below A definition of calculus its tangent line, then the slopes of the nearby tangent lines will There is an assumption that holds in math courses that the decrease and the curve turns downward. Smoothly increasing author of the text has a clear idea of what constitutes calculus slopes implies that the derivative of the slope is positive. If which certainly the student will also have by the end of the the original curve is described by the equation y = f(x) and the course. Is calculus the study of functions? Some very slope of the tangent line is described by the derivative function intelligent men have dreamed up examples of fantastic m(x) then we will need the derivative of the derivative to functions which most students do not need at the moment and describe whether the curve is turning up or turning down.