VISUALIZING SPECIAL FUNCTIONS AND APPLICATIONS IN CALCULUS 1

Modern calculus courses are incorporating many more applications and special functions into their content. Using and applying these special functions is oftentimes a complete mystery to our students. To help them better explore real-life applications and the prop-erties of special functions, we have developed a special set of visual tools that we use with our students. In this paper, we present a collection of examples that demonstrate the use of these visual tools. Defining a Special Function In the first week of our calculus courses when we first introduce the concept of a fun tion to our students, we talk about input, output, and the rule describing the function. TEMATH’s Rectangular Tracker tool is used to help students visualize the concept of inputting a value of x, outputting a value of y, and using a graph to represent this process (see Figure 1). A few weeks later, we introduce the concept of the derivative as a func-tion. From a purely algebraic point of view, this is quite bewildering to our students. We use TEMATH’s Dynamic Tangent tool to help students visualize the derivative as a function where the input is a value of x and the output is the slope of the tangent line to the curve. As the tangent dynamically moves along the curve of the function f (x) , the value of the slope of the tangent line is plotted creating the graph of the derivative func-tion f (x) (see Figure 2). This visual demonstration helps students develop an intuition about the derivative as a function, and more importantly, as a function that measures the instantaneous rate of change of f (x). x