# Tools and techniques for visualizing concepts in Calculus and higher math

When I first started teaching Calculus in the fall of 2012, I found that my students were having problems visualizing certain topics we covered in class. The first, and one of the most noticeable topics, was the Squeeze Theorem. So I asked myself how I could show my students how certain theorems or equations could be visually seen, and not just talked about in an equation sense. So I turned to using WinPlot.

WinPlot is a free graphing utility for Windows. It can be used for Windows XP and higher, including Windows 10. WinPlot is an easy to learn software that can advance the visual aid of learning, and it will be demonstrated for calculus courses and differential equations. We will look at both two and three dimensional graphs including volumes, solids, and vector fields from differential equations, and how I use many of the capabilities in my own courses.

In my own calculus courses, I use WinPlot in a variety of ways. First, as I mentioned before, in differential calculus, I first use the software when we discuss the Squeeze Theorem. I can visually show how a function, such as f(x) = x2 sin (1x), can be bounded by two functions, f(x) = –x2 and f(x) = x2. I will also show how tangent lines move as the slope changes along a curve.

Second, in integral calculus, I use the software to demonstrate how Riemann sums can estimate the area underneath the curve. We can visually see the estimations using left, right, and midpoint rectangles. The software also will do parabolic (Simpson’s Rule) and trapezoidal estimations for the area underneath the curve.

Third, I tend to also use the software to show who the area between two functions can be rotated about an axis to form a solid of revolution. It is quite simple to animate this kind of graph, where drawing something like that by hand can be quite difficult.

Something new that I am trying this semester is to incorporate visual aspects into differential equations. Since this is my first semester teaching the course, I am slowly finding ways to show first and second order differential equations in a visual sense. Currently, I am working through families of solutions of first order differential equations, and I am showing the students the vector fields with initial value trajectories on top of the vector field. Each initial value problem can be shown quickly by clicking anywhere on the vector field.

I am also starting to create examples for the second order differential equation problems. Graphically, it is a little more difficult to get a second order problem in the software since there is no command for second order problems d2y/dx2 like there is for the first order problems dy/dx. It is possible to do use a t-parameter to do dx/dt to graph a second order differential equation, and I will demonstrate this as well.

Another positive for the software is the cost. Since it is free, students can also download the software to enhance their own learning. When I show the software in my larger sections, I will have students come up and want to download it for themselves. Since I am demonstrating the software in class, they can see how easy it is to work.

In conclusion, WinPlot is a powerful graphing software with major visual benefits for my students. It requires no previous computer coding skills, and best of all, it is a free software that works with any Windows PC. So come and see how you may be able to also use this in your own classes.

Professor Banik will be speaking at ICTCM 2017. Register today and join your colleagues either in Chicago or virtually to discuss many mathematics topics. 