**Subject Area: **Mathematics

**Course: **Calculus and Vectors

**Topic: **Many Topics

**Task: **Curve Sketching

**Technology Tool: **Desmos

**Description**

One of the fundamental skills that students are expected to develop in the study of calculus is the ability to sketch any given function. More specifically, students are expected to be able to determine the behavior of a function at various key points, through function analysis and differentiation. Curve sketching is a large part of the course, and often times, students find themselves clueless as to whether or not the curve they have sketched is even remotely accurate. This is particularly true when the function involves logarithms, long polynomials, and fractions. Often times, students check their work by plugging in certain values into their calculator, replacing ‘x’ in the function with random values.

**Suggested Use of Technology Tool**

It is recommended that Desmos is only used after students have already tried their best to sketch a function. There is no need to worry about typing complicated functions, as you can simply type in ‘sin’, ‘arctan’, ‘e’, ‘x^3’, ‘d/dx’, ‘log’, or ‘sqrt’. Don’t have a laptop with you in class? There’s an app for your phone!

*Sketching related functions (ex: revenue and cost): *Desmos allows you to graph many functions as a time. The functions are color coded, making it very easy to see the point of intersection between two function. In addition, students can define a function in terms of other functions, a very useful feature in calculus.

*Checking your steps: *When finding the max/min/inflection points of a graph (as required during curve sketching), students must find the function’s first and second derivatives. Desmos can graph not only the function written in the textbook, but the derivative of the function as well! Just type (d/dx) before the name of the function. In the event that a student realizes that his final curve does not look right, he or she can go back and check the accuracy of the first/second derivatives.

*Checking your key points: *Desmos also provides the coordinates of a function’s key points (max/min, inflection, roots). They are labelled as dots on the graph, which you can press to reveal coordinates.

**Why and how would this technology tool improve student learning?**

The benefit of being able graph functions on Desmos extends far beyond the fact that it can save lots of time when doing calculus homework. When students check with their calculators, they often only check points that they consider essential to the function. For instance, a student may notice that (2, 30) is a max point on the graph, and check that setting x to 2 will result in 30. However, he or she may not realize that another max exists in the function. Perhaps (2, 30) is a point on the function, but not a max! Thus, Desmos not only saves time, but provides 100% accuracy (given that the function is inputted correctly).

There is no doubt that Desmos is a very useful tool for calculus students who are perfecting the art of curve sketching! It will improve their confidence (and mark!) on quizzes, tests, and exams!