# Autograph Newsletter 4 – Calculus: Differentiation

 Welcome Welcome to the fourth Autograph newsletter! Each jam-packed edition will look at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better. Hope you enjoy it! Craig Barton Advanced Skills Teacher, creator of www.mrbartonmaths.com and TES Secondary Maths Adviser. Follow me on Twitter: @TESMaths
 Introduction I have found that students are aware of calculus long before they study it in a mathematics classroom. Often they have heard it mentioned in movies or TV shows, and witnessed seeming incomprehensible scribbles on blackboards by bearded brain-boxes. Bearing this in mind, there is a danger that students can become overawed by the concept and methods involved with calculus before they have even got going. If we are not careful, it can become a rather abstract and daunting topic, which is clearly a major problem as the majority of advanced mathematics is built upon its foundations. Fortunately, help is at hand! By using dynamic geometry software we can bring the topic to life, looking at where it all comes from, right up to the most advanced features. In this newsletter we look at how we might use Autograph to introduce and study the first part of calculus – differentiation.
 Diagnostic Question Diagnostic questions are ideal to use at the start of the lesson to enable you to get a quick and accurate picture of your students’ levels of understanding. They are designed in such a way that common misconceptions that your students may hold should steer them to one of the incorrect answers, thus allowing you to learn where the problems lie from their responses. Typically I give my class 30 seconds thinking time and then ask them to hold up their fingers: 1 for A, 2 for B, etc. Free Online Autograph Activity The Rules of Differentiation Can you discover the rules for differentiating quadratic functions? Can you then apply this to other polynomials? These Autograph activities do not require the full version of Autograph to run them. You just need to install the free Autograph Player (you will be guided through how to do this), which means you can use these activities in the classroom or set them for your students to do at home.
Ideas for Extension
The following ideas for extending the topic of differentiation require the full version of Autograph. Click on the image to download the individual Autograph files.
Idea 1 – Differentiation from First Principles
Download 1.firstprinc.agg
Demonstrate to your students where the concept of differentiation comes from by using the concept of a limit.
 • The file shows a calculation for the gradient of the line segment joining the two points • The second point has an x value of “1+h” • Use the Constant Controller to gradually reduce the value of h • Drag and zoom in around the two points as h gets smaller • Let the students see that the gradient of the line segment is approaching that of the curve • What value does this gradient tend towards? • You can then try your points in a different position • You can also double-click on the curve and type in a new equation
Idea 2 – The Gradient Function
Download 2.gradientfunction.agg
Autograph has a lovely feature that allows you to instantly plot the gradient function of any function
 • Challenge your students to predict what the gradient function of the given curve will look like • What shape will it have? • Where will it cross the x-axis? • How about the y-axis? • Make sure Slow Plot mode is on • Simply left-click on the curve and click on Gradient Function from the top toolbar and click OK • Why does the gradient function pause at a certain point? • Can anyone give me a function that would lead to a straight line gradient function? • How about a negative quadratic? • How about a gradient function that passes through the origin? • When you are ready, click on the gradient function, press delete, and then double-click on the function and pick another one to investigate!
Idea 3 – Discovering e
Download 3.ex.agg
Instead of telling your students about one of the most important numbers in mathematics, let them discover it for themselves!
 • The file shows the graph of y = ax and its gradient function • The value of a is set to 2 • What will happen to the graph if I increase the value of a to 3? • What will happen to the gradient function? • Use the Constant Controller to change the value of a • Encourage your students to speculate about a function whose graph is identical to that of its gradient function • Use the Constant Controller and Zoom-in to try to find that special value of a • If y = ex, dy/dx = …
Idea 4 – The Chain Rule
Download 4.chain.agg
We can use the graphs of trigonometric functions to discover the chain rule
 • This is the graph of y = sin(bx – c), with b set to 1 and c to 0, so we have y = sin(x) • What would the gradient function look like? What shape would it take? • Left-click on the curve and click on Gradient Function from the top toolbar and click OK • What do you think will happen to the graph if we increase b to 2? • What about the effect on the gradient function? • Use the Constant Controller to change the value of b • Can you guess what the equation of the gradient function is? • Enter the equations of your students’ guesses to see if any of them match the gradient function • What do you think will happen to both graphs if I change the value of c? • Now you can experiment by starting with the graph of y = cos(x) or even y = tan(x)
 Video Tutorials The following video takes you through, step-by-step, how you can use tangents to investigate why some curves have the same gradient function Handy Autograph Tip
Slow Plot is an extremely useful feature of Autograph. It enables you to challenge your students to predict the shape and key features of a function before drawing it. Here are a few ways of using it. Open Autograph in Standard Level Turn on Slow Plot mode Challenge your students to Scribble on the key points that the graph of y = x³ – 2x² + x will pass through Now enter the equation and watch it slowly take shape To pause the plot at key points, either press the Pause button or hit Spacebar on the keyboard To immediately jump to the end of the plot, press Fast Forward or turn off Slow Plot mode To watch the plot again, just click replot You can then do exactly the same process to try and figure out what the gradient function looks like. Note: The gradient function immediately pauses at stationary points and points of inflection.