![]() In addition, Sketchpad contains a computer algebra system capable of symbolic differentiation ( Number | Define Derivative Function). The common Sketchpad tools used in calculus are the basic geometry tools of the Toolbox and the Construct menu, combined with functions ( Graph | New Function), their numeric evaluation for specific or calculated x-values (via the Calculator), and their graph on a given coordinate system ( Graph | Plot Function). Think of editing a function’s equation as calculus’s dynamic equivalent of dragging an independent point in elementary geometry!) (As a shortcut to Edit | Edit Function, just double-click a function. Therefore, repeatedly changing a function’s expression in a Sketchpad model can allow you to see or demonstrate how that single model applies to an entire class of mathematical behaviors. For example, if f' is the derivative of f, then f'( x 0) determines the slope of the tangent to f at x 0 whether f is defined as 2 x or x 2 or sin x. Importantly, in many dynamic models of general definitions, the exact equation of a function is less important than its relation to other objects in the model. This experience gives them a concrete image-and an accessible, tangible process for extending it-as the foundation for more abstract concepts and definitions. With Sketchpad, students can build and see these physical representations-and they can see how formal definitions emerge by reasoning from a few successive approximations, through a few dozen, and then through hundreds or thousands of better approximations all approaching the limit. Similar dynamic visualizations-of numeric integration through Riemann sums, or of slope fields and differential equations-repeatedly demonstrate how continuous calculus definitions emerge from finite processes taken to the limit as key distances approach zero. But if ∆ x were zero, A and B would be the same point-and so the secant would no longer be defined. Motivate the derivative through the realization that ∆ x can’t actually be zero for purely geometric reasons: two points define a line. But as you drag the two points closer to each other-as ∆ x approaches zero-the secant approaches the tangent, since at small enough scale the function is locally straight (and thus, locally collinear to its tangent). Depending on where you place your points, this secant will not be a very good approximation of a tangent. Now pursue this idea to think about tangency: zoom out to see the curvature again, and construct a secant to your function. Zoom in by dragging the unit point on the coordinate system away from the origin, and keep zooming until the function appears linear. Sketchpad excels at modeling and illustrating many ideas based on graphical analysis, beginning with a core idea of calculus: at a sufficiently small scale, almost everything appears linear.įor example, choose Graph | Plot New Function and plot a nonlinear function that passes through the origin, perhaps f( x) = x( x – 1)( x + 1). Sketchpad can offer unique insight into any topic for which mathematical visualization is relevant-classical and modern geometries, graphing, complex analysis, statistical charts and diagrams, physical simulations, topology, and so on-with its connection between direct manipulation and continuous visualization. ![]() ![]() Because Sketchpad’s functionality focuses on fundamental mathematical objects and operations, there’s no upper bound to the type of mathematics you can explore and model. ![]()
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