TIME 2012 Technology and its Integration in Mathematics Education 10 th Conference for CAS in Education & Research July 10-14, Tartu, Estonia

Overview  Important Features of 2D plots.  What About 2D plots in Nspire CAS?  2D and 3D plots (DERIVE, Nspire).  Piecewise Continuous Functions.  Show Step Simplification.  Compact/Simplified Answers.  Controlling Precision.  Domain Definition. 2

Some Important Features of 2D Plots Given an equation of the form f(x) = g(x), one can select the LHS, the RHS or the whole equation: DERIVE plots what is highlighted when you open a 2D plot window.  This feature of DERIVE has many teaching advantages.  Example:. What do you want to plot? 3

Some Important Features of 2D Plots 4

Implicit plots and plots of two columns matrices.  Example: the system A few seconds and you see both curves, the intersection points (and the “Gröbner basis” function informs you how DERIVE solved this system!!!). 5

Some Important Features of 2D Plots This last example is a nice illustration of “Point. Click. Derive.” 6

Some Important Features of 2D Plots  Complicated equation(s) can be easier solved numerically if a starting point is given (with the help of a graph!).  Example: find one complex solution to the equation (The “solve” command fails in DERIVE: the “csolve” command of Npsire CAS finds no complex solutions but finds the 3 real solutions!) 7

Some Important Features of 2D Plots We simply need to define f(z) to be and to substitute for z a complex number x + iy. Then plot the real and imaginary parts of f(x + iy) = 0 in the same windows. The fact that DERIVE has a “Simplify Before Plotting” command is very useful (TI should add this in Nspire CAS). 8

Some Important Features of 2D Plots Now you plot #2 and #3 in the same window (“Simplify Before Plotting” being used: no necessity to simplify #2 or #3): 9

What About 2D Plots in Nspire CAS?  (In OS 3.2) The 2D plot window graph Entry/Edit accepts up to 7 different types but 2D implicit plots are still missing.  Slider bars, animations, dynamic geometry, styles and colors make each of these 2D plot windows very attractive and useful for teaching mathematics and sciences. 10

What About 2D Plots in Nspire CAS?  The plotting algorithm of Nspire CAS seems to be different from the one of Voyage 200.  In fact, in V200, if we use the Y= Editor and type y1(x) = zeros(expr, y) where expr is an expression in variables x and y, it takes a VERY long time but the implicit graph comes out!!! 11

What About 2D Plots in Nspire CAS? Let’s consider the 2D implicit curve given by the equation. There is no way to solve for y, so a real 2D implicit plotter is required for the graph. Here are the graphs showed by DERIVE, Nspire CAS and V200: 12

What About 2D Plots in Nspire CAS? DERIVE Nspire CAS Voyage

2D and 3D Plots: Important Features A fast and accurate 2D implicit plotter is a must in multiple variable calculus for plotting level curves.  Here is an example where 3D plotting will also be used.  We will plot some level curves of the function 14

2D and 3D Plots: Important Features  The autoscaling 3D plot command can be used to find the maximum value.  The highest point on the surface can be plotted because DERIVE can plot 3 columns matrices in various ways.  Splitting windows allows the teacher to show the relationship between level curves in 2D and level curves on the surface. 15

2D and 3D Plots: Important Features 16

What About 3D Plotting in Nspire CAS?  Major improvements with David Parker on board: in OS 3.2, parametric 3D plotting has been added.  Here is the graph of the function from the former example performed with Nspire CAS (I don’t know how to plot the maximum point): 17

Piecewise Continuous Functions Indicator (“CHI”), signum (“SIGN”), and Heaviside (“STEP”) functions are built- in; in Nspire CAS, only “sign” is built- in.  The symbolic integration of products with other functions works correctly.  The reason: DERIVE knows the rule: 18

Piecewise Continuous Functions  This rule is applied when you want to compute integrals some integrals (this is the case when someone needs to find a Fourier series): let’s try And let’s compute the Fourier series of a square wave. 19

Piecewise Continuous Functions DERIVE not only knows the rule but can SHOW it to the user with the “Display Step” button. Let’s talk about this now. 20

Show Step Simplification For integrals, this is comparable to be looking into a book on Standard Mathematical Tables and Formulae.  Each of the following examples has a particularity that would be lost or unknown without the Show Step Simplification option. 21

Show Step Simplification  Sometimes DERIVE is unable to get a closed form but you learn something! 22

Show Step Simplification  Sometimes, it is just amazing! 23

Show Step Simplification  Sometimes, DERIVE does exactly what a math teacher would do: 24

The Importance of Compact/Simplified Answers Solving cubic polynomial equations: when all 3 solutions are real, DERIVE uses trigonometric substitutions (Maple is using Cardano’s formula and complex numbers appear; Nspire CAS is also using trig answers but not in a very attractive way…):  For example: 25

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The Importance of Compact/Simplified Answers Solving cubic polynomial equations: when only one solution is real, all systems are using Cardano’s formula. But DERIVE’s answer is the nicest one: NO radicals in the denominator!  For example: 27

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The Importance of Compact/Simplified Answers Can your favorite CAS simplify these?  Nested radicals:  Complex numbers written in various forms: 29

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The Importance of Compact/Simplified Answers Everything should be made as simple as possible, but not simpler. Albert Einstein 31

The Importance of Controlling Precision It is important of being able to increase the number of digits of precision: Nspire CAS software is restricted to 14 digits (as the handheld), not DERIVE.  Example, the number e: let’s compute and observe what is going on. 32

The Importance of Controlling Precision 33

The Importance of Domain Definition  In Nspire CAS, one can restrict the domain of a variable INSIDE a command or definition.  An arbitrary variable x is real by default while x_ is complex and n1 is an integer.  In DERIVE, the domain of a variable can be set by a command. Moreover, the domain can also be vector, set and logical. 34

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