# JavaSketchpad Calculus Applets

**Construction Details**

One way to document a Geometer's Sketchpad construction is to ** Show All Hidden** from the

**Display**menu,

**from the**

*Select All***Edit**menu, then

**from the**

*Create New Tool***Custom Tool**menu. The tool's script can be displayed and printed, and with experience, the steps in the script can be associated with the proper Sketchpad constructions.

The details below elaborate upon a few features of the GSP constructions, and how the JavaSketchpad code was edited to produce the final applet.

Secant/Tangent Applet | Construction Details | |

1. | ## Setup a coordinate system with labeled axes.
The GSP coordinate system has labeled axes with tick marks but the labels and tick marks will not be displayed by JSP, so we must create our own.
Make tick marks by | |

2. | ## Graph the curve y = .02x | |

3. | ## Construct the line tangent to the curve at (a, f(a)).The tangent line is the linearization at x = a consisting of points (q, f(a) + m
| |

4. | ## Construct the secant line from (a, f(a)) to (a + h, f(a + h)).Again, Calculate m | |

5. | ## Add ButtonsIn GSP one button can toggle between hide and show, but in JSP separate buttons are required. | |

6. | ## Label the tick marks.In Sketchpad, when a point is hidden its label is also hidden. To label the tick marks without showing the points, JSP code was added to create hidden captions for each point with the |

Derivative Definitions Applet | Construction Details |

The The |

Area Applet | Construction Details | |

1. | ## Graph the piecewise functionDefine a new coordinate system and construct a segment from –4 to 4 to be the domain of f(x), as done for the Construct a point on this segment and label it " Calculate and label it y1, and label it y2, and label it y3.
y = y1 + y2 + y3 is the equation for the piecewise function. Select parameters w and y and | |

2. | ## Display the region between the piecewise function and the x axis over [a, b].Construct points a and
Measure the ratio of | |

3. | ## Calculate the area of the region.A continuous antiderivative of the piece–wise function is In the sketch, A = F(a) and B = F(b). JSP does not support function definitions, so A and B are calculations. then A = a | |

4. | ## Note that expressions such as have been used as "indicator" or "characteristic" functions because .This technique is recommended in Erroneous values for the piecewise function at the endpoints of a subinterval can be overcome by rounding the value of the characteristic function. This rounding was done for the JSP version of the area calculation. |

Area Function Applet | Construction Details |

The area function applet is an extension of the area applet. Plot the point (-9, 0) and use it as the origin for a second coordinate system. Then construct the graph of the area function relative to this new coordinate system as the locus of a plotted point (x, Area(x)). The point |

Inverse Function Applet | Construction Details | |

1. | ## Construct a rectangular coordinate system and the interval [–π/2, π/2] on the x-axis to serve as the principle domain of the sine function.
Select measurement π/2 and parameter t
Select measurement –π/2 and parameter t
It might also be useful to plot points (–1, 0), (0, 1) and (0, -1) if you wish to see these points on the axes.
Similarly plot points (0, –π/2) and (0, π/2) on the y–axis joined by a segment as the domain of arcsin x then hide the y-axis. This construction also plots the points on the x- and y-axis to correspond to the common fractional parts of π. | |

2. | ## Sketch the graph of sin x as the locus of a plot.Select a point with on the interval [–π/2, π/2] and label it " From the edit menu's preferences set units for angles as radian and for distance as pixels. From the
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3. | ## Draw the tangent to the curve at point (x, sin(x) )
| |

4. | ## Draw arcsin(x) and its tangentSelect the calculations sin(x) + (dy/dx)(k–x) and h in this order, and | |

5. | ## Modify the JSP code to label the axis tick marks.The "π" symbol from the GSP construction will display as the letter "p" in the JSP applet. In order to label these points, expressions for the common fractional parts of π were created with an equation editor and stored as gif files in the It is possible to insert the asci character number into the label format for a point. For example, in Windows' |

Volume of a Cone Applet | Construction Details | |

1. | ## Construct the baseTo give perspective to the cone, its base will be an ellipse . In this sketch, a = 1 and the ellipse passes through point P(c, d) to control the appearance of perspective. So , and the ellipse is the locus of point (x, y) where . The interior of the ellipse can be shaded like the area in the previous piecewise function, as the locus of a segment. | |

2. | ## Construct the Cross-sectionThis is constructed as the base of the cone but proportional to the distance x. | |

3. | ## The Layer CommandTo control the foreground/appearance as the cross-section obscures the base and the rear surface of the cone, the The cross section, its perimeter, and its radius are all given the value "layer(50)," sufficient to obscure the base. The top segment of the cone's altitude and the front surface of the cone are given layer(60) so they will appear on top of the cross section. | |

## Pyramid Challenge | ||

## This applet was constructed utilizing Cathi Sanders' |