JavaSketchpad Calculus Applets
One way to document a Geometer's Sketchpad construction is to Show All Hidden from the Display menu, Select All from the Edit menu, then Create New Tool from the 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.
Setup a coordinate system with labeled axes.
Define a coordinate system, then Select the coordinate system and set the line width to Thick so JSP will show a grid instead of merely "dot paper."
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. Calculate parameters t1 = 1, t2 = 2,… , t10 = 10, t11 = –1,… , t20 = –10, and t21 = 0. Select parameters 0 and –10 and Plot as (x, y). Label this point "–10." Likewise Plot as (x,y) and label all integer points on the axes. The labels will be hidden along with the points, but new labels will be attached later.
Construct the segment connecting points (–2, 0) and (6, 0), then Construct the segment connecting points (0, –10) and (0, 10). These segments are the displayed x– and y–axis. Hide the original coordinate system axes.
Make tick marks by translating every point on each axis ±5 pixels perpendicular to the axis, then construct segments connecting the axis points to their translations. Hide the points. Tick marks are easy to construct if a custom tool is made.
Graph the curve y = .02x4 – x2 + 3x + 1 as the locus of plotted points.
Construct a domain for the function. The segment connecting points (–2, 0) and (6, 0) as constructed above will serve as the domain. If you do not construct a domain, then the graph produced by JSP may be less than you expected.
Select the interval [–2, 6] and Construct a point on this segment and label it "p." Select the points (0, 0), (0, 1) and p in that order and Measure the ratio. This ratio will be the coordinate of p; label the ratio "p." Calculate 0.02p4 – p2 + 3p + 1, and, if you wish, label the calculation "f(p)." Select p and f(p) and Plot as (x, y). Then select the plotted point (p, f (p)), the point p on the x-axis and the interval [–2, 6] and Construct the locus. You should see the graph of f(x) = .02x4 – x2 + 3x + 1. It will be useful to create a new custom tool based upon the value p and the calculation f(p).
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) + mtan(q – a)) for all q ∈ [–2, 6].
Select the interval [–2, 6] and Construct a point on this segment and label it "a." Select the points (0, 0), (0, 1) and a in that order and Measure the ratio. This ratio will be the coordinate of a; label the ratio "a." Calculate 0.02a4 – a2 + 3a + 1, and label the calculation "f(a)." Select a and f(a) and Plot as (x, y). Label this point as (a, f(a)).
Calculate 0.08a3 – 2a + 3 and label the measurement mtan , which label is typed as "m[tan]." Now construct the tangent line as a locus. Select the interval [–2, 6] and Construct a point on this segment and label it "q." Select the points (0, 0), (0, 1) and q in that order and Measure the ratio. This ratio will be the coordinate of q. Label the ratio "q." Calculate f(a) + mtan(q – a). Select the ratio q and the latest calculations and Plot as (x, y). Then Select the plotted point, the point q on the x-axis and the interval [–2, 6] and Construct the locus. You should see the graph of the tangentline.
Construct the secant line from (a, f(a)) to (a + h, f(a + h)).
Again, Select the interval [–2, 6] and Construct a point on this segment and label it "a + h." Select the points (0, 0), (0, 1) and a + h in that order and Measure the ratio and label the ratio "a + h." Calculate 0.02(a+h)4 – (a + h)2 + 3(a + h) + 1, and label the calculation "f(a + h)." Select a + h and f(a + h) and Plot as (x, y).
Calculate msec using a, f(a), a + h and f(a+h). The secant line will be the locus of points (r, y) such that y = f(a) + msec(r – a) for all r ∈ [–2, 6]. Construct this as a locus of points the same way the tangent line was constructed.
In GSP one button can toggle between hide and show, but in JSP separate buttons are required.
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 FixedText construction, then attach each caption with a point using the PeggedText construction.
Derivative Definitions Applet
The Derivative Definitions applet is a copy of the Secant/Tangent applet, but adds another set of calculations and labels to display the difference quotient in the form . These additions are lines 230–238 of the JSP code. You will notice, for example, that lines 230–234 are repeats of lines 160–164, but with different labels. The Derivative applet also adds another pair of buttons to Show/Hide the labels for the second form of the quotient.
The Derivative applet then adds a pair of simultaneous buttons, a feature new to JSP4, one to show the first form while hiding the second, the other button to hide the first form and show the second. Another feature new to JSP4 allows these simultaneous buttons to have GIF images displayed. The images were written with an equation editor, saved as .gif images, and are stored in the images subdirectory of the JSP folder.
Graph the piecewise function
Define a new coordinate system and construct a segment from –4 to 4 to be the domain of f(x), as done for the Derivative applet.
Construct a point on this segment and label it "w." Select O, 1 and w in that order and measure their ratio. Label the ratio "w." This is the abscissa of the point w and will be the independent variable of the piecewise function.
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 Plot as (x, y). Select this plotted point, the point w and the interval [–4, 4] and construct the locus. This will display the piecewise function.
Display the region between the piecewise function and the x axis over [a, b].
Construct points a and b on the interval [-4, 4], construct the segment joining a and b, and construct a point x on the segment joining a and b.
Measure the ratio of O, 1 and x , and label the ratio "x." Calculate f(x) = z1+z2+z3 as above, then select x and f(x) and Plot as (x,y). Construct a segment joining x to this point (x, f(x)). The segment should be thick and of some appropriate color. Select this segment, the point x and the segment joining a and b then construct the locus. The locus of the segment will display as the region between the curve and the x-axis over [a, b]
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 = a1+a2+a3=F(a). Likewise calculate B=b1+b2+b3=F(b). The area of the region between f(x) and the x-axis is B – A = F(b) – F(a).
Note that expressions such as have been used as "indicator" or "characteristic" functions because .
This technique is recommended in Geometer's Sketchpad's "Help" system.
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
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 a has been renamed x. Tick marks and labels are added as before.
Inverse Function Applet
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.
Define Coordinate System from the Graph menu. (Grid Form should be set to Square Grid.) Select the coordinate and make the line width thick if you wish to see the grid, or Hide the coordinate system if the grid lines will not be important for the graph of sin x and arcsin x.
Calculate a new parameter t1 = 0 from the Measure|Calculate|value menu.
Calculate π/2 and –π/2 from the Measure menu
Select measurement π/2 and parameter t1 = 0 then Plot as (x, y) from the Graph menu. Label this point as π/2 and from the style dialogue box choose font Symbol. The character "p" will display as "π" in the Symbol font.
Select measurement –π/2 and parameter t1 = 0 then Plot as (x, y) from the Graph menu. Label this point as –π/2 with style as before.
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.
Construct a segment with endpoints –π/2 and π/2 which will serve as the principle domain for the sin function then hide the x-axis.
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.
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 "x." Select the origin (0, 0), the unit point (1, 0) and x in this order then Measure the ratio. Label this ratio measurement "x." This ratio will serve as the abscissa for the point x.
From the edit menu's preferences set units for angles as radian and for distance as pixels. From the Measure menu Calculate the value of the function sin() at the ratio measurement labeled "x." Note that as point x moves along the interval [–π/2, π/2] the value of sin(x) will range from –1 to 1.
Select measurements x and sin(x) and plot as (x, y). Select this point, the point x, and the segment from –π/2 to π/2 and Construct the locus. The graph of sin(x) should appear.
Draw the tangent to the curve at point (x, sin(x) )
Construct a point on segment [–π/2, π/2] and display its label, which we will call "H" here. Select the origin, the unit point and H in that order and measure the ratio. Label the ratio "h", the abscissa of point H.
Calculate cos(x) and label it as dy/dx. This will be the slope of y = sin(x) at x.
Calculate sin(x) + (dy/dx)*(h–x). Select h and this last calculation and Plot as (x, y). Select the point H, the segment [–π/2, π/2] and this plotted point and construct the locus. This locus should be the tangent to curve sin(x) at point (x, sin(x)).
Draw arcsin(x) and its tangent
Select the calculations sin(x) + (dy/dx)(k–x) and h in this order, and Plot as (x, y). Then select this plotted point, H, and the segment [–π/2, π/2] then construct the locus. This will be the tangent to arcsin(x).
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 images subdirectory of the jsp folder. The ImageOnPoint command, new to JSP4, was then used to attach the images to axis tick points. Of course, the actual point can be hidden, resulting in what appears as labeled tick marks.
It is possible to insert the asci character number into the label format for a point. For example, in Windows' Courier and Times fonts, the asci number for π is 960, so "label(‘π/2')" will display the expression "π/2" as a label. However, this will not be recognized in other operating systems.
Volume of a Cone Applet
Construct the base
To 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.
Construct the Cross-section
This is constructed as the base of the cone but proportional to the distance x.
The Layer Command
To control the foreground/appearance as the cross-section obscures the base and the rear surface of the cone, the layer command new to JSP4 was used.
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.
This applet was constructed utilizing Cathi Sanders' Perspective Drawing with the Geometer's Sketchpad.