Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
User Guide
Approximation of Arc Length of Quadratic Function
GeoGebra Applet: ArcLength Approximation ProjectileMotion sol.2.ggb
Link: http://geogebrawiki.wikispaces.com/Arc+Length+Approximation
GeoGebra tools:
Insert text: Creates static and dynamic text or LaTeX formulas in the Graphics
View.
Move: Drag and drop free objects with the mouse.
Slider: Creates a slider for a number or an angle.
Move: Move Graphics View.
GeoGebra commands:
Root[Polynomial]: Yields all roots of the polynomial as intersection points of the
function graph and the x‐axis.
Function[f, Number a, Number b]: Yields a function graph, that is equal to f on the
interval [a, b] and not defined outside of [a, b].
Segment[Point A, Point B]: Creates a segment between two points A and B.
Sequence[Expression, Variable i, Number a, Number b, Increment]: Yields
a list of objects created using the given expression and the index i that ranges
from number a to number b with given increment.
Element[List, Number n]: Yields the n element of the list.
Sum[List, Number n of Elements]: Calculates the sum of the first n list elements.
Length[Function, Number a, Number b]: Yields the length of the function graph in the
interval [a, b].
Problem:
A projectile is launched upward at 10 m/s from a platform that is 50 meters high. Find the
distance the projectile travelled (Fig.1).
Fig.1
__________________________________________________________________________________________________________
Biljana Janakievska
1
Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
Solution:
The distance the projectile travelled can be calculated accurately with calculus and the formula
for "arc length". If f(x) is continous function on interval [a,b], then the arth length of the graph of the
f(x) between [a,b] is:
1 dx
Another way to calculate the arc length is to use the fact that a curve in the plane can be
approximated by connecting a finite number of points on the curve using line segments to create a
polygonal path (Fig.2). Since it is straightforward to calculate the length of each linear segment
(using the Pythagorean theorem in Euclidean space (Fig.3)) the total length of the approximation can
be found by summing the lengths of each linear segment.
Fig.2
Fig.3
Polygonal approximations are linearly dependent on the curve in a few select cases. One of these
cases is when the curve is simply a point function as is its polygonal approximation. Another case
where the polygonal approximation is linearly dependent on the curve is when the curve is linear.
This would mean the approximation is also linear and the curve and its approximation overlap. Both
of these two circumstances result in an eigenvalue equal to one. There are also a set of
circumstances where the polygonal approximation is still linearly dependent but the eigenvalue is
equal to zero. This case is a function with petals where all points for the polygonal approximation
are at the origin.
If the curve is not already a polygonal path, better approximations to the curve can be obtained
by following the shape of the curve increasingly more closely. The approach is to use an
increasingly larger number of segments of smaller lengths. The lengths of the successive
approximations do not decrease and will eventually keep increasing—possibly indefinitely, but for
smooth curves this will tend to a limit as the lengths of the segments get arbitrarily small.
That's now calculate / approximate arclength with GeoGebra.
Construction protocol:
Plot the graph of the quadratic function f(x).
Restrict it to the arc which length is calculated (determine points on the curve of f(x) for
x = a from x = b if you to evaluate the arc-length in the interval x ∈ [a, b]).
Generate the points on the arc.
Constitute the line segments, attaching every successive point on the curve.
Calculate the sum of the length of the line segments.
Compare the approximated sum with the exact length of the arc (calculated with
command Length).
__________________________________________________________________________________________________________
Biljana Janakievska
2
Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
1.
Manage the view of the GeoGebra window
Open Geogebra window. From the View menu choose Algebra (or click Ctrl+Shift+A),
Spreadsheet (Ctrl+Shift+S) and Graphisc (Ctrl+Shift+1) to make Algebra, Speadsheet and
Graphics view visible (if they are not).
2.
Insert text
Click on the
Insert text tool and then in Graphics window to insert text. A dialog window
opens (Fig.4). In the Edit window type text "Number of segments". Click Ok. The text is
inserted in the Graphics window.
Use the
Move tool to change the position of the text in the Graphics window. Click on the
tool then on the text and drag it to new position.
To change the color of the text right click on the text and choose
on the Color tab and choose the color.
Fig.4
3.
Object Properties. Click
Fig.5
Insert slider for the number of segments
Click on the
Slider tool and then in Graphics window. A dialog window opens (Fig.5).
GeoGebra suggests a for name of the slider. Insert n in the Name bar for the name of the slider
and the number of segments. Insert 1 for min, 20 for max and 1 for Increment. Now slider n can
get value that is one of the numbers from 1 to 20. Click on the Slider tab and uncheck the
button
to make the slider moveable on the screen. If you want you can change the slider
position to vertical (default is horizontal) and the width of the slider (default is 100). Click on the
Animation tab to change the speed of moving the point on the slider (default is 1) and the way
of moving the point on the slider (Oscillating, Increasing, Decreasing and Increrasing (Once)).
We'll use the deafault settings in the construction.
Moving the point on the slider will cause drawing of the segements. The current number on the
slider will correspodent to the number of the segments. The sum of the length of the segments
will be the approximated length of the arc (this will be set later in the construction). You will see
that by increasing of the number of the segments, the approximated sum will be getting more
closer to the exact length of the arc.
__________________________________________________________________________________________________________
Biljana Janakievska
3
Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
4.
Insert sliders for the initial velocity and the initial height
Repeat the procedure from steps 3 and 4 to insert sliders for initial velocity and initial height. Set
the minimum and maximum value of initial velocity to 1 and 20, and the minimum and
maximum value of initial height to 1 and 100. Change the color of each of the sliders. Double
click or right click on the slider in the Graphics window or right click on the name of the slider
in the Algebra window and then click on the Color tab. Choose the color and click on the Close
tab.
5.
Draw the quadratic function
Insert f(x)=4.9x v0x h0 in the Input bar in the down left bottom of the GeoGebra
window (see Fig.6) and press the Enter key. Use the symbol menu in right side od the input bar
to insert grade 2 (Fig.7).
Fig.6
Fig.7
The curve is drawn in Graphics window and equation of the curve is written in the Algebra
window (Fig.8). To change the look of the curve right click on the equation of the curve in the
Algebra window or graph in the Graphics window and choose Options. Click on the Color tab
to change the color and on the Style tab to change style and thickness of the curve.
Fig. 8
6.
Find the roots of the quadratic function
Use the command Root[ <Polynomial> ] to find the roots of the function. Type root(f) in the
Input bar and press on the Enter key. The roots, points A an B (the names are suggested by the
__________________________________________________________________________________________________________
Biljana Janakievska
4
Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
application) are marked in the Graphics window, and their coordinates are written in the Algebra
window (Fig.9).
Fig.9
7.
Restrict the quadratic function
We'll restrict original function to new function f&'(&x. f&'(& x=f(x) where x gets values from
0 to the x-coordinate of the second root B.
Insert a=0 and b=x(B) (the x-coordinate of point B) in the Input bar and press on the Enter key.
Use the command Function[ Function, Start x-Value, End x-Value] to restrict a defined
function in interval [Start x-Value, End x-Value].
Write f&'(&x=Function[f,a,b] in the Input bar and press the Enter key. Now function f(x) is
restricted on interval [a,b], where a=0 and b=x(B) (Fig.10).
Fig.10
To make the points A and B and function f(x) invisible in the Graphics window simply uncheck
the mark left of the object (or right click on the object in the Algebra or Graphics window,
choose the Basic tab and unclick option Show Object).
__________________________________________________________________________________________________________
Biljana Janakievska
5
Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
8.
Generate the coordinates of points
First we'll find the step of change of the x coordinates of the points. ∆x=(b-a)/n, where n is the
number of segments (slider n), and the [a,b] is interval on which the function f(x) is restricted.
Insert ∆x=(b-a)/n in the Input bar and press on the Enter key.
Use the command Sequence[Expression, Variable i, Number a, Number b, Increment] to
create
a list of objects using the given expression and the index i that ranges from number a to number
b with given increment.
Type ListXCoord =Sequence[a + i ∆x, i, 0, n, 1] in the Input bar and press on the Enter key.
Since the left margin of the restriction of the function is a=0, the first point will have x coordinate 0, the second 0+∆x, the third 0+2∆x, and the last 0+n∆x.
Type ListYCoord =Sequence[f(a + i ∆x), i, 0, n, 1] in the Input bar and press on the Enter key.
So the first point will have y – coordinate f(0), the second f(0+∆x, the third f(0+2∆x, and the
last f(0+n∆x.
9.
Generate the points
Use the command Element[List, Number n] to yield the nth element of the list.
Type ListPoints = Sequence[(Element[ListXCoord, i], Element[ListYCoord, i]), i, 1, n + 1, 1] in
the Input bar and press on the Enter key. The number of generated points will be n+1, where n is
the number of segments.
A list of points is written in the Algebra window and marked in the Graphics window (Fig.11).
To change the color of the points right click on the text and choose
on the Color tab and choose the color.
Object Properties. Click
Fig.11
10. Generate the segments
Use the command Segment[Point A, Point B] to create a segment between two points A and B.
Type ListSegments=Sequence[Segment[Element[ListPoints, i], Element[ListPoints, i + 1]], i, 1,
n,1] in the Input bar and press on the Enter key.
__________________________________________________________________________________________________________
Biljana Janakievska
6
Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
The command creates list of n segments that are created between two succesive points.
The sequence of segments is written in the Algebra Window and drawn in the Graphics
window(Fig.12).
To change the color and line thickness of the segments right click on the list of segments in the
Algebra window and choose
Object Properties. Click on the Color tab to choose the color
and on the Style tab to change the thickness.
Fig.12
11. Calculate the sum of the segments
Use the command Length[Function, Number a, Number b] to find the length of the function
graph in the interval [a, b].
In the Input bar type ArcLengthExact=Length[f, a, b] and press on the Enter key.
Use the command Sum[List, Number n of Elements] to calculates the sum of the first n list
elements.
In the Input bar type ArcLengthApprox=Sum[ListSegments, n] and press on the Enter key.
The exact and approximated length are written in the Algebra window.
Click on the
Insert text tool and then in Graphics window to insert text. A dialog window
opens . In the Edit window insert text "The length of the arc is" and then click on the number
ArcLengthExact in the Algebra window . Click Ok. The text is inserted in the Graphics window.
Make the color of the text same as the color of the restricted function.
Now, click again on the
Insert text tool, in the Edit window insert text "The
approximated length of the arc is" and then click on the number ArcLengthApprox in the
Algebra window . Click Ok. The text is inserted in the Graphics window. Make the color of the
text same as the color of the segments.
__________________________________________________________________________________________________________
Biljana Janakievska
7
Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
12. Test the construction
Move the point on slider v0 to change the initial velocity.
Move the point on slider h0 to change the initial height.
Move the point on slider n to change the number of segments. See that increasing the value of n
(decreasing the value of ∆x) indicates the approximated length to come closer to exact length
calculated with command Length (Fig.13).
Fig.13a) n=3
Fig.13b) n=18
13. View construction protocol
From the View menu choose Construction protocol (or press Ctrl+Shift+L) to view protocol of
the construction (Fig.14).
__________________________________________________________________________________________________________
Biljana Janakievska
8
Approximation of ArcLength of Quadratic Function
__________________________________________________________________________________________________________
Fig.14
From the View menu choose Navigation Bar for Construction Steps >Show to view protocol
of the construction step by step.
The Navigation Bar provides a set of navigation buttons and displays the number of construction
steps (e. g., 2 / 20 means that currently the second step of a total of 20 construction steps is
displayed):
• button
: go back to step 1.
• button
: go back step by step.
• button
: go forward step by step.
• button
: go to the last step.
• Play
: automatically play the construction step by step.
Note: You may change the speed of this automatic play feature using the text box to
the right of the Play button.
• Pause
: pause the automatic play feature.
__________________________________________________________________________________________________________
Biljana Janakievska
9