Applets for Further Mathematics
This online book compiles the applets used for teaching and learning of H2 Further Mathematics (Singapore).
Table of Contents
- Conics Sections and Polar Curves
- DEF Definitions of Conics
- Reflection Property of an Elliptical Mirror
- Reflection Property of a Hyperbolic Mirror
- Reflection Property of a Parabolic Mirror
- A Way to Generate Parabola with Two Circles
- A Way to Generate Ellipse and Hyperbola with Two Circles
- Further Applications of Integration
- Further Differential Equations
- Recurrence Relations
- Mathematical Induction
- Numerical Methods
- Matrices and Linear Spaces
- Visualisation of eigenvectors and eigenvalues
- Complex Numbers II
DEF Definitions of Conics
Any conic section can be defined as the locus of points whose distances to a point (the [b]focus[/b]) and a line (the [b]directrix[/b]) are in a constant ratio (the [b]eccentricity[/b]). [br][br]In this activity, we shall explore how [b]eccentricity [/b]affects the shape of the conics.
Instructions:
1. Drag to set the fixed point [i]F[/i]; drag the red dot to set the fixed line [i]d[/i].[br]2. Use the slider to adjust the eccentricity [i]e[/i].[br]3. Click the checkbox to construct the locus of the variable point [i]P[/i] satisfying [b][i]PF[/i]/[i]Pd[/i] = [i]e[/i][/b].[br]4. Click the 'Erase' button before constructing a new locus.
Questions:
1. How does the eccentricity [i]e[/i] affect the locus of [i]P[/i]?[br]2. Predict the locus when [i]e[/i] = 0.
Visualisation of eigenvectors and eigenvalues
This applet aims to help visualize the geometrical interpretation of the eigenvector(s) and eigenvalue(s) of a 2-by-2 matrix.[br][br]In this applet, users may[br]- define the 2-by-2 matrix by entering the values of the elements,[br]- drag the point V to view the vector [b]v[/b] and the vector [i]Av[/i] in the same diagram,[br]- receive a notification when an eigenvalue that satisfies [i]Av=kv[/i] is found.
1. What are the eigenvectors and the corresponding eigenvalues of [math]{{2,3},{4,1}}[/math]?[br]2. How many eigenvalues can a 2-by-2 matrix possibly have?[br]3. How many eigenvectors can a 2-by-2 matrix possibly have?[br]4. What can you say about the eigenvector(s) and eigenvalue(s) of a 2-by-2 matrix whose determinant is 0?