GeoGebra Apps for A-Level Further Pure Mathematics

Mark Willis, 2020年8月22日

目錄

  1. Matrices and transformations:
    1. Matrix representation for a rotation θ degrees anticlockwise about (0, 0)
    2. The matrix representation for a reflection in the line y = mx.
    3. The matrix representation of a shear.
    4. Matrix representation of a reflection in 3D.
    5. Finding invariant lines under a transformation given by a matrix.
    6. A line of invariant points
    7. Eigenvalues and eigenvectors
    8. Eigenvalues and eigenvectors
    9. Eigenvalues and eigenvectors in 3D
    10. Repeated Transformations of a matrix
  2. Introduction to complex numbers:
    1. Solutions of a quadratic equation.
    2. Argand diagram representation of complex numbers.
    3. Modulus-argument form of a complex number.
  3. Roots of polynomials:
    1. The roots of polynomials.
    2. Forming new cubic equations using sum and product of roots.
  4. Sequences and series:
  5. Complex numbers and geometry:
    1. Writing a complex number in modulus-argument form.
    2. Multiplication and division of complex numbers in modulus-argument form.
    3. Loci in the Argand diagram introduction.
    4. Loci of the form |z - a| =r.
    5. Loci of the form |z - a| < r.
    6. Loci of the form |z - a| < r.
    7. Drawing loci of the form arg(z - a) = θ.
    8. Inequalities on an Argand diagram of form β ≤ arg(z - a) ≤ α.
    9. Loci of Iz-z_1I=Iz-z_2I on an argand diagram.
    10. Loci of the form |z - z_1| > |z - z_2|.
    11. Combination of loci on an Argand diagram.
    12. Complex roots of a quadratic
  6. Matrices and their inverses:
    1. The determinant of a 2 x 2 matrix.
    2. Solving simultaneous equations using matrices.
    3. Intersection of three planes introduction
  7. Vectors and 3D space:
    1. The scalar product proof
    2. Scalar product 3d
    3. How many points are required to define a plane?
    4. The vector equation of the plane
    5. The normal vector equation of the plane
    6. Converting the vector to the Cartesian plane equation
    7. Finding the Cartesian equation of a plane through 3 points
    8. A plane containing a point and a line
    9. Determining whether four points lie in a plane
    10. The angle between two planes.
    11. The intersection of two planes
    12. The intersection of three planes.
    13. The intersection of 3 planes at a point
    14. The intersection of three planes.
    15. Intersection of 3 planes using the inverse of a matrix
    16. The intersection of three planes using the inverse matrix
    17. Intersection of three planes by row elimination
    18. Direction cosines
    19. Proof of the scalar product to find the angle between 2 vectors
    20. The area of a parallelogram in 3D space
    21. Scalar triple product of three vectors
    22. Vector triple product of 3 vectors
  8. Vectors 1:
    1. The cross product or vector product
    2. The vector equation of the line in 2-dimensions
    3. The vector equation of a line in 3D.
    4. The point of intersection and angle between two lines in 3D
    5. Parallel, perpendicular and skew lines in 3D
    6. The distance between parallel lines in space
    7. The shortest distance between skew lines
    8. The intersection of a line and a plane
    9. The intersection of two planes
    10. The distance between two parallel planes
    11. The shortest distance between a line and a parallel plane.
    12. Intersection of 3 planes by row elimination including non-unique solutions
  9. Further calculus:
    1. Improper integrals introduction
    2. Using improper integrals to find areas
    3. y=arcsinx from y=sinx
    4. y=arccosx from y=cosx
    5. y=arctanx from y=tanx
    6. Parametric integration 01
    7. Parametric integration 02
  10. Sketching polar curves:
    1. An introduction to polar coordinates
    2. Copy of Sketching curves with polar equations introduction
    3. Sketching polar curves: Circle centre (0, 1), radius 1
    4. Sketching polar curves: Cardioid
    5. Sketching polar curves: A four-leaved rose
    6. Sketching polar curves: Parabola
    7. Sketching polar curves: Limaçon
    8. Sketching polar curves: An eight-leaved rose
    9. Sketching polar curves: Lemniscate
    10. Sketching polar curves: Advanced
    11. Sketching polar curves: Conchoid
    12. Sketching polar curves: Spiral
    13. Finding the area enclosed by a polar curve
  11. Maclaurin series:
    1. Maclaurin series: Exponential function
    2. Maclaurin series: The sine function
    3. Maclaurin series: The cosine function
    4. Maclaurin series: The function ln(1 + x)
    5. Maclaurin series: f(x) = (1 + x)^n
  12. Hyperbolic functions:
    1. Properties of the graph of y = cosh x
    2. Properties of the graph of y = sinh x
    3. The graph of y = tanh x
    4. The graph of y = arsinh x
    5. The graph of y = artanh x
    6. The graph of y = arcosh x
  13. Applications of integration:
    1. Volume of revolution introduction.
    2. A volume of revolution around the x-axis
    3. Volume of revolution of the area bounded by two functions
    4. A volume of revolution around the y-axis
    5. Area and volume using parametric curves.
    6. Gabriel's Horn
    7. Gabriel's Horn 01
    8. The arc length of a curve.
    9. The area of a surface of revolution.
    10. Volume of revolution by cylindrical shells
  14. Complex numbers - De Moivre's theorem:
    1. The nth root of a complex number
    2. The nth root of unity
  15. Extra pure Mathematics: Multivariable calculus:
    1. The domain of the graph for a surface.
    2. Sketching contour curves of a function of two variables 01.
    3. Sketching contour curves of a function of two variables 02.
    4. The geometrical meaning of partial derivatives.
    5. Finding stationary points of surfaces 01
    6. Finding stationary points of surfaces 02.
    7. Tangent planes and normals to a point introduction.
    8. The tangent plane and normal line to a surface 01.
    9. The tangent plane and normal line to a surface 02.
    10. Cylindrical coordinates.
    11. Spherical coordinates
    12. Triple integral to determine volume
    13. Using a triple integral to determine volume
  16. 1st order differential equations
    1. Solution of a 1st order differential equation with slope field.
    2. Solving a differential equation for an electrical circuit.
    3. The limiting value of a differential equation 02.
    4. The limiting value of a differential equation 01.
    5. Differential equation solution of a mass being projected upwards
  17. Euler's method
    1. Euler's method for f'(t, x) = x-2t, x(0) = 1
    2. Euler's method for f'(t, x)= xt, x(0) = 2
    3. Euler's method for f'(t, x)=x/(2sqrt(t + x)), x(0.5) = 1
    4. Euler's method for f'(t, x)=(4 - t)/(t + x)
    5. Euler's method for f'(t, x)=1/xt, x(1)=1
    6. Euler's method for f'(t, x)=xt/(x^2+2), x(1)=2
    7. Euler's method for f'(t, x)=1/lnx, x(1)=1.2
    8. Numerical solution of a third-order differential equation 01
    9. Numerical solution of a third-order differential equation 02
    10. Numerical solution of a second-order differential equation
  18. 2nd order differential equations
    1. Second order differential equations damped oscillations
    2. Non-homogeneous 2nd order differential equations
    3. Modelling a bungee jump with a 2nd order differential equation.
    4. The steady state response of a damped second-order system
    5. Solving a 2nd-order differential equation using Laplace transformations
    6. Graphical solution of a 2nd-order differential equation
    7. A system of first-order differential equations
  19. Fourier series
    1. Basic sine wave
    2. Integrals of periodic functions
    3. Square wave using Fourier terms
    4. Sawtooth wave using Fourier terms
    5. Triangular wave using Fourier terms
    6. Fourier series for a periodic function 01
    7. Fourier series for a periodic function 02
    8. Fourier series for a periodic function 03
    9. Fourier series for a periodic function 04
    10. Fourier series problem
    11. Fourier series for a piecewise periodic function 01
    12. Fourier series for a piecewise periodic function 02
    13. Fourier series for a piecewise periodic function 03
    14. Fourier series for a piecewise periodic function 04
    15. Fourier series for a piecewise periodic function 05
    16. Fourier series for a piecewise function 01
    17. Fourier series for a piecewise function 02
    18. Fourier series for a piecewise function 03
    19. Fourier series for a piecewise function 04
    20. Fourier series for a piecewise function 05
    21. Fourier series for a piecewise function 06
    22. An even function
    23. An odd function
    24. Fourier half-range series
    25. Significance of the constant term in Fourier series
    26. How to change the period of a sine graph
    27. How to change the period of a cosine graph
    28. Fourier series with an arbitrary period of 2
    29. Fourier series with an arbitrary period of 4
    30. Fourier half-range cosine or sine series
  20. Vector calculus
    1. Vector field 2D
    2. Vector field 3D
    3. Line integrals
    4. A scalar field
    5. A vector field
    6. A scalar field and vector field

1.

Matrices and transformations:

  • 1. Matrix representation for a rotation θ degrees anticlockwise about (0, 0)
  • 2. The matrix representation for a reflection in the line y = mx.
  • 3. The matrix representation of a shear.
  • 4. Matrix representation of a reflection in 3D.
  • 5. Finding invariant lines under a transformation given by a matrix.
  • 6. A line of invariant points
  • 7. Eigenvalues and eigenvectors
  • 8. Eigenvalues and eigenvectors
  • 9. Eigenvalues and eigenvectors in 3D
  • 10. Repeated Transformations of a matrix

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1.4.

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1.5.

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1.6.

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1.7.

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1.8.

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1.10.

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2.

Introduction to complex numbers:

  • 1. Solutions of a quadratic equation.
  • 2. Argand diagram representation of complex numbers.
  • 3. Modulus-argument form of a complex number.

2.1.

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2.2.

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2.3.

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3.

Roots of polynomials:

  • 1. The roots of polynomials.
  • 2. Forming new cubic equations using sum and product of roots.

3.1.

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3.2.

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4.

Sequences and series:


    本章節尚無任何資源。

5.

Complex numbers and geometry:

  • 1. Writing a complex number in modulus-argument form.
  • 2. Multiplication and division of complex numbers in modulus-argument form.
  • 3. Loci in the Argand diagram introduction.
  • 4. Loci of the form |z - a| =r.
  • 5. Loci of the form |z - a| < r.
  • 6. Loci of the form |z - a| < r.
  • 7. Drawing loci of the form arg(z - a) = θ.
  • 8. Inequalities on an Argand diagram of form β ≤ arg(z - a) ≤ α.
  • 9. Loci of Iz-z_1I=Iz-z_2I on an argand diagram.
  • 10. Loci of the form |z - z_1| > |z - z_2|.
  • 11. Combination of loci on an Argand diagram.
  • 12. Complex roots of a quadratic

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6.

Matrices and their inverses:

  • 1. The determinant of a 2 x 2 matrix.
  • 2. Solving simultaneous equations using matrices.
  • 3. Intersection of three planes introduction

6.1.

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6.3.

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7.

Vectors and 3D space:

  • 1. The scalar product proof
  • 2. Scalar product 3d
  • 3. How many points are required to define a plane?
  • 4. The vector equation of the plane
  • 5. The normal vector equation of the plane
  • 6. Converting the vector to the Cartesian plane equation
  • 7. Finding the Cartesian equation of a plane through 3 points
  • 8. A plane containing a point and a line
  • 9. Determining whether four points lie in a plane
  • 10. The angle between two planes.
  • 11. The intersection of two planes
  • 12. The intersection of three planes.
  • 13. The intersection of 3 planes at a point
  • 14. The intersection of three planes.
  • 15. Intersection of 3 planes using the inverse of a matrix
  • 16. The intersection of three planes using the inverse matrix
  • 17. Intersection of three planes by row elimination
  • 18. Direction cosines
  • 19. Proof of the scalar product to find the angle between 2 vectors
  • 20. The area of a parallelogram in 3D space
  • 21. Scalar triple product of three vectors
  • 22. Vector triple product of 3 vectors

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8.

Vectors 1:

  • 1. The cross product or vector product
  • 2. The vector equation of the line in 2-dimensions
  • 3. The vector equation of a line in 3D.
  • 4. The point of intersection and angle between two lines in 3D
  • 5. Parallel, perpendicular and skew lines in 3D
  • 6. The distance between parallel lines in space
  • 7. The shortest distance between skew lines
  • 8. The intersection of a line and a plane
  • 9. The intersection of two planes
  • 10. The distance between two parallel planes
  • 11. The shortest distance between a line and a parallel plane.
  • 12. Intersection of 3 planes by row elimination including non-unique solutions

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9.

Further calculus:

  • 1. Improper integrals introduction
  • 2. Using improper integrals to find areas
  • 3. y=arcsinx from y=sinx
  • 4. y=arccosx from y=cosx
  • 5. y=arctanx from y=tanx
  • 6. Parametric integration 01
  • 7. Parametric integration 02

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10.

Sketching polar curves:

  • 1. An introduction to polar coordinates
  • 2. Copy of Sketching curves with polar equations introduction
  • 3. Sketching polar curves: Circle centre (0, 1), radius 1
  • 4. Sketching polar curves: Cardioid
  • 5. Sketching polar curves: A four-leaved rose
  • 6. Sketching polar curves: Parabola
  • 7. Sketching polar curves: Limaçon
  • 8. Sketching polar curves: An eight-leaved rose
  • 9. Sketching polar curves: Lemniscate
  • 10. Sketching polar curves: Advanced
  • 11. Sketching polar curves: Conchoid
  • 12. Sketching polar curves: Spiral
  • 13. Finding the area enclosed by a polar curve

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11.

Maclaurin series:

  • 1. Maclaurin series: Exponential function
  • 2. Maclaurin series: The sine function
  • 3. Maclaurin series: The cosine function
  • 4. Maclaurin series: The function ln(1 + x)
  • 5. Maclaurin series: f(x) = (1 + x)^n

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12.

Hyperbolic functions:

  • 1. Properties of the graph of y = cosh x
  • 2. Properties of the graph of y = sinh x
  • 3. The graph of y = tanh x
  • 4. The graph of y = arsinh x
  • 5. The graph of y = artanh x
  • 6. The graph of y = arcosh x

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13.

Applications of integration:

  • 1. Volume of revolution introduction.
  • 2. A volume of revolution around the x-axis
  • 3. Volume of revolution of the area bounded by two functions
  • 4. A volume of revolution around the y-axis
  • 5. Area and volume using parametric curves.
  • 6. Gabriel's Horn
  • 7. Gabriel's Horn 01
  • 8. The arc length of a curve.
  • 9. The area of a surface of revolution.
  • 10. Volume of revolution by cylindrical shells

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14.

Complex numbers - De Moivre's theorem:

  • 1. The nth root of a complex number
  • 2. The nth root of unity

14.1.

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14.2.

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15.

Extra pure Mathematics: Multivariable calculus:

  • 1. The domain of the graph for a surface.
  • 2. Sketching contour curves of a function of two variables 01.
  • 3. Sketching contour curves of a function of two variables 02.
  • 4. The geometrical meaning of partial derivatives.
  • 5. Finding stationary points of surfaces 01
  • 6. Finding stationary points of surfaces 02.
  • 7. Tangent planes and normals to a point introduction.
  • 8. The tangent plane and normal line to a surface 01.
  • 9. The tangent plane and normal line to a surface 02.
  • 10. Cylindrical coordinates.
  • 11. Spherical coordinates
  • 12. Triple integral to determine volume
  • 13. Using a triple integral to determine volume

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16.

1st order differential equations

  • 1. Solution of a 1st order differential equation with slope field.
  • 2. Solving a differential equation for an electrical circuit.
  • 3. The limiting value of a differential equation 02.
  • 4. The limiting value of a differential equation 01.
  • 5. Differential equation solution of a mass being projected upwards

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17.

Euler's method

  • 1. Euler's method for f'(t, x) = x-2t, x(0) = 1
  • 2. Euler's method for f'(t, x)= xt, x(0) = 2
  • 3. Euler's method for f'(t, x)=x/(2sqrt(t + x)), x(0.5) = 1
  • 4. Euler's method for f'(t, x)=(4 - t)/(t + x)
  • 5. Euler's method for f'(t, x)=1/xt, x(1)=1
  • 6. Euler's method for f'(t, x)=xt/(x^2+2), x(1)=2
  • 7. Euler's method for f'(t, x)=1/lnx, x(1)=1.2
  • 8. Numerical solution of a third-order differential equation 01
  • 9. Numerical solution of a third-order differential equation 02
  • 10. Numerical solution of a second-order differential equation

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18.

2nd order differential equations

  • 1. Second order differential equations damped oscillations
  • 2. Non-homogeneous 2nd order differential equations
  • 3. Modelling a bungee jump with a 2nd order differential equation.
  • 4. The steady state response of a damped second-order system
  • 5. Solving a 2nd-order differential equation using Laplace transformations
  • 6. Graphical solution of a 2nd-order differential equation
  • 7. A system of first-order differential equations

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19.

Fourier series

  • 1. Basic sine wave
  • 2. Integrals of periodic functions
  • 3. Square wave using Fourier terms
  • 4. Sawtooth wave using Fourier terms
  • 5. Triangular wave using Fourier terms
  • 6. Fourier series for a periodic function 01
  • 7. Fourier series for a periodic function 02
  • 8. Fourier series for a periodic function 03
  • 9. Fourier series for a periodic function 04
  • 10. Fourier series problem
  • 11. Fourier series for a piecewise periodic function 01
  • 12. Fourier series for a piecewise periodic function 02
  • 13. Fourier series for a piecewise periodic function 03
  • 14. Fourier series for a piecewise periodic function 04
  • 15. Fourier series for a piecewise periodic function 05
  • 16. Fourier series for a piecewise function 01
  • 17. Fourier series for a piecewise function 02
  • 18. Fourier series for a piecewise function 03
  • 19. Fourier series for a piecewise function 04
  • 20. Fourier series for a piecewise function 05
  • 21. Fourier series for a piecewise function 06
  • 22. An even function
  • 23. An odd function
  • 24. Fourier half-range series
  • 25. Significance of the constant term in Fourier series
  • 26. How to change the period of a sine graph
  • 27. How to change the period of a cosine graph
  • 28. Fourier series with an arbitrary period of 2
  • 29. Fourier series with an arbitrary period of 4
  • 30. Fourier half-range cosine or sine series

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20.

Vector calculus

  • 1. Vector field 2D
  • 2. Vector field 3D
  • 3. Line integrals
  • 4. A scalar field
  • 5. A vector field
  • 6. A scalar field and vector field

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