2. Binomial expansions
• General binomial expansion, for :
• When the series is infinite, this is only valid if |x|<1
• (a+x)n should be written as
3. Partial fractions
• A proper algebraic fraction with a denominator which factorises can be
decomposed into a sum of partial fractions. Following forms should be
used:
• e.g.
9. Parametric Equations
• To draw a graph from parametric equations, plot the points on the curve
given by different values of the parameter.
• Eliminate the parameter to give the cartesian equation of the curve.
• Parametric equations of circles:
- Circle centre (0,0) and radius r
- Circle centre (a,b) and radius r
•
10. Techniques for integration
Volumes of revolution
• About the x axis:
• About the y axis:
• Trapezium rule, with n strips of width h:
Note that this gives an overestimation of the area under the curve.
11. Vectors
• Magnitude-direction form: (2 dimensions)
measured
anticlockwise
• Component form: from
horizontal
• Position vector is from origin to point P
• Vector
• ‘r’ denotes position vector of a general point
12. Vector equations
• Vector equation of the line through A with direction u is given by:
• Vector equation through points A and B is given by:
• Equation of line through in direction is given by:
Cartesian form
Vector form
13. Angle between two vectors
• The angle between a and b is given by
• Where
in two dimensions
in three dimensions
14. • Cartesian equation of a plane perpendicular to is:
• Equation of the plane through the point with position vector a, and
perpendicular to n, is given by
(r-a).n=0.
15. Differential equations
• A differential equation involves derivatives such as
• First-order differential equation involves only a first derivative
• Some first-order differential equations can be solved by separating the
variables
• In a general solution you leave the constant of integration in the
solution, and in a particular solution you use additional information to
calculate the constant of integration.
