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Mathematics

Mathematics. Higher Revision Notes. Notes on Points. straight line equations gradient points of intersection parallel lines and perpendicular lines vectors and directed line segments scalar product. gradient is vertical / horizontal. Straight line equations. perpendicular lines.

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Mathematics

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  1. Mathematics • Higher Revision Notes

  2. Notes on Points • straight line equations • gradient • points of intersection • parallel lines and perpendicular lines • vectors and directed line segments • scalar product

  3. gradient isvertical /horizontal Straight line equations perpendicular lines parallel lines midpoint point of intersection -solve equations simultaneously Distance Formula

  4. P Three Dimensions: Distance Formula position vector of P magnitude of u p component form Scalar product (dot product) perpendicular

  5. C B • n : m A medians altitudes drop perpendicular from vertex angles cut in half join vertex to midpoint of opposite side bisectors sides B divides AC in the ratio......

  6. Notes on Trigonometry • trigonometric functions • radians • trigonometric graphs • solve trigonometric equations • compound angles • wave function

  7. • 360˚ • 360˚ sin • tan = cos 360˚ 180˚ sin cos sin2  + cos2  = 1 tan r 1 r r 1 radian

  8. A S C T two values in 1 complete turn sin = n  = sin-1(n) sin(A+B) = sinAcosB+ cosA sinB sin(2A) = 2sinA cosA sin(A-B) = sinA cosB- cosA sinB cos(A+B) = cosA cosB- sinA sinB cos(A-B) = cosA cosB+ sinA sinB cos(2A) = cos2A- sin2A cos(2A) = 2cos2A- 1 cos(2A) = 1 - 2sin2A

  9. A S C T in form also in form Reminder: Maximum and Minimum values of sinx or cosx are 1 and -1 SohCahToa for exact values

  10. Notes on Calculus • Differerentiation • Integration • polynomials • trigonometric functions • Area / Rate of change / Curve sketching • chain rule

  11. rate of change Area under / between curves gradient ‘Undoing’ differentiation gradient of tangent stationary points: maximum, minimum, inflexion sketch the curve displacement / velocity / acceleration

  12. Basic functions x in radians

  13. Always check your integration by differentiating! x in radians Reminder:

  14. Y X giving turning points at turning points solve equation to give maximum?/minimum?

  15. Notes on Parabolae / Circles • Geometry /Symmetry • minimum / maximum • centre, radius • standard equations • points of intersection • tangents

  16. Parabola Circles polynomial of degree 2 Centre O(0,0) radius r radius Centre minimum at (a, b) maximum at (a, b) cuts the X-axis at (a,0) and (b,0)

  17. Sketching graphs Given f(x)..... k stretches - f(x) reflection in X-axis b periods in 360˚ or 2π ↕ stretch k f(x) -a horizontal shift f(x) + b ↑ move up +a horizontal shift <- f(x - a) → move right amplitude k f(x + a) ← move left period f(-x) reflection in Y-axis

  18. Points of intersection: Solve simultaneous equations (by substitution). It is a Tangent if two solutions are equal. Reminder: find discriminant for a quadratic equation. if zero, then equal roots => tangent if less than 0, then no roots => no points of intersection A tangent to a circle meets the radius at 90˚ (perpendicular). and remember right angles in semicircle.

  19. Notes onRecurrence RelationsLogarithms / Indices • Those bacteria! • Napiers shortcuts! / focus on indices

  20. Limit Find how ‘long’ til ..... After 1 after 2 after 3..... State that: Limit exists if Make sure you make most efficient use of your calculator.

  21. Logarithms = Indices can use calculator for base e and base 10 non-calculator for other bases

  22. Examination Techniques Do read each question carefully. Re-read each question once you have finished to make sure you have answered all parts appropriately. Make sure you leave enough time to attempt all questions. Show all working steps. (particularly the substitution of numbers into formulae) Having prepared thoroughly, get a good night’s sleep before your exam!

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