Additional Mathematics for 5th Form Exams

Addmath 5th Form

1. Add Maths 5th Form

2. SEQUENCES & SERIES

3. Sequences & Series A set of numbers where each successive term differs from the preceding term by the same constant. Eg. 20, 18, 16, 14, 12, 10, … [ Differs by the constant -2 ] Terms used: a n n th term l last term a first term d common difference r common ratio

4. Arithmetic Progression A progression in which there is a common difference between each term and the consecutive term. a n = a + (n - 1) d S n = n (a + l ) 2 = n (2a + (n - 1)d) 2

5. Geometric Progression A progression in which there is a common ratio between each term and the consecutive term. a n = ar (n - 1) S n = a(1 - r n ) when |r| < 1 1 - r S n = a(r n - 1) when |r| > 1 r - 1

6. Sum to Infinity For GPs with |r|<1 there exists a sum to infinity since such series tend to zero, i.e. they are convergent series. S n = a 1 - r

7. Sigma ( Σ ) Notation b Σ (a n ) = S n n = a = a + a a+1 + a a+2 +... a b

8. DIFFERENTIATION

9. First and Second Derivatives For a function y = ax n + bx m First derivative, f’(x) or y’ or dy = anx (n - 1) + bmx (m - 1) dx Second Derivative, f”(x) or y” or d 2 y = d( dy / dx ) dx 2 dx

10. Derivatives of Trigonometric Ratios d(sin(x)) = cos(x) dx d(cos(x)) = -sin(x) dx d(tan(x)) = sec 2 (x) dx

11. Chain, Product and Quotient Rules CHAIN RULE For y in terms of x, where x can be written in terms of t dy = dy × dt dx dt dx Shortcut method: f’(x) = (g(h(x)))’ = g’(h(x))h’(x) PRODUCT RULE For y = uv where u and v are in terms of x y’ = uv’ + vu’ QUOTIENT RULE For y = u / v where u and v are in terms of x y’ = vu’ - uv’ v 2

12. Stationary Points A stationary point is where the first derivative, y’ = 0. The nature of a stationary point can be determined by the second derivative test: if y” < 0 then it is a maximum if y” > 0 then it is a minimum

13. Rates of Change RATE - a measure of change of a quantity in relation to another quantity CHANGE - an increase or decrease in value of a quantity Rates of Change can often be expressed as the derivative of a quantity with respect to another, i.e. dy/dx

14. Small Changes

15. STATISTICS

16. Percentiles

17. Quartiles

18. Measures of Central Tendency

19. Standard Deviation and Variance

20. Box and Whisker Plot

21. Stem and Leaf Plot

22. Skewness

23. KINEMATICS