where ax^2 + bx + c = 0
Completing the Square:
Combinations - Number of possible outcomes of an event where the ORDER in which the events occur DOES NOT MATTER!
Permutations - Number of possible outcomes of an event where the ORDER in which the events occur DOES MATTER!
Binomial Expansion - Describes the algebraic expansion of exponents of a binomial. It expands the power (x + y)^n into a sum involving terms in the form of ax^b.y^c. The Binomial expansion is performed as following:
where is a binomial and can also be written as nCk. It is equal to the number of combinations in which n events result in k outcomes (where the ORDER DOES'NT MATTER!!!)
The probability of event X occurring k times out of n trials where p is theprobability of the event X occurring is represented by the following mathematical formula:
Furthermore, the probability of an event X occurring k times is equal to p^k. Note that can also be written as nCk.
Example Questions and Answers
Q1: The probability that a pen drawn at random from a large box of pens is defective is 0.1. A sample of eight of these pens are taken. Find the probability (to 2 d.p.) that it contains (i) no defective pens (ii) one defective pen (iii) at least two defective pens A1:n=8, p=0.1, in (i) k=0, (ii) k=1, (iii) k≥2, ∴
(i) P(X=0) = 8C0*(0.1^0)(0.9^8) = 0.43
(ii) P(X=1) = 8C1(0.1^1)(0.9^7) = 0.38
(iii) P(X≥2) = P(X=2) + P(X=3) +...+P(X+8) = 1 - [P(X=0)+P(X=1)] = 1 - [(8C0(0.1^0)(0.9^8))+(8C1(0.1^1)*(0.9^7))] = 1 - 0.43 - 0.38 = 0.18.
Differentiation - used to find the rate of change of a function (the gradient). For a function f(x) = axⁿ, f'(x) = naxⁿ⁻¹. For example, the function y = 3x²+ 24x + 10 differentiates into dy/dx = 6x + 24. The rule is: Multiply the coefficient by the exponent, then decrease the exponent by 1.
Integration - used to find the area under a curve. By integrating a differentiated function, you return to the original function. Therefore, the method of integration is quite logically following. Increase the exponent by 1, then divide by the new exponent.
∫f'(x) dx = f(x) + C
where C is the constant of integration. It cannot be determined unless we know one coordinate through which f(x) passes.
This is called the fundamental theorem of calculus
Application of Calculus to Kinematics
Calculus can be applied to almost anything in the entire universe which contains rates of change. One example in the FSMQ Add Maths syllabus is Kinematics. Kinematics is a branch of mathematical physics concerned with the equations of motion. These are used to find:
u - Initial velocity
v - Final velocity
a - Acceleration
s - Displacement
t - Time
Please note that all of these parameters except for time are in bold because they are vectors. They have both magnitude (length etc.), and direction (e.g. 24° above of horizontal).
The Constant Acceleration Equations
Below are the equations of motion: