Ungiven Formulae - LC Maths

This set of flashcards covers essential mathematical concepts and formulas relevant for reviewing Algebra, Functions, Differentiation, Integration, Sequences, Statistics, Complex Numbers, and Geometry.

Difference of two cubes

a³ - b³ = (a - b)(a² + ab + b²).

Sum of two cubes

a³ + b³ = (a + b)(a² - ab + b²)

Two equal roots

When b² - 4ac = 0

Two distinct real roots

b² - 4ac > 0

Imaginary roots

When b² - 4ac < 0

Forming Quadratic equations

x² - (sum of roots)x + (product of roots) = 0

Laws of surds

√(ab) = √a√b and √(a/b) = √a/√b.

Completed square

y = a(x - h)² + k, turning point = (h,k)

Injective functions proof

Horizontal Line Test

Surjective functions proof

Codomain = domain.

Finding slope using differentiation

Differentiate and sub in x

Turning point

Let dy/dx = 0 and solve for x; substitute into the original equation to get y.

Local maximum

d²y/dx² < 0

Local minimum

d²y/dx² > 0

Point of inflection

d²y/dx² = 0

Area under a curve

$$\int_{a}^{b}\!f\left(x\right)\,dx$$

Average value of a function

$$\frac{1}{b-a}\int_{a}^{b}\!f\left(x\right)\,dx$$

General term of a quadratic sequence

Tn = an² + bn + c, where 2a = 2nd difference.

Proof of arithmetic sequence

Tn - Tn-1 = d

Proof of geometric sequence

Tn/Tn-1 = r

Complex number modulus

|a + bi| = √(a² + b²).

Polar Form

$$r\left(\cos\theta+i\sin\theta\right)$$

Probability of Union of Events

P(A) + P(B) - P(A ∩ B)

Independent event

P(A) × P(B) = P(A ∩ B)

Conditional probability

P(A | B) = P(A ∩ B) / P(B)

Mutually exclusive events

A ∩ B = 0

Expected value

E(x) = ΣxP(x)

95% Confidence interval (standard deviation & mean)

x̅ ± 1.96(σ/√n)

95% confidence interval (population proportion)

$$\overline{p}\pm1.96\sqrt{\overline{p}\frac{\left(1-\overline{p}\right)}{n}}$$

Data % within Standard Deviations (1,2,3)

68%, 95%, 99.7%

Range

(a-b, a+b)

Function Period

$$\frac{2\pi}{c}$$

Trigonometric functions

a + bCos(cθ); a= horizontal midway line, b = amplitude