Junkao Mathematics - independent shared foundations cheat sheet

Remainder theorem: dividing by (x − a) leaves f(a); a zero remainder means a factor.

f⁻¹ exists only on a one-to-one domain, and its domain is the range of f.

y = f(x + a) shifts the graph a units to the left, not right.

Repeated linear factors need two partial-fraction terms, one for each power.

Always verify a factorisation by expanding back to the original polynomial.

AP sum: n/2 × (first + last) or n/2(2a + (n − 1)d).

GP sum to infinity exists only for |r| < 1 and equals a/(1 − r).

The general binomial term C(n, r)a^(n−r)b^r gives any coefficient directly.

Count the number of terms carefully when the index starts at zero.

A negative common ratio alternates signs, so check the parity of the power.

Arc length rθ and sector area ½r²θ require θ in radians.

sin²θ + cos²θ = 1 generates the sec–tan and cosec–cot identities.

For a sin θ + b cos θ, the amplitude is √(a² + b²).

Periodic solutions repeat every 2π for sine and cosine, every π for tangent.

Never divide by a trigonometric factor that could be zero; factorise instead.

Chain rule: dy/dx = dy/du × du/dx; never omit the inner derivative.

Quotient rule numerator is u′v − uv′, in that order.

Implicit differentiation attaches dy/dx to every y-term differentiated.

Parametric gradient: dy/dx = (dy/dt) ÷ (dx/dt).

Second derivative greater than zero at a stationary point means a minimum.

|a| = √(x² + y² + z²), and the unit vector is a divided by |a|.

a·b = |a||b|cos θ and also the sum of component products.

Perpendicular vectors satisfy a·b = 0.

Line equation: r = a + λb, where a is a point and b a direction.

Skew lines require all three components checked for consistency.

Complete the square to convert a general conic equation into standard form.

The constants added during completing the square must also be added to the other side.

For ellipses c^2 = a^2 - b^2; for hyperbolas c^2 = a^2 + b^2.

Center coordinates have the opposite sign of the numbers inside the squared binomials.

Perpendicular slopes multiply to -1, so the negative reciprocal is required.

The larger denominator in an ellipse equation identifies the direction of the major axis.

Independencia: P(A∩B) = P(A)P(B); incompatibles: P(A∩B) = 0.

Binomial: media np y desviación típica √(npq).

np > 5 y nq > 5 es una regla orientativa de aproximación binomial-normal; revisar precisión y corrección ±0,5.

Con σ poblacional conocida y normalidad o aproximación adecuada: IC de la media x̄ ± z·σ/√n; z = 1,96 al 95 %.

Tamaño de muestra: n = (zσ/E)², redondeando siempre hacia arriba.