Equations

Formulae booklet, and the ones to memorise

Every result used across the notes, linked back to the lesson it came from and marked according to whether the Edexcel formulae booklet prints it. Each one carries the conditions it needs to be true as well, because a result lifted out of its lesson loses the sentence that said when it applies.

The 98 to learnStart here if revision time is short98 results

Indices and surds

a^x × a^y = a^x+y$$a^{x} \times a^{y} = a^{x+y}$$a^x × a^y = a^(x + y) NOT IN THE BOOKLET — LEARN ITConditionsSame base throughout, with a > 0 whenever the indices are not integers.

Indices and surds

a^x ÷ a^y = a^x-y$$a^{x} \div a^{y} = a^{x-y}$$a^x ÷ a^y = a^(x - y) NOT IN THE BOOKLET — LEARN ITConditionsSame base throughout, a ≠ 0, and a > 0 whenever the indices are not integers.

Indices and surds

(a^x)^y = a^xy$$(a^{x})^{y} = a^{xy}$$(a^x)^y = a^xy NOT IN THE BOOKLET — LEARN ITConditionsa > 0. For a negative base the rule breaks as soon as the indices stop being integers.

Indices and surds

√xy = √x√y$$\sqrt{xy} = \sqrt{x}\sqrt{y}$$√(xy) = √(x)√(y) NOT IN THE BOOKLET — LEARN ITConditionsx ≥ 0 and y ≥ 0. It fails for negative radicands: √(−1)√(−1) is not √1.

Quadratic functions

x = −b ± √b² − 4ac2a$$x = \frac{−b \pm \sqrt{b^{2} − 4ac}}{2a}$$x = (−b ± √(b^2 − 4ac))/2a NOT IN THE BOOKLET — LEARN ITConditionsa ≠ 0, and real roots need b² − 4ac ≥ 0.

Polynomials and the factor theorem

f(a) = 0 ⇔ (x − a) is a factor of f(x)$$f(a) = 0 \iff (x − a) \text{ is a factor of } f(x)$$f(a) = 0 ⇔ (x − a) is a factor of f(x) NOT IN THE BOOKLET — LEARN ITConditionsf(x) a polynomial.

Straight lines

y − y₁ = m(x − x₁)$$y − y_{1} = m(x − x_{1})$$y − y_1 = m(x − x_1) NOT IN THE BOOKLET — LEARN ITConditionsThe line is not vertical. A vertical line has no gradient m; write x = x₁ instead.

Straight lines

m₁ m₂ = −1$$m_{1} m_{2} = −1$$m_1 m_2 = −1 NOT IN THE BOOKLET — LEARN ITConditionsBoth lines have a gradient, so neither is vertical. A vertical and a horizontal line are perpendicular without satisfying this.

Circles

(x − a)² + (y − b)² = r²$$(x − a)^{2} + (y − b)^{2} = r^{2}$$(x − a)^2 + (y − b)^2 = r^2 NOT IN THE BOOKLET — LEARN ITConditionsr > 0, with centre (a, b).

Arithmetic series

u_n = a + (n − 1)d$$u_{n} = a + (n − 1)d$$u_n = a + (n − 1)d NOT IN THE BOOKLET — LEARN ITConditionsAn arithmetic sequence with common difference d; n a positive integer.

Geometric series

u_n = arⁿ⁻¹$$u_{n} = ar^{n−1}$$u_n = ar^(n − 1) NOT IN THE BOOKLET — LEARN ITConditionsA geometric sequence with common ratio r; n a positive integer.

Triangles and the sine and cosine rules

asin A = bsin B = csin C$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$a/sin A = b/sin B = c/sin C NOT IN THE BOOKLET — LEARN ITConditionsAny triangle. Finding an angle is ambiguous: the calculator returns the acute value and the obtuse supplement may fit as well.

Triangles and the sine and cosine rules

a² = b² + c² − 2bc cos A$$a^{2} = b^{2} + c^{2} − 2bc \cos A$$a^2 = b^2 + c^2 − 2bc cos A NOT IN THE BOOKLET — LEARN ITConditionsAny triangle, with A the angle opposite the side a.

Triangles and the sine and cosine rules

Area = 12ab sin C$$\text{Area} = \frac{1}{2}ab \sin C$$Area = (1/2)ab sin C NOT IN THE BOOKLET — LEARN ITConditionsC the angle between the two sides a and b.

Trigonometric graphs and equations

tan θ = sin θcos θ$$\tan θ = \frac{\sin θ}{\cos θ}$$tan θ = sin θ/cos θ NOT IN THE BOOKLET — LEARN ITConditionscos θ ≠ 0, so θ is not an odd multiple of 90°.

Trigonometric graphs and equations

sin ² θ + cos ² θ = 1$$\sin^{2} θ + \cos^{2} θ = 1$$sin^2 θ + cos^2 θ = 1 NOT IN THE BOOKLET — LEARN ITConditionsEvery angle θ; no restriction.

Radians, arcs and small angles

s = rθ$$s = rθ$$s = rθ NOT IN THE BOOKLET — LEARN ITConditionsθ in radians.

Radians, arcs and small angles

A = 12r²θ$$A = \frac{1}{2}r^{2}θ$$A = (1/2)r^2θ NOT IN THE BOOKLET — LEARN ITConditionsθ in radians.

Reciprocal and inverse trigonometric functions

sec² θ = 1 + tan² θ$$\text{sec}^{2} θ = 1 + \text{tan}^{2} θ$$sec^2 θ = 1 + tan^2 θ NOT IN THE BOOKLET — LEARN ITConditionscos θ ≠ 0.

Reciprocal and inverse trigonometric functions

cosec² θ = 1 + cot² θ$$\text{cosec}^{2} θ = 1 + \text{cot}^{2} θ$$cosec^2 θ = 1 + cot^2 θ NOT IN THE BOOKLET — LEARN ITConditionssin θ ≠ 0.

Compound angles and the harmonic form

sin 2A = 2 sin A cos A$$\sin 2A = 2 \sin A \cos A$$sin 2A = 2 sin A cos A NOT IN THE BOOKLET — LEARN ITConditionsAll A; no restriction.

Compound angles and the harmonic form

cos 2A = cos ²A − sin ²A = 2cos ²A − 1 = 1 − 2sin ²A$$\cos 2A = \cos^{2}A − \sin^{2}A = 2\cos^{2}A − 1 = 1 − 2\sin^{2}A$$cos 2A = cos^2A − sin^2A = 2cos^2A − 1 = 1 − 2sin^2A NOT IN THE BOOKLET — LEARN ITConditionsAll A; no restriction.

Compound angles and the harmonic form

tan 2A = 2tan A1 − tan ²A$$\tan 2A = \frac{2\tan A}{1 − \tan^{2}A}$$tan 2A = 2 tan A/(1 − tan^2A) NOT IN THE BOOKLET — LEARN ITConditionstan A defined and tan²A ≠ 1.

Exponential functions and e

gradient of e^kx = ke^kx$$\text{gradient of } e^{kx} = ke^{kx}$$gradient of e^kx = ke^kx NOT IN THE BOOKLET — LEARN ITConditionsAll x; no restriction.

Logarithms and their laws

x = aⁿ ⇔ n = log_a x$$x = a^{n} \iff n = \text{log}_{a} x$$x = a^n ⇔ n = log_a x NOT IN THE BOOKLET — LEARN ITConditionsa > 0, a ≠ 1 and x > 0.

Logarithms and their laws

log_a x + log_a y = log_a xy$$\text{log}_{a} x + \text{log}_{a} y = \text{log}_{a} xy$$log_a x + log_a y = log_a xy NOT IN THE BOOKLET — LEARN ITConditionsa > 0, a ≠ 1, x > 0 and y > 0.

Logarithms and their laws

log_a x − log_a y = log_a xy$$\text{log}_{a} x − \text{log}_{a} y = \text{log}_{a} \frac{x}{y}$$log_a x − log_a y = log_a (x/y) NOT IN THE BOOKLET — LEARN ITConditionsa > 0, a ≠ 1, x > 0 and y > 0.

Logarithms and their laws

k log_a x = log_a x^k$$k \text{log}_{a} x = \text{log}_{a} x^{k}$$k log_a x = log_a x^k NOT IN THE BOOKLET — LEARN ITConditionsa > 0, a ≠ 1 and x > 0.

Differentiating powers of x

y = xⁿ ⇒ dydx = nxⁿ⁻¹$$y = x^{n} \, ⇒ \, \frac{dy}{dx} = nx^{n−1}$$y = x^n ⇒ dy/dx = nx^(n − 1) NOT IN THE BOOKLET — LEARN ITConditionsAny real n, taking x > 0 where a non-integer power needs it.

Differentiating trig, exponentials and logs

sin kx → kcos kx, cos kx → −ksin kx$$\sin kx → k\cos kx, \, \cos kx → −k\sin kx$$sin kx → k cos kx, cos kx → − k sin kx NOT IN THE BOOKLET — LEARN ITConditionsx in radians. Both derivatives are wrong in degrees.

Differentiating trig, exponentials and logs

e^kx → ke^kx, ln x → 1x$$e^{kx} → ke^{kx}, \, \text{ln } x → \frac{1}{x}$$e^kx → ke^kx, ln x → 1/x NOT IN THE BOOKLET — LEARN ITConditionsln x needs x > 0.

The product, quotient and chain rules

dydx = dydu × dudx$$\frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dx}$$dy/dx = dy/du × du/dx NOT IN THE BOOKLET — LEARN ITConditionsy differentiable in u and u differentiable in x.

Integration as antidifferentiation

∫ xⁿ dx = xⁿ⁺¹n + 1 + c (n ≠ −1)$$\text{∫} x^{n} \, dx = \frac{x^{n+1}}{n + 1} + c \, (n ≠ −1)$$∫ x^n dx = x^(n + 1)/(n + 1) + c (n ≠ − 1) NOT IN THE BOOKLET — LEARN ITConditionsn ≠ −1, as printed, and an interval that avoids x = 0 when n is negative.

Integrating standard functions

∫ cos kx dx = 1ksin kx + c$$\text{∫} \cos kx \, dx = \frac{1}{k}\sin kx + c$$∫ cos kx dx = (1/k) sin kx + c NOT IN THE BOOKLET — LEARN ITConditionsk ≠ 0, x in radians.

Integrating standard functions

∫ sin kx dx = −1kcos kx + c$$\text{∫} \sin kx \, dx = −\frac{1}{k}\cos kx + c$$∫ sin kx dx = −(1/k) cos kx + c NOT IN THE BOOKLET — LEARN ITConditionsk ≠ 0, x in radians.

Integrating standard functions

∫ e^kx dx = 1ke^kx + c$$\text{∫} e^{kx} \, dx = \frac{1}{k}e^{kx} + c$$∫ e^kx dx = (1/k)e^kx + c NOT IN THE BOOKLET — LEARN ITConditionsk ≠ 0.

Integrating standard functions

∫ 1x dx = ln|x| + c$$\text{∫} \frac{1}{x} \, dx = \text{ln}|x| + c$$∫ (1/x) dx = ln|x| + c NOT IN THE BOOKLET — LEARN ITConditionsx ≠ 0, and the constant may differ on each side of zero.

Integration by substitution and by parts

∫ f'(x)f(x) dx = ln|f(x)| + c$$\text{∫} \frac{f'(x)}{f(x)} \, dx = \text{ln}|f(x)| + c$$∫ (f′(x)/f(x)) dx = ln|f(x)| + c NOT IN THE BOOKLET — LEARN ITConditionsf(x) ≠ 0 right across the interval of integration.

Areas, parametric curves and the limit of a sum

area = ∫ y dxdt dt$$\text{area} = \text{∫} y \, \frac{dx}{dt} \, dt$$area = ∫ y (dx/dt) dt NOT IN THE BOOKLET — LEARN ITConditionsx and y differentiable in t, limits taken as the t values at the ends, and y ≥ 0 for the integral to measure the area.

Vectors in two dimensions

AB = b − a$$\overrightarrow{AB} = \text{b} − \text{a}$$vec AB = b − a NOT IN THE BOOKLET — LEARN ITConditionsa and b the position vectors of A and B, measured from the same origin.

Vectors in three dimensions

|xi + yj + zk| = √x² + y² + z²$$|x\text{i} + y\text{j} + z\text{k}| = \sqrt{x^{2} + y^{2} + z^{2}}$$|x i + y j + z k| = √(x^2 + y^2 + z^2) NOT IN THE BOOKLET — LEARN ITConditionsComponents taken along mutually perpendicular axes.

Sampling and the large data set

stratum sizepopulation size × sample size$$\frac{\text{stratum size}}{\text{population size}} \times \text{sample size}$$stratum size/population size × sample size NOT IN THE BOOKLET — LEARN ITConditionsNon-overlapping strata that between them cover the population; the numbers are then rounded to whole members.

Measures of location and spread

mean: Σ xn$$\text{mean: } \, \frac{Σ x}{n}$$mean: ((Σ x)/n) NOT IN THE BOOKLET — LEARN ITConditionsn ≥ 1.

Measures of location and spread

mean(x) = a + b mean(y),     σ_x = |b|σ_y$$\text{mean}(x) = a + b \, \text{mean}(y), \qquad σ_{x} = |b|σ_{y}$$mean(x) = a + b mean(y), σ_x = |b|σ_y NOT IN THE BOOKLET — LEARN ITConditionsCoding of the form x = a + by. The spread relation needs the magnitude of b: the standard deviation of x is |b| times that of y, and the form printed here assumes b > 0.

Representing and interpreting data

frequency density = frequencyclass width$$\text{frequency density} = \frac{\text{frequency}}{\text{class width}}$$frequency density = frequency/class width NOT IN THE BOOKLET — LEARN ITConditionsGrouped continuous data drawn to the equal-area convention, with class widths in the same units throughout.

Representing and interpreting data

outlier: beyond Q₁ - 1.5 × IQR or Q₃ + 1.5 × IQR$$\text{outlier: beyond } Q_{1} - 1.5 \times \text{IQR} \text{ or } Q_{3} + 1.5 \times \text{IQR}$$outlier: beyond Q_1 - 1.5 × IQR or Q_3 + 1.5 × IQR NOT IN THE BOOKLET — LEARN ITConditionsA convention rather than a definition. Use whichever rule the question states.

Conditional probability

P(A|B) = P(A ∩ B)P(B)$$P(A|B) = \frac{P(A ∩ B)}{P(B)}$$P(A|B) = (P(A ∩ B))/P(B) NOT IN THE BOOKLET — LEARN ITConditionsP(B) > 0.

The binomial distribution

P(X = x) = 1n$$\text{P}(X = x) = \frac{1}{n}$$P(X = x) = 1/n NOT IN THE BOOKLET — LEARN ITConditionsA discrete uniform distribution: n listed outcomes, every one equally likely. Equal probabilities are the whole condition, so a total of two dice does not qualify.

The normal distribution

Z = X - μσ$$Z = \frac{X - μ}{σ}$$Z = (X - μ)/σ NOT IN THE BOOKLET — LEARN ITConditionsX ~ N(μ, σ²) with σ > 0.

The normal distribution

X ~ B(n, p) ≈ N(np, np(1 - p))$$X ~ B(n, p) \, \approx \, N(np, \, np(1 - p))$$X ~ B(n, p) ≈ N(np, np(1 - p)) NOT IN THE BOOKLET — LEARN ITConditionsn large with p not close to 0 or 1, so that np and n(1 − p) both comfortably exceed 5. A continuity correction is needed on the way across.

Hypothesis testing: correlation and the normal

mean of n observations: N(μ, σ²/n)$$\text{mean of } n \text{ observations: } \, N(μ, \, σ^{2}/n)$$mean of n observations: N(μ, σ^2/n) NOT IN THE BOOKLET — LEARN ITConditionsA random sample of n independent observations from N(μ, σ²).

Kinematics with variable acceleration

v = dxdt,     a = dvdt$$v = \frac{dx}{dt}, \qquad a = \frac{dv}{dt}$$v = (dx/dt), a = dv/dt NOT IN THE BOOKLET — LEARN ITConditionsMotion along a straight line, with x and v differentiable in t.

Forces and Newton's laws

F = ma$$F = ma$$F = ma NOT IN THE BOOKLET — LEARN ITConditionsConstant mass, F the resultant force, measured in an inertial frame and in consistent units.

Friction and inclined planes

F ≤ μR$$F ≤ μR$$F ≤ μR NOT IN THE BOOKLET — LEARN ITConditionsR the normal reaction. Equality holds only on the point of slipping or while sliding.

Moments

moment = force × perpendicular distance$$\text{moment} = \text{force} × \text{perpendicular distance}$$moment = force × perpendicular distance NOT IN THE BOOKLET — LEARN ITConditionsDistance measured perpendicular from the pivot to the line of action of the force.

Modulus, argument and loci

z = r(cos θ + i sin θ)$$z = r(\cos θ + \text{i} \sin θ)$$z = r(cos θ + i sin θ) NOT IN THE BOOKLET — LEARN ITConditionsr = |z| ≥ 0 and θ = arg z, conventionally taken in (−π, π].

Roots of polynomials

α + β + γ = -ba, αβ + αγ + βγ = ca, αβγ = -da$$α + β + γ = -\frac{b}{a}, αβ + αγ + βγ = \frac{c}{a}, αβγ = -\frac{d}{a}$$α + β + γ = -(b/a), αβ + αγ + βγ = (c/a), αβγ = -d/a NOT IN THE BOOKLET — LEARN ITConditionsα, β, γ the roots of ax³ + bx² + cx + d = 0 with a ≠ 0, counted with multiplicity.

Summing series and the method of differences

Σ r = n(n+1)2$$Σ r = \frac{n(n+1)}{2}$$Σ r = (n(n + 1))/2 NOT IN THE BOOKLET — LEARN ITConditionsSummed from r = 1 to n, with n a positive integer.

Lines and planes in three dimensions

r = a + λb$$\text{r} = \text{a} + λ\text{b}$$r = a + λ b NOT IN THE BOOKLET — LEARN ITConditionsb ≠ 0, and a the position vector of a point on the line.

Lines and planes in three dimensions

r · n = a · n = d$$\text{r} · \text{n} = \text{a} · \text{n} = d$$r · n = a · n = d NOT IN THE BOOKLET — LEARN ITConditionsn a non-zero normal to the plane, a the position vector of a point on it.

Volumes of revolution

V = π ∫_a^b y² dx$$V = π \text{∫}_{a}^{b} y^{2} \, dx$$V = π ∫_a^b y^2 dx NOT IN THE BOOKLET — LEARN ITConditionsA full turn about the x-axis of the region between the curve and that axis, with the curve not crossing the axis between a and b.

Mean values and improper integrals

mean value = 1b - a ∫_a^b f(x) dx$$\text{mean value} = \frac{1}{b - a} \text{∫}_{a}^{b} \text{f}(x) \, dx$$mean value = (1/(b - a)) ∫_a^b f(x) dx NOT IN THE BOOKLET — LEARN ITConditionsb ≠ a, and f integrable across [a, b].

Hyperbolic functions and identities

cosh x = e^x + e^-x2, sinh x = e^x - e^-x2$$\text{cosh } x = \frac{e^{x} + e^{-x}}{2}, \text{sinh } x = \frac{e^{x} - e^{-x}}{2}$$cosh x = ((e^x + e^(-x))/2), sinh x = (e^x - e^(-x))/2 NOT IN THE BOOKLET — LEARN ITConditionsDefinitions rather than results, so they hold for every real x.

Polar curves

x = r cos θ, y = r sin θ, r² = x² + y²$$x = r \cos θ, y = r \sin θ, r^{2} = x^{2} + y^{2}$$x = r cos θ, y = r sin θ, r^2 = x^2 + y^2 NOT IN THE BOOKLET — LEARN ITConditionsr ≥ 0, with the quadrant of θ chosen to match the signs of x and y rather than taken straight from a calculator.

First order equations and integrating factors

IF = e^{∫ P dx}$$\text{IF} = e^{\text{∫} P \, dx}$$IF = e^(∫ P dx) NOT IN THE BOOKLET — LEARN ITConditionsThe equation must first be written as dy/dx + P(x)y = Q(x), with the coefficient of dy/dx equal to 1.

Second order equations

am² + bm + c = 0$$am^{2} + bm + c = 0$$am^2 + bm + c = 0 NOT IN THE BOOKLET — LEARN ITConditionsConstant coefficients with a ≠ 0, for the homogeneous equation ay″ + by′ + cy = 0.

The t-formulae

sin θ = 2t1 + t², cos θ = 1 - t²1 + t², tan θ = 2t1 - t²$$\sin θ = \frac{2t}{1 + t^{2}}, \cos θ = \frac{1 - t^{2}}{1 + t^{2}}, \tan θ = \frac{2t}{1 - t^{2}}$$sin θ = (2t/(1 + t^2)), cos θ = ((1 - t^2)/(1 + t^2)), tan θ = 2t/(1 - t^2) NOT IN THE BOOKLET — LEARN ITConditionst = tan(θ/2), so θ/2 must avoid odd multiples of 90°; the tan form additionally needs t² ≠ 1.

Taylor series

f(x) = f(a) + f'(a)(x - a) + f''(a)2!(x - a)² + …$$\text{f}(x) = \text{f}(a) + \text{f}'(a)(x - a) + \frac{\text{f}''(a)}{2!}(x - a)^{2} + …$$f(x) = f(a) + f′(a)(x - a) + (f′′(a)/2!)(x - a)^2 + … NOT IN THE BOOKLET — LEARN ITConditionsf differentiable to the order used at a, and x inside the interval where the series converges.

Leibnitz's theorem and the Weierstrass substitution

(fg)⁽ⁿ⁾ = Σ ⁿC_r f^(r) g^(n-r)$$(\text{fg})^{(n)} = Σ \, ^{n}C_{r} \, \text{f}^{(r)} \text{g}^{(n-r)}$$(fg)^(n) = Σ ^nC_r f^(r) g^((n - r)) NOT IN THE BOOKLET — LEARN ITConditionsf and g both n times differentiable.

Numerical methods for differential equations

dydx ≈ y_n+1 - y_nh or y_n+1 - y_n-12h$$\frac{dy}{dx} \approx \frac{y_{n+1} - y_{n}}{h} \text{ or } \frac{y_{n+1} - y_{n-1}}{2h}$$dy/dx ≈ ((y_(n + 1) - y_n)/h) or ((y_(n + 1) - y_(n - 1))/2h) NOT IN THE BOOKLET — LEARN ITConditionsBoth improve as the step h shrinks, and the central form is the more accurate for the same h, but it needs a value on each side of the point.

Numerical methods for differential equations

∫_a^b y dx ≈ h3(y₀ + 4y₁ + 2y₂ + … + 4y_n-1 + y_n)$$\text{∫}_{a}^{b} y \, dx \approx \frac{h}{3}(y_{0} + 4y_{1} + 2y_{2} + … + 4y_{n-1} + y_{n})$$∫_a^b y dx ≈ (h/3)(y_0 + 4y_1 + 2y_2 + … + 4y_(n - 1) + y_n) NOT IN THE BOOKLET — LEARN ITConditionsAn even number of strips of equal width h, so the points come in triples. With an odd number of strips the rule does not apply.

Contingency tables

E_ij = row total × column totalgrand total$$E_{ij} = \frac{\text{row total} × \text{column total}}{\text{grand total}}$$E_ij = (row total × column total)/grand total NOT IN THE BOOKLET — LEARN ITConditionsComputed under the null hypothesis that the two classifications are independent; each expected frequency should be at least 5.

Eigenvalues and eigenvectors

det(M - λI) = 0$$\text{det}(\text{M} - λ\text{I}) = 0$$det(M - λ I) = 0 NOT IN THE BOOKLET — LEARN ITConditionsM a square matrix.

Diagonalisation and the Cayley-Hamilton theorem

P^-1MP = D$$\text{P}^{-1}\text{MP} = \text{D}$$P^(-1) MP = D NOT IN THE BOOKLET — LEARN ITConditionsM must have a full set of independent eigenvectors; P is then invertible with those eigenvectors as its columns.

Modular arithmetic and Fermat's little theorem

a^p-1 ≡ 1 (mod p)$$a^{p-1} \equiv 1 \, (\text{mod } p)$$a^(p - 1) ≡ 1 (mod p) NOT IN THE BOOKLET — LEARN ITConditionsp prime, and p must not divide a.

Continuous random variables: density and distribution functions

P(a < X ≤ b) = ∫_a^b f(x) dx$$\text{P}(a < X \le b) = \text{∫}_{a}^{b} \text{f}(x) \, dx$$P(a < X ≤ b) = ∫_a^b f(x) dx NOT IN THE BOOKLET — LEARN ITConditionsX continuous with density f, where f ≥ 0 and integrates to 1 across the whole range.

Continuous random variables: density and distribution functions

f(x) = dF(x)dx$$\text{f}(x) = \frac{d\text{F}(x)}{dx}$$f(x) = d F(x)/dx NOT IN THE BOOKLET — LEARN ITConditionsWherever F is differentiable.

The continuous uniform distribution

F(x) = 0 for x < a,      x - ab - a for a ≤ x ≤ b,      1 for x > b$$\text{F}(x) = 0 \text{ for } x < a, \qquad \frac{x - a}{b - a} \text{ for } a \le x \le b, \qquad 1 \text{ for } x > b$$F(x) = 0 for x < a, ((x - a)/(b - a)) for a ≤ x ≤ b, 1 for x > b NOT IN THE BOOKLET — LEARN ITConditionsa < b. Defined for every real x: the ramp holds on [a, b], and the two flat branches carry F to 0 below a and to 1 above b.

Combinations of normal random variables

aX ± bY is normal, with mean aμ_x ± bμ_y$$aX ± bY \text{ is normal, with mean } aμ_{x} ± bμ_{y}$$aX ± bY is normal, with mean aμ_x ± bμ_y NOT IN THE BOOKLET — LEARN ITConditionsX and Y independent, and each normally distributed.

Estimators, standard error and confidence intervals

s² = 1n - 1(Σ x² - (Σ x)²n)$$s^{2} = \frac{1}{n - 1}(Σ x^{2} - \frac{(Σ x)^{2}}{n})$$s^2 = (1/(n - 1))(Σ x^2 - ((Σ x)^2)/n) NOT IN THE BOOKLET — LEARN ITConditionsA random sample of n ≥ 2 observations; algebraically the same estimator as the one above.

Estimators, standard error and confidence intervals

sample mean ± z × σ√n,     z = 1.96 at 95 per cent$$\text{sample mean} ± z \times \frac{σ}{\sqrt{n}}, \qquad z = 1.96 \text{ at 95 per cent}$$sample mean ± z × (σ/√(n)), z = 1.96 at 95 per cent NOT IN THE BOOKLET — LEARN ITConditionsA random sample with σ known, and either a normal population or a sample large enough for the central limit theorem.

Momentum and impulse

I = Ft = mv - mu$$\text{I} = \text{F}t = m\text{v} - m\text{u}$$I = F t = m v - m u NOT IN THE BOOKLET — LEARN ITConditionsA constant force acting for time t on a constant mass, with one positive direction fixed along the line.

Momentum and impulse

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂$$m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 NOT IN THE BOOKLET — LEARN ITConditionsNo external impulse on the pair during the collision, and all velocities measured along the same line with one sign convention.

Impulse and momentum as vectors

I = mv - mu$$\text{I} = m\text{v} - m\text{u}$$I = m v - m u NOT IN THE BOOKLET — LEARN ITConditionsConstant mass, with the vectors resolved in the same pair of directions throughout.

Work, energy and power

KE = 12mv²,      GPE = mgh$$\text{KE} = \frac{1}{2}mv^{2}, \qquad \text{GPE} = mgh$$KE = (1/2)mv^2, GPE = mgh NOT IN THE BOOKLET — LEARN ITConditionsGPE is measured from a stated reference level; KE uses speed, so direction does not enter it.

Work, energy and power

P = Fv$$P = Fv$$P = Fv NOT IN THE BOOKLET — LEARN ITConditionsF the driving force along the direction of motion, and v the instantaneous speed.

Hooke's law and elastic strings

T = λ xl$$T = \frac{λ x}{l}$$T = (λ x)/l NOT IN THE BOOKLET — LEARN ITConditionsAn elastic string or spring of natural length l with x the extension. A string goes slack rather than pushing, so T = 0 once the extension would be negative.

Elastic potential energy

EPE = λ x²2l$$\text{EPE} = \frac{λ x^{2}}{2l}$$EPE = (λ x^2)/2l NOT IN THE BOOKLET — LEARN ITConditionsThe same string or spring, x the extension or compression, and the deformation within the elastic limit.

Direct impact and Newton's law of restitution

e = speed of separationspeed of approach$$e = \frac{\text{speed of separation}}{\text{speed of approach}}$$e = speed of separation/speed of approach NOT IN THE BOOKLET — LEARN ITConditionsDirect impact along the line of centres, with 0 ≤ e ≤ 1 and speeds measured along that line.

Angular speed and horizontal circular motion

tan θ = v²rg$$\tan θ = \frac{v^{2}}{rg}$$tan θ = v^2/rg NOT IN THE BOOKLET — LEARN ITConditionsA banked track or conical pendulum with no friction, moving in a horizontal circle at constant speed v. Once friction acts, this design speed no longer applies.

Centre of mass of a discrete distribution

mean x = Σ m_ix_iΣ m_i,      mean y = Σ m_iy_iΣ m_i$$\text{mean x} = \frac{Σ m_{i}x_{i}}{Σ m_{i}}, \qquad \text{mean y} = \frac{Σ m_{i}y_{i}}{Σ m_{i}}$$mean x = ((Σ m_ix_i)/(Σ m_i)), mean y = (Σ m_iy_i)/(Σ m_i) NOT IN THE BOOKLET — LEARN ITConditionsA non-zero total mass, with the particles coplanar and all coordinates measured from one origin.

Centres of mass by integration

mean x = ∫ xy dx∫ y dx,      mean y = ∫ 12y² dx∫ y dx$$\text{mean x} = \frac{\text{∫} xy \, dx}{\text{∫} y \, dx}, \qquad \text{mean y} = \frac{\text{∫} \frac{1}{2}y^{2} \, dx}{\text{∫} y \, dx}$$mean x = ((∫ xy dx)/(∫ y dx)), mean y = (∫ (1/2)y^2 dx)/(∫ y dx) NOT IN THE BOOKLET — LEARN ITConditionsA uniform lamina bounded by y = f(x), the x-axis and the limits, with y ≥ 0 across the interval.

Centres of mass by integration

mean x = ∫ x y² dx∫ y² dx$$\text{mean x} = \frac{\text{∫} x y^{2} \, dx}{\text{∫} y^{2} \, dx}$$mean x = (∫ x y^2 dx)/(∫ y^2 dx) NOT IN THE BOOKLET — LEARN ITConditionsA uniform solid of revolution, formed by rotating y = f(x) through a full turn about the x-axis.

Newton's laws with a variable force

F(x) = mvdvdx,     F(t) = mdvdt$$F(x) = mv\frac{dv}{dx}, \qquad F(t) = m\frac{dv}{dt}$$F(x) = mv(dv/dx), F(t) = m(dv/dt) NOT IN THE BOOKLET — LEARN ITConditionsMotion in a straight line. Use the dv/dx form when the force is given in terms of x or v, and the dv/dt form when it is given in terms of t or v.

Simple harmonic motion

d²xdt² = -ω²x$$\frac{d^{2}x}{dt^{2}} = -ω^{2}x$$d^2x/dt^2 = -ω^2x NOT IN THE BOOKLET — LEARN ITConditionsSimple harmonic motion: the acceleration must be proportional to the displacement from a fixed point and directed towards it, with ω > 0.

Simple harmonic motion

v² = ω²(a² - x²)$$v^{2} = ω^{2}(a^{2} - x^{2})$$v^2 = ω^2(a^2 - x^2) NOT IN THE BOOKLET — LEARN ITConditionsThe same simple harmonic motion, a the amplitude and x measured from the centre of the oscillation, so |x| ≤ a.

Further kinematics: acceleration as a function of x, t or v

dvdt = f(t) or f(v),     vdvdx = f(v) or f(x)$$\frac{dv}{dt} = \text{f}(t) \text{ or f}(v), \qquad v\frac{dv}{dx} = \text{f}(v) \text{ or f}(x)$$dv/dt = f(t) or f(v), v(dv/dx) = f(v) or f(x) NOT IN THE BOOKLET — LEARN ITConditionsMotion in a straight line, choosing whichever form matches the variable the acceleration is given in.

The stepping-stone method

improvement index = cost - R - K$$\text{improvement index} = \text{cost} - R - K$$improvement index = cost - R - K NOT IN THE BOOKLET — LEARN ITConditionsShadow costs R and K fixed from the occupied cells of a balanced problem, where total supply equals total demand.

Algebra and functionsYears 12-137 results

Indices and surds

a^x × a^y = a^x+y$$a^{x} \times a^{y} = a^{x+y}$$a^x × a^y = a^(x + y) NOT IN THE BOOKLET — LEARN ITConditionsSame base throughout, with a > 0 whenever the indices are not integers.

a^x ÷ a^y = a^x-y$$a^{x} \div a^{y} = a^{x-y}$$a^x ÷ a^y = a^(x - y) NOT IN THE BOOKLET — LEARN ITConditionsSame base throughout, a ≠ 0, and a > 0 whenever the indices are not integers.

(a^x)^y = a^xy$$(a^{x})^{y} = a^{xy}$$(a^x)^y = a^xy NOT IN THE BOOKLET — LEARN ITConditionsa > 0. For a negative base the rule breaks as soon as the indices stop being integers.

√xy = √x√y$$\sqrt{xy} = \sqrt{x}\sqrt{y}$$√(xy) = √(x)√(y) NOT IN THE BOOKLET — LEARN ITConditionsx ≥ 0 and y ≥ 0. It fails for negative radicands: √(−1)√(−1) is not √1.

Quadratic functions

ax² + bx + c = a(x + b2a)² + (c − b²4a)$$ax^{2} + bx + c = a(x + \frac{b}{2a})^{2} + (c − \frac{b^{2}}{4a})$$ax^2 + bx + c = a(x + b/2a)^2 + (c − b^2/4a) NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsa ≠ 0.

x = −b ± √b² − 4ac2a$$x = \frac{−b \pm \sqrt{b^{2} − 4ac}}{2a}$$x = (−b ± √(b^2 − 4ac))/2a NOT IN THE BOOKLET — LEARN ITConditionsa ≠ 0, and real roots need b² − 4ac ≥ 0.

Polynomials and the factor theorem

f(a) = 0 ⇔ (x − a) is a factor of f(x)$$f(a) = 0 \iff (x − a) \text{ is a factor of } f(x)$$f(a) = 0 ⇔ (x − a) is a factor of f(x) NOT IN THE BOOKLET — LEARN ITConditionsf(x) a polynomial.

Coordinate geometryYears 12-133 results

Straight lines

y − y₁ = m(x − x₁)$$y − y_{1} = m(x − x_{1})$$y − y_1 = m(x − x_1) NOT IN THE BOOKLET — LEARN ITConditionsThe line is not vertical. A vertical line has no gradient m; write x = x₁ instead.

m₁ m₂ = −1$$m_{1} m_{2} = −1$$m_1 m_2 = −1 NOT IN THE BOOKLET — LEARN ITConditionsBoth lines have a gradient, so neither is vertical. A vertical and a horizontal line are perpendicular without satisfying this.

Circles

(x − a)² + (y − b)² = r²$$(x − a)^{2} + (y − b)^{2} = r^{2}$$(x − a)^2 + (y − b)^2 = r^2 NOT IN THE BOOKLET — LEARN ITConditionsr > 0, with centre (a, b).

Sequences and seriesYears 12-139 results

The binomial expansion

ⁿC_r = n!r!(n − r)!$$^{n}C_{r} = \frac{n!}{r!(n − r)!}$$^nC_r = n!/(r!(n − r)!) IN THE FORMULAE BOOKLETConditionsn and r non-negative integers with r ≤ n.

(a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + … + bⁿ$$(a + b)^{n} = a^{n} + \,^{n}C_{1}a^{n−1}b + … + b^{n}$$(a + b)^n = a^n + ^nC_1a^(n − 1)b + … + b^n IN THE FORMULAE BOOKLETConditionsn a positive integer, so that the expansion terminates.

The general binomial expansion

(1 + x)ⁿ = 1 + nx + n(n − 1)2!x² + …$$(1 + x)^{n} = 1 + nx + \frac{n(n − 1)}{2!}x^{2} + …$$(1 + x)^n = 1 + nx + ((n(n − 1))/2!)x^2 + … IN THE FORMULAE BOOKLETConditions|x| < 1 unless n is a non-negative integer, in which case the expansion terminates and holds for every x.

(a + bx)ⁿ = aⁿ(1 + bx/a)ⁿ$$(a + bx)^{n} = a^{n}(1 + bx/a)^{n}$$(a + bx)^n = a^n(1 + bx/a)^n NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsa ≠ 0, and the expansion that follows is valid only for |bx/a| < 1.

Arithmetic series

u_n = a + (n − 1)d$$u_{n} = a + (n − 1)d$$u_n = a + (n − 1)d NOT IN THE BOOKLET — LEARN ITConditionsAn arithmetic sequence with common difference d; n a positive integer.

S_n = n2(2a + (n − 1)d) = n2(a + l)$$S_{n} = \frac{n}{2}(2a + (n − 1)d) = \frac{n}{2}(a + l)$$S_n = (n/2)(2a + (n − 1)d) = (n/2)(a + l) IN THE FORMULAE BOOKLETConditionsAn arithmetic series, with l the nth term.

Geometric series

u_n = arⁿ⁻¹$$u_{n} = ar^{n−1}$$u_n = ar^(n − 1) NOT IN THE BOOKLET — LEARN ITConditionsA geometric sequence with common ratio r; n a positive integer.

S_n = a(1 − rⁿ)1 − r$$S_{n} = \frac{a(1 − r^{n})}{1 − r}$$S_n = (a(1 − r^n))/(1 − r) IN THE FORMULAE BOOKLETConditionsr ≠ 1. When r = 1 the sum is simply na.

S_∞ = a1 − r$$S_{∞} = \frac{a}{1 − r}$$S_∞ = a/(1 − r) IN THE FORMULAE BOOKLETConditions|r| < 1. Outside that the series diverges and there is no sum to infinity.

TrigonometryYears 12-1316 results

Triangles and the sine and cosine rules

asin A = bsin B = csin C$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$a/sin A = b/sin B = c/sin C NOT IN THE BOOKLET — LEARN ITConditionsAny triangle. Finding an angle is ambiguous: the calculator returns the acute value and the obtuse supplement may fit as well.

a² = b² + c² − 2bc cos A$$a^{2} = b^{2} + c^{2} − 2bc \cos A$$a^2 = b^2 + c^2 − 2bc cos A NOT IN THE BOOKLET — LEARN ITConditionsAny triangle, with A the angle opposite the side a.

Area = 12ab sin C$$\text{Area} = \frac{1}{2}ab \sin C$$Area = (1/2)ab sin C NOT IN THE BOOKLET — LEARN ITConditionsC the angle between the two sides a and b.

Trigonometric graphs and equations

tan θ = sin θcos θ$$\tan θ = \frac{\sin θ}{\cos θ}$$tan θ = sin θ/cos θ NOT IN THE BOOKLET — LEARN ITConditionscos θ ≠ 0, so θ is not an odd multiple of 90°.

sin ² θ + cos ² θ = 1$$\sin^{2} θ + \cos^{2} θ = 1$$sin^2 θ + cos^2 θ = 1 NOT IN THE BOOKLET — LEARN ITConditionsEvery angle θ; no restriction.

Radians, arcs and small angles

s = rθ$$s = rθ$$s = rθ NOT IN THE BOOKLET — LEARN ITConditionsθ in radians.

A = 12r²θ$$A = \frac{1}{2}r^{2}θ$$A = (1/2)r^2θ NOT IN THE BOOKLET — LEARN ITConditionsθ in radians.

sin θ ≈ θ, cos θ ≈ 1 − θ²2, tan θ ≈ θ$$\sin θ ≈ θ, \, \cos θ ≈ 1 − \frac{θ^{2}}{2}, \, \tan θ ≈ θ$$sin θ ≈ θ, cos θ ≈ 1 − (θ^2/2), tan θ ≈ θ IN THE FORMULAE BOOKLETConditionsθ small and measured in radians. In degrees all three are false.

Reciprocal and inverse trigonometric functions

sec² θ = 1 + tan² θ$$\text{sec}^{2} θ = 1 + \text{tan}^{2} θ$$sec^2 θ = 1 + tan^2 θ NOT IN THE BOOKLET — LEARN ITConditionscos θ ≠ 0.

cosec² θ = 1 + cot² θ$$\text{cosec}^{2} θ = 1 + \text{cot}^{2} θ$$cosec^2 θ = 1 + cot^2 θ NOT IN THE BOOKLET — LEARN ITConditionssin θ ≠ 0.

Compound angles and the harmonic form

sin (A ± B) = sin A cos B ± cos A sin B$$\sin (A ± B) = \sin A \cos B ± \cos A \sin B$$sin(A ± B) = sin A cos B ± cos A sin B IN THE FORMULAE BOOKLETConditionsAll A and B; no restriction.

cos (A ± B) = cos A cos B ∓ sin A sin B$$\cos (A ± B) = \cos A \cos B ∓ \sin A \sin B$$cos(A ± B) = cos A cos B ∓ sin A sin B IN THE FORMULAE BOOKLETConditionsAll A and B; no restriction.

tan (A ± B) = tan A ± tan B1 ∓ tan A tan B$$\tan (A ± B) = \frac{\tan A ± \tan B}{1 ∓ \tan A \tan B}$$tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B) IN THE FORMULAE BOOKLETConditionstan A, tan B and tan(A ± B) must all be defined, and 1 ∓ tan A tan B ≠ 0.

sin 2A = 2 sin A cos A$$\sin 2A = 2 \sin A \cos A$$sin 2A = 2 sin A cos A NOT IN THE BOOKLET — LEARN ITConditionsAll A; no restriction.

cos 2A = cos ²A − sin ²A = 2cos ²A − 1 = 1 − 2sin ²A$$\cos 2A = \cos^{2}A − \sin^{2}A = 2\cos^{2}A − 1 = 1 − 2\sin^{2}A$$cos 2A = cos^2A − sin^2A = 2cos^2A − 1 = 1 − 2sin^2A NOT IN THE BOOKLET — LEARN ITConditionsAll A; no restriction.

tan 2A = 2tan A1 − tan ²A$$\tan 2A = \frac{2\tan A}{1 − \tan^{2}A}$$tan 2A = 2 tan A/(1 − tan^2A) NOT IN THE BOOKLET — LEARN ITConditionstan A defined and tan²A ≠ 1.

Exponentials and logarithmsYear 125 results

Exponential functions and e

gradient of e^kx = ke^kx$$\text{gradient of } e^{kx} = ke^{kx}$$gradient of e^kx = ke^kx NOT IN THE BOOKLET — LEARN ITConditionsAll x; no restriction.

Logarithms and their laws

x = aⁿ ⇔ n = log_a x$$x = a^{n} \iff n = \text{log}_{a} x$$x = a^n ⇔ n = log_a x NOT IN THE BOOKLET — LEARN ITConditionsa > 0, a ≠ 1 and x > 0.

log_a x + log_a y = log_a xy$$\text{log}_{a} x + \text{log}_{a} y = \text{log}_{a} xy$$log_a x + log_a y = log_a xy NOT IN THE BOOKLET — LEARN ITConditionsa > 0, a ≠ 1, x > 0 and y > 0.

log_a x − log_a y = log_a xy$$\text{log}_{a} x − \text{log}_{a} y = \text{log}_{a} \frac{x}{y}$$log_a x − log_a y = log_a (x/y) NOT IN THE BOOKLET — LEARN ITConditionsa > 0, a ≠ 1, x > 0 and y > 0.

k log_a x = log_a x^k$$k \text{log}_{a} x = \text{log}_{a} x^{k}$$k log_a x = log_a x^k NOT IN THE BOOKLET — LEARN ITConditionsa > 0, a ≠ 1 and x > 0.

DifferentiationYears 12-137 results

The derivative from first principles

f'(x) = lim_{h → 0} f(x + h) − f(x)h$$f'(x) = \text{lim}_{h → 0} \frac{f(x + h) − f(x)}{h}$$f′(x) = lim_(h → 0) ((f(x + h) − f(x))/h) IN THE FORMULAE BOOKLETConditionsThe limit must exist, which is what it means for f to be differentiable at x.

Differentiating powers of x

y = xⁿ ⇒ dydx = nxⁿ⁻¹$$y = x^{n} \, ⇒ \, \frac{dy}{dx} = nx^{n−1}$$y = x^n ⇒ dy/dx = nx^(n − 1) NOT IN THE BOOKLET — LEARN ITConditionsAny real n, taking x > 0 where a non-integer power needs it.

Differentiating trig, exponentials and logs

sin kx → kcos kx, cos kx → −ksin kx$$\sin kx → k\cos kx, \, \cos kx → −k\sin kx$$sin kx → k cos kx, cos kx → − k sin kx NOT IN THE BOOKLET — LEARN ITConditionsx in radians. Both derivatives are wrong in degrees.

e^kx → ke^kx, ln x → 1x$$e^{kx} → ke^{kx}, \, \text{ln } x → \frac{1}{x}$$e^kx → ke^kx, ln x → 1/x NOT IN THE BOOKLET — LEARN ITConditionsln x needs x > 0.

The product, quotient and chain rules

dydx = dydu × dudx$$\frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dx}$$dy/dx = dy/du × du/dx NOT IN THE BOOKLET — LEARN ITConditionsy differentiable in u and u differentiable in x.

(uv)' = u'v − uv'v²$$(\frac{u}{v})' = \frac{u'v − uv'}{v^{2}}$$(u/v)′ = (u′v − uv′)/v^2 IN THE FORMULAE BOOKLETConditionsu and v differentiable, and v ≠ 0.

Implicit and parametric differentiation

dydx = 1dx/dy = 1cos y$$\frac{dy}{dx} = \frac{1}{dx/dy} = \frac{1}{\cos y}$$dy/dx = 1/(dx/dy) = 1/cos y NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsdx/dy must be non-zero and finite; here that means cos y ≠ 0.

IntegrationYears 12-1311 results

Integration as antidifferentiation

∫ xⁿ dx = xⁿ⁺¹n + 1 + c (n ≠ −1)$$\text{∫} x^{n} \, dx = \frac{x^{n+1}}{n + 1} + c \, (n ≠ −1)$$∫ x^n dx = x^(n + 1)/(n + 1) + c (n ≠ − 1) NOT IN THE BOOKLET — LEARN ITConditionsn ≠ −1, as printed, and an interval that avoids x = 0 when n is negative.

Integrating standard functions

∫ cos kx dx = 1ksin kx + c$$\text{∫} \cos kx \, dx = \frac{1}{k}\sin kx + c$$∫ cos kx dx = (1/k) sin kx + c NOT IN THE BOOKLET — LEARN ITConditionsk ≠ 0, x in radians.

∫ sin kx dx = −1kcos kx + c$$\text{∫} \sin kx \, dx = −\frac{1}{k}\cos kx + c$$∫ sin kx dx = −(1/k) cos kx + c NOT IN THE BOOKLET — LEARN ITConditionsk ≠ 0, x in radians.

∫ e^kx dx = 1ke^kx + c$$\text{∫} e^{kx} \, dx = \frac{1}{k}e^{kx} + c$$∫ e^kx dx = (1/k)e^kx + c NOT IN THE BOOKLET — LEARN ITConditionsk ≠ 0.

∫ 1x dx = ln|x| + c$$\text{∫} \frac{1}{x} \, dx = \text{ln}|x| + c$$∫ (1/x) dx = ln|x| + c NOT IN THE BOOKLET — LEARN ITConditionsx ≠ 0, and the constant may differ on each side of zero.

Integration by substitution and by parts

∫ f'(x)f(x) dx = ln|f(x)| + c$$\text{∫} \frac{f'(x)}{f(x)} \, dx = \text{ln}|f(x)| + c$$∫ (f′(x)/f(x)) dx = ln|f(x)| + c NOT IN THE BOOKLET — LEARN ITConditionsf(x) ≠ 0 right across the interval of integration.

∫ u dvdx dx = uv − ∫ v dudx dx$$\text{∫} u \frac{dv}{dx} \, dx = uv − \text{∫} v \frac{du}{dx} \, dx$$∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx IN THE FORMULAE BOOKLETConditionsu and v differentiable on the interval.

Integrating rational functions

∫ 23x + 5 dx = 23ln|3x + 5| + c$$\text{∫} \frac{2}{3x + 5} \, dx = \frac{2}{3}\text{ln}|3x + 5| + c$$∫ (2/(3x + 5)) dx = (2/3) ln|3x + 5| + c NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsThe interval of integration must not contain x = −5/3.

Areas, parametric curves and the limit of a sum

area = ∫ y dxdt dt$$\text{area} = \text{∫} y \, \frac{dx}{dt} \, dt$$area = ∫ y (dx/dt) dt NOT IN THE BOOKLET — LEARN ITConditionsx and y differentiable in t, limits taken as the t values at the ends, and y ≥ 0 for the integral to measure the area.

lim_{δx → 0} Σ f(x) δx = ∫_a^b f(x) dx$$\text{lim}_{δx → 0} \, Σ \, f(x) \, δx = \text{∫}_{a}^{b} f(x) \, dx$$lim_(δx → 0) Σ f(x) δx = ∫_a^b f(x) dx NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsf integrable on [a, b], with the strips covering the interval and their width tending to zero.

Solving differential equations

∫ 1g(y) dy = ∫ f(x) dx$$\text{∫} \frac{1}{g(y)} \, dy = \text{∫} f(x) \, dx$$∫ (1/g(y)) dy = ∫ f(x) dx NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsThe equation must separate as dy/dx = f(x)g(y), and g(y) ≠ 0. Values with g(y) = 0 are constant solutions that dividing throws away.

Numerical methodsYear 133 results

Locating roots and iteration

x_n+1 = (x_n + 1)^1/3$$x_{n+1} = (x_{n} + 1)^{1/3}$$x_(n + 1) = (x_n + 1)^(1/3) NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsOne rearrangement of x³ − x − 1 = 0. Convergence needs |g′(x)| < 1 near the root, and another rearrangement of the same equation may not converge at all.

Newton-Raphson and the trapezium rule

x_n+1 = x_n − f(x_n)f'(x_n)$$x_{n+1} = x_{n} − \frac{f(x_{n})}{f'(x_{n})}$$x_(n + 1) = x_n − f(x_n)/f′(x_n) IN THE FORMULAE BOOKLETConditionsf differentiable, f′(xₙ) ≠ 0, and a starting value close enough to the root; a turning point nearby throws it off.

T = h2(y₀ + 2(y₁ + … + y_n−1) + y_n)$$T = \frac{h}{2}(y_{0} + 2(y_{1} + … + y_{n−1}) + y_{n})$$T = (h/2)(y_0 + 2(y_1 + … + y_(n − 1)) + y_n) IN THE FORMULAE BOOKLETConditionsn strips of equal width h spanning [a, b].

VectorsYears 12-132 results

Vectors in two dimensions

AB = b − a$$\overrightarrow{AB} = \text{b} − \text{a}$$vec AB = b − a NOT IN THE BOOKLET — LEARN ITConditionsa and b the position vectors of A and B, measured from the same origin.

Vectors in three dimensions

|xi + yj + zk| = √x² + y² + z²$$|x\text{i} + y\text{j} + z\text{k}| = \sqrt{x^{2} + y^{2} + z^{2}}$$|x i + y j + z k| = √(x^2 + y^2 + z^2) NOT IN THE BOOKLET — LEARN ITConditionsComponents taken along mutually perpendicular axes.

StatisticsYears 12-1313 results

Sampling and the large data set

stratum sizepopulation size × sample size$$\frac{\text{stratum size}}{\text{population size}} \times \text{sample size}$$stratum size/population size × sample size NOT IN THE BOOKLET — LEARN ITConditionsNon-overlapping strata that between them cover the population; the numbers are then rounded to whole members.

Measures of location and spread

mean: Σ xn$$\text{mean: } \, \frac{Σ x}{n}$$mean: ((Σ x)/n) NOT IN THE BOOKLET — LEARN ITConditionsn ≥ 1.

σ² = Σ x²n - (Σ xn)²$$σ^{2} = \frac{Σ x^{2}}{n} - (\frac{Σ x}{n})^{2}$$σ^2 = (Σ x^2)/n - ((Σ x)/n)^2 IN THE FORMULAE BOOKLETConditionsThe variance of all n values treated as the population. An unbiased estimate of a population variance from a sample divides by n − 1 instead.

mean(x) = a + b mean(y),     σ_x = |b|σ_y$$\text{mean}(x) = a + b \, \text{mean}(y), \qquad σ_{x} = |b|σ_{y}$$mean(x) = a + b mean(y), σ_x = |b|σ_y NOT IN THE BOOKLET — LEARN ITConditionsCoding of the form x = a + by. The spread relation needs the magnitude of b: the standard deviation of x is |b| times that of y, and the form printed here assumes b > 0.

Representing and interpreting data

frequency density = frequencyclass width$$\text{frequency density} = \frac{\text{frequency}}{\text{class width}}$$frequency density = frequency/class width NOT IN THE BOOKLET — LEARN ITConditionsGrouped continuous data drawn to the equal-area convention, with class widths in the same units throughout.

outlier: beyond Q₁ - 1.5 × IQR or Q₃ + 1.5 × IQR$$\text{outlier: beyond } Q_{1} - 1.5 \times \text{IQR} \text{ or } Q_{3} + 1.5 \times \text{IQR}$$outlier: beyond Q_1 - 1.5 × IQR or Q_3 + 1.5 × IQR NOT IN THE BOOKLET — LEARN ITConditionsA convention rather than a definition. Use whichever rule the question states.

Probability and Venn diagrams

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)$$P(A ∪ B) = P(A) + P(B) − P(A ∩ B)$$P(A ∪ B) = P(A) + P(B) − P(A ∩ B) IN THE FORMULAE BOOKLETConditionsAny two events. For mutually exclusive events P(A ∩ B) is zero.

Conditional probability

P(A|B) = P(A ∩ B)P(B)$$P(A|B) = \frac{P(A ∩ B)}{P(B)}$$P(A|B) = (P(A ∩ B))/P(B) NOT IN THE BOOKLET — LEARN ITConditionsP(B) > 0.

The binomial distribution

P(X = x) = 1n$$\text{P}(X = x) = \frac{1}{n}$$P(X = x) = 1/n NOT IN THE BOOKLET — LEARN ITConditionsA discrete uniform distribution: n listed outcomes, every one equally likely. Equal probabilities are the whole condition, so a total of two dice does not qualify.

P(X = x) = ⁿC_x p^x (1 - p)^{n - x}$$P(X = x) = \, ^{n}C_{x} \, p^{x} (1 - p)^{n - x}$$P(X = x) = ^nC_x p^x (1 - p)^(n - x) IN THE FORMULAE BOOKLETConditionsn fixed independent trials, two outcomes each, constant p; x = 0, 1, …, n.

The normal distribution

Z = X - μσ$$Z = \frac{X - μ}{σ}$$Z = (X - μ)/σ NOT IN THE BOOKLET — LEARN ITConditionsX ~ N(μ, σ²) with σ > 0.

X ~ B(n, p) ≈ N(np, np(1 - p))$$X ~ B(n, p) \, \approx \, N(np, \, np(1 - p))$$X ~ B(n, p) ≈ N(np, np(1 - p)) NOT IN THE BOOKLET — LEARN ITConditionsn large with p not close to 0 or 1, so that np and n(1 − p) both comfortably exceed 5. A continuity correction is needed on the way across.

Hypothesis testing: correlation and the normal

mean of n observations: N(μ, σ²/n)$$\text{mean of } n \text{ observations: } \, N(μ, \, σ^{2}/n)$$mean of n observations: N(μ, σ^2/n) NOT IN THE BOOKLET — LEARN ITConditionsA random sample of n independent observations from N(μ, σ²).

MechanicsYears 12-138 results

Kinematics with constant acceleration

v = u + at,     s = (u + v)2t$$v = u + at, \qquad s = \frac{(u + v)}{2}t$$v = u + at, s = (((u + v))/2)t IN THE FORMULAE BOOKLETConditionsConstant acceleration in a straight line.

s = ut + 12at²,     v² = u² + 2as$$s = ut + \frac{1}{2}at^{2}, \qquad v^{2} = u^{2} + 2as$$s = ut + (1/2)at^2, v^2 = u^2 + 2as IN THE FORMULAE BOOKLETConditionsConstant acceleration in a straight line.

s = vt - 12at²$$s = vt - \frac{1}{2}at^{2}$$s = vt - (1/2)at^2 IN THE FORMULAE BOOKLETConditionsConstant acceleration in a straight line.

Kinematics with variable acceleration

v = dxdt,     a = dvdt$$v = \frac{dx}{dt}, \qquad a = \frac{dv}{dt}$$v = (dx/dt), a = dv/dt NOT IN THE BOOKLET — LEARN ITConditionsMotion along a straight line, with x and v differentiable in t.

Forces and Newton's laws

F = ma$$F = ma$$F = ma NOT IN THE BOOKLET — LEARN ITConditionsConstant mass, F the resultant force, measured in an inertial frame and in consistent units.

Projectiles

y = x tan θ - gx²2u²cos ² θ$$y = x \tan θ - \frac{gx^{2}}{2u^{2}\cos^{2} θ}$$y = x tan θ - gx^2/(2u^2cos^2 θ) NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsLaunch at angle θ with speed u from the origin, no air resistance, constant g, and the projectile treated as a particle.

Friction and inclined planes

F ≤ μR$$F ≤ μR$$F ≤ μR NOT IN THE BOOKLET — LEARN ITConditionsR the normal reaction. Equality holds only on the point of slipping or while sliding.

Moments

moment = force × perpendicular distance$$\text{moment} = \text{force} × \text{perpendicular distance}$$moment = force × perpendicular distance NOT IN THE BOOKLET — LEARN ITConditionsDistance measured perpendicular from the pivot to the line of action of the force.

Further proofFurther Maths1 results

Proof by induction: sums and series

true for n = 1, and true for k + 1 whenever true for k$$\text{true for } n = 1, \text{ and true for } k + 1 \text{ whenever true for } k$$true for n = 1, and true for k + 1 whenever true for k NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsBoth parts are needed: the base case proved outright, and the k + 1 case derived from the assumption at k rather than asserted.

Complex numbersFurther Maths2 results

Modulus, argument and loci

z = r(cos θ + i sin θ)$$z = r(\cos θ + \text{i} \sin θ)$$z = r(cos θ + i sin θ) NOT IN THE BOOKLET — LEARN ITConditionsr = |z| ≥ 0 and θ = arg z, conventionally taken in (−π, π].

De Moivre's theorem and trigonometric identities

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ$$(\cos θ + \text{i} \sin θ)^{n} = \cos nθ + \text{i} \sin nθ$$(cos θ + i sin θ)^n = cos nθ + i sin nθ IN THE FORMULAE BOOKLETConditionsn an integer. For fractional n the right-hand side gives only one of the several values.

Further algebra and seriesFurther Maths5 results

Roots of polynomials

α + β + γ = -ba, αβ + αγ + βγ = ca, αβγ = -da$$α + β + γ = -\frac{b}{a}, αβ + αγ + βγ = \frac{c}{a}, αβγ = -\frac{d}{a}$$α + β + γ = -(b/a), αβ + αγ + βγ = (c/a), αβγ = -d/a NOT IN THE BOOKLET — LEARN ITConditionsα, β, γ the roots of ax³ + bx² + cx + d = 0 with a ≠ 0, counted with multiplicity.

Summing series and the method of differences

Σ r = n(n+1)2$$Σ r = \frac{n(n+1)}{2}$$Σ r = (n(n + 1))/2 NOT IN THE BOOKLET — LEARN ITConditionsSummed from r = 1 to n, with n a positive integer.

Σ r² = n(n+1)(2n+1)6, Σ r³ = n²(n+1)²4$$Σ r^{2} = \frac{n(n+1)(2n+1)}{6}, Σ r^{3} = \frac{n^{2}(n+1)^{2}}{4}$$Σ r^2 = ((n(n + 1)(2n + 1))/6), Σ r^3 = (n^2(n + 1)^2)/4 IN THE FORMULAE BOOKLETConditionsSummed from r = 1 to n, with n a positive integer.

Maclaurin series

f(x) = f(0) + f'(0)x + f''(0)2!x² + …$$\text{f}(x) = \text{f}(0) + \text{f}'(0)x + \frac{\text{f}''(0)}{2!}x^{2} + …$$f(x) = f(0) + f′(0)x + (f′′(0)/2!)x^2 + … IN THE FORMULAE BOOKLETConditionsf and all its derivatives must exist at 0, and the series must converge to f(x) over the range used.

ln(1 + x) = x - x²2 + x³3 - x⁴4 + … (-1 < x ≤ 1)$$\text{ln}(1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + … (-1 < x ≤ 1)$$ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + … (-1 < x ≤ 1) IN THE FORMULAE BOOKLETConditions−1 < x ≤ 1, as printed. Outside that the series diverges.

Further vectorsFurther Maths3 results

Lines and planes in three dimensions

r = a + λb$$\text{r} = \text{a} + λ\text{b}$$r = a + λ b NOT IN THE BOOKLET — LEARN ITConditionsb ≠ 0, and a the position vector of a point on the line.

r · n = a · n = d$$\text{r} · \text{n} = \text{a} · \text{n} = d$$r · n = a · n = d NOT IN THE BOOKLET — LEARN ITConditionsn a non-zero normal to the plane, a the position vector of a point on it.

Intersections, angles and distances

distance = |n₁x₀ + n₂y₀ + n₃z₀ - d||n|$$\text{distance} = \frac{|n_{1}x_{0} + n_{2}y_{0} + n_{3}z_{0} - d|}{|\text{n}|}$$distance = (|n_1x_0 + n_2y_0 + n_3z_0 - d|)/| n| IN THE FORMULAE BOOKLETConditionsThe plane written as r·n = d with n = (n₁, n₂, n₃) and |n| ≠ 0.

Further calculusFurther Maths4 results

Volumes of revolution

V = π ∫_a^b y² dx$$V = π \text{∫}_{a}^{b} y^{2} \, dx$$V = π ∫_a^b y^2 dx NOT IN THE BOOKLET — LEARN ITConditionsA full turn about the x-axis of the region between the curve and that axis, with the curve not crossing the axis between a and b.

Mean values and improper integrals

mean value = 1b - a ∫_a^b f(x) dx$$\text{mean value} = \frac{1}{b - a} \text{∫}_{a}^{b} \text{f}(x) \, dx$$mean value = (1/(b - a)) ∫_a^b f(x) dx NOT IN THE BOOKLET — LEARN ITConditionsb ≠ a, and f integrable across [a, b].

Calculus with inverse trigonometric functions

ddx(arcsin x) = 1√1 - x²$$\frac{d}{dx}(\text{arcsin } x) = \frac{1}{\sqrt{1 - x^{2}}}$$(d/dx)(arcsin x) = 1/√(1 - x^2) IN THE FORMULAE BOOKLETConditions|x| < 1.

ddx(arctan x) = 11 + x²$$\frac{d}{dx}(\text{arctan } x) = \frac{1}{1 + x^{2}}$$(d/dx)(arctan x) = 1/(1 + x^2) IN THE FORMULAE BOOKLETConditionsAll x; no restriction.

Hyperbolic functionsFurther Maths2 results

Hyperbolic functions and identities

cosh x = e^x + e^-x2, sinh x = e^x - e^-x2$$\text{cosh } x = \frac{e^{x} + e^{-x}}{2}, \text{sinh } x = \frac{e^{x} - e^{-x}}{2}$$cosh x = ((e^x + e^(-x))/2), sinh x = (e^x - e^(-x))/2 NOT IN THE BOOKLET — LEARN ITConditionsDefinitions rather than results, so they hold for every real x.

Calculus with hyperbolic functions

∫ 1√x² + a² dx = arsinhxa + c$$\text{∫} \frac{1}{\sqrt{x^{2} + a^{2}}} \, dx = \text{arsinh}\frac{x}{a} + c$$∫ (1/√(x^2 + a^2)) dx = arsinh (x/a) + c IN THE FORMULAE BOOKLETConditionsa > 0.

Polar coordinatesFurther Maths2 results

Polar curves

x = r cos θ, y = r sin θ, r² = x² + y²$$x = r \cos θ, y = r \sin θ, r^{2} = x^{2} + y^{2}$$x = r cos θ, y = r sin θ, r^2 = x^2 + y^2 NOT IN THE BOOKLET — LEARN ITConditionsr ≥ 0, with the quadrant of θ chosen to match the signs of x and y rather than taken straight from a calculator.

Areas with polar coordinates

A = 12 ∫_α^β r² dθ$$A = \frac{1}{2} \text{∫}_{α}^{β} r^{2} \, dθ$$A = (1/2) ∫_α^β r^2 dθ IN THE FORMULAE BOOKLETConditionsThe region swept between θ = α and θ = β, traced once, with r ≥ 0 throughout.

Differential equationsFurther Maths2 results

First order equations and integrating factors

IF = e^{∫ P dx}$$\text{IF} = e^{\text{∫} P \, dx}$$IF = e^(∫ P dx) NOT IN THE BOOKLET — LEARN ITConditionsThe equation must first be written as dy/dx + P(x)y = Q(x), with the coefficient of dy/dx equal to 1.

Second order equations

am² + bm + c = 0$$am^{2} + bm + c = 0$$am^2 + bm + c = 0 NOT IN THE BOOKLET — LEARN ITConditionsConstant coefficients with a ≠ 0, for the homogeneous equation ay″ + by′ + cy = 0.

Further Pure 1Further Maths7 results

The t-formulae

sin θ = 2t1 + t², cos θ = 1 - t²1 + t², tan θ = 2t1 - t²$$\sin θ = \frac{2t}{1 + t^{2}}, \cos θ = \frac{1 - t^{2}}{1 + t^{2}}, \tan θ = \frac{2t}{1 - t^{2}}$$sin θ = (2t/(1 + t^2)), cos θ = ((1 - t^2)/(1 + t^2)), tan θ = 2t/(1 - t^2) NOT IN THE BOOKLET — LEARN ITConditionst = tan(θ/2), so θ/2 must avoid odd multiples of 90°; the tan form additionally needs t² ≠ 1.

Taylor series

f(x) = f(a) + f'(a)(x - a) + f''(a)2!(x - a)² + …$$\text{f}(x) = \text{f}(a) + \text{f}'(a)(x - a) + \frac{\text{f}''(a)}{2!}(x - a)^{2} + …$$f(x) = f(a) + f′(a)(x - a) + (f′′(a)/2!)(x - a)^2 + … NOT IN THE BOOKLET — LEARN ITConditionsf differentiable to the order used at a, and x inside the interval where the series converges.

Leibnitz's theorem and the Weierstrass substitution

(fg)⁽ⁿ⁾ = Σ ⁿC_r f^(r) g^(n-r)$$(\text{fg})^{(n)} = Σ \, ^{n}C_{r} \, \text{f}^{(r)} \text{g}^{(n-r)}$$(fg)^(n) = Σ ^nC_r f^(r) g^((n - r)) NOT IN THE BOOKLET — LEARN ITConditionsf and g both n times differentiable.

Conic sections

x²a² + y²b² = 1, x = a cos t, y = b sin t$$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, x = a \cos t, y = b \sin t$$x^2/a^2 + y^2/b^2 = 1, x = a cos t, y = b sin t IN THE FORMULAE BOOKLETConditionsa > 0 and b > 0; the parametrisation traces the whole ellipse as t runs over 2π.

The vector product and the scalar triple product

a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)$$\text{a} × \text{b} = (a_{2}b_{3} - a_{3}b_{2}, \, a_{3}b_{1} - a_{1}b_{3}, \, a_{1}b_{2} - a_{2}b_{1})$$a × b = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) IN THE FORMULAE BOOKLETConditionsComponents taken in a right-handed set of axes.

Numerical methods for differential equations

dydx ≈ y_n+1 - y_nh or y_n+1 - y_n-12h$$\frac{dy}{dx} \approx \frac{y_{n+1} - y_{n}}{h} \text{ or } \frac{y_{n+1} - y_{n-1}}{2h}$$dy/dx ≈ ((y_(n + 1) - y_n)/h) or ((y_(n + 1) - y_(n - 1))/2h) NOT IN THE BOOKLET — LEARN ITConditionsBoth improve as the step h shrinks, and the central form is the more accurate for the same h, but it needs a value on each side of the point.

∫_a^b y dx ≈ h3(y₀ + 4y₁ + 2y₂ + … + 4y_n-1 + y_n)$$\text{∫}_{a}^{b} y \, dx \approx \frac{h}{3}(y_{0} + 4y_{1} + 2y_{2} + … + 4y_{n-1} + y_{n})$$∫_a^b y dx ≈ (h/3)(y_0 + 4y_1 + 2y_2 + … + 4y_(n - 1) + y_n) NOT IN THE BOOKLET — LEARN ITConditionsAn even number of strips of equal width h, so the points come in triples. With an odd number of strips the rule does not apply.

Further Statistics 1Further Maths13 results

Discrete random variables and expectation

E(X) = Σ x P(X = x)$$\text{E}(X) = Σ \, x \, \text{P}(X = x)$$E(X) = Σ x P(X = x) IN THE FORMULAE BOOKLETConditionsX discrete, summed over every value in its range, and the sum must converge.

Var(X) = Σ x² P(X = x) - μ²$$\text{Var}(X) = Σ \, x^{2} \, \text{P}(X = x) - μ^{2}$$Var(X) = Σ x^2 P(X = x) - μ^2 IN THE FORMULAE BOOKLETConditionsX discrete with μ = E(X), and E(X²) must exist.

The Poisson distribution

P(X = x) = e^-λ λ^xx!, E(X) = Var(X) = λ$$\text{P}(X = x) = e^{-λ} \frac{λ^{x}}{x!}, \text{E}(X) = \text{Var}(X) = λ$$P(X = x) = e^(-λ) (λ^x/x!), E(X) = Var(X) = λ IN THE FORMULAE BOOKLETConditionsEvents occurring singly, independently and at a constant average rate λ > 0; x = 0, 1, 2, …

Geometric and negative binomial distributions

P(X = x) = p(1 - p)^x-1, μ = 1p, σ² = 1 - pp²$$\text{P}(X = x) = p(1 - p)^{x-1}, μ = \frac{1}{p}, σ^{2} = \frac{1 - p}{p^{2}}$$P(X = x) = p(1 - p)^(x - 1), μ = (1/p), σ^2 = (1 - p)/p^2 IN THE FORMULAE BOOKLETConditionsIndependent trials with constant p, 0 < p ≤ 1, X counting the trial of the first success; x = 1, 2, 3, …

P(X = x) = ^x-1C_r-1 p^r(1 - p)^x-r$$\text{P}(X = x) = \, ^{x-1}C_{r-1} \, p^{r}(1 - p)^{x-r}$$P(X = x) = ^(x - 1)C_(r - 1) p^r(1 - p)^(x - r) IN THE FORMULAE BOOKLETConditionsIndependent trials with constant p, X counting the trial of the rth success; x = r, r + 1, …

μ = rp, σ² = r(1 - p)p²$$μ = \frac{r}{p}, σ^{2} = \frac{r(1 - p)}{p^{2}}$$μ = (r/p), σ^2 = (r(1 - p))/p^2 IN THE FORMULAE BOOKLETConditionsThe negative binomial above, with p > 0.

The Central Limit Theorem

sample mean ≈ N(μ, σ²n)$$\text{sample mean} \approx \text{N}(μ, \frac{σ^{2}}{n})$$sample mean ≈ N(μ, (σ^2/n)) IN THE FORMULAE BOOKLETConditionsA random sample of n independent, identically distributed observations with a finite mean and variance, and n large. The parent distribution need not be normal.

Goodness-of-fit tests

χ² = Σ (O_i - E_i)²E_i$$χ^{2} = Σ \frac{(O_{i} - E_{i})^{2}}{E_{i}}$$χ^2 = Σ (((O_i - E_i)^2)/E_i) IN THE FORMULAE BOOKLETConditionsExpected frequencies all at least 5, pooling classes where they are not, and the observations independent.

Contingency tables

E_ij = row total × column totalgrand total$$E_{ij} = \frac{\text{row total} × \text{column total}}{\text{grand total}}$$E_ij = (row total × column total)/grand total NOT IN THE BOOKLET — LEARN ITConditionsComputed under the null hypothesis that the two classifications are independent; each expected frequency should be at least 5.

Probability generating functions

G(t) = E(t^X) = Σ t^x P(X = x)$$\text{G}(t) = \text{E}(t^{X}) = Σ \, t^{x} \, \text{P}(X = x)$$G(t) = E(t^X) = Σ t^x P(X = x) IN THE FORMULAE BOOKLETConditionsX taking non-negative integer values, with the series convergent, which it is for |t| ≤ 1.

Var(X) = G''(1) + G'(1) - [G'(1)]²$$\text{Var}(X) = \text{G}''(1) + \text{G}'(1) - [\text{G}'(1)]^{2}$$Var(X) = G′′(1) + G′(1) - [ G′(1)]^2 IN THE FORMULAE BOOKLETConditionsG differentiable twice at t = 1, so the mean and variance must exist.

B(n, p): (q + pt)ⁿ,      Po(λ): e^{λ(t - 1)}$$\text{B}(n, p): (q + pt)^{n}, \qquad \text{Po}(λ): e^{λ(t - 1)}$$B(n, p): (q + pt)^n, Po(λ): e^(λ(t - 1)) IN THE FORMULAE BOOKLETConditionsB(n, p) with q = 1 − p; Po(λ) with λ > 0.

Geo(p): pt1 - qt,      Neg B(r, p): (pt1 - qt)^r$$\text{Geo}(p): \frac{pt}{1 - qt}, \qquad \text{Neg B}(r, p): (\frac{pt}{1 - qt})^{r}$$Geo(p): (pt/(1 - qt)), Neg B(r, p): (pt/(1 - qt))^r IN THE FORMULAE BOOKLETConditionsq = 1 − p, and |qt| < 1 for the series to converge.

Further Pure 2Further Maths5 results

Reduction formulae

n I_n = (n - 1) I_n-2, I_n = ∫₀^π/2 sin ⁿ x dx$$n I_{n} = (n - 1) I_{n-2}, I_{n} = \text{∫}_{0}^{π/2} \sin^{n} x \, dx$$n I_n = (n - 1) I_(n - 2), I_n = ∫_0^(π/2) sin^n x dx NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsThis formula belongs to Iₙ = ∫₀^π/2 sinⁿ x dx with n ≥ 2. Every reduction formula is tied to its own integral and its own limits.

Arc length and surface area

s = ∫ √1 + (dydx)² dx$$s = \text{∫} \sqrt{1 + (\frac{dy}{dx})^{2}} \, dx$$s = ∫ √(1 + (dy/dx)^2) dx IN THE FORMULAE BOOKLETConditionsy differentiable with a continuous derivative across the interval.

Eigenvalues and eigenvectors

det(M - λI) = 0$$\text{det}(\text{M} - λ\text{I}) = 0$$det(M - λ I) = 0 NOT IN THE BOOKLET — LEARN ITConditionsM a square matrix.

Diagonalisation and the Cayley-Hamilton theorem

P^-1MP = D$$\text{P}^{-1}\text{MP} = \text{D}$$P^(-1) MP = D NOT IN THE BOOKLET — LEARN ITConditionsM must have a full set of independent eigenvectors; P is then invertible with those eigenvectors as its columns.

Modular arithmetic and Fermat's little theorem

a^p-1 ≡ 1 (mod p)$$a^{p-1} \equiv 1 \, (\text{mod } p)$$a^(p - 1) ≡ 1 (mod p) NOT IN THE BOOKLET — LEARN ITConditionsp prime, and p must not divide a.

Further Statistics 2Further Maths19 results

Least squares regression and residuals

b = S_xyS_xx,     a = mean of y - b × mean of x$$b = \frac{S_{xy}}{S_{xx}}, \qquad a = \text{mean of y} - b \times \text{mean of x}$$b = (S_xy/S_xx), a = mean of y - b × mean of x IN THE FORMULAE BOOKLETConditionsThe least squares line of y on x, with Sxx not zero. It is not the line to use for predicting x from y.

RSS = S_yy - (S_xy)²S_xx$$\text{RSS} = S_{yy} - \frac{(S_{xy})^{2}}{S_{xx}}$$RSS = S_yy - (S_xy)^2/S_xx IN THE FORMULAE BOOKLETConditionsThe least squares line of y on x, with Sxx not zero.

Continuous random variables: density and distribution functions

P(a < X ≤ b) = ∫_a^b f(x) dx$$\text{P}(a < X \le b) = \text{∫}_{a}^{b} \text{f}(x) \, dx$$P(a < X ≤ b) = ∫_a^b f(x) dx NOT IN THE BOOKLET — LEARN ITConditionsX continuous with density f, where f ≥ 0 and integrates to 1 across the whole range.

f(x) = dF(x)dx$$\text{f}(x) = \frac{d\text{F}(x)}{dx}$$f(x) = d F(x)/dx NOT IN THE BOOKLET — LEARN ITConditionsWherever F is differentiable.

Mean, variance and skewness of continuous variables

E(X) = ∫ x f(x) dx,      Var(X) = E(X²) - [E(X)]²$$\text{E}(X) = \text{∫} x \, \text{f}(x) \, dx, \qquad \text{Var}(X) = \text{E}(X^{2}) - [\text{E}(X)]^{2}$$E(X) = ∫ x f(x) dx, Var(X) = E(X^2) - [ E(X)]^2 IN THE FORMULAE BOOKLETConditionsIntegrated across the whole range on which f is non-zero, and both integrals must converge.

The continuous uniform distribution

F(x) = 0 for x < a,      x - ab - a for a ≤ x ≤ b,      1 for x > b$$\text{F}(x) = 0 \text{ for } x < a, \qquad \frac{x - a}{b - a} \text{ for } a \le x \le b, \qquad 1 \text{ for } x > b$$F(x) = 0 for x < a, ((x - a)/(b - a)) for a ≤ x ≤ b, 1 for x > b NOT IN THE BOOKLET — LEARN ITConditionsa < b. Defined for every real x: the ramp holds on [a, b], and the two flat branches carry F to 0 below a and to 1 above b.

E(X) = a + b2,      Var(X) = (b - a)²12$$\text{E}(X) = \frac{a + b}{2}, \qquad \text{Var}(X) = \frac{(b - a)^{2}}{12}$$E(X) = ((a + b)/2), Var(X) = ((b - a)^2)/12 IN THE FORMULAE BOOKLETConditionsX uniform on [a, b] with a < b.

Correlation coefficients: product moment and Spearman

r = S_xy√S_xx S_yy$$r = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}$$r = S_xy/√(S_xx S_yy) IN THE FORMULAE BOOKLETConditionsSxx and Syy both non-zero. It measures linear association only, so a strong curved relationship can still give r near zero.

r_s = 1 - 6 Σ d²n(n² - 1)$$r_{s} = 1 - \frac{6 Σ d^{2}}{n(n^{2} - 1)}$$r_s = 1 - (6 Σ d^2)/(n(n^2 - 1)) IN THE FORMULAE BOOKLETConditionsNo tied ranks. With ties, rank the tied values by their mean rank and compute the product moment coefficient of those ranks instead.

Combinations of normal random variables

aX ± bY is normal, with mean aμ_x ± bμ_y$$aX ± bY \text{ is normal, with mean } aμ_{x} ± bμ_{y}$$aX ± bY is normal, with mean aμ_x ± bμ_y NOT IN THE BOOKLET — LEARN ITConditionsX and Y independent, and each normally distributed.

Var(aX ± bY) = a²σ_x² + b²σ_y²$$\text{Var}(aX ± bY) = a^{2}σ_{x}^{2} + b^{2}σ_{y}^{2}$$Var(aX ± bY) = a^2σ_x^2 + b^2σ_y^2 IN THE FORMULAE BOOKLETConditionsX and Y independent. Without independence a covariance term appears, whichever sign the combination carries.

Estimators, standard error and confidence intervals

s² = 1n - 1 Σ (x_i - sample mean)²$$s^{2} = \frac{1}{n - 1} Σ (x_{i} - \text{sample mean})^{2}$$s^2 = (1/(n - 1)) Σ (x_i - sample mean)^2 IN THE FORMULAE BOOKLETConditionsA random sample of n ≥ 2 observations; this is the unbiased estimator of the population variance.

s² = 1n - 1(Σ x² - (Σ x)²n)$$s^{2} = \frac{1}{n - 1}(Σ x^{2} - \frac{(Σ x)^{2}}{n})$$s^2 = (1/(n - 1))(Σ x^2 - ((Σ x)^2)/n) NOT IN THE BOOKLET — LEARN ITConditionsA random sample of n ≥ 2 observations; algebraically the same estimator as the one above.

sample mean ± z × σ√n,     z = 1.96 at 95 per cent$$\text{sample mean} ± z \times \frac{σ}{\sqrt{n}}, \qquad z = 1.96 \text{ at 95 per cent}$$sample mean ± z × (σ/√(n)), z = 1.96 at 95 per cent NOT IN THE BOOKLET — LEARN ITConditionsA random sample with σ known, and either a normal population or a sample large enough for the central limit theorem.

Comparing two normal means

(difference of means) - (μ_x - μ_y)√σ_x²n_x + σ_y²n_y is N(0, 1)$$\frac{(\text{difference of means}) - (μ_{x} - μ_{y})}{\sqrt{\frac{σ_{x}^{2}}{n_{x}} + \frac{σ_{y}^{2}}{n_{y}}}} \text{ is N}(0, 1)$$(((difference of means) - (μ_x - μ_y))/√(σ_x^2/n_x + σ_y^2/n_y)) is N(0, 1) IN THE FORMULAE BOOKLETConditionsIndependent random samples from normal populations with known variances; for non-normal populations both samples must be large.

Testing variances: chi-squared and the F-distribution

(n - 1)S²σ² is χ² on n - 1 degrees of freedom$$\frac{(n - 1)S^{2}}{σ^{2}} \text{ is } χ^{2} \text{ on } n - 1 \text{ degrees of freedom}$$(((n - 1)S^2)/σ^2) is χ^2 on n - 1 degrees of freedom IN THE FORMULAE BOOKLETConditionsA random sample from a normal population. The result is sensitive to departures from normality in a way the t test is not.

S₁²S₂² is F on n₁ - 1 and n₂ - 1 degrees of freedom$$\frac{S_{1}^{2}}{S_{2}^{2}} \text{ is } F \text{ on } n_{1} - 1 \text{ and } n_{2} - 1 \text{ degrees of freedom}$$(S_1^2/S_2^2) is F on n_1 - 1 and n_2 - 1 degrees of freedom IN THE FORMULAE BOOKLETConditionsIndependent random samples from two normal populations, equal variances under the null hypothesis, and the larger estimate placed on top.

Confidence intervals and tests with the t-distribution

sample mean - μS√n is t on n - 1 degrees of freedom$$\frac{\text{sample mean} - μ}{\frac{S}{\sqrt{n}}} \text{ is } t \text{ on } n - 1 \text{ degrees of freedom}$$((sample mean - μ)/(S/√(n))) is t on n - 1 degrees of freedom IN THE FORMULAE BOOKLETConditionsA random sample from a normal population with σ unknown, S being the sample standard deviation with divisor n − 1.

s² = (n₁ - 1)s₁² + (n₂ - 1)s₂²n₁ + n₂ - 2$$s^{2} = \frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2}$$s^2 = ((n_1 - 1)s_1^2 + (n_2 - 1)s_2^2)/(n_1 + n_2 - 2) IN THE FORMULAE BOOKLETConditionsTwo independent random samples from normal populations assumed to share a common variance.

Further Mechanics 1Further Maths8 results

Momentum and impulse

I = Ft = mv - mu$$\text{I} = \text{F}t = m\text{v} - m\text{u}$$I = F t = m v - m u NOT IN THE BOOKLET — LEARN ITConditionsA constant force acting for time t on a constant mass, with one positive direction fixed along the line.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂$$m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 NOT IN THE BOOKLET — LEARN ITConditionsNo external impulse on the pair during the collision, and all velocities measured along the same line with one sign convention.

Impulse and momentum as vectors

I = mv - mu$$\text{I} = m\text{v} - m\text{u}$$I = m v - m u NOT IN THE BOOKLET — LEARN ITConditionsConstant mass, with the vectors resolved in the same pair of directions throughout.

Work, energy and power

KE = 12mv²,      GPE = mgh$$\text{KE} = \frac{1}{2}mv^{2}, \qquad \text{GPE} = mgh$$KE = (1/2)mv^2, GPE = mgh NOT IN THE BOOKLET — LEARN ITConditionsGPE is measured from a stated reference level; KE uses speed, so direction does not enter it.

P = Fv$$P = Fv$$P = Fv NOT IN THE BOOKLET — LEARN ITConditionsF the driving force along the direction of motion, and v the instantaneous speed.

Hooke's law and elastic strings

T = λ xl$$T = \frac{λ x}{l}$$T = (λ x)/l NOT IN THE BOOKLET — LEARN ITConditionsAn elastic string or spring of natural length l with x the extension. A string goes slack rather than pushing, so T = 0 once the extension would be negative.

Elastic potential energy

EPE = λ x²2l$$\text{EPE} = \frac{λ x^{2}}{2l}$$EPE = (λ x^2)/2l NOT IN THE BOOKLET — LEARN ITConditionsThe same string or spring, x the extension or compression, and the deformation within the elastic limit.

Direct impact and Newton's law of restitution

e = speed of separationspeed of approach$$e = \frac{\text{speed of separation}}{\text{speed of approach}}$$e = speed of separation/speed of approach NOT IN THE BOOKLET — LEARN ITConditionsDirect impact along the line of centres, with 0 ≤ e ≤ 1 and speeds measured along that line.

Further Mechanics 2Further Maths18 results

Angular speed and horizontal circular motion

a = rω² = v²r$$a = rω^{2} = \frac{v^{2}}{r}$$a = rω^2 = v^2/r IN THE FORMULAE BOOKLETConditionsCircular motion of radius r. This is the component towards the centre; if the speed varies there is a tangential component as well.

tan θ = v²rg$$\tan θ = \frac{v^{2}}{rg}$$tan θ = v^2/rg NOT IN THE BOOKLET — LEARN ITConditionsA banked track or conical pendulum with no friction, moving in a horizontal circle at constant speed v. Once friction acts, this design speed no longer applies.

Motion in a vertical circle

T - mgcos θ = mv²r$$T - mg\cos θ = \frac{mv^{2}}{r}$$T - mg cos θ = mv^2/r NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsResolving towards the centre for a particle on a light inextensible string of length r, θ measured from the lowest point, and valid only while the string stays taut.

Centre of mass of a discrete distribution

mean x = Σ m_ix_iΣ m_i,      mean y = Σ m_iy_iΣ m_i$$\text{mean x} = \frac{Σ m_{i}x_{i}}{Σ m_{i}}, \qquad \text{mean y} = \frac{Σ m_{i}y_{i}}{Σ m_{i}}$$mean x = ((Σ m_ix_i)/(Σ m_i)), mean y = (Σ m_iy_i)/(Σ m_i) NOT IN THE BOOKLET — LEARN ITConditionsA non-zero total mass, with the particles coplanar and all coordinates measured from one origin.

Centres of mass of plane figures and frameworks

Triangular lamina: 23 along the median from the vertex$$\text{Triangular lamina: } \tfrac{2}{3} \text{ along the median from the vertex}$$Triangular lamina: (2/3) along the median from the vertex IN THE FORMULAE BOOKLETConditionsA uniform triangular lamina.

Circular arc, radius r, angle at centre 2α: r sin αα from the centre$$\text{Circular arc, radius } r, \text{ angle at centre } 2α: \frac{r \sin α}{α} \text{ from the centre}$$Circular arc, radius r, angle at centre 2α: ((r sin α)/α) from the centre IN THE FORMULAE BOOKLETConditionsA uniform circular arc, with α in radians.

Sector of a circle, radius r, angle at centre 2α: 2r sin α3α from the centre$$\text{Sector of a circle, radius } r, \text{ angle at centre } 2α: \frac{2r \sin α}{3α} \text{ from the centre}$$Sector of a circle, radius r, angle at centre 2α: ((2r sin α)/3α) from the centre IN THE FORMULAE BOOKLETConditionsA uniform sector of a circle, with α in radians.

Solid hemisphere, radius r: 38r from the centre$$\text{Solid hemisphere, radius } r: \tfrac{3}{8}r \text{ from the centre}$$Solid hemisphere, radius r: (3/8)r from the centre IN THE FORMULAE BOOKLETConditionsA uniform solid hemisphere.

Hemispherical shell, radius r: 12r from the centre$$\text{Hemispherical shell, radius } r: \tfrac{1}{2}r \text{ from the centre}$$Hemispherical shell, radius r: (1/2)r from the centre IN THE FORMULAE BOOKLETConditionsA uniform hemispherical shell.

Solid cone or pyramid, height h: 14h above the base, on the line from the centre of the base to the vertex$$\text{Solid cone or pyramid, height } h: \tfrac{1}{4}h \text{ above the base, on the line from the centre of the base to the vertex}$$Solid cone or pyramid, height h: (1/4)h above the base, on the line from the centre of the base to the vertex IN THE FORMULAE BOOKLETConditionsA uniform solid cone or pyramid.

Conical shell, height h: 13h above the base, on the same line$$\text{Conical shell, height } h: \tfrac{1}{3}h \text{ above the base, on the same line}$$Conical shell, height h: (1/3)h above the base, on the same line IN THE FORMULAE BOOKLETConditionsA uniform conical shell.

Centres of mass by integration

mean x = ∫ xy dx∫ y dx,      mean y = ∫ 12y² dx∫ y dx$$\text{mean x} = \frac{\text{∫} xy \, dx}{\text{∫} y \, dx}, \qquad \text{mean y} = \frac{\text{∫} \frac{1}{2}y^{2} \, dx}{\text{∫} y \, dx}$$mean x = ((∫ xy dx)/(∫ y dx)), mean y = (∫ (1/2)y^2 dx)/(∫ y dx) NOT IN THE BOOKLET — LEARN ITConditionsA uniform lamina bounded by y = f(x), the x-axis and the limits, with y ≥ 0 across the interval.

mean x = ∫ x y² dx∫ y² dx$$\text{mean x} = \frac{\text{∫} x y^{2} \, dx}{\text{∫} y^{2} \, dx}$$mean x = (∫ x y^2 dx)/(∫ y^2 dx) NOT IN THE BOOKLET — LEARN ITConditionsA uniform solid of revolution, formed by rotating y = f(x) through a full turn about the x-axis.

Newton's laws with a variable force

F(x) = mvdvdx,     F(t) = mdvdt$$F(x) = mv\frac{dv}{dx}, \qquad F(t) = m\frac{dv}{dt}$$F(x) = mv(dv/dx), F(t) = m(dv/dt) NOT IN THE BOOKLET — LEARN ITConditionsMotion in a straight line. Use the dv/dx form when the force is given in terms of x or v, and the dv/dt form when it is given in terms of t or v.

Simple harmonic motion

d²xdt² = -ω²x$$\frac{d^{2}x}{dt^{2}} = -ω^{2}x$$d^2x/dt^2 = -ω^2x NOT IN THE BOOKLET — LEARN ITConditionsSimple harmonic motion: the acceleration must be proportional to the displacement from a fixed point and directed towards it, with ω > 0.

v² = ω²(a² - x²)$$v^{2} = ω^{2}(a^{2} - x^{2})$$v^2 = ω^2(a^2 - x^2) NOT IN THE BOOKLET — LEARN ITConditionsThe same simple harmonic motion, a the amplitude and x measured from the centre of the oscillation, so |x| ≤ a.

Oscillations on strings and springs

md²xdt² = -λ xl,     ω² = λml$$m\frac{d^{2}x}{dt^{2}} = -\frac{λ x}{l}, \qquad ω^{2} = \frac{λ}{ml}$$m(d^2x/dt^2) = -((λ x)/l), ω^2 = λ/ml NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionsx measured from the equilibrium position rather than the natural length, and the string must stay taut throughout; a spring may be compressed, a string may not.

Further kinematics: acceleration as a function of x, t or v

dvdt = f(t) or f(v),     vdvdx = f(v) or f(x)$$\frac{dv}{dt} = \text{f}(t) \text{ or f}(v), \qquad v\frac{dv}{dx} = \text{f}(v) \text{ or f}(x)$$dv/dt = f(t) or f(v), v(dv/dx) = f(v) or f(x) NOT IN THE BOOKLET — LEARN ITConditionsMotion in a straight line, choosing whichever form matches the variable the acceleration is given in.

Decision Mathematics 1Further Maths1 results

Linear programming: formulation and graphical solution

3x + 2y ≤ 20 becomes 3x + 2y + s₁ = 20$$3x + 2y \le 20 \text{ becomes } 3x + 2y + s_{1} = 20$$3x + 2y ≤ 20 becomes 3x + 2y + s_1 = 20 NOT IN THE BOOKLET — DERIVED IN THE LESSONConditionss₁ ≥ 0, added to a ≤ constraint. A ≥ constraint takes a subtracted surplus variable and then an artificial variable as well.

Decision Mathematics 2Further Maths1 results

The stepping-stone method

improvement index = cost - R - K$$\text{improvement index} = \text{cost} - R - K$$improvement index = cost - R - K NOT IN THE BOOKLET — LEARN ITConditionsShadow costs R and K fixed from the occupied cells of a balanced problem, where total supply equals total demand.

177 results in total, 98 of them to learn outright. A result badged derived in the lesson is a working line the notes build in front of you rather than a standard result to memorise. See also the key ideas and key definitions.