Panhellenic Exact Sciences - independent subject practice cheat sheet

Chain rule: never drop the inner derivative g'(x).

Integration by parts: choose u so du is simpler than u.

Change limits immediately when substituting in a definite integral.

Improper integrals require an explicit limit, not direct substitution.

Implicit differentiation collects dy/dx terms after differentiating both sides.

State domain restrictions before simplifying; cancelled factors still exclude values.

Composition f(g(x)) is evaluated inside-out and is generally not commutative.

Inverses require one-to-one functions; domain and range swap.

Horizontal transformations act opposite to the sign or factor shown.

Log laws convert sums to products, enabling solution of log equations.

Always check candidate solutions against original domain restrictions.

Similarity scales lengths by k, areas by k², volumes by k³.

Inscribed angle equals half the intercepted central arc.

Sine law requires a known side–opposite-angle pair; watch the ambiguous case.

Cosine law handles SAS and SSS configurations.

Arc length and sector area formulas require radians.

Pythagorean identity sin²θ + cos²θ = 1 underpins most simplifications.

Bayes: posterior ∝ likelihood × prior; always compute P(+) first.

Standard error of the mean is σ/√n, not σ.

p-value = P(data at least as extreme | H₀), never P(H₀ true).

Binomial variance np(1 − p); Poisson mean and variance both λ.

Sample variance uses n − 1 to be unbiased.

Resolve vectors before applying F = ma or the SUVAT equations

Momentum is a vector: conserve each component only when the corresponding net external impulse is zero

Equilibrium requires both zero resultant force and zero resultant moment

The centripetal component points to the centre; uniform circular motion has constant speed but changing velocity

SHM signature: a = −ω²x, giving T = 2π/ω

State g and sign conventions explicitly at the start of a solution

EMF is per unit charge from the source; terminal p.d. is E − Ir

Kirchhoff's laws follow from conservation of charge and energy

Capacitor energy is ½CV²; discharge is exponential with time constant RC

Magnetic-force magnitude is |q|vB sinθ and its vector is perpendicular to velocity, so magnetic force does no work

n = m/M, n = cV (V in dm3) and n = V/24.0 for gases at RTP are the three key conversions.

The limiting reagent is the one with the smallest moles ÷ coefficient; product amount depends on it.

Percentage yield measures practical success; atom economy measures how much reactant mass is useful.

Dilution uses c1V1 = c2V2 because the moles of solute do not change.

Always balance the equation before applying any molar ratio.

ΔH is negative for exothermic reactions; standard enthalpy of formation of an element in its standard state is zero.

q = mcΔT uses the total mass of solution, and ΔH per mole = −q/n.

Hess's law lets you combine any route; combustion data are used in the reverse order to formation data.

ΔH = Σ bonds broken − Σ bonds formed, and mean bond enthalpies are averages.

Feasibility is determined by ΔG = ΔH − TΔS, not by ΔH alone.

Orders are determined experimentally, not from stoichiometric coefficients.

Rate constant units depend on overall order: for second order overall, dm3 mol−1 s−1; for third order overall, dm6 mol−2 s−1.

Relative atomic mass is an abundance-weighted mean of isotope masses, calculated from mass-spectrum data.

Across a period effective nuclear charge rises, so radius falls and ionisation energy and electronegativity rise.

Large jumps in successive ionisation energies reveal the number of electrons in the outer shell.

Cr and Cu adopt 3d5 4s1 and 3d10 4s1 configurations; 4s electrons are removed first on ionisation.

Transition metals require a partially filled d subshell in at least one common oxidation state.

VSEPR: count all electron domains, then apply lone pair–lone pair > lone pair–bond pair > bond pair–bond pair repulsion.

Molecular polarity requires both polar bonds and an asymmetric arrangement of those bonds.

Higher bond order gives a shorter, stronger bond; resonance spreads the order over equivalent positions.

Only period 3 elements and below can expand their octet; N, O, F and C cannot.

Ionic lattice strength increases with ion charge and decreases with ion radius.

Convert to K and m3 before using PV = nRT; R = 8.314 J mol−1 K−1.

Real gases deviate from ideality at high pressure and low temperature.

Functional group priority determines the suffix and numbering in IUPAC names.

2,4-DNP is used for suitable aldehyde/ketone carbonyl tests; reducing tests have exceptions and must be interpreted with the supplied structure and conditions.

Primary alcohols oxidise to aldehyde then acid; secondary to ketone; tertiary alcohols are resistant.

For acyclic-alkene E/Z isomerism, each carbon needs two different groups; a tetrahedral carbon with four different groups is one common, not the only, source of chirality.

Markovnikov: the hydrogen adds to the carbon already carrying more hydrogens.

A linear noncyclic n-residue peptide has n−1 peptide links in the stated model.

Protein folding changes do not necessarily hydrolyze the primary peptide backbone.

Triglycerides contain ester links; saponification gives glycerol and carboxylate salts in excess base.

A permanent crosslinked network does not simply remelt like an ordinary thermoplastic.

Bio-based origin, fragmentation and verified biodegradation are different claims.