AP ECET cheat sheet

Limit definition of the derivative: f′(x) = lim(h→0) [f(x+h) − f(x)]/h. Differentiability at a point implies continuity there, but continuity does not imply differentiability (for example |x| at x = 0).

Standard derivatives: d/dx(xⁿ) = nxⁿ⁻¹; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x; d/dx(sin x) = cos x; d/dx(tan x) = sec²x; d/dx(arcsin x) = 1/√(1−x²).

Product, quotient and chain rules: (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²; dy/dx = (dy/du)(du/dx).

Integration by parts: ∫u dv = uv − ∫v du; pick u in the LIATE order (logarithmic, inverse trigonometric, algebraic, trigonometric, exponential). Also ∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ −1 and ∫dx/x = ln|x| + C.

Quadratic formula: x = [−b ± √(b²−4ac)]/2a. Discriminant D > 0 gives real distinct roots, D = 0 real equal roots, D < 0 a conjugate complex pair; sum of roots = −b/a and product = c/a.

Binomial theorem: (a+b)ⁿ = Σ nCr·aⁿ⁻ʳ·bʳ with general term T(r+1) = nCr·aⁿ⁻ʳ·bʳ and nCr = n!/[r!(n−r)!]; the expansion contains n+1 terms.

Vectors: a·b = |a||b|cos θ, a scalar that is zero when the vectors are perpendicular; a×b = |a||b|sin θ·n̂, a vector that is zero when they are parallel, and |a×b| equals the area of the parallelogram they span.

Matrices: (AB)ᵀ = BᵀAᵀ; A⁻¹ = adj(A)/|A| exists only when |A| ≠ 0; a homogeneous system AX = 0 has non-trivial solutions exactly when |A| = 0.

Probability: P(A∪B) = P(A) + P(B) − P(A∩B); P(A|B) = P(A∩B)/P(B) for P(B) > 0; Bayes gives P(Aᵢ|B) = P(B|Aᵢ)P(Aᵢ) / Σ P(B|Aⱼ)P(Aⱼ).

Newton's laws: F = dp/dt, which becomes F = ma for constant mass; action–reaction pairs act on different bodies and never cancel on the same body; impulse J = ∫F dt = Δp.

Work–energy theorem: W_net = ΔKE = ½mv² − ½mu²; for a conservative force F = −dU/dx and total mechanical energy is conserved when only conservative forces do work.

Circular motion: a_c = v²/r = ω²r and centripetal force F = mv²/r; for a frictionless banked road tan θ = v²/(rg).

Gravitation: g = GM/R²; orbital speed v = √(GM/r); escape speed v_e = √(2GM/R) ≈ 11.2 km/s from the Earth's surface; Kepler's third law T² ∝ a³.

Simple harmonic motion: x = A sin(ωt + φ), v = Aω cos(ωt + φ), a = −ω²x; T = 2π√(m/k) for a spring and T = 2π√(L/g) for a simple pendulum of small amplitude.

First law of thermodynamics: ΔQ = ΔU + ΔW; isothermal means ΔU = 0, adiabatic means ΔQ = 0; Carnot efficiency η = 1 − T_c/T_h with both temperatures in kelvin.

Waves: v = fλ; speed on a stretched string v = √(T/μ) where μ is mass per unit length; beat frequency = |f₁ − f₂|; Doppler shift f′ = f(v ± v_o)/(v ∓ v_s).

Ray optics: 1/f = 1/v + 1/u for mirrors and 1/f = 1/v − 1/u for lenses under the Cartesian sign convention; Snell's law n₁sin θ₁ = n₂sin θ₂; critical angle sin C = 1/n.

Current electricity: V = IR; series R = R₁ + R₂ + …; parallel 1/R = 1/R₁ + 1/R₂ + …; power P = VI = I²R = V²/R; Kirchhoff's junction rule conserves charge and the loop rule conserves energy.

Mole concept: one mole contains 6.022×10²³ entities; molar mass in g/mol equals atomic or molecular mass in u; one mole of an ideal gas occupies 22.4 L at STP (273.15 K, 1 atm).

Ideal gas equation PV = nRT with R = 8.314 J mol⁻¹K⁻¹ = 0.0821 L atm mol⁻¹K⁻¹; at constant T, P₁V₁ = P₂V₂; at constant P, V₁/T₁ = V₂/T₂.

Concentration terms: molarity = moles of solute per litre of solution; molality = moles of solute per kg of solvent; normality = molarity × n-factor; dilution obeys M₁V₁ = M₂V₂.

pH = −log[H⁺] and pOH = −log[OH⁻]; at 25 °C, pH + pOH = 14 and Kw = [H⁺][OH⁻] = 1×10⁻¹⁴; weak acids require Ka to find [H⁺], strong acids dissociate completely.

Le Chatelier's rule: raising pressure shifts equilibrium toward the side with fewer gaseous moles; raising temperature favours the endothermic direction; a catalyst changes neither K nor the equilibrium position.

Equilibrium constants: Kp = Kc(RT)^Δn where Δn = gaseous product moles − gaseous reactant moles; if the reaction quotient Q < K the reaction proceeds in the forward direction.

Electrochemistry: Nernst equation E = E° − (0.0591/n) log Q at 25 °C; ΔG° = −nFE°; oxidation always occurs at the anode and reduction at the cathode.

VSEPR and hybridization: steric number 2 gives sp (linear, 180°), 3 gives sp² (trigonal planar, 120°), 4 gives sp³ (tetrahedral, 109.5°); lone-pair repulsion follows lp–lp > lp–bp > bp–bp.

Thermochemistry: ΔU = q + w; ΔH = ΔU + Δn_g·RT; Hess's law permits summing step enthalpies to obtain the overall ΔH; bond breaking absorbs energy and bond formation releases it.

Nominal concrete mixes (IS 456): M10 = 1:3:6, M15 = 1:2:4, M20 = 1:1.5:3; M25 and above must be design mixes; minimum cement content and maximum water–cement ratio govern durability.

Workability tests: slump test is the most common and suits medium workability; compaction factor test suits low workability mixes; Vee-Bee consistometer is used for very dry mixes.

Abram's law: concrete strength is inversely related to the water–cement ratio; mix design targets a mean strength of f_ck + 1.65σ for the specified grade.

Bending theory: M/I = σ/y = E/R, so σ = My/I and M = σZ with section modulus Z = I/y_max. For a simply supported beam under UDL w over span L, maximum BM = wL²/8 at midspan and maximum SF = wL/2 at the supports.

Transverse shear stress in a beam: τ = VQ/(Ib), zero at the extreme fibres and maximum at the neutral axis; shear reinforcement is required wherever τ_v exceeds the design shear strength τ_c.

Levelling: height of instrument HI = RL of station + backsight; RL of next station = HI − foresight; rise = BS − FS (a fall if negative); arithmetic check ΣBS − ΣFS = last RL − first RL.

Theodolite traverse: latitude = l·cos θ and departure = l·sin θ; closing error = √[(ΣL)² + (ΣD)²]; Bowditch correction is proportional to length, transit correction proportional to latitude or departure.

Soil relations: void ratio e = V_v/V_s; porosity n = e/(1+e); degree of saturation S = V_w/V_v; dry unit weight γ_d = γ/(1+w); γ = (G + Se)γ_w/(1+e).

Open-channel flow: Manning's V = (1/n)R^(2/3)S^(1/2); Chezy's V = C√(RS); hydraulic radius R = A/P; the most economical rectangular section has depth equal to half the width.