FE Exam cheat sheet

Quadratic formula: for ax² + bx + c = 0, x = [−b ± √(b² − 4ac)]/(2a); discriminant b² − 4ac > 0 gives two real roots, = 0 gives one repeated root, < 0 gives a complex-conjugate pair.

Differentiation rules: d/dx(xⁿ) = n·xⁿ⁻¹; product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v − uv')/v²; chain rule d/dx f(g(x)) = f'(g)·g'.

Integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫eᵃˣ dx = eᵃˣ/a + C; integration by parts ∫u dv = uv − ∫v du.

L'Hôpital's rule: if lim f/g gives 0/0 or ∞/∞, then lim f/g = lim f'/g', applied repeatedly as needed and valid only for those indeterminate forms.

Area between curves = ∫(upper − lower) dx between the intersection limits; solid of revolution (disk method) V = π∫[f(x)]² dx.

Vectors: A·B = |A||B|cos θ = AₓBₓ + A_yB_y + A_zB_z; |A×B| = |A||B|sin θ; A·B = 0 when perpendicular, A×B = 0 when parallel.

Triangle relations: law of sines a/sin A = b/sin B = c/sin C; law of cosines c² = a² + b² − 2ab cos C.

2×2 matrix inverse: [[a,b],[c,d]]⁻¹ = (1/(ad − bc))·[[d,−b],[−c,a]], valid only when the determinant ad − bc ≠ 0.

Taylor series about x = a: f(x) = f(a) + f'(a)(x − a) + f''(a)(x − a)²/2! + f'''(a)(x − a)³/3! + …

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero; for independent events P(A ∩ B) = P(A)P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B); Bayes' theorem P(A|B) = P(B|A)P(A)/P(B).

Binomial: P(X = k) = C(n,k)pᵏ(1 − p)ⁿ⁻ᵏ; mean = np, variance = np(1 − p).

Poisson: P(X = k) = e^(−λ)λᵏ/k!; mean and variance both equal λ, used for counts of rare events in a fixed interval.

Exponential distribution: f(t) = λe^(−λt) for t ≥ 0; mean = 1/λ, variance = 1/λ², used for times between Poisson events.

Normal distribution: z = (x − μ)/σ; approximately 68.3%, 95.4%, and 99.7% of values lie within ±1σ, ±2σ, and ±3σ of the mean.

Confidence interval for a mean with known σ: x̄ ± z_(α/2)·σ/√n; the standard error of the mean is s/√n and the sample variance uses n − 1 in the denominator.

Hypothesis tests: α = P(Type I error) = rejecting a true null; β = P(Type II error) = failing to reject a false null; power = 1 − β.

Linear regression: slope b = Sxy/Sxx and intercept a = ȳ − b·x̄; the correlation coefficient r lies between −1 and +1, and r² is the fraction of variance explained.

Single-payment compounding: F = P(1 + i)ⁿ and P = F(1 + i)⁻ⁿ, with i per period and n the number of periods.

Uniform series: F = A[((1 + i)ⁿ − 1)/i]; P = A[((1 + i)ⁿ − 1)/(i(1 + i)ⁿ)]; A = P[i(1 + i)ⁿ/((1 + i)ⁿ − 1)].

Effective annual rate: i_a = (1 + r/m)^m − 1 for nominal rate r compounded m times per year; continuous compounding gives i_a = e^r − 1.

Net present worth: NPW = Σ CF_t/(1 + i)^t; a project is acceptable when NPW ≥ 0 at the minimum attractive rate of return, and alternatives are compared by NPW at equal analysis periods.

Annual worth: AW = NPW·[i(1 + i)ⁿ/((1 + i)ⁿ − 1)]; for equal lives, maximizing AW is equivalent to maximizing NPW.

Straight-line depreciation: D = (B − S)/n per year, where B is first cost, S salvage value, and n recovery period; book value after t years = B − t·D.

MACRS uses 200% declining balance switching to straight line for 3-, 5-, 7-, and 10-year property and 150% declining balance for 15- and 20-year property, with the half-year convention unless otherwise elected.

Simple payback period is the time for cumulative undiscounted cash inflows to equal the initial investment; it ignores cash flows after payback and the time value of money.

Inflation: market (combined) rate i_f relates to real rate i_r and inflation f by (1 + i_f) = (1 + i_r)(1 + f); use constant-value dollars with the real rate for consistent analysis.

NSPE Fundamental Canons: hold paramount the safety, health, and welfare of the public; perform services only in areas of competence; issue public statements only in an objective and truthful manner.

Additional canons: act for each employer or client as faithful agents or trustees, avoid deceptive acts, and conduct themselves honorably and lawfully so as to enhance the honor, reputation, and usefulness of the profession.

Engineers may sign and seal only plans and documents they prepared or that were prepared under their direct supervisory control, and only within their competence.

Conflicts of interest, including financial interests in suppliers or contractors, must be disclosed to the client or employer; compensation from more than one party for the same service requires full disclosure and consent.

Confidential information of clients and employers may not be revealed without consent, except where required by law or where public safety is endangered.

Engineers shall not accept, directly or indirectly, any gift, gratuity, or compensation from contractors, suppliers, or parties dealing with the employer in connection with work for which the engineer is responsible.

The standard of care is that of a reasonably prudent engineer under similar circumstances; it is not a warranty of perfection, and negligence is judged against that professional standard.

When an engineer's judgment is overruled and public safety is endangered, the engineer must inform the employer or client and, if necessary, notify the appropriate authority.

Professional obligations attach to the individual licensee; a supervisor's approval or a company policy does not relieve an engineer of personal responsibility for sealed work.

Static equilibrium of a rigid body: ΣF = 0 and ΣM = 0 about any point; in two dimensions this yields three independent equations (ΣFx, ΣFy, ΣM).

Dry friction: static friction satisfies f_s ≤ μ_s·N, with impending motion at f_s = μ_s·N; kinetic friction is f_k = μ_k·N and μ_k is generally less than μ_s.

Normal and shear stress: σ = F/A perpendicular to the area; τ = V/A parallel to the area; average bearing stress uses the projected contact area.

Hooke's law for uniaxial loading: σ = Eε, and axial deformation δ = FL/(AE) for a prismatic, linearly elastic member.

Torsion of a circular shaft: τ_max = Tc/J, and the angle of twist θ = TL/(JG), where J = πd⁴/32 for a solid circular shaft.

Beam bending: σ = Mc/I with c the distance from the neutral axis; the elastic curve satisfies d²y/dx² = M/(EI).

Parallel axis theorem: I = I_c + A·d², where d is the perpendicular distance between the centroidal axis and the parallel axis of interest.

Trusses: analyze with the method of joints or sections. At a joint with only two non-collinear members and no load, both members are zero-force members.

Projectile motion with no drag: horizontal range = v²sin(2θ)/g, maximum height = v²sin²θ/(2g), and time of flight = 2v·sin θ/g for level ground.

Engineering stress-strain curve: the slope in the linear region is the elastic modulus E; the proportional limit marks the end of linearity, followed by yield, ultimate strength, and fracture.

Ductility is reported as percent elongation or percent reduction in area; a material with less than about 5% elongation is generally considered brittle.

Poisson's ratio ν = −ε_lateral/ε_axial; for isotropic materials −1 < ν ≤ 0.5, and most metals fall near 0.25 to 0.35.

Isotropic elastic constants: G = E/[2(1 + ν)] and K = E/[3(1 − 2ν)]; specifying any two determines the third along with the others.

Fatigue: repeated cycling below the yield strength can cause failure; steel exhibits an endurance limit, while aluminum and most nonferrous alloys do not.

Creep is time-dependent deformation under sustained stress, significant at temperatures above roughly 0.4 of the absolute melting temperature.

Phase diagrams: use the lever rule to find phase fractions, mass fraction of a phase = (length of tie-line segment opposite that phase)/(total tie-line length).

Heat treatment: rapid quenching of austenite forms hard, brittle martensite; tempering after quench reduces hardness and restores toughness.

Corrosion: in a galvanic couple the more anodic metal corrodes; cathodic protection supplies electrons with a sacrificial anode or an impressed current.

Continuity for steady incompressible flow: A₁V₁ = A₂V₂; in general mass flow ṁ = ρAV is conserved.

Bernoulli along a streamline (steady, incompressible, frictionless): P/γ + V²/(2g) + z = constant; add head-loss and pump-head terms for real systems.

Reynolds number Re = ρVD/μ = VD/ν; in pipe flow Re < 2100 is laminar, Re > 4000 is turbulent, and 2100–4000 is transitional.

Darcy-Weisbach head loss: h_f = f·(L/D)·(V²/2g), with f from the Moody chart using relative roughness ε/D and Reynolds number.

Laminar pipe flow (Hagen-Poiseuille): h_f = 32μLV/(γD²) and f = 64/Re, independent of roughness.

Hydrostatics: P = γh; the resultant force on a plane submerged surface is F = γ·h_c·A acting at the center of pressure below the centroid.

Buoyancy: F_b = γ·V_displaced by Archimedes' principle; a floating body displaces its own weight in fluid.

Minor losses: h_m = K·V²/(2g), with K depending on the fitting, valve, or entrance/exit geometry.

Pump power required: P = γQH/η, where H is the total head added, Q the volume flow rate, and η the pump efficiency.

First law for a closed system: Q − W = ΔU; for a quasi-equilibrium process the boundary work is W = ∫P dV.

Ideal gas: PV = nRT with R = 8.314 J/(mol·K) or PV = mRT with R specific to the gas; also c_p − c_v = R and k = c_p/c_v.

Steady-flow energy equation: Q̇ − Ẇ = ṁ[(h₂ − h₁) + (V₂² − V₁²)/2 + g(z₂ − z₁)].

Second law: no heat engine operating between reservoirs at T_H and T_L can exceed the Carnot efficiency η = 1 − T_L/T_H, with absolute temperatures.

Entropy: ΔS = ∫δQ_rev/T, and the entropy change of an isolated system (ΔS_universe) must be greater than or equal to zero.

Refrigerators and heat pumps: COP_R = Q_L/W_in, COP_HP = Q_H/W_in, and COP_HP = COP_R + 1 for the same cycle.

Isentropic ideal-gas relations: T₂/T₁ = (P₂/P₁)^((k−1)/k) = (v₁/v₂)^(k−1), and Pv^k = constant.

Saturated mixture properties: quality x = (v − v_f)/(v_g − v_f), and h = h_f + x·h_fg; quality is undefined in the compressed-liquid and superheated-vapor regions.

Ohm's law V = IR and power P = VI = I²R = V²/R; resistive elements convert electrical energy to heat.

Kirchhoff's laws: KCL states the algebraic sum of currents at a node is zero; KVL states the algebraic sum of voltages around a closed loop is zero.

Series resistances add directly, R_eq = ΣRᵢ; parallel resistances combine as 1/R_eq = Σ(1/Rᵢ).

Thevenin's theorem: any linear two-terminal network equals a voltage source V_th (open-circuit voltage) in series with R_th, the resistance seen with independent sources deactivated.

Capacitors: i = C·dv/dt with impedance 1/(jωC); inductors: v = L·di/dt with impedance jωL. Time constants are τ = RC and τ = L/R.

Steady-state sinusoidal quantities: V_rms = V_peak/√2; complex power S = V·I* = P + jQ, with power factor cos θ equal to P/|S|.

Balanced three-phase power: P = √3·V_L·I_L·cos θ, where V_L and I_L are line quantities and θ is the phase angle between phase voltage and phase current.

Ideal transformer: V₁/V₂ = N₁/N₂ and I₁/I₂ = N₂/N₁, with input apparent power equal to output apparent power.

Digital logic: NAND and NOR gates are functionally complete; De Morgan's laws state (A·B)' = A' + B' and (A + B)' = A'·B'.

Hierarchy of controls, most to least effective: elimination, substitution, engineering controls, administrative controls, then personal protective equipment.

OSHA noise: permissible exposure limit 90 dBA as an 8-hour time-weighted average, action level 85 dBA, and a 5 dB exchange rate (each 5 dB increase halves allowable exposure time).

Decibel arithmetic: doubling the number of identical sound sources increases the level by about 3 dB; adding a source 10 dB lower changes the total by only about 0.4 dB.

Flammability: a vapor-air mixture burns only between its lower and upper explosive limits; below the LEL the mixture is too lean, above the UEL too rich.

Fall protection thresholds: 4 ft in general industry, 6 ft in construction, and 10 ft for scaffolding, with guardrails, nets, or personal fall-arrest systems.

Permit-required confined spaces have a hazardous atmosphere, engulfment potential, or other serious hazards; entry requires atmospheric testing, isolation, and an attendant.

Exposure limits: TLV-TWA is an 8-hour average, TLV-STEL a 15-minute average, and TLV-C a ceiling that must never be exceeded; IDLH means immediate danger to life or health.

Environmental statutes: the Clean Air Act regulates emissions, the Clean Water Act regulates discharges through NPDES permits, and RCRA governs hazardous waste handling.

Hazard communication requires labels, safety data sheets, and training for each hazardous chemical in the workplace; lockout/tagout controls hazardous energy during servicing.