Your FREE FE Exam Flashcards 2026 – 200+ Cards

cos(x). The derivative measures the instantaneous rate of change of the function.

A scalar has magnitude only (e.g., temperature, mass). A vector has both magnitude and direction (e.g., force, velocity).

A·B = |A||B|cos θ. A scalar result; zero when the vectors are perpendicular.

A×B = |A||B|sin θ, giving a vector perpendicular to both A and B; its magnitude is the area of the parallelogram they form.

−sin(x).

eˣ. The exponential function is its own derivative.

1/x.

ln|x| + C.

d/dx f(g(x)) = f′(g(x)) · g′(x). Differentiate the outer function, then multiply by the derivative of the inner function.

d/dx (uv) = u′v + uv′.

d/dx (u/v) = (u′v − uv′) / v².

The signed area between the curve and the x-axis over the interval of integration.

Integration and differentiation are inverse operations: ∫ₐᵇ f(x) dx = F(b) − F(a), where F′ = f.

The order is the highest derivative that appears. y″ + 3y′ + 2y = 0 is second-order.

P = P₀ eᵏᵗ — exponential growth (k > 0) or decay (k < 0), where P₀ is the initial value.

Linear: the dependent variable and its derivatives appear only to the first power and are not multiplied together. Otherwise it is nonlinear.

Homogeneous: the forcing term (right-hand side) is zero. Nonhomogeneous: it has a nonzero forcing function.

ad − bc.

When the coefficient matrix is square and its determinant is nonzero (the matrix is nonsingular / invertible).

A scalar λ for which Av = λv has a nonzero vector v; found from det(A − λI) = 0.

c² = a² + b² − 2ab·cos C. Generalizes the Pythagorean theorem to any triangle.

1. The fundamental Pythagorean trigonometric identity.

e raised to the power iθ equals cos θ + i·sin θ. It links the exponential and trigonometric functions for complex numbers.

√(a² + b²).

(y₂ − y₁) / (x₂ − x₁) — rise over run.

Approximates a function near a point as a power series using the function's derivatives at that point.

A vector ∇f pointing in the direction of steepest increase, with magnitude equal to the maximum rate of change.

Divergence (∇·F) measures a field's net outflow (a scalar). Curl (∇×F) measures its rotation (a vector).

log a + log b. Multiplication becomes addition in log space.

x = (−b ± √(b² − 4ac)) / (2a) — the quadratic formula.

b² − 4ac. Positive = two real roots; zero = one repeated root; negative = two complex roots.

Mean is the arithmetic average (sum ÷ count). Median is the middle value when data are ordered; it resists outliers.

The variance's positive root (√variance); a measure of how spread out data are around the mean.

The average of the squared deviations from the mean. Standard deviation squared.

Multiply: P(A and B) = P(A) × P(B).

Add: P(A or B) = P(A) + P(B).

P(A) + P(B) − P(A and B). Subtract the overlap so it is not double-counted.

P(A and B) / P(B) — the probability of A given that B has occurred.

Two events are independent if P(A|B) = P(A); knowing B happened does not change the probability of A.

A symmetric, bell-shaped curve fully described by its mean and standard deviation.

For a normal distribution, about 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations of the mean.

(x − mean) / standard deviation. The number of standard deviations a value lies from the mean.

Models the number of successes in n independent trials, each with the same success probability p.

The probability-weighted sum of its possible values: E(X) = Σ xᵢ P(xᵢ).

Permutations count ordered arrangements (order matters); combinations count selections (order does not matter).

C(n,r) = n! / [r!(n − r)!].

−1 to +1. It measures the strength and direction of a linear relationship between two variables.

Find the line that minimizes the sum of the squared vertical distances between the data points and the line.

Type I rejects a true null hypothesis (false positive); Type II fails to reject a false null (false negative).

A range that, for a given confidence level (e.g., 95%), is expected to contain the true population parameter.

The most frequently occurring value in a data set.

The set of all possible outcomes of a random experiment.

A dollar today is worth more than a dollar in the future because money can earn a return over time.

F = P(1 + i)ⁿ, where P is present value, i is the interest rate per period, and n is the number of periods.

P = F / (1 + i)ⁿ — discount a future amount back to today.

Simple interest is earned only on the principal. Compound interest is earned on principal plus accumulated interest.

Nominal is the stated annual rate. Effective accounts for compounding within the year: i_eff = (1 + r/m) raised to the m-th power, minus 1.

Accept a project if its NPV (sum of discounted cash flows minus initial cost) is positive.

The discount rate that makes a project's NPV equal to zero; compare it to the minimum acceptable rate (MARR).

A series of equal payments made at regular intervals.

The present value of an asset assumed to last forever: P = A / i, where A is the annual cost and i the interest rate.

Equal annual depreciation = (initial cost − salvage value) / useful life.

A past cost that has already been incurred and cannot be recovered; it should be ignored in current decisions.

Minimum Attractive (Acceptable) Rate of Return — the lowest return a project must earn to be worthwhile.

A project is economically justified when the benefit-cost ratio is greater than or equal to 1.

Real rate ≈ nominal rate − inflation rate. Inflation erodes the purchasing power of future cash flows.

The time required for a project's cumulative cash inflows to recover its initial investment; ignores the time value of money.

Converts a present amount into an equivalent uniform series of annual payments over n periods.

To hold paramount the safety, health, and welfare of the public.

The engineer must meet the higher standard — protecting the public can require action beyond the legal minimum.

Engineers must perform services only in their areas of competence and not sign or seal work outside their expertise.

Disclose all known or potential conflicts of interest to clients or employers promptly and in writing.

Confidentiality yields to the duty to protect the public; a danger to public safety can require disclosure to the proper authority.

Engineers must act for each employer or client as faithful agents or trustees, avoiding deception.

Engineers must be objective and truthful in professional reports, statements, and testimony, including all relevant information.

Engineers must not solicit or accept gratuities, directly or indirectly, to influence their professional judgment.

Engineers must give credit for work to those to whom credit is due and recognize the proprietary interests of others.

If a judgment is overruled where public safety is endangered, the engineer must notify the proper authorities and may withdraw.

Engineers should continue professional development throughout their careers and keep current in their field.

Engineers may seal only work prepared by them or under their responsible charge; sealing others' work is unethical and illegal.

Engineers are encouraged to adhere to principles of sustainable development to protect the environment for future generations.

To protect the public by ensuring only qualified individuals may offer engineering services to the public.

A body is in static equilibrium when the sum of all forces equals zero (ΣF = 0) and the sum of all moments equals zero (ΣM = 0).

A body at rest stays at rest, and a body in motion stays in motion, unless acted on by a net external force (inertia).

M = F × d, where d is the perpendicular distance from the pivot to the line of action of the force.

Two equal, opposite, parallel forces that produce a pure moment (no net force) independent of the reference point.

A sketch of a body isolated from its surroundings showing all external forces and moments acting on it.

Two force components (horizontal and vertical); a pin resists translation but allows rotation, so no moment reaction.

A single force perpendicular to the rolling surface.

Two force components plus a moment — it resists translation in both directions and rotation.

A member loaded at only two points carries force only along the line joining those points (pure tension or compression).

Apply ΣFx = 0 and ΣFy = 0 at each joint to solve for member forces; start at a joint with two unknowns.

Cut through the members of interest and apply equilibrium to one portion to find specific member forces directly.

The geometric center of an area or volume — where the first moment of area equals zero.

F ≤ μₛN. It opposes impending motion and reaches a maximum of μₛN just before sliding starts.

Kinetic friction (object sliding) is usually less than the maximum static friction (object about to slide).

Replaced by a single force equal to the area under the load curve, acting at the centroid of that area.

A truss member carrying no load; common at unloaded joints with two collinear members and one non-collinear member.

A geometric property describing resistance to bending; depends on cross-sectional shape and distance from the neutral axis.

I = I_c + Ad², shifting a moment of inertia from the centroidal axis to a parallel axis a distance d away.

F = ma. Net force equals mass times acceleration.

Kinematics describes motion (position, velocity, acceleration) without forces; kinetics relates motion to the forces causing it.

v = v₀ + at.

s = s₀ + v₀t + ½at².

v² = v₀² + 2a(s − s₀).

Horizontal and vertical motions are independent; horizontal velocity is constant while vertical motion accelerates at g.

a = v² / r, directed toward the center of the circular path.

KE = ½mv².

PE = mgh, relative to a chosen reference height.

The net work done on a body equals its change in kinetic energy: W = ΔKE.

p = mv. A vector quantity conserved when no net external force acts.

Impulse (force × time) equals the change in momentum: FΔt = Δ(mv).

In a collision with no external forces, total momentum before equals total momentum after.

Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve momentum but lose kinetic energy.

The ratio of relative separation velocity to relative approach velocity; 1 for perfectly elastic, 0 for perfectly plastic.

v = ωr for a point at radius r on a rotating body.

Torque τ = Iα, where I is the mass moment of inertia and α is angular acceleration.

Angular frequency ω = √(k/m); period T = 2π√(m/k).

A force whose work is path-independent (e.g., gravity, springs); energy is conserved when only conservative forces act.

σ = P / A — force divided by the cross-sectional area resisting it.

ε = ΔL / L — the change in length divided by the original length (dimensionless).

σ = Eε. Within the elastic region, stress is proportional to strain; E is the modulus of elasticity.

The slope of the linear elastic portion of the stress-strain curve; a measure of stiffness.

τ = V / A — force acting parallel to a surface divided by the area.

The ratio of lateral strain to axial strain (typically 0.25–0.35 for metals); describes transverse contraction under axial load.

The stress at which a material begins to deform permanently (plastically).

The maximum stress a material can withstand before failure.

Elastic deformation is recoverable when the load is removed; plastic deformation is permanent.

δ = PL / (AE) — elongation of a bar under axial load.

σ = Mc / I, where M is the bending moment, c the distance to the outer fiber, and I the moment of inertia.

τ = Tr / J, where T is torque, r the radius, and J the polar moment of inertia.

ε = αΔT, where α is the coefficient of thermal expansion and ΔT the temperature change.

The ratio of a material's failure (or yield) strength to the actual applied stress.

P_cr = π²EI / (KL)², the critical axial load that buckles a slender column; K depends on end conditions.

A local rise in stress at a geometric discontinuity such as a hole, notch, or fillet.

Ductile materials deform significantly before fracture (warning); brittle materials fracture suddenly with little deformation.

A graphical method to find principal stresses and maximum shear stress from a known stress state.

The line in a beam's cross-section where bending stress is zero; it passes through the centroid for symmetric sections.

A map showing the stable phases of a material as a function of temperature and composition.

Ferrous metals contain iron (steel, cast iron); nonferrous do not (aluminum, copper, titanium).

Higher carbon increases hardness and strength but reduces ductility and weldability.

A heat treatment that softens metal, relieves internal stresses, and improves ductility by slow cooling.

Rapid cooling (quench) hardens steel; reheating (temper) restores some toughness and reduces brittleness.

A material made by combining a metal with one or more elements to improve properties such as strength or corrosion resistance.

A material's resistance to localized plastic deformation, such as indentation or scratching.

The ability to absorb energy and deform plastically before fracturing; the area under the stress-strain curve.

Failure under repeated cyclic loading at stresses below the static strength; cracks initiate and propagate over many cycles.

The stress amplitude below which some materials (notably steel) can endure essentially infinite load cycles.

Slow, permanent deformation under constant load over time, accelerated by high temperature.

The gradual degradation of a material, usually metal, by chemical or electrochemical reaction with its environment.

Accelerated corrosion when two dissimilar metals are in electrical contact in an electrolyte; the more active metal corrodes.

Smaller grains generally increase strength and hardness (Hall-Petch relationship).

Metals are ductile conductors; ceramics are hard, brittle, heat-resistant insulators; polymers are light, flexible, low-melting.

A material combining two or more constituents (e.g., fiber + matrix) to achieve properties neither could alone.

A₁V₁ = A₂V₂. Volumetric flow rate Q is constant, so velocity rises where the cross-sectional area shrinks.

Density ρ = mass/volume. Specific weight γ = ρg = weight/volume.

A fluid's resistance to shear or flow; the proportionality between shear stress and velocity gradient (τ = μ du/dy).

P = ρgh. Gauge pressure increases linearly with depth in a static fluid.

A pressure change applied to an enclosed fluid is transmitted undiminished to every point in the fluid.

F_b = ρ_fluid × g × V_displaced — equal to the weight of the fluid displaced.

For incompressible flow, A₁V₁ = A₂V₂; flow rate is conserved, so velocity increases where area decreases.

Along a streamline, P/ρ + V²/2 + gz = constant — pressure, kinetic, and potential energy trade off in ideal flow.

Re = ρVD/μ. A dimensionless ratio of inertial to viscous forces that predicts laminar vs turbulent flow.

Laminar (smooth, layered) for Re below ~2300; turbulent (chaotic, mixing) for Re above ~4000.

Q = AV — cross-sectional area times average velocity.

ṁ = ρAV = ρQ.

Energy lost to friction, found with the Darcy-Weisbach equation h_f = f(L/D)(V²/2g).

Measures pressure difference from the height difference of a liquid column: ΔP = ρg Δh.

An ideal fluid has no viscosity and no friction losses; real fluids have viscosity and dissipate energy.

P = ρgQH / η, where H is the head added and η the pump efficiency.

The ratio of a substance's density to the density of water (1000 kg/m³); dimensionless.

The pressure when a flowing fluid is brought to rest; static pressure plus dynamic pressure (½ρV²).

Energy is conserved: the change in internal energy ΔU = Q − W, where Q is heat added to the system and W is work done by the system.

If two systems are each in thermal equilibrium with a third, they are in equilibrium with each other — the basis of temperature.

The total entropy of an isolated system never decreases; heat flows spontaneously from hot to cold.

A measure of a system's disorder or unavailable energy; it increases in any real (irreversible) process.

η = 1 − T_cold/T_hot (temperatures in kelvin) — the maximum possible efficiency of a heat engine between two reservoirs.

Carnot efficiency uses absolute temperature; only the kelvin scale starts at absolute zero, so ratios are physically meaningful.

H = U + PV. A property combining internal energy with flow work; useful for constant-pressure processes.

PV = nRT (or PV = mRT). Relates pressure, volume, and absolute temperature for an ideal gas.

The energy required to raise the temperature of a unit mass by one degree: Q = mcΔT.

cp is specific heat at constant pressure; cv at constant volume. For ideal gases cp − cv = R, and cp > cv.

Isothermal: constant temperature (heat exchanged). Adiabatic: no heat transfer (Q = 0).

Energy absorbed or released during a phase change (melting, vaporization) at constant temperature.

A device that converts heat into work by operating in a cycle between a hot and a cold reservoir.

For refrigerators/heat pumps, the ratio of useful heat moved to the work input; can exceed 1.

Conduction (through a solid), convection (fluid motion), and radiation (electromagnetic waves).

q = −kA(dT/dx) — heat flows down the temperature gradient; k is thermal conductivity.

q = hA(T_surface − T_fluid), where h is the convective heat transfer coefficient.

Radiated power ∝ εσAT⁴; emitted energy rises with the fourth power of absolute temperature.

A material property describing how readily it conducts heat; metals are high, insulators low.

V = I × R. Voltage equals current times resistance.

The rate of flow of electric charge: I = Q/t, measured in amperes (coulombs per second).

The energy per unit charge between two points: V = energy/charge, measured in volts (joules per coulomb).

P = VI = I²R = V²/R, measured in watts.

Add directly: R_total = R₁ + R₂ + … The same current flows through each.

1/R_total = 1/R₁ + 1/R₂ + … The same voltage appears across each; total resistance is less than the smallest.

The sum of currents entering a node equals the sum leaving it (charge is conserved).

The sum of voltage rises and drops around any closed loop equals zero (energy is conserved).

Q = CV. A capacitor stores energy in an electric field; energy = ½CV².

Stores energy in a magnetic field and opposes changes in current: V = L(di/dt); energy = ½LI².

X_C = 1/(2πfC). It decreases as frequency increases; a capacitor blocks DC, passes high-frequency AC.

X_L = 2πfL. It increases with frequency; an inductor passes DC, opposes high-frequency AC.

V_rms = V_peak / √2 ≈ 0.707 × V_peak — the equivalent DC value delivering the same power.

The cosine of the phase angle between voltage and current; the ratio of real power to apparent power.

The total opposition to AC current, combining resistance and reactance: Z = √(R² + X²).

Real (W) does useful work; reactive (VAR) oscillates in reactive elements; apparent (VA) is their vector sum.

Resistors dissipate energy as heat; capacitors and inductors store and return it.

V₁/V₂ = N₁/N₂ — the voltage ratio equals the turns ratio.

The SI unit for amount of substance; one mole contains Avogadro's number (6.022 × 10²³) of particles.

The study of the quantitative relationships between reactants and products in a chemical reaction.

Adjust coefficients so each element has equal atoms on both sides — conservation of mass.

pH = −log[H⁺]. Below 7 is acidic, 7 is neutral, above 7 is basic; each unit is a tenfold change.

An acid donates protons (H⁺) or accepts electrons; a base accepts protons or donates hydroxide (OH⁻).

Concentration in moles of solute per liter of solution (mol/L).

PV = nRT, relating pressure, volume, moles, and absolute temperature of a gas.

Exothermic reactions release heat (negative ΔH); endothermic reactions absorb heat (positive ΔH).

Oxidation is loss of electrons; reduction is gain of electrons (OIL RIG).

The reactant that is fully consumed first, determining the maximum amount of product formed.

Ionic bonds transfer electrons between metal and nonmetal; covalent bonds share electrons between nonmetals.

A substance that speeds a reaction by lowering activation energy and is not consumed in the process.

Atomic number = number of protons (defines the element); mass number = protons + neutrons.

6.022 × 10²³ — the number of particles in one mole of a substance.

Matter is neither created nor destroyed; the total mass of reactants equals the total mass of products.