Calculus, Differential Equations, and Linear Algebra Review

Mathematics on the FE Mechanical Exam

The FE Mechanical exam (110 questions, 6 hours, computer-based at Pearson VUE) devotes roughly 6–9 questions to Mathematics and 4–6 to Probability & Statistics. Math is rarely tested for its own sake; it is the language used to set up mechanics, thermodynamics, and dynamics problems. The fastest path to points is model recognition—deciding whether a problem needs a derivative, an integral, an ODE, or a matrix—then executing with the NCEES FE Reference Handbook, which supplies every formula you may use. Memorize where formulas live in the Handbook, not the formulas themselves.

Analytic Geometry

The general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 classifies conics by the discriminant B² − 4AC: negative → ellipse (circle if A = C, B = 0); zero → parabola; positive → hyperbola. The straight line y = mx + b has slope m; two lines are perpendicular when m₁m₂ = −1. The distance between points is √[(x₂−x₁)² + (y₂−y₁)²]. These appear in geometry, statics, and dynamics setups.

Differential Calculus

A derivative gives an instantaneous rate or slope. The core rules in the Handbook are the power rule d/dx(xⁿ) = nxⁿ⁻¹, the product rule (uv)' = u'v + uv', the quotient rule (u/v)' = (u'v − uv')/v², and the chain rule d/dx[f(g(x))] = f'(g)·g'(x). Trig and exponential derivatives—d/dx(sin x) = cos x, d/dx(eˣ) = eˣ—round out the tested set.

Optimization is the most common derivative question: set f'(x) = 0 to locate critical points, then use the second derivative—f''(x) < 0 is a local maximum, f''(x) > 0 a local minimum, and f''(x) = 0 a possible inflection point.

Worked Example — Maximum

Find the maximum of f(x) = −2x² + 12x − 7. Then f'(x) = −4x + 12 = 0 → x = 3. Since f'' = −4 < 0, x = 3 is a maximum, and f(3) = −18 + 36 − 7 = 11. This same procedure sizes minimum-cost or maximum-deflection problems elsewhere on the exam.

Integral Calculus

A definite integral ∫ₐᵇ f(x) dx accumulates a quantity: area under a curve, work W = ∫F dx, impulse ∫F dt, or fluid force. Key Handbook integrals include ∫xⁿ dx = xⁿ⁺¹/(n+1) + C and ∫(1/x) dx = ln|x| + C.

The average value of f over [a, b] is (1/(b−a))·∫ₐᵇ f(x) dx—a favorite for average velocity or mean ordinate. The centroid of an area uses x̄ = (∫x dA)/A. For example, ∫₀² 3x² dx = [x³]₀² = 8, which could represent work done by a position-dependent force from 0 to 2 m.

Differential Equations

Mechanical problems—cooling, mixing, RC/RL circuits, vibration—are governed by ordinary differential equations (ODEs).

First-order linear: dy/dx + P(x)y = Q(x) solves with integrating factor μ = e^∫P dx. A pure decay form dy/dt = −ky has solution y = y₀e^(−kt)—exactly Newton's law of cooling T(t) = T_∞ + (T₀ − T_∞)e^(−kt).

Second-order, constant coefficients: y'' + ay' + by = 0 uses the characteristic equation r² + ar + b = 0. Two real roots → y = C₁e^(r₁t) + C₂e^(r₂t); complex roots r = α ± βi → underdamped oscillation y = e^(αt)(C₁cos βt + C₂sin βt). This is the math behind the spring–mass–damper model mₓ'' + cx' + kx = 0.

Linear Algebra

Simultaneous equations are written Ax = b. A 2×2 determinant is |a b; c d| = ad − bc; Cramer's rule gives each unknown as a ratio of determinants. Eigenvalues come from det(A − λI) = 0, which underlies natural-frequency and principal-stress problems.

Your approved calculator solves 2×2 and 3×3 systems and finds determinants directly—practice the keystrokes so setup, not arithmetic, is the only thinking on exam day.

How These Tools Connect Across the Exam

The value of this math review is that the same handful of tools resurfaces throughout the FE Mechanical blueprint, so recognizing them quickly pays off well beyond the Mathematics section itself.

Derivatives appear whenever a problem asks for a rate of change: velocity is the derivative of position, acceleration the derivative of velocity, and the slope of a stress–strain curve is the elastic modulus. Maxima and minima show up as maximum bending moment along a beam (where the shear, the derivative of moment, equals zero) and as the optimum design that minimizes cost or weight.

Integrals accumulate quantities: the area under a force–displacement curve is work, the area under a velocity–time curve is distance, and centroids of areas feed directly into the moment-of-inertia calculations used in Mechanics of Materials. Knowing that ∫ is "add up the little pieces" lets you set up these problems even when the formula sheet does not state them explicitly.

Differential equations describe anything that changes over time in proportion to its current value or its derivatives—radioactive-style decay, charging and discharging, transient heat transfer, and free vibration. Linear algebra organizes any problem with several unknowns and several equations, from method-of-joints truss analysis to resistor networks. The lesson for exam day: read the problem, name the mathematical structure, locate the matching formula in the Handbook, and let the approved calculator do the arithmetic. Spending ten seconds classifying the problem usually saves a minute of misdirected algebra.