Mathematics
Mathematics supplies the analytical machinery every other FE topic runs on, so points here compound across the whole exam. Coverage spans calculus, differential equations, linear algebra, analytic geometry, trigonometry, and vectors — mostly single-step evaluations where knowing the right rule is the whole battle.
Differential calculus
The derivative is an instantaneous rate of change. The power rule d/dx(xⁿ) = n·xⁿ⁻¹ handles polynomials: for f(x) = 3x³ − 5x, f'(x) = 9x² − 5, so f'(2) = 9·4 − 5 = 31. Memorize the standard derivatives: d/dx(eˣ) = eˣ, d/dx(e^(ax)) = a·e^(ax) so f'(0) = 2 for e^(2x); d/dx(ln x) = 1/x so f'(2) = 0.5; d/dx(sin x) = cos x and d/dx(cos x) = −sin x, giving zero slope for cos x at x = 0. The product, quotient, and chain rules combine these. Setting the derivative to zero locates maxima and minima.
Integral calculus
The definite integral gives net area and is evaluated by the Fundamental Theorem of Calculus: find an antiderivative F and compute F(b) − F(a). For polynomials the power rule reverses to ∫xⁿ dx = xⁿ⁺¹/(n+1); ∫₀² x² dx = [x³/3]₀² = 8/3 ≈ 2.67. Know the standard forms: ∫sin x dx = −cos x, so ∫₀^π sin x dx = [−cos x]₀^π = −(−1) − (−1) = 2; and ∫(1/x) dx = ln|x|, so ∫₁^e (1/x) dx = ln e − ln 1 = 1. Integration also yields averages, volumes, and centroids.
Linear algebra and matrices
Matrices encode systems of linear equations. The determinant of a 2×2 matrix [[a, b], [c, d]] is a·d − b·c; for [[4, 3], [2, 5]] it is 4·5 − 3·2 = 14. A nonzero determinant means the system has a unique solution and the matrix is invertible; a zero determinant signals dependent equations or no unique solution. Matrix multiplication is not commutative. Eigenvalues λ satisfy det(A − λI) = 0 and appear in vibration modes, principal stresses, and stability analysis.
Analytic geometry and trigonometry
The straight line y = m·x + b has slope m and intercept b; perpendicular lines have slopes whose product is −1. The distance between two points is √((x₂−x₁)² + (y₂−y₁)²). Core trig identities include sin²θ + cos²θ = 1 and the law of cosines c² = a² + b² − 2ab·cos C for non-right triangles. Radians (2π = 360°) are the natural angle measure in calculus. These tools support vector resolution and the geometry threaded through statics, dynamics, and surveying.
Vectors and complex numbers
A vector adds component-by-component; its magnitude is √(a² + b² + c²). The dot product A·B = |A||B|cos θ returns a scalar and detects perpendicularity (zero when orthogonal); the cross product A×B returns a vector perpendicular to both, with magnitude |A||B|sin θ, used for moments and areas. Complex numbers combine a real and imaginary part, a + b·i; in polar form r·e^(iθ) they simplify AC-circuit and rotation problems, with r = √(a² + b²) and θ = arctan(b/a).
Differential equations and numerical methods
A first-order linear equation like dy/dt = k·y models exponential growth or decay, y = y₀·e^(k·t); it describes radioactive decay, cooling, and RC circuits. Second-order equations such as m·y'' + c·y' + k·y = 0 govern oscillation, with the roots of the characteristic equation determining whether the response is underdamped, critically damped, or overdamped. When no closed form exists, numerical methods step in: Newton's method refines roots, and the trapezoidal or Simpson's rule approximates integrals. Recognizing the equation's order and type points directly to the solution form.
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