Mathematics
Mathematics
The FE Mechanical Mathematics section tests algebra, trigonometry, analytic geometry, differential and integral calculus, linear algebra, differential equations, and numerical methods. Items are usually short calculations; the NCEES FE Reference Handbook supplies formulas, so the challenge is selecting the right relation and applying it cleanly.
Core rules and concepts
- Quadratic formula: x = (-b ± sqrt(b² - 4ac))/(2a). The discriminant b² - 4ac tells whether roots are real, repeated, or complex.
- Trigonometry: sin²θ + cos²θ = 1; law of sines a/sin A = b/sin B; law of cosines c² = a² + b² - 2ab cos C. Use radians in calculus.
- Derivatives: product, quotient, and chain rules. d/dx sin x = cos x; d/dx e^(ax) = a e^(ax); d/dx ln x = 1/x.
- Integrals: power rule, substitution, and integration by parts ∫u dv = uv - ∫v du. A definite integral is a number; include the constant only for indefinite integrals.
- Linear algebra: for a 2x2 matrix, determinant = ad - bc. A matrix is singular when determinant = 0; then no inverse exists.
- First-order linear ODE y' + P(x)y = Q(x): integrating factor μ = e^(∫P dx), solution y = (1/μ)∫μQ dx + C/μ.
Worked example
Find f'(x) for f(x) = x² ln(3x). Apply the product rule with u = x² and v = ln(3x): f' = 2x ln(3x) + x²(1/x) = 2x ln(3x) + x. The chain rule contributes the 1/x from the inner function 3x; forgetting it gives x²/(3x), a common error.
Traps that cost marks
- Dropping the inner derivative in a chain-rule problem.
- Using degrees in a calculus derivative or integral.
- Treating a definite integral as a family of functions rather than a scalar.
- Reversing matrix multiplication order; AB is generally not BA.
- Losing the constant of integration in an initial-value problem.
- Accepting an extraneous root after squaring an equation.