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Single-Variable Calculus

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FE Fundamentals of Engineering Exam

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1 Introduction to the FE Exam

About the FE Exam FE Exam Study Strategies Navigating the FE Reference Handbook

2 Mathematics (8–12 Questions)

Analytic Geometry and Trigonometry Single-Variable Calculus Differential Equations Linear Algebra and Matrix Operations Numerical Methods

3 Probability and Statistics (6–9 Questions)

Probability Fundamentals Probability Distributions Statistical Estimation and Hypothesis Testing

4 Chemistry (5–8 Questions)

Chemical Reactions and Stoichiometry Acids, Bases, and pH Oxidation-Reduction and Corrosion

5 Ethics and Professional Practice (5–8 Questions)

Engineering Ethics Fundamentals

6 Engineering Economics (6–9 Questions)

Time Value of Money Economic Analysis Methods

7 Statics (9–14 Questions)

Force Systems and Equilibrium Trusses, Frames, and Machines Centroids and Moments of Inertia Static Friction

8 Dynamics (9–14 Questions)

Particle Kinematics Kinetics: Force, Mass, and Acceleration Work, Energy, Impulse, and Momentum Rigid-Body Rotation and Mechanical Vibrations

9 Strength of Materials (9–14 Questions)

Stress and Strain Fundamentals Beam Analysis: Shear and Moment Diagrams Bending Stress and Torsion Stress Transformation and Material Failure

10 Materials Science (6–9 Questions)

Material Properties and Selection Phase Diagrams and Alloy Systems

11 Fluid Mechanics (12–18 Questions)

Fluid Properties and Statics Fluid Dynamics and Bernoulli's Equation Pumps, Turbines, and Flow Measurement

12 Thermodynamics and Heat Transfer (9–14 Questions)

Laws of Thermodynamics and Properties Thermodynamic Cycles and Applications Heat Transfer: Conduction, Convection, and Radiation

13 Basic Electrical Engineering (6–9 Questions)

Electrical Fundamentals and DC Circuits AC Circuits and Three-Phase Power

14 Safety, Health, and Environment (6–9 Questions)

Industrial Safety and Hazard Management

15 Instrumentation and Controls (4–6 Questions)

Sensors, Data Acquisition, and Control Systems

16 Computational Tools and Computer Science

Spreadsheets, Structured Programming, and Flowcharts

Single-Variable Calculus

Key Takeaways Explain This Topic AI

The power rule d/dx[xⁿ] = nxⁿ⁻¹ and chain rule d/dx[f(g(x))] = f′(g(x))·g′(x) are the most-used FE differentiation tools.

Critical points occur where f′(x) = 0; the second-derivative test classifies them (f″ > 0 minimum, f″ < 0 maximum).

Integration reverses differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1, and ∫(1/x) dx = ln|x| + C.

L'Hôpital's Rule resolves 0/0 and ∞/∞ limits by differentiating numerator and denominator separately.

Disk method gives volume of revolution V = π∫[f(x)]² dx; arc length L = ∫√(1+[f′(x)]²) dx.

The Maclaurin series of eˣ is 1 + x + x²/2! + x³/3! + …; sin x and cos x have alternating-sign series.

Last updated: June 2026

Calculus is the analytical engine behind statics, dynamics, fluid mechanics, and engineering economics. On the FE you must differentiate, integrate, and find extrema quickly , then plug into Handbook formulas (centroids, moments of inertia, work integrals) that are themselves integrals.

Limits and Continuity

f(x) is continuous at x = a when f(a) exists, lim(x→a) f(x) exists, and the two are equal. L'Hôpital's Rule: if lim f(x)/g(x) gives the indeterminate forms 0/0 or ∞/∞, then lim f(x)/g(x) = lim f′(x)/g′(x).

Worked example — L'Hôpital. Evaluate lim(x→0) sin x / x. Direct substitution gives 0/0, so differentiate top and bottom: lim cos x / 1 = cos 0 = 1. This limit (= 1) is worth memorizing; it underlies the derivative of sine.

Differentiation Rules

  • Rule Formula Example
  • Power d/dx[xⁿ] = nxⁿ⁻¹ d/dx[x³] = 3x²
  • Product (fg)′ = f′g + fg′ d/dx[x sin x] = sin x + x cos x
  • Quotient (f/g)′ = (f′g − fg′)/g² —
  • Chain d/dx[f(g(x))] = f′(g)·g′ d/dx[sin 3x] = 3 cos 3x
  • Common Derivatives
  • f(x) f′(x) f(x) f′(x)
  • sin x cos x eˣ eˣ
  • cos x −sin x ln x 1/x
  • tan x sec²x arctan x 1/(1+x²)

Worked example — product + chain. Differentiate f(x) = x³ sin x. By the product rule, f′(x) = (3x²)(sin x) + (x³)(cos x) = 3x² sin x + x³ cos x. A frequent trap is treating it as a simple product of derivatives (3x²·cos x), which drops two terms.

Applications of Derivatives — Maxima and Minima

The FE loves optimization (minimize cost, maximize area). Procedure: (1) write the quantity as a function of one variable, (2) set f′(x) = 0 to find critical points , (3) classify with the second-derivative test — f″ > 0 gives a local minimum (concave up), f″ < 0 gives a local maximum (concave down), f″ = 0 is inconclusive. Inflection points are where f″(x) = 0 and concavity changes.

Worked example — maxima/minima. A rectangular pen of fixed perimeter P = 40 m is built against a wall (only three sides fenced). Maximize area. Let x = the two sides perpendicular to the wall; then the parallel side is 40 − 2x, and A(x) = x(40 − 2x) = 40x − 2x². Set A′(x) = 40 − 4x = 0 → x = 10 m. Since A″ = −4 < 0, this is a maximum. The dimensions are 10 m × 20 m and A = 200 m². Checking endpoints (x = 0 or x = 20 give A = 0) confirms the interior critical point is the true maximum.

Integration

Integration accumulates quantities and reverses differentiation.

  • f(x) ∫f(x) dx
  • xⁿ (n ≠ −1) xⁿ⁺¹/(n+1) + C
  • 1/x ln|x| + C
  • eˣ eˣ + C
  • sin x −cos x + C
  • cos x sin x + C
  • Techniques

u-substitution: ∫f(g(x))·g′(x) dx = ∫f(u) du with u = g(x).

Integration by parts: ∫u dv = uv − ∫v du. Choose u by LIATE (Logarithmic, Inverse-trig, Algebraic, Trig, Exponential).

Partial fractions: split a rational function such as 1/[(x−1)(x+2)] into A/(x−1) + B/(x+2) before integrating.

Worked example — definite integral. Evaluate ∫₀¹ 2x dx = [x²]₀¹ = 1² − 0² = 1. Always evaluate the antiderivative at the upper limit minus the lower limit; forgetting to subtract the lower bound is the classic error.

Applications of Integrals

  • Quantity Formula
  • Area between curves A = ∫ₐᵇ [f(x) − g(x)] dx, f ≥ g
  • Volume (disk, about x-axis) V = π∫ₐᵇ [f(x)]² dx
  • Volume (shell, about y-axis) V = 2π∫ₐᵇ x·f(x) dx
  • Arc length L = ∫ₐᵇ √(1 + [f′(x)]²) dx
  • Average value f̄ = (1/(b−a)) ∫ₐᵇ f(x) dx

Worked example — volume of revolution. Rotate y = √x for 0 ≤ x ≤ 4 about the x-axis. Disk method: V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = π(16/2) = 8π ≈ 25.1. Squaring √x to get x is the step most often missed.

Series and Sequences

The Taylor series about x = a is f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ. The Maclaurin series is the special case a = 0.

  • Function Maclaurin series
  • eˣ 1 + x + x²/2! + x³/3! + …
  • sin x x − x³/3! + x⁵/5! − …
  • cos x 1 − x²/2! + x⁴/4! − …
  • ln(1+x) x − x²/2 + x³/3 − …
  • 1/(1−x) 1 + x + x² + x³ + … (|x| < 1)

Worked example — series approximation. Estimate e^0.1 using the first three terms: 1 + 0.1 + 0.1²/2 = 1 + 0.1 + 0.005 = 1.105. The true value is 1.10517, so three terms already give four-figure accuracy — illustrating why low-order Taylor terms dominate near the expansion point. Recognize that the alternating series with x³/3! belongs to sin x and the one starting 1 − x²/2! belongs to cos x; mixing them up is a common distractor on the FE.

L'Hôpital's Rule and Curve Behavior

When a limit gives the indeterminate form 0/0 or ∞/∞, L'Hôpital's rule lets you differentiate numerator and denominator separately: lim f/g = lim f′/g′. For example, lim(x→0) sin x / x = lim cos x / 1 = 1. Reapply if the new limit is still indeterminate.

The first and second derivative tests classify curve behavior, a frequent FE item:

  • Condition Meaning
  • f′(x) = 0 Critical point (candidate max/min)
  • f″(x) > 0 Concave up → local minimum
  • f″(x) < 0 Concave down → local maximum
  • f″(x) = 0 (sign change) Inflection point

To find a maximum, set f′(x)=0, solve, then check the sign of f″. Optimization problems (largest area, least cost) reduce to exactly this procedure after writing the quantity as a function of one variable.