FE Exam: Mathematics Review

Mathematics is one of the highest-leverage areas on the Fundamentals of Engineering (FE) exam because it supports many other topics, from statics and dynamics to fluid mechanics and circuits. A strong math foundation does not require memorizing obscure tricks. It requires comfort with core tools: algebraic manipulation, trigonometry, calculus, differential equations, linear algebra, and basic numerical methods. This review focuses on what you are most likely to need and how to apply it efficiently under exam conditions.

Algebra: the language of engineering problems

Algebra shows up everywhere on the FE exam, often disguised inside word problems and unit conversions. The goal is speed and accuracy in rearranging expressions and solving for unknowns.

Core skills to sharpen

• Solving linear equations and systems, including quick elimination and substitution.
• Manipulating exponents and radicals, especially with scientific notation and unit scaling.
• Factoring and simplifying rational expressions, including identifying restrictions (denominators not equal to zero).
• Logarithms and exponentials, including change of base and properties:
• $\log (ab)=\log a+\log b$
• $\log (a^n)=n\log a$
• Quadratics: factoring when possible, and using the quadratic formula when not:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Practical exam insight

Many FE math errors come from algebra in the middle of an otherwise straightforward calculation. When you solve for a variable, pause to check:

• Are the units consistent?
• Does the sign make physical sense?
• Is the result in a reasonable magnitude range?

Trigonometry: angles, components, and periodic behavior

Trigonometry appears in vector decomposition, geometry, waves, and rotating systems. The exam typically favors practical identities over heavy derivations.

Must-know relationships

• Right-triangle definitions for sine, cosine, tangent.
• Pythagorean identity: ${\sin }^2\theta +{\cos }^2\theta =1$
• Angle sum formulas are helpful but less frequently required than basics:
• $\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$
• $\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

Common engineering usage: components

If a force $F$ is applied at angle $\theta$ from the $x$-axis, then:

• $F_x=F\cos \theta$
• $F_y=F\sin \theta$

Be consistent about the reference angle. A frequent pitfall is mixing “from the horizontal” and “from the vertical.”

Radians vs degrees

Calculus formulas assume radians. If a problem involves derivatives of $\sin$ or $\cos$ and the angle is in degrees, convert first. Remember $180^{\circ }=\pi$ radians.

Calculus: rates of change and accumulation

Calculus on the FE exam is usually applied, not theoretical. You should be comfortable taking derivatives and integrals quickly, recognizing common forms, and interpreting what the result means.

Differentiation essentials

• Power rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Product rule and quotient rule.
• Chain rule for composite functions.
• Derivatives of common functions:
• $\frac{d}{dx}\sin x=\cos x$
• $\frac{d}{dx}\cos x=-\sin x$
• $\frac{d}{dx}{e}^x=e^x$
• $\frac{d}{dx}\ln x=\frac{1}{x}$

Integration essentials

• Power rule: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for $n\ne -1$
• $\int \frac{1}{x}dx=\ln |x|+C$
• Definite integrals as net area/accumulation.

Interpreting derivatives and integrals

Many FE questions test meaning more than mechanics:

• A derivative represents a rate, such as velocity from position.
• An integral represents accumulated quantity, such as total mass from a density function.

Optimization and extrema

Know the standard workflow:

1. Write the objective function.
2. Compute derivative and set to zero.
3. Check critical points (and boundaries if applicable).
4. Use second derivative or reasoning to confirm max/min.

Differential equations: modeling change

Differential equations (DEs) often appear in first-order forms tied to physical systems, such as mixing tanks, charging capacitors, or heat transfer approximations.

First-order linear DEs

A standard form is: $$\frac{dy}{dx}+P(x)y=Q(x)$$ Solve using an integrating factor: $$\mu (x)=e^{\int P(x)dx}$$ Then: $$\frac{d}{dx}\left(\mu y\right)=\mu Q$$

Separable equations

If the DE can be written as: $$\frac{dy}{dx}=g(x)h(y)$$ then separate variables: $$\frac{1}{h(y)}dy=g(x)dx$$ and integrate both sides.

What the exam tends to emphasize

• Recognizing whether a DE is separable or linear.
• Applying initial conditions to find the constant.
• Understanding qualitative behavior, for example exponential growth/decay.

Exponential decay appears often as $\frac{dy}{dt}=-ky$, with solution $y=y_0{e}^{-kt}$.

Linear algebra: vectors, matrices, and systems

Linear algebra shows up in solving simultaneous equations, analyzing linear transformations, and working with vectors in 2D and 3D. The FE exam typically prioritizes computational competence.

Systems of linear equations

Be comfortable solving $Ax=b$ using:

• Elimination (by hand for small systems)
• Matrix methods when appropriate (especially if formulas are provided)

Know the meaning of:

• Unique solution (full rank, nonzero determinant for square matrices)
• No solution (inconsistent system)
• Infinitely many solutions (dependent equations)

Determinants and invertibility

For a square matrix, a nonzero determinant implies an inverse exists and the system has a unique solution. For a $2\times 2$ matrix: If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, then $\det (A)=ad-bc$.

Eigenvalues (conceptual)

Some FE questions touch eigenvalues in a basic way: solve $\det (A-\lambda I)=0$ for $\lambda$. The key is recognizing that eigenvalues relate to how a matrix scales special directions (eigenvectors). Keep it practical, not abstract.

Numerical methods: approximate solutions with control

Numerical methods appear when exact algebra is inconvenient or when iterative procedures are part of engineering workflows. The FE exam usually focuses on interpreting or performing simple iterations.

Root-finding basics

• Bisection method: reliable, slower; requires a sign change.
• Newton’s method: faster near the root; uses derivative:

$$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$$

Know what can go wrong: Newton’s method can diverge if the initial guess is poor or if $f^{\prime }(x)$ is near zero.

Numerical integration

Common ideas include trapezoidal approximation and Simpson’s rule. Even if you do not memorize every formula, you should understand what these methods do: approximate area under a curve by summing geometric shapes.

Rounding and error awareness

Pay attention to significant figures and intermediate rounding. Premature rounding can shift a final answer enough to miss a multiple-choice option.

How to use this review in FE exam prep

Build speed on fundamentals

Create a short daily set that rotates:

• 5 algebra manipulations
• 5 trig component problems
• 5 derivatives/integrals
• 2 first-order differential equations
• 2 small linear systems
• 1 numerical iteration problem

Focus on recognition

The FE exam rewards fast identification of the tool:

• “Rate of change” points to derivatives.
• “Accumulated” or “total from density” points to integrals.
• “System of equations” points to matrices or elimination.
• “Model with proportional rate” points to exponential DEs.

Keep a personal error log

Track mistakes by category: sign errors, unit issues, misread angles, dropped constants, or calculator mode problems (degrees vs radians). Fixing two recurring error types can improve your score more than learning new techniques.

Mathematics on the FE exam is less about advanced theory and more about dependable execution. If you can simplify cleanly, compute accurately, and choose the right method quickly, you will not only score well in the math section but also make the rest of the exam feel more manageable.