FE Mathematics: Calculus Review

Success on the FE exam hinges on executing core mathematical procedures quickly and accurately under pressure. A strong command of calculus, which forms a significant portion of the mathematics section, is not about theoretical exploration but about reliable computational fluency. This review distills the essential differentiation and integration techniques you need, focusing on the application and problem-solving patterns most likely to appear on the test.

1. Limits and Continuity: The Foundation

Understanding limits is crucial because they formally define both derivatives and integrals. For the FE exam, you must be adept at evaluating limits algebraically, especially those that initially yield indeterminate forms like $0/0$ or $\infty /\infty$. The key techniques are factoring, rationalizing, or applying L'Hôpital's Rule, which states that for indeterminate forms $0/0$ or $\infty /\infty$, the limit of a quotient is equal to the limit of the quotient of their derivatives: ${\lim }_{x\to c}\frac{f(x)}{g(x)}={\lim }_{x\to c}\frac{f^{\prime }(x)}{g^{\prime }(x)}$, provided the latter limit exists.

A function is continuous at a point if the limit as you approach the point equals the function's value there: ${\lim }_{x\to a}f(x)=f(a)$. Intuitively, you can draw it without lifting your pencil. On the exam, you might be asked to find a constant that makes a piecewise function continuous, which involves setting the left-hand and right-hand limits at the boundary equal to each other.

Exam Tip: When you see a limit problem, first try direct substitution. If it yields an indeterminate form, immediately factor or consider L'Hôpital's Rule. These problems are designed for a quick solution.

2. Differentiation: Rules and Applications

Differentiation measures the instantaneous rate of change. You must have the following rules at your fingertips:

• Power Rule: $\frac{d}{dx}[x^n]=n{x}^{n-1}$
• Product Rule: $\frac{d}{dx}[f(x)g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$
• Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$
• Chain Rule: $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$

The chain rule is the most frequently applied and often requires you to identify an "inner" and "outer" function. For example, to differentiate $\sin (x^2)$, the outer function is $\sin (u)$ and the inner function is $u=x^2$. The derivative is $\cos (x^2)\cdot 2x$.

A common exam application is related rates or optimization within a simple geometric context (e.g., maximizing the area of a rectangle with a fixed perimeter). Your task is to set up the correct derivative equation from the given word problem.

Exam Tip: For complex derivatives, break the function into parts. The chain rule is often used multiple times in a single problem. Speed is gained through practice, not just memorization.

3. Integration Techniques: The Toolkit

Integration is the inverse process of differentiation, finding the antiderivative or the accumulated quantity. The FE exam tests your ability to select and execute the correct technique swiftly.

• Basic Antiderivatives & Substitution: Know the integrals of common functions like $x^n$, $\sin x$, $\cos x$, and $e^x$. $u$-substitution is your primary tool for reversing the chain rule. You look for a composite function where the derivative of the inner part is also present. For $\int 2x\cos (x^2)dx$, you'd let $u=x^2$, so $du=2xdx$. The integral simplifies to $\int \cos udu=\sin (x^2)+C$.
• Integration by Parts: Used for integrals of products, typically where $u$-substitution fails. The formula is derived from the product rule: $\int udv=uv-\int vdu$. The strategic choice of $u$ and $dv$ is critical. Remember the LIATE heuristic (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to guide your choice for $u$.
• Partial Fractions: A key technique for integrating rational functions (a polynomial divided by a polynomial). You algebraically decompose a complex fraction into a sum of simpler ones. For example, you would express $\frac{1}{(x-1)(x+2)}$ as $\frac{A}{x-1}+\frac{B}{x+2}$, solve for constants $A$ and $B$, and then integrate each simple term.

4. Definite, Improper Integrals, and Applications

A definite integral, ${\int }_a^{b}f(x)dx$, computes the net area between the curve $f(x)$ and the x-axis from $x=a$ to $x=b$. The Fundamental Theorem of Calculus connects differentiation and integration: ${\int }_a^{b}f(x)dx=F(b)-F(a)$, where $F(x)$ is any antiderivative of $f(x)$.

An improper integral has infinite limits of integration (e.g., ${\int }_1^{\infty }\frac{1}{x^2}dx$) or an integrand that approaches infinity within the interval. You evaluate them by taking a limit: ${\int }_a^{\infty }f(x)dx={\lim }_{t\to \infty }{\int }_a^{t}f(x)dx$.

Application problems are formula-driven on the FE:

• Area between curves: $A={\int }_a^b[f(x)-g(x)]dx$, where $f(x)$ is the top curve.
• Volume of revolution (Disk/Washer): Rotating around the x-axis: $V=\pi {\int }_a^b[R(x){]}^{2}dx$ (disk) or $V=\pi {\int }_a^b\left([R(x){]}^2-[r(x){]}^2\right)dx$ (washer for a hollow solid).
• Arc Length: For a smooth curve $y=f(x)$ from $a$ to $b$, $L={\int }_a^b\sqrt{1+[f^{\prime }(x){]}^2}dx$.

Exam Tip: For application problems, focus on correctly setting up the integral. The exam reference handbook provides these formulas, but you must know which one to use and how to identify the limits of integration ($a$ and $b$) from the problem description or graph.

Common Pitfalls

1. Misapplying the Fundamental Theorem: Forgetting the constant of integration $C$ is fine for definite integrals, but a fatal error is to mis-evaluate $F(b)-F(a)$. A common sign mistake is computing $F(a)-F(b)$. Always compute the antiderivative at the top limit first.
2. Chain Rule vs. $u$-Substitution Confusion: When differentiating, you multiply by the derivative of the inner function. When integrating via $u$-substitution, you must account for the derivative of the inner function via $du$. If it's not present, you cannot directly substitute. Forgetting to change the limits of integration when performing $u$-substitution on a definite integral is another frequent error. You can either change the limits to be in terms of $u$, or substitute back to $x$ before evaluating.
3. Ignoring Absolute Value in Certain Integrals: The integral $\int \frac{1}{x}dx=\ln |x|+C$, not just $\ln x$. While the context of an application problem might imply positive $x$, it's a good habit to include the absolute value to maintain mathematical correctness, especially in indefinite integrals.
4. Setup Errors in Application Problems: Using the disk formula when the washer formula is needed is a classic mistake. Always sketch a quick representative slice (a disk or a washer) to identify the correct outer radius $R(x)$ and inner radius $r(x)$. For arc length, a very common error is to forget the "+1" inside the square root, instead writing $\int f^{\prime }(x)dx$.

Summary

• Master the core rules: Have the product, quotient, and chain rules for differentiation, and substitution, parts, and partial fractions for integration committed to memory for instant recall.
• Application = Correct Setup: The hardest part of area, volume, and arc length problems is translating the word problem or diagram into the correct integral. The computation itself is often straightforward.
• Speed through algebra: Exam problems are designed to be solved in minutes. Efficient algebraic manipulation, especially when evaluating limits or simplifying integrals, is as important as knowing the calculus concepts.
• Know your reference handbook: Be intimately familiar with where the calculus formulas are located in the NCEES FE Reference Handbook to avoid wasting time searching during the exam.
• Practice under timed conditions: Fluency is your primary goal. Work through many practice problems to build the muscle memory needed to identify problem types and execute solutions without hesitation.