Calculus Review for the FE Exam
Calculus is one of the most heavily tested mathematical topics on the FE exam. Whether you are solving optimization problems, computing areas and volumes, or analyzing rates of change, a solid command of derivatives and integrals is essential. This review focuses on the specific calculus skills you need for the exam.
Derivatives: The Fundamentals
A derivative measures the instantaneous rate of change of a function. On the FE exam, you will use derivatives to find slopes, optimize functions, and analyze motion.
Essential Derivative Rules
These rules should be second nature by exam day:
- Power rule: The derivative of x^n is n*x^(n-1). This is the most frequently used rule.
- Product rule: The derivative of f(x)*g(x) is f'(x)*g(x) + f(x)*g'(x).
- Quotient rule: The derivative of f(x)/g(x) is [f'(x)*g(x) - f(x)*g'(x)] / [g(x)]^2.
- Chain rule: The derivative of f(g(x)) is f'(g(x))*g'(x). This is essential for composite functions.
Common Derivatives to Know
- Derivative of sin(x) = cos(x)
- Derivative of cos(x) = -sin(x)
- Derivative of e^x = e^x
- Derivative of ln(x) = 1/x
- Derivative of tan(x) = sec^2(x)
While these are in the reference handbook, knowing them from memory saves valuable time during the exam.
Applications of Derivatives on the FE Exam
Optimization: Find the maximum or minimum of a function by setting the first derivative equal to zero and solving. Use the second derivative test to determine whether the critical point is a maximum (negative second derivative) or minimum (positive second derivative).
Related rates: These problems involve finding the rate of change of one quantity in terms of another. Set up the equation relating the quantities, differentiate both sides with respect to time, and solve.
Tangent lines: The derivative at a point gives the slope of the tangent line. The equation of the tangent line at point (a, f(a)) is y - f(a) = f'(a)(x - a).
Integrals: The Fundamentals
Integration is the reverse of differentiation and is used to compute accumulated quantities: areas, volumes, total work, and more.
Essential Integration Techniques
- Power rule for integration: The integral of x^n is x^(n+1)/(n+1) + C (for n not equal to -1).
- Substitution (u-substitution): Transform a complex integral into a simpler one by substituting u = g(x). This is the reverse of the chain rule.
- Integration by parts: For integrals of products: integral of u dv = uv - integral of v du. Choose u and dv using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).
- Partial fractions: Decompose a rational function into simpler fractions that can be integrated individually.
Definite Integrals and the Fundamental Theorem
The Fundamental Theorem of Calculus connects derivatives and integrals: the definite integral from a to b of f(x) dx equals F(b) - F(a), where F is an antiderivative of f. On the FE exam, definite integrals are used for:
- Computing the area under a curve
- Finding the area between two curves
- Calculating volumes of revolution (disk and shell methods)
- Determining centroids and moments of inertia
- Computing total work done by a variable force
Numerical Integration
When a function cannot be integrated analytically, numerical methods provide approximate solutions. The FE exam may test:
- Trapezoidal rule: Approximates the area under a curve using trapezoids. More accurate with more subdivisions.
- Simpson's rule: Uses parabolic segments for better accuracy than the trapezoidal rule with the same number of subdivisions. Requires an even number of intervals.
Multivariable Calculus
Some FE exam problems involve functions of multiple variables. Key concepts include:
- Partial derivatives: Differentiate with respect to one variable while treating others as constants.
- Gradient: A vector of partial derivatives that points in the direction of steepest increase.
- Double integrals: Used for computing volumes and areas in two dimensions.
Practice Strategy for Calculus
Calculus problems on the FE exam are typically straightforward applications of these core concepts. The challenge is speed and accuracy under time pressure. Practice by:
- Solving 10 to 15 calculus problems per study session during your math review week
- Timing yourself to build speed (aim for 2 to 3 minutes per calculus problem)
- Practicing with the reference handbook open to build lookup speed
- Focusing on the types of problems that appear most frequently: optimization, area, and rate of change problems
Calculus is a topic where practice truly makes perfect. The more problems you solve, the faster you will recognize patterns and apply the correct techniques on exam day.