Overview
This lecture reviews core math concepts and problem-solving strategies for the FE (Fundamentals of Engineering) exam, focusing on topics such as analytic geometry, calculus, vectors, statistics, and calculator use.
FE Math Exam Topics & Approach
- Core math topics: analytic geometry, single variable calculus, vector operations, statistics.
- Focus is on fundamental concepts that underpin more advanced topics.
- Differential equations are not emphasized on the current exam.
Equations of Lines and Circles
- Equation of a line: slope-intercept form (y = mx + b) and point-slope form (y - y₁ = m(x - x₁)).
- Equation of a circle: (x - h)² + (y - k)² = r², where (h, k) is the center, r is the radius.
- Pythagoras’ theorem used to find the distance (radius) between two points.
Algebraic Problem Solving & Calculator Use
- Systems of equations can be solved by elimination, substitution, or calculator functions.
- Knowing how to use the approved calculator (TI-36X Pro, Casio FX-115) can save significant time.
Function Manipulation & Logarithms
- To find the inverse of an exponential function, swap x and y and solve.
- Logarithmic identities: logₐx = y ↔ aʸ = x; log(xy) = log x + log y; log(x/y) = log x - log y.
Trigonometry & Analytic Geometry
- Know basic trig identities: tan x = sin x / cos x, sec x = 1 / cos x, csc x = 1 / sin x.
- Familiarity with the Pythagorean identity: sin²x + cos²x = 1.
- Be careful with calculator modes (radians vs. degrees).
Similar Triangles & Geometry Applications
- Similar triangles have proportional sides and are useful in geometry and truss analysis.
- Ratio relationships can solve for unknown sides in similar triangles.
Areas: Circles & Segments
- Area of a trapezoid: ½(base₁+base₂) × height.
- Area of a circular segment can require formulas from both geometry and transportation sections.
- Always check if angle input should be in radians or degrees.
Calculus: Derivatives & Integrals
- Derivative rules: product rule d(uv)/dx = u dv/dx + v du/dx; chain rule applies for trig/exponential functions.
- For tangency/parallelism in curves, set derivatives/slope equal and solve for the point.
- Definite integrals find area between curves: ∫ₐᵇ [f₁(x) – f₂(x)] dx.
Vectors: Dot & Cross Products
- Dot product: a·b = a₁b₁ + a₂b₂ + a₃b₃; result is a scalar.
- Cross product: produces a vector perpendicular to both inputs.
- Unit vector: v/|v|, where |v| = √(x² + y² + z²).
Statistics & Probability
- Mean (x̄): sum of values / number of samples.
- Standard deviation (Sₓ): measures data spread.
- Normal distribution: use z-score (z = (μ – x) / σ) and z-tables to find probabilities.
- Binomial probability: probability of k successes in n trials, P(X = k) = nCk·pᵏ·qⁿ⁻ᵏ.
Regression & Data Analysis
- Linear regression finds best-fit lines, typically y = ax + b.
- Calculators and Excel expedite regression analysis and statistics calculations.
Key Terms & Definitions
- Slope-intercept form — y = mx + b, where m is slope, b is y-intercept.
- Point-slope form — y - y₁ = m(x - x₁), alternate equation for a line.
- Circle equation — (x - h)² + (y - k)² = r².
- Dot product — Scalar result: a·b = Σa_ib_i.
- Cross product — Vector result perpendicular to both vectors.
- Standard deviation (Sₓ) — Measure of data spread in a sample.
- Z-score — (μ – x) / σ, standardizes a value in a normal distribution.
- Binomial distribution — Probability of k successes in n independent trials.
Action Items / Next Steps
- Review the reference handbook’s formulas for lines, circles, trig, and probability.
- Practice using your calculator for equations, regression, statistics, and integrals.
- Study and memorize basic trig identities and calculus derivative rules.
- Prepare for next week’s lecture on statics.